Wigner Functions in Optics- Describing Beams as Ray Bundles and Pulses as Particl Ensembles

7/31/2019 Wigner Functions in Optics- Describing Beams as Ray Bundles and Pulses as Particl Ensembles

http://slidepdf.com/reader/full/wigner-functions-in-optics-describing-beams-as-ray-bundles-and-pulses-as-particl 1/94

Wigner functions in optics: describingbeams as ray bundles and pulses as

particle ensembles

Miguel A. Alonso

The Institute of Optics, University of Rochester, Rochester, New York 14627, USA

[email protected]

Received June 3, 2011; revised October 4, 2011; accepted October 5, 2011; published

November 21, 2011 (Doc. ID 148673)

This tutorial gives an overview of the use of the Wigner function as a tool for

modeling optical field propagation. Particular emphasis is placed on the spatial

propagation of stationary fields, as well as on the propagation of pulses throughdispersive media. In the first case, the Wigner function gives a representation

of the field that is similar to a radiance or weight distribution for all the rays

in the system, since its arguments are both position and direction. In cases in

which the field is paraxial and where the system is described by a simple linear

relation in the ray regime, the Wigner function is constant under propagation

along rays. An equivalent property holds for optical pulse propagation

in dispersive media under analogous assumptions. Several properties and

applications of the Wigner function in these contexts are discussed, as is its

connection with other common phase-space distributions like the ambiguity

function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek

functions. Also discussed are modifications to the definition of the Wigner

function that allow extending the property of conservation along paths to a

wider range of problems, including nonparaxial field propagation and pulse

propagation within general transparent dispersive media. c 2011 Optical

Society of America

OCIS codes: 070.7425, 070.7345, 080.5084, 030.5620, 070.2590, 080.2730

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

2. The Fourier Transform and Other Unitary Linear Transformations 277

2.1. The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2782.2. The Fractional Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

2.3. Linear Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . .280

2.3a. Fractional Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . 282

2.3b. Scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

2.3c. Fresnel Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .283

2.3d. Chirping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

2.4. Phase-Space Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Advances in Optics and Photonics 3, 272–365 (2011) doi:10.1364/AOP.3.0002721943-8206/11/040272-94/$15.00 c OSA

272



3. Simple Phase-Space Distributions: the Windowed Fourier Transformand the Spectrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

4. The Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .286

4.1. Reality and Possible Negativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .288

4.2. Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .289

4.3. Linear Mapping under Linear Canonical Transformations andPhase-Space Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

4.3a. Fractional Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . 291

4.3b. Scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

4.3c. Fresnel Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .292

4.3d. Chirping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

4.3e. Phase-Space Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

4.4. Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

4.5. Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295

4.6. Oscillatory Character of the Wigner Function . . . . . . . . . . . . . . . . .297

5. Partial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

5.1. Correlations and Coherent Mode Expansions . . . . . . . . . . . . . . . . .299

5.2. Wigner Function and Spectrogram for Mixed States . . . . . . . . . . 301

5.3. Overall Degree of Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

5.4. Quasi-homogeneous Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .302

5.5. Smoothness of the Wigner Function for Highly Mixed States . 302

6. Other Phase-Space Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

6.1. Distributions Related to the Spectrogram . . . . . . . . . . . . . . . . . . . . .305

6.2. The Kirkwood–Rihaczek and Margenau–Hill Distributions . . . 306

6.3. The Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

6.4. The Cohen Class of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .308

7. Wigner-Based Modeling of Beam Propagation . . . . . . . . . . . . . . . . . . . .314

7.1. Paraxial ABCD Systems in the Ray Domain . . . . . . . . . . . . . . . . . . 314

7.2. Radiometry and the Radiance in the Paraxial Regime . . . . . . . . .317

7.3. Paraxial ABCD Systems in the Wave Domain . . . . . . . . . . . . . . . . .3187.4. Wave Propagation Through ABCD Systems in Terms of theWigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

7.5. Phase-Space Interpretation of Measurement . . . . . . . . . . . . . . . . . .322

7.6. Applications of the Wigner and Ambiguity Functions in theParaxial Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

7.6a. Imaging Systems and the Optical Transfer Function . . . . . 324

7.6b. Propagation-Invariant Beams and Extended Depth of Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

7.6c. The Talbot Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

7.6d. Sampling and Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

7.7. Partially Coherent Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331

7.8. Wigner Functions that Account for Polarization in the ParaxialRegime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

8. The Use of the Wigner Function for More Complicated Systemsand/or Nonparaxial Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

8.1. Generalized Radiometry for Nonparaxial Fields . . . . . . . . . . . . . .334

9. Short Pulses and Propagation in Linear Dispersion Media . . . . . . . . 335

9.1. Wigner Function of Time versus Frequency . . . . . . . . . . . . . . . . . . 337

9.2. Wigner Function of Position versus Wavenumber . . . . . . . . . . . . 338

Advances in Optics and Photonics 3, 272–365 (2011) doi:10.1364/AOP.3.000272 273



10. Phase-Space Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33910.1. Phase-Space Tomography in Quantum Optics . . . . . . . . . . . . . . .33910.2. Phase-Space Tomography of Classical Fields . . . . . . . . . . . . . . . . 341

11. Generalized Wigner Functions: Achieving Exact Conservation . . . 34211.1. General Procedure for Constructing Conserved Wigner

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34311.2. Nonparaxial Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34511.3. Pulse Propagation Through Transparent Dispersive Media . . 349

11.3a. Lorentz Model Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35011.3b. Waveguide Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .350

11.4. Reduction to the Standard Wigner Function for ParabolicManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

12. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355




Wigner functions in optics: describingbeams as ray bundles and pulses as

particle ensembles

Miguel A. Alonso

1. Introduction

One of the most used mathematical tools in physics and engineeringis the Fourier transformation, which converts a function of a given

variable into a function of another, Fourier conjugate variable. Particularlyin situations where the mathematical description of the physical processis independent of the choice of origin for the initial variable, the Fourierconjugate variable has clear physical significance. Consider, for example,a function of time representing the local air pressure for a sound orthe electric field for optical radiation. Fourier transformation allows usto express this function in terms not of time but of frequency, whichour senses perceive as pitch or color. When considering the spatialdistribution of a monochromatic beam, on the other hand, Fouriertransformation replaces the spatial variable with one associated withdirection of propagation. In quantum mechanics, Fourier transformationover the position coordinates of a particle’s wave function leads to a

distribution in momentum, while Fourier transformation over time givesan energy distribution.

In any of these specific contexts, the well-known mathematical proper-ties of the Fourier transformation acquire a physical interpretation. Inparticular, a function and its Fourier transform cannot be simultaneouslyarbitrarily narrow, since the product of the widths of their squaredmoduli (defined as the standard deviations of their variables) must beequal to or greater than a given constant. This purely mathematicalproperty, sometimes referred to as the space–bandwidth producttheorem, has profound physical implications in the different casesmentioned earlier. Within the context of sound, it tells us that a musical

note cannot be arbitrarily short and yet have a well-defined pitch. Whenapplied to monochromatic wave beams over a transverse coordinate,it implies that a beam cannot be simultaneously arbitrarily thin andwell collimated. In quantum mechanics (according to the Copenhageninterpretation), this relation gives rise to the famous Heisenberguncertainty principle that limits the simultaneous knowledge about theposition and momentum of a particle. Due to this latter interpretation,many authors refer to the general space–bandwidth relation of Fouriertransform pairs as the uncertainty relation.




In physical applications, the Fourier transformation provides a changein the representation of a system, from a description in terms of onephysically meaningful variable (e.g., time or position) into one in termsof another (e.g., frequency, direction, or momentum). Nevertheless,in many of these contexts, our intuition is more inclined towards arepresentation in terms of both Fourier conjugate variables. For example,we think of a musical piece as a sound distribution in terms of bothtime and frequency, represented as a musical score. Similarly, when

designing and modeling macroscopic imaging or illumination opticalsystems, one thinks of light as a distribution over a bundle of rays, eachof these specified by both its position and direction. The same is true inmechanics, where the classical model, closer than the quantum one to oureveryday perception of the world, requires the specification of both theposition and momentum of a particle. Notice that these three simplisticmodels (musical notation, ray optics, and classical mechanics) disregardthe limitations imposed by the space–bandwidth product theorem oruncertainty relation.

There is a family of mathematical transformations that allow thedescription of a function in terms of both a variable and its Fourierconjugate. When applied to the physical situations mentioned earlier,

these transformations mimic the so-called phase-space distributions usedin the simplified physical models (e.g., a musical score, a radiancedistribution giving the weight of all rays in an optical system, ora distribution of classical particles in terms of both their positionand velocities). In this context, the term phase space refers to theextended space composed of both sets of variables (time–frequency,position–direction, position–momentum). For this reason, these math-ematical transformations are sometimes referred to as phase-spacequasi-probability distributions or mock-phase-space distributions [1], althoughmost authors drop the cautionary qualifiers and refer to them simply as phase-space distributions, as we will do here.

One of the most widely used distributions of this type is the Wignerfunction, proposed by Wigner [2] to represent a quantum state as adistribution in terms of both position and momentum, reminiscentof a statistical–mechanical distribution of classical particles. The samemathematical transformation was independently proposed in the signalanalysis community by Ville [3] for studying time signals, and by both Dolin [4] and Walther [5] to represent optical wave fields as raydistributions analogous to what is known in the theory of radiometryas the radiance or specific intensity [6] and in image science as the light field [7]. Excellent reviews on the Wigner function and other phase-spacedistributions are given in the books by Cohen [8] and Mecklenbraukerand Hlawatsch [9], and the review articles by Lee [10], Agarwal andWolf [11–13], Balazs and Jennings [1], and Hillery et al. [14], to name afew. For applications in optics, the reader can also consult, for example,the book edited by Testorf et al. [15], the book by Torre [16], and thereview articles by Dragoman [17] and Sheridan et al. [18].

This tutorial gives a review of the Wigner function and its use withinseveral contexts in classical optics. While many other phase-space repre-sentations have also been proposed (as discussed in Sections 3 and 6),the Wigner function presents two properties that make it unique inmany applications: (i) its so-called marginal projections yield observable




physical quantities, and (ii) under suitable situations, it satisfies the sameconservation rules under spatial propagation or temporal evolution asthe simpler physical model it mimics. For example, in the case of paraxialfields propagating through simple optical systems, the Wigner functionplays the role of a distribution of weights for the rays, because its valuefor each ray remains constant under propagation across the system, and because its integral over all rays passing through a point leads to thelocal optical intensity. The main emphasis of this work is precisely on the

property of conservation along paths under propagation or evolution, itsusefulness in practical applications, and range of validity. Also presentedhere are generalizations to the standard definition of the Wigner functionthat allow extending this property to more general situations, includingnonparaxial propagation and pulse propagation in arbitrary transparentdispersive media.

The outline of this article is as follows. Some unitary transformations(including the Fourier transformation) that are useful in the descriptionof certain simple optical systems are described in Section 2. A verysimple phase-space representation, the spectrogram, is introduced inSection 3. The definition and some properties of the Wigner function are

presented in Section 4 for the case of deterministic functions, and themore general definition of this representation for the case of correlationsis given in Section 5. An overview of several phase-space distributionsis given in Section 6. The Wigner function’s marginal and conservationproperties are discussed in Section 7 for the case of classical paraxialoptical fields propagating through simple systems and in Section 9for the case of optical pulses propagating in media presenting lineardispersion. In both these contexts (as well as in quantum optics), amethod can be used to recover the coherence properties of the field basedon the measurement of the marginal projections of the Wigner functionat different stages. This scheme, known as phase-space tomography, isdiscussed briefly in Section 10. The conservation property of the Wigner

function under classical propagation, however, is valid only undercertain limiting cases, as discussed in Section 8. In Section 11 we discuss aprocedure for defining generalizations to the Wigner function that satisfythe desired propagation and marginal properties for other systems,including nonparaxial propagation through homogeneous media andpulse propagation through more general dispersive media. Finally, some brief concluding remarks are given in Section 12.

2. The Fourier Transform and Other Unitary LinearTransformations

Before introducing the Wigner function and other phase-space distribu-tions, let us first review some simple integral transformations that appearnaturally in the approximate description of certain optical systems.All these transformations depend linearly on the function they areapplied to.




2.1. The Fourier Transform

Let us start by introducing the generic definition of Fourier transfor-mation that is used in this article. Consider a scalar, square-integrablefunction f (q), where q = (q1, q2, . . . , q N ) is an N -dimensional variable.The Fourier transform ˜ f = F f of this function is defined here as

˜ f (p) = { ˆF

f }(p) = |K

|2π N /2

f (q) exp(−iK q · p) d

N

q, (1)

where p = ( p1, p2, . . . , p N ) is the Fourier conjugate variable and K is areal constant [with units of (q · p)−1] dictated by the physical context.Throughout, integrals run from minus to plus infinity unless otherwisespecified. The inverse Fourier transformation has the form

f (q) = {F −1˜ f }(q) = |K |

2π

N /2 ˜ f (p) exp(iK q · p) d N p. (2)

In this article, we consider Fourier transforms connecting

(a) time t and frequency ω, for which N

=1, K

= −1, q

=t and p

=ω;

(b) the transverse position and transverse direction of a beam at a fixedfrequency ω, where N = 2, K = k = ω/c (the free-space wavenumber,given by the ratio of the frequency and the speed of light c), q =x = ( x, y) (the components of the position vector perpendicular tothe optical axis), and p is the transverse part of a vector indicating adirection of propagation;

(c) two quantized field quadratures, i.e., the “real” and “imaginary”parts of the complex representation of a Cartesian component of thequantized electric field, with N = K = 1 and q and p being the twoquadratures.

In the nonrelativistic quantum-mechanical description of a massiveparticle (not discussed here), q and p are the position and momentum

variables, respectively, K = h−1 (the inverse of the reduced Planckconstant), and N is the spatial dimensionality of the problem.

There are several well-known properties of the Fourier transformation(see, e.g., Ref. [19] for a comprehensive list). Let us mention here only acouple:

(i) Parseval–Plancherel theorem. This theorem states that the inner productof two functions f and g is the same in either representation, due to theunitarity of the Fourier transformation, i.e.,

f ∗(q)g(q)d

N

q = ˜ f ∗(p)g(p)d N

p. (3)

(ii) Space–bandwidth product theorem or uncertainty relation. The standard

deviations of | f (q)|2 and |˜ f (p)|2 for each Cartesian component of theirvariables cannot simultaneously be arbitrarily small, since their productmust satisfy

qi pi

≥ 1

2|K | , (4)




for i = 1, 2, . . . , N , where the standard deviations are definedaccording to

2qi

=

(qi − qi)2| f (q)|2d N q | f (q)|2d N q

, qi =

qi| f (q)|2d N q | f (q)|2d N q, (5)

2 pi

=

( pi − ¯ pi)2|˜ f (p)|2d N p

|˜ f (p)|2d N p, ¯ pi =

pi|˜ f (p)|2d N p

|˜ f (p)|2d N p. (6)

As is well known, only real Gaussian distributions (multiplied by anarbitrary linear phase factor) achieve all the lower bounds in relation (4).More general forms of the multidimensional uncertainty relation existthat account for misalignment of the axes of symmetry of the Gaussianwith respect to the coordinate axes, as well as for chirping (complexGaussian widths), but these are beyond the scope of this article. Thereader can consult, e.g., Ref. [20] for more on the uncertainty relation.

2.2. The Fractional Fourier Transform

A generalization of the Fourier transformation that gives access to acontinuum of functions joining the original function and its Fouriertransform was proposed by Condon [21–24]. This transformation,referred to as the fractional Fourier transformation of degree θ , is definedhere as

f θ (s) = {F θ f }(s)

= |K |(tan θ − i)

2π σ tan θ

N /2 f (q) exp

iK

(q2 + σ 2s2) cos θ − 2σ q · s

2σ 2 sin θ

d N q,

(7)

where a scaling constant σ (assumed to be positive) with units of √

q/ p

was introduced. In applications where q and s have the same units, wecan set σ = 1. The fractional Fourier transformation is unitary, so ananalog of the Parseval–Plancherel theorem also exists for it:

f ∗θ (s)gθ (s) d N s =

f ∗(q)g(q) d N q. (8)

Notice that, for θ = 0 and θ = π/2, the fractional Fourier transformreduces, respectively, to scaled versions of the original function and itsFourier transform, i.e.,

f 0(s) = σ N /2 f (σ s), (9a)

f π/2(s) = σ − N /2 ˜ f (σ −1s). (9b)

(The verification of the first relation requires the use of the methodof stationary phase [25] under the limit θ → 0.) The fractionalFourier transform presents an additivity property, which takes aparticularly simple form in cases where σ can be set to unity: twoconsecutive transformations with degrees θ 1 and θ 2, respectively, areequivalent to a single transformation with degree θ 1 + θ 2. As will bediscussed later in this article, the spatial field propagation through




a quadratic gradient index waveguide or the temporal evolution of the quadratures of a quantized field are described mathematically by fractional Fourier transformation. In these contexts, the additivityproperty has a straightforward physical interpretation, since θ isproportional to the propagation distance or time. Due to both Eqs. (9a)and (9b) and to this additivity property, many authors use, instead of thedegree θ , the order γ = 2θ /π , so that γ = 1 corresponds to the standardFourier transformation [24]. This way, γ can be thought of as the power

to which the Fourier transformation operator F is raised before acting ona function.

It must be noted that a more general version (sometimes calledanamorphic) of the fractional Fourier transformation can be writtenin the multidimensional case, where a different θ is used for eachCartesian component of the transformation variables [26]. This moregeneral definition is in turn a special case of an even more general familyof transformations, described in what follows.

2.3. Linear Canonical Transformations

A more general class of unitary transformations that includes theregular and fractional Fourier transformtions is now described. Thesetransformations, known as linear canonical transformations, are usedin paraxial optics and nonrelativistic quantum mechanics to describesimple systems [27–30]. Consider a linear transformation whose kernelhas constant amplitude, like those of the standard and fractional Fouriertransforms:

{C f }(q) = a

f (q) exp

iKv(q, q)

d N q, (10)

where a is a constant that ensures that the transformation is unitary,and v(q, q) is a real function. Let us assume that this function is well

approximated by its second-order Taylor expansion in both variables,i.e.,

v(q, q) ≈ v0 + v · q + v · q + q · Vq + 2q · Vq + q · Vq

2, (11)

where the scalar v0, the vectors v and v, and the matrices V, V,and V, are constant expansion coefficients. Clearly, V and V must besymmetric matrices. Since the only effect of the constant component v0

is an unimportant global phase factor of the transform in Eq. (10), wewill set this constant to zero. Further, it can be shown that the analog of Eq. (3) is given by

{C f 1}∗(q){C f 2}(q) d N q = |a|2

2π

K

N 1

Det(V)

f ∗1 (q) f 2(q) d N q, (12)

so in order to make the transformation unitary, we must choose thenormalization constant as

a = exp(iφ0)

K

2π

N /2 Det(V), (13)

where φ0 is an arbitrary constant phase.




Let us now assume that v satisfies the symmetry property v(−q, −q) =v(q, q), so that v = v = 0 (where 0 denotes a zero vector). It turns out to be convenient to rewrite Eq. (11) as

v(q, q) = q · B−1Aq − 2q · B−1q + q · DB−1q

2, (14)

where A, B, and D, are N × N real matrices. The justification for thisunintuitive choice of representing the matricial coefficients will be givensoon. The symmetry of V = B−1A and V = DB−1 implies that

B−1

A = (B−1

A)T, DB−1 = (DB

−1)T, (15)

or, equivalently,

ABT = BA

T, BT

D = DT

B. (16)

Also, V = −B−1 implies that Det(V) = (−1) N /Det(B). The resultingdefinition for a linear canonical transformation is then the following:

{ ˆC (S) f

}(q)

= −iK

2π N /2

1

√ Det(B) f (q

)

× exp

iK

q · B−1Aq − 2q · B−1q + q · DB−1q

2

d N q,

(17)

where we chose φ0 = N π/4. (The reason this choice will be apparent inwhat follows.)

Note that in Eq. (17), the transformation C depends on S, which is a2 N × 2 N matrix in which A, B, and D are embedded:

S

= A B

C D .

(18)

The remaining N × N submatrix, C, must be chosen such that S issymplectic, i.e., that it satisfies the property

O I

−I O

S

T

O −I

I O

= S

−1, (19)

where I and O are, respectively, the N × N identity and zero matrices. Bymultiplying on the left both sides of this equation by S and using Eq. (16),it is easy to see that the remaining submatrix, C, must be chosen such that

DA

T

− CB

T

= I, (20)i.e.,

C = DB−1

A − (B−1)T = (DA

T − I)(B−1)T, (21)

where the symmetry of B−1A was used in the last step. Finally, the factthat S is symplectic implies that its determinant is unity:

Det(S) = 1. (22)




At a first glance, characterizing the linear canonical transformation interms of the 2 N × 2 N matrix S instead of the three N × N matricesV, V, and V might seem unnecessarily convoluted. However, thisparametrization is convenient, since it confers on the linear canonicaltransformation a group property similar to (and more general than) thatof the fractional Fourier transformation: two consecutive linear canonicaltransformations are equivalent to a single one whose parameter is theproduct of the parameters of the individual transformations, i.e.,

{C (S2)C (S1) f }(q) = {C (S2S1) f }(q), (23)

for any two symplectic matrices S1 and S2. [Note that the choice φ0 = N π/4 in Eq. (17) guarantees that the right-hand side of Eq. (23) has nophase factor in front of it.] It is easy to see from Eq. (19) that the productof two symplectic matrices is itself symplectic.

Let us now discuss four special cases that appear often in the study of optical systems, starting with the fractional Fourier transform.

2.3a. Fractional Fourier Transformation

The particular case of the fractional Fourier transformation corresponds

to {F θ f }(s) = exp(iθ/2){C [SFrFT( θ , σ )] f }(σ s), where the symplectic matrixis given by

SFrFT( θ , σ ) =

σ −1I cos θ σ I sin θ

−σ −1I sin θ σ I cos θ

. (24)

That is, the matrix that describes fractional Fourier transformation is arotation matrix, up to a scaling factor. The relation between the groupproperties of the fractional Fourier (for σ = 1) and linear canonicaltransformations is then trivial, since SFrFT(θ 2, 1)SFrFT(θ 1, 1)

=SFrFT(θ 1

+θ 2, 1).

2.3b. Scalings

Norm-preserving scalings can be defined according to

{S (b) f }(q) = b− N /2 f (q/b), (25)

where b is a real nonzero constant. This is a linear canonicaltransformation corresponding to the matrix

SSc(b) = bI OO b−1

I

. (26)

Note that, since in this case B = O, the substitution of this matrixinto Eq. (17) does not lead directly to Eq. (25). One way to show thisrelation is to notice that SSc(b) = SFrFT(0, 1/b). Therefore, the fact thatnorm-preserving scalings are linear canonical transformations can beshown from taking the limit θ → 0 of a fractional Fourier transform withσ = 1/b, and using the method of stationary phase.




It is straightforward to see that SSc(b2)SSc(b1) = SSc(b1b2). More general,anamorphic versions of this transformation can be written where adifferent scaling applies in each direction. If these directions are alignedwith the coordinate axes, this anamorphic scaling corresponds to a linearcanonical transformation where S is diagonal.

2.3c. Fresnel Transformation

The Fresnel transformation, used to model paraxial free propagation inoptics or nonrelativistic free evolution in quantum mechanics, is given by

{H(τ ) f }(q) =−iK

2π τ

N /2 f (q) exp

−iK

|q − q|2

2τ

d N q, (27)

which corresponds to a linear canonical transformation with symplecticmatrix given by

SFresnel(τ ) = I τ I

O I

. (28)

2.3d. Chirping

Chirping corresponds to a simple multiplication by a quadratic phasefactor:

{V (τ ) f }(q) = exp

i

K τ |q|2

2

f (q). (29)

This operation can be shown to correspond to a linear canonical

transformation with the matrix

SChirp(τ ) =

I O

τ I I

. (30)

Again, showing this equivalence is not straightforward, given that B =O; the proof requires writing this matrix as a limit of a matrix with B = O,for example,

SChirp(τ ) = lim→0

I I

τ I I + τ I

. (31)

2.4. Phase-Space Shifts

A different type of linear unitary transformation corresponds tocombinations of shifts in both sets of Fourier conjugate variables. Let usdefine an operator that causes a shift in the origin of a function as

{T (q0, 0) f }(q) = f (q − q0). (32)




Similarly, due to the shift-phase property of the Fourier transformation,we can define an operator that causes a shift in the Fourier transform of the function through multiplication by a linear phase:

{T (0, p0) f }(q) = exp(iK q · p0) f (q), (33)

so that the Fourier transform of the right-hand side becomes ˜ f (p −p0). These two types of shift can be combined to define what we call

henceforth a phase-space shift. Note, however, that these two types of shiftdo not commute:

{T (0, p0)T (q0, 0) f }(q) = exp(iK q0 · p0){T (q0, 0)T (0, p0) f }(q). (34)

For symmetry, we then define the composite phase-space shift as

{T (q0, p0) f }(q) = exp−iK

q0 · p0

2

exp(iK q · p0) f (q − q0). (35)

It can be shown that this transformation introduces a pure shift (withouta phase factor in front of it) in an anamorphic fractional Fouriertransform of f (q) with degrees θ i

=arctan(σ 2 pi/qi).

The action of many simple systems is expressible as a combination of linear canonical transformations and phase-space shifts. This includesthe general linear transformation in Eqs. (10) and (11) in which the linearcoefficients v and v do not vanish.

3. Simple Phase-Space Distributions: the WindowedFourier Transform and the Spectrogram

In this and the next section, we discuss different mathematical definitionsof phase-space representations, i.e., representations of a function in

terms of both sets of Fourier conjugate variables. Perhaps the mostintuitive representation of this type is the windowed Fourier transform.Here, the part of a function f around a prescribed location q is isolatedthrough multiplication by a window function that differs significantlyfrom zero only within a small neighborhood around q. This product isthen Fourier-transformed, in order to give the “local frequency content”of the function at that region. The windowed Fourier transform isthen a function of q, the central position of the window, and p, theFourier variable. Mathematically, the windowed Fourier transform can be written as

˜ f w(q, p)

=exp iK q · p

2 |K |

2π

N /2

×

f (q)w(q − q) exp(−iK q · p) d N q, (36)

where w(q) is the window function, assumed to be significantly differentfrom zero only within a finite region around q = 0. For simplicity,we consider only real window functions. Notice that we chose toinclude a linear phase factor outside the integral in Eq. (36), so that thecorresponding expression for this transformation in terms of the Fourier




transforms of f and w has a similar form:

˜ f w(q, p) = exp

−iK q · p

2

|K |2π

N /2

×

˜ f (p)w(p − p) exp(iK q · p) d N p. (37)

Note that we can also write Eq. (36) as the overlap of f with a

phase-space-shifted version of the window function by using thephase-space-shift operation defined in Eq. (35):

˜ f w(q, p) = |K |

2π

N /2 {T (q, p)w}∗(q) f (q) d N q, (38)

The windowed Fourier transform is linear in f and complex. For manypractical applications, it is more useful to use the squared modulus of this transformation, referred to in the signal analysis community as thespectrogram:

S f ,w

(q, p)= |˜

f w

(q, p)|

2

= |K |

2π

N

f (q)w(q − q) exp(−iK q · p) d N q

2

. (39)

The spectrogram is explicitly real and non-negative. Notice that thephase factor inserted outside the integral in Eq. (36) has no effect on thespectrogram.

In order to visualize the definition of the windowed Fourier transformand the spectrogram, consider a one-dimensional ( N = 1) function given by the sum of two identical Gaussians, one shifted by the action of

T (q0, p0) and multiplied by a constant phase factor exp(i):

f (q) = f 1(q) + f 2(q), (40a)

f 1(q) = π 1/4 exp

−q2

2

, (40b)

f 2(q) = exp(i){T (q0, p0) f 1}(q). (40c)

Consider the case q0 = 8, p0 = 4 and = 0. Also, let us use K = 1

for simplicity, and employ a Gaussian window, w(q) = π−1/4 exp(−q2/2).The first row of the movie frame (Media 1) in Fig. 1 shows the real (blue)and imaginary (red) parts of f (q) for q ∈ [−16, 16]. The second rowshows the window function w(q − q) over the same range, for varyingq. The third row shows the products of each of the curves in the firstrow with that in the second row, i.e., the real and imaginary parts of alocal detail of f isolated by the window function. The spectrogram for thecorresponding value of q, given by the squared modulus of the Fouriertransform of the function in the third row, is shown in the fourth row. Itis clear that the spectrogram has significant contributions only when thewindow overlaps the regions where f is significant.

The movie frames (Media 2 and Media 3) in Figs. 2 and 3 presentthe spectrogram of this function over phase space for several values


http://www.opticsinfobase.org/aop/viewmedia.cfm?uri=aop-3-4-272-1








Figure 1

q'

q'

q'

p

Illustration of the definition of the spectrogram. The first row shows the real

(blue) and imaginary (red) parts of the function in Eqs. (40) for q ∈ [−16, 16].

The second row shows the Gaussian window function w(q − q) over the same

range, for varying q. The third row shows the products of each of the curves in

the first row with that in the second row. The fourth row shows the spectrogram

for the corresponding value of q, given by the squared modulus of the Fourier

transform of the function in the third row. In the movie ( Media 1), q varies from−12 to 12.

of q0, p0 and , for a rectangular window w(q) = rect(2q)/√

2 (which

equals 1/√

2 for |q| ≤ 1 and vanishes otherwise) and for a Gaussianwindow w(q) = π−1/4 exp(−q2/2), respectively. The blue dot indicatesthe coordinates (q0, p0). Unless both q0 and p0 are of the order of unityor smaller, the spectrogram is composed of two separate and similardistributions, one centered at the origin (associated with f 1) and one at

(q0, p0) (associated with f 2). This example shows that the action of T is

indeed to shift a distribution in phase space by the amounts specified by its two arguments. The movies also show that the effect of thephase between the two separate contributions on the spectrogramis negligible except when the two contributions are so close that theyoverlap significantly.

One drawback of the windowed-Fourier transform and the spectrogramas representations of local frequency content is that they do not have aunique definition, since they depend on the choice of a window function.In some cases, the window function is dictated by the physics of theproblem, as we will discuss later. In general, though, the choice of window function limits the range of frequencies that can be described.For this reason, it can be useful to have a phase-space distribution thatdoes not rely on the choice of an ancillary function. One such distributionis the Wigner function, described in the next section.

4. The Wigner Function

We now introduce the Wigner function as a way to represent the function f jointly in terms of both q and p, without the need for an ancillary






Figure 2

Spectrogram of the superposition of two Gaussians in Eqs. (40) over the rangeq ∈ [−4, 10], p ∈ [−4, 10], for a square window function. Here, black indicates

zero, and brighter shades of green indicate higher values. The blue dot indicates

the coordinates (q0, p0). The movie (Media 2) shows, for (q0, p0) = (2, 2), the

visible effect of varying the phase difference over a complete cycle. Then, q0

is varied from 2 to 8, and this is followed by a similar variation in p0. Finally,

the phase is varied again over a complete cycle, even though the effect of this

variation is not noticeable due to the separation of the two components.

function:

W f (q, p) = |K |

2π

N f ∗

q − q

2

f

q + q

2

exp(−iK q · p) d N q. (41)

(The more general definition of the Wigner function, applicable tostochastic functions, will be given in Section 5.) It is easy to show thatthe Wigner function can be written in terms of the Fourier transform of f

in a very similar form:

W f (q, p) = |K |

2π

N ˜ f ∗

p − p

2

˜ f

p + p

2

exp(iK q · p) d N p. (42)

The definition of the Wigner function in Eq. (41) is illustrated for the case N = 1 in Media 4 (Fig. 4), for the same function as in Fig. 1, namely, thefunction in Eqs. (40) with (q0, p0) = (8, 4) and = 0. The first and secondrows show plots, respectively, of f (q+q/2) and f ∗(q−q/2) as functions of q ∈ [−16, 16]. The value of q is varied from −12 to 12 in Media 4. In theseplots, the real and imaginary parts are shown as blue and red curves,respectively. Note that, since they are plotted as functions of q, thesefunctions are stretched by a factor of 2. The third row shows the productof these two functions, and the fourth row shows the Fourier transform


http://www.opticsinfobase.org/aop/viewmedia.cfm?uri=aop-3-4-272-s2








Figure 3

Spectrogram of the superposition of two Gaussians in Eqs. (40) over the rangeq ∈ [−4, 10], p ∈ [−4, 10], for a Gaussian window function. In the movie

(Media 3), the variation of , q0 and p0 isas in Fig. 2. Notice that here the bumps

representing the two Gaussian contributions are more symmetric and localized

than in Fig. 2.

of this product, corresponding to the Wigner function as a function of p

for the specified value of q. A comparison of the graphic representationin Fig. 4 with that for the spectrogram in Fig. 1 suggests that, in the caseof the Wigner function, f ∗(q − q/2) is somewhat analogous to a window

function acting on f (q+q/2). Note that the product f (q+q/2) f ∗(q−q/2)has the same local frequency (i.e., the same rate of change of the phase interms of q) for small q as f (q + q). Note also that, for this choice of f , theWigner function is significant in three regions: when the contributionsdue to f 1 in the first two rows overlap (i.e., for q ≈ 0), when thecontributions due to f 2 in the first two rows overlap (i.e., for q ≈ q0),and when the contributions due to f 1 in one row overlap with those dueto f 2 in the other row (for q ≈ q0/2). This third contribution (visible in thestatic frame of Fig. 4) will be discussed in more detail later.

Comprehensive summaries of the properties of the Wigner function aregiven in Refs. [1,8–14]. Here we concentrate on a few of them that are of particular interest for the applications mentioned in what follows.

4.1. Reality and Possible Negativity

The Wigner function is always real, because it is the Fourier transformover q of the Hermitian function f (q+q/2) f ∗(q−q/2). That is, changingthe sign of the variable of integration q in the integrand in Eq. (41) isequivalent to complex conjugation, so the Wigner function equals itscomplex conjugate. It must be noted, however, that the Wigner function






Figure 4

q'

q'

q'

p

Illustration of the definition of the Wigner function (Media 4). The first and

second rows show plots, respectively, of f (q + q/2) and f ∗(q − q/2) (real parts

in blue, imaginary parts in red) as functions of q ∈ [−16, 16] for different

values of q. The third row shows the product of these functions, and the fourth

row shows the Fourier transform of this product, equal to the Wigner function

for the corresponding value of q.

is not necessarily non-negative; for example, the Wigner function plottedin the fourth row of Fig. 4 presents positive and negative oscillationswhen q ≈ q0/2 = 4. The possible negativity of the Wigner function forsome values of its arguments is sometimes seen as a conceptual problemwhen trying to ascribe to it a literal physical meaning as a power orprobability density. However, as will be discussed later, these negativevalues are needed for the description of coherent effects. Note also thatmultiplying f (q) by a global constant phase factor does not change theresulting Wigner function, given its bilinear definition.

4.2. Marginals

Among the main properties of the Wigner function are the so-calledmarginal relations: the integral over one set of variables of the Wignerfunction gives the square modulus of the function in the representationassociated with the remaining variable:

W f (q, p) d N p = | f (q)|2, (43a)

W f (q, p) d N q = |˜ f (p)|2. (43b)

In fact, more general marginal properties exist. For example, considera projection in a “rotated” direction in the phase space (q, p). Sincein general the physical parameters q and p have different units, weintroduce a positive scaling constant σ with units of

√ q/ p. Then,

the marginal projection over the rotated direction in each (σ −1qi, σ pi)

subspace at an angle θ from the σ pi axis gives the squared modulus of






the fractional Fourier transform defined in Eq. (7):

W f [σ (s cos θ − t sin θ ),σ −1(t cos θ + s sin θ )] d N t = | f θ (s)|2, (44)

so that the marginals in Eqs. (43a) and (43b) correspond to θ = 0 andθ = π/2, respectively. This projection property was found independently by Mustard [31] and Lohmann [32], and it is sometimes referred

to as the Radon–Wigner transform [33,34]. (A more general relationwas found by Bertrand and Bertrand [35].) As will be discussed inSection 10, this marginal projection property is the basis for a techniquecalled phase-space tomography, used in classical optics in both theposition–direction [36] and time–frequency [37] phase spaces, as wellas in quantum optics [35,38,39]. In fact, the formulas for calculating theWigner function in Eqs. (41) and (42) can be generalized in terms of thefractional Fourier transform as

W f [σ (s cos θ − t sin θ ),σ −1(t cos θ + s sin θ )]

= |K |2π

N

f ∗θ s − s

2 f θ s + s

2 exp(−iK s · t) d N s. (45)

Of course, different angles can be used for each Cartesian componentin the last two equations, providing a connection between theWigner function and the anamorphic fractional Fourier transform. Thisconnection is the basis of a generalization of phase-space tomography tofour-dimensional phase space [36], also discussed in Section 10.

From any of the marginal relations given earlier, it is easy to see that theintegral of W f over all phase space gives the squared norm of f :

W f (q, p) d N pd N q =

| f (q)|2d N q. (46)

The right-hand side of this equation is strictly positive (except in thetrivial case f = 0). Therefore, while the Wigner function might benegative at some regions of phase space, the integral over all positiveregions outweighs that over all negative regions. That is, the Wignerfunction is globally “more positive than negative.”

4.3. Linear Mapping under Linear Canonical Transformationsand Phase-Space Shifts

The Wigner representation for the result of a linear canonical transforma-

tion on a given function is just a simple mapped version of the Wignerrepresentation of the original function, namely,

W C(S) f

(q, p) = W f (Aq + B

p, Cq + D

p), (47a)

where A, B, C, and D are the submatrices of S−1. This relation can bewritten in the equivalent form

W f (q, p) = W C(S) f

(Aq + Bp, Cq + Dp). (47b)




Figure 5

(a) (b)

The effect of a fractional Fourier transformation (with σ = 1) is a clockwise

rotation of the Wigner function by an angle θ . In the static frame, (a) shows

the Wigner function of the original function, and (b) shows the Wigner function

following a transformation with θ = π/3. The range shown in both cases is

[−4, 4] for both p and q. In Media 5, θ varies from 0 to π .

Given the fact that the Jacobian between the arguments (Aq + Bp, Cq +Dp) and (q, p) is given by Det(S) = 1, this mapping preservesphase-space volume. We now discuss the form that this mapping takesfor the particular cases described in Subsection 2.3. In the graphicexamples, we consider a one-dimensional function ( N = 1) and use K = 1

for simplicity. The function we will use is a Gaussian, given by

f (q) =√

2 π 1/4 exp(−2q2), (48)

for which the Wigner function is given by

W f (q, p) = exp

−4q2 − p2

4

. (49)

In all plots, a grid is overlaid over phase space to better illustrate thegeometrical transformation.

4.3a. Fractional Fourier Transformation

The substitution of Eq. (24) into Eqs. (47) gives

W f θ (s, t) = W f [σ (s cos θ − t sin θ ),σ −1(t cos θ + s sin θ )], (50a)

W f (q, p) = W f θ (σ −1q cos θ + σ p sin θ, σ p cos θ − σ −1q sin θ ). (50b)

Therefore, fractional Fourier transformation can be associated with aclockwise rotation (after the appropriate scaling of phase space) by anangle θ of the Wigner function in phase space. This is illustrated Media 5(Fig. 5) for the Wigner function in Eq. (49) with σ = 1.









Figure 6

(a) (b)

The effect of a scaling on the Wigner function. In the static frame, (a) shows

the Wigner function of the original function, and (b) shows the Wigner function

following a transformation with b = 1.4. The range shown in both cases is

[−4, 4] for both p and q. In Media 6, b varies from 1 to 1.4.

4.3b. Scalings

Similarly, a scaling transformation causes a phase-space volume-preserving scaling of the Wigner function:

W S b f

(q, p) = W f (q/b, bp). (51)

This mapping is illustrated in Media 6 (Fig. 6). For anamorphic scalings(for N ≥ 2), each component of q and p is divided and multiplied,respectively, by the corresponding scaling factor.

4.3c. Fresnel Transformation

Fresnel transformation corresponds to a horizontal shearing of theWigner function:

W Hτ f

(q, p) = W f (q − τ p, p). (52)

(We chose the symbol Hτ for this transformation to stand for“horizontal.”) This shearing is illustrated by Media 7 (Fig. 7).

4.3d. Chirping

Similarly, chirping corresponds to a vertical phase-space shearing of theWigner function:

W V τ f

(q, p) = W f (q, p − τ q). (53)

(The symbol V τ here stands for “vertical.”) This vertical shearing isshown in Media 8 (Fig. 8).













Figure 7

(a) (b)

The effect of a Fresnel transformation is a horizontal shearing of the Wigner

function. In the static frame, (a) shows the Wigner function of the original

function, and (b) shows the Wigner function following a transformation with

τ = 1. The range shown in both cases is [−4, 4] for both p and q. In Media 7, τ

varies from 0 to 1.

Figure 8

(a) (b)

The effect of chirping is a vertical shearing of the Wigner function. In the static

frame, (a) shows the Wigner function of the original function, and (b) shows the

Wigner function following a transformation with τ = 1. The range shown in

both cases is [−4, 4] for both p and q. In Media 8, τ varies from 0 to 1.

4.3e. Phase-Space Shifts

A similar simple phase-space rearrangement occurs in the case of

phase-space shifts: the Wigner function of {T (q0, p0) f }(q) is given by

W T q0,p0

f (q, p) = W f (q − q0, p − p0). (54)

That is, the Wigner function is shifted rigidly in phase space, asillustrated by Media 9 in Fig. 9. Note that using a different ordering


http://www.opticsinfobase.org/aop/viewmedia.cfm?uri=aop-3-4-272-s007.MOV









Figure 9

(a) (b)

The effect of a phase-space shift transformation is precisely to shift the Wigner

function in phase space in the specified amounts q0, p0. In the static frame, (a)

shows the Wigner function of the original function, and (b) shows the Wigner

function following a transformation with (q0, p0) = (1.2, 2). The range shown

in both cases is [−4, 4] for both p and q. Media 9 shows displacements from

(0, 0) to (0, 3) to (3, 3).

convention in the definition of a phase-space shift would only lead toa global constant phase factor that would have no effect on the Wignerfunction.

4.4. Inner Product

Suppose that we have two arbitrary square-integrable functions f (q)

andg(q)

. The integral over all phase space of the product of theircorresponding Wigner functions is always a non-negative real number,given by

W f (q, p)W g(q, p) d N qd N p =

|K |2π

N

g∗(q) f (q) d N q

2

. (55)

In quantum mechanics, if f is the wave function for the state of a systemand g is that for a measuring instrument, then the expression on theright-hand side of this equation is related to the result of a measurementof the wave function with the corresponding instrument [40,41].A similar interpretation exists for the classical measurement of a

monochromatic wave field [42–45]. In this context, the left-hand sideof Eq. (55) has an intuitive interpretation, as will be discussed inSubsection 7.5.

Notice that for f = g, we get

W 2 f (q, p) d N qd N p =

|K |2π

N | f (q)|2d N q

2

. (56)






4.5. Superposition

Given the bilinear character of this transformation, the Wigner functionof the sum of two functions, f = f 1 + f 2, is not, in general, the sum of thecorresponding two Wigner functions. Instead we get

W f 1+ f 2 (q, p) = W f 1 (q, p) + 2[W f 1, f 2 (q, p)] + W f 2 (q, p), (57)

where W f 1, f 2 is the (complex) cross-Wigner function defined as

W f 1, f 2 (q, p) = W ∗ f 2, f 1(q, p)

= |K |

2π

N f ∗1

q − q

2

f 2

q + q

2

exp(−iK q · p) d N q.

(58)

Note that this interference term can be highly oscillatory if W f 1 and W f 2

are localized at different regions of phase space. Consider, for example,the functions defined in Eqs. (40) . The three contributions to the Wignerfunction are easily shown to be given by

W f 1 (q, p) = exp(−q2 − p2), (59a)

W f 2 (q, p) = exp[−(q − q0)2 − ( p − p0)2], (59b)

2[W f 1, f 2 (q, p)] = 2 exp

−

q − q0

2

2

− p − p0

2

2

× cos (qp0 − q0 p + ) . (59c)

That is, the interference term is located half-way between thecontributions for the two independent Gaussians, and its oscillationfrequency increases linearly with the separation of the two contributions.The phase of the oscillations depends on the relative phase () betweenthe two contributions. The origin of this interference contribution can be appreciated from Media 4 (Fig. 4): the main contributions to theWigner function occur not only for values of q at which a significantpart of f overlaps with itself within the two top rows of that figure, but also when one significant part of f in the first row overlaps witha different significant part in the second row. At these values of q, theproduct shown in the third row of the figure is composed of two identical(up to an inversion and complex conjugation) contributions, roughlyseparated by the same distance as that between the two contributionsof f . The Fourier transform of this product (i.e., the Wigner function) isthen oscillatory, with frequency proportional to this distance.

The dependence of the Wigner function on the location of the second

contribution and the phase difference between the two contributions isillustrated in Media 10 (Fig. 10) for the same variation in q0, p0, and

as in Figs. 2 and 3. It is seen there that the oscillation fringes are parallelto the straight line joining the two contributions. The fringe structureindicates the phase between the two contributions, regardless of theirseparation: the central fringe is positive if the two contributions are inphase, and negative if they are completely out of phase. The role of thesefringes is clear from considering the marginal properties in Eq. ( 43a)and (43b), shown in Fig. 11(a) for the case f (q) = π 1/4 exp[−(q + 4)2/2]








Figure 10

0

1

–1

Wigner function of the superposition of two Gaussians in Eqs. (40) over the

range q ∈ [−4, 10], p ∈ [−4, 10]. The blue dot indicates the coordinates of (q0, p0). Media 10 shows, for (q0, p0) = (2, 2), the effect of varying the phase

difference over a complete cycle. Then, q0 is varied from 2 to 8, and this is

followed by a similar variation in p0. Finally, the phase is varied again over a

complete cycle.

+ π 1/4 exp[−(q − 4)2/2]: the interference term cancels almost entirelyfollowing integration in p, leading to a marginal distribution with twomaxima at ±q0; on the other hand, the interference term is responsiblefor the oscillations of the marginal in q. That is, the negative valuesof the Wigner function are needed to account for coherent effects likedestructive interference. Figure 11(b) shows the Wigner function of asuperposition of three Gaussians, one of them multiplied by a linearphase factor in order to displace it from the q axis. Notice that there is aninterference term between any two contributions, and that the frequencyof oscillations increases with the distance between them.

It was discussed in Subsection 4.4 that the effect of a linear canonicaltransformation and/or phase-space shift is a simple linear mappingof the Wigner function over phase space. These mappings are theonly ones that not only preserve phase-space area, but also map anystraight line in phase space onto another straight line. In other words,these mappings are the only ones that guarantee that the half-pointinterference contributions of the Wigner function remain as half-points

following the transformation. Therefore, transformations that are notcombinations of linear canonical transformations and phase-space shiftscannot correspond to mappings of the Wigner function in general. Noticethat following a linear canonical transformation and/or phase-spaceshift, the fringes of an interference contribution remain aligned withthe straight line joining the two contributions giving rise to it, and anincrease/decrease in the distance between two contributions brings withit the corresponding decrease/increase in the transverse spacing of thefringes, given the constraint of phase-space area conservation.






Figure 11

p

q

(a)

(b)

p

q

Wigner function of (a) the sum of two Gaussians of unit width, given by

f (q) = π 1/4 exp[−(q + 4)2/2] + π 1/4 exp[−(q − 4)2/2], and (b) the sum of

three Gaussians of unit width, corresponding to f (q) = π 1/4{exp[−(q−4)2/2]+exp[−(q + 4)2/2]+ exp[−(q − 4)2/2] exp(i5q)}. Both q and p are shown within

the interval [−8, 8]. The marginal projections given by | f (q)|2 and |˜ f ( p)|2 are

shown on the top and left margins, respectively.

4.6. Oscillatory Character of the Wigner Function

The interference terms described in Subsection 4.5 can cause the Wignerfunction to be highly oscillatory within some regions of phase space.From a computational point of view, this sometimes constitutes a majorshortcoming of this representation: the Wigner function represents afunction of N variables in terms of a potentially complicated, highlyoscillatory function of 2 N variables. Therefore, except for a limited

number of cases that admit of an analytic description, the numericalmodeling of a system by using the Wigner function requires bothdoubling the number of variables over which to sample and potentiallyincreasing significantly the sampling rate in each direction in order to beable to resolve the fast oscillations.

The problem with oscillations is particularly critical for functions whoseWigner functions extend over phase space regions much larger than theminimum-uncertainty-limit cell size, |2K |− N . One such case of particularrelevance in optics and quantum mechanics is that of functions of theform f (q) = A(q) exp[iK (q)], where both A and are real, and A variesslowly compared to the oscillations of exp[iK (q)]. In this case, onewould expect the Wigner function to be localized in phase space around

the N -dimensional manifold described by the equation

p = ∂

∂q. (60)

For an analytic time signal (where K = −1, q = t , and p = ω)this expression corresponds to the so-called instantaneous frequency. Inclassical wave optics (where K = k ) and quantum mechanics (whereK = 1/h), this type of function can be associated with the ray-optical




Figure 12

(a) (b)

p

q

p

q

(a) Plot of the phase-space curve p = (q) (black line) as well as several

half-scaled replicas p = (q0)/2 − (2q − q0)/2 (green, red, and purple

lines), for (q) = q5 − 4q3 and different values of q0. The anchor point

of the scaling (indicated by small circles in the case of the red and purple

curves) is the point along the curve

[q0, (q0)

]where the replica and the curve

coincide. This construction is also illustrated in Media 11. (b) Wigner functionfor f (q) = A(q) exp[iK (q)], with A(q) = exp[−(q/4)4] and K = 2. Notice

the correlation between the areas presenting oscillations and those occupied by

the scaled replicas in (a). In both figures, the ranges shown are q ∈ [−2, 2]and p ∈ [−10, 10]. Also shown in part (a) is a straight line segment joining

two anchor points (for the red and purple lines), whose midpoint coincides with

the intersection of the two corresponding replicas. At this point, also shown in

part (b) as a yellow dot, the fringes of the Wigner function are approximately

parallel to this line segment, and the phase of the oscillations is proportional

to K times the phase-space area enclosed by this segment and the phase-space

curve p = (q). In the region below the phase-space curve between the two

minima, the pattern is more complicated, since each point is a half-point for

two pairs of points, i.e., four replicas cross each point. Therefore the pattern isformed by the interference of two sets of fringes.

and classical limits, respectively. The phase function then corresponds tothe Eikonal in optics or the action in mechanics, and its derivative is thestandard definition of (optical or mechanical) momentum.

As described in detail by Berry and collaborators [46–48], W f does indeedpresent a positive ridge structure that mimics the manifold defined by Eq. (60), but due to the interference contributions discussed in the

previous section, it also displays an intricate pattern of fast oscillationsinvolving large values, both positive and negative, at many other regionsof phase space. These regions are composed of all phase space locationsthat are midway between any two points along the manifold. Letus consider the N = 1 case for simplicity, and use the constructionproposed in Ref. [49] to visualize the location of these contributions.Consider a plot of the curve p = (q), shown as a black curve inFig. 12(a). Now superpose on this plot a series of one-half-scale replicasof the same curve, where, for each replica, the anchor point for the







scaling is a point along the curve, as illustrated in this figure and theaccompanying movie. Let the density of these replicas be proportional to| A(q)|2, where q is the location of the anchor point. The region occupied by these replicas indicates the region where the Wigner function issignificant, as can be seen from the comparison of this picture with theWigner function in Fig. 12(b). As explained in Ref. [49], the amplitude,phase, and orientation of these oscillations can be inferred from thisgeometrical construction: the amplitude is proportional to the density

of replica intersections. The phase of the oscillations, on the other hand,is proportional to K times the area enclosed by the phase space curveand a straight line joining the anchor points of the two replicas crossingat the corresponding point, and the local direction of the fringes isapproximately parallel to this line, as shown in Fig. 12. The complexityof the pattern depends on the geometry of the curve; for example, morecomplicated patterns result for curves including inflection points (likethe one shown in Fig. 12), for which more than one pair of points alongthe curve can have their half-point at the same phase-space location.

The same type of construction can be used for functions associatedwith phase-space manifolds that are not single-valued in q. Considerthe simplest such case, given by a one-dimensional Hermite–Gaussian

function of the form

f (q) = 1

π 1/4√

2nn! H n(q) exp

−q2

2

, (61)

where H n is an nth-order Hermite polynomial. For simplicity, we considerK = 1. These Hermite–Gaussian functions are associated with ellipticcontours in the (q, p) phase space, described by the equation

q2 + p2 = 2n + 1. (62)

The Wigner representation of this function can be found to be given by

W f (q, p) = (−1)n

π Ln[2(q2 + p2)] exp(−q2 − p2), (63)

where Ln is an nth-order Laguerre polynomial. That is, with theappropriate scaling of phase space, the Wigner function is formed of concentric fringes, where n corresponds to the number of zeros in theradial direction. As shown in Fig. 13, this result is consistent with thereplica-based construction described above. In particular, the amplitudeof the oscillations is related to the density of replica intersections, with amaximum at the origin where all replicas cross.

5. Partial Coherence

5.1. Correlations and Coherent Mode Expansions

An advantage of bilinear representations like the Wigner function andthe spectrogram over linear ones like the windowed Fourier transform isthat they can be used to represent not only deterministic functions f (q), but also the bilinear autocorrelation of stochastic functions at two values




Figure 13

q

p(a) (b)

(a) Plot of the phase-space curve q2 + p2 = 2n+ 1 (black line) as well as several

half-scaled replicas of this curve (green lines). (b) Wigner function in Eq. (63)

for n = 8, with the curve q2 + p2 = 2n + 1 overlaid (white line).

of their argument, i.e.,

F (q1, q2) = f ∗(q1) f (q2), (64)

where the angular brackets denote a correlation or average (over anensemble or over time). Note that F is Hermitian, i.e., F (q1, q2) =F ∗(q2, q1). Correlations of this type are used, for example, in quantummechanics and quantum optics to describe mixed states, or in classicaloptics to describe partially coherent fields.

A useful representation of F is given in terms of its eigenvalues n andeigenfunctions f n(q), resulting from considering the integral Fredholmequation

f n(q)F (q, q) d N q = n f n(q). (65)

Because F is Hermitian, its eigenvalues n must be real, and itseigenfunctions f n(q) can be made to be orthonormal and form a completeset:

f ∗n (q) f n(q) d N q = δn,n , (66)

n

f ∗n

(q) f n

(q)=

δ(q −

q). (67)

(Note that, for many of the eigenfunctions, the eigenvalue might be zero.)By multiplying both sides of Eq. (65) (with q = q2) by f ∗n (q1), summingover all n, and using Eq. (67), F can be written as a diagonal sum knownas a Mercer or coherent mode expansion [50]:

F (q1, q2) =

n

n f ∗n (q1) f n(q2). (68)




In addition to being real, the eigenvalues n must be non-negative inorder to guarantee that F (q, q) =

nn| f n(q)|2 be non-negative for any q,regardless of the functional form of the eigenfunctions f n.

5.2. Wigner Function and Spectrogram for Mixed States

The Wigner function for a partially coherent field or mixed state

described by F is given by

W F (q, p) = |K |

2π

N f ∗

q − q

2

f

q + q

2

exp(−iK q · p) d N q

= |K |

2π

N F

q − q

2, q + q

2

exp(−iK q · p) d N q. (69)

It must be noted that the original definition by Wigner [2] was preciselyfor mixed quantum-mechanical states, so Eq. (69) is the general definitionof the Wigner function. Similarly, the spectrogram for a mixed state isgiven by

S F ,w(q, p) = |K |

2π

N

F (q1, q

2)w(q1 − q)w(q

2 − q)

× exp[−iK (q2 − q

1) · p] d N q1d N q

2, (70)

where it was assumed that the window function is real. By using thecoherent mode superposition, it is easy to see that the Wigner functionand spectrogram can be written as a weighted superposition of thecorresponding representations for the coherent modes:

W F (q, p) =

n

nW f n (q, p), (71)

S F ,w(q, p) =n

nS f n,w(q, p). (72)

5.3. Overall Degree of Purity

A measure of the overall level of coherence or purity of a field is given by [51]

ν =

n 2n

1/2

n n

= |F (q1, q2)|2 d N q1d N q2

1/2

F (q, q) d N q

. (73)

Notice that, if the field were composed of m equally weighted coherentmodes, this measure would be ν = m−1/2. Therefore, 1/ν2 can be thoughtof as a measure of the effective number of significant coherent modes inF [52]; ν = 1 implies that there is only one coherent mode, that is, that Fcan be factored out as the product f ∗(q1) f (q2). The analog of Eq. (56) formixed states is given by

W 2F (q, p) d N qd N p =

|K |2π

N |F (q1, q2)|2 d N q1d N q2, (74)




so the degree of purity can be written in terms of the Wigner function as

ν =

2π

|K | N /2

W 2F (q, p) d N qd N p

1/2 W F (q, p) d N qd N p

. (75)

5.4. Quasi-homogeneous Fields

As will be discussed later, there is a type of optical field known asquasi-homogeneous, for which the following factorization exists for thecorrelation:

F QH(q1, q2) = F 0

q1 + q2

2

µ(q2 − q1), (76)

where F 0 is real and non-negative, and µ is dimensionless and mustsatisfy the properties

|µ(q)| ≤ µ(0) = 1, µ(q) = µ∗(−q). (77)

Also, if µ is significantly different from zero only over a range of valuesof its arguments of size qc (the coherence width), then F 0 must haveinsignificant variation over changes of its arguments that are of the orderof qc. Note that, for this type of field, the overall degree of purity can bewritten as

ν =

F 20(q) d N q |µ(q)|2 d N q

1/2 F 0(q) d N q

, (78)

which is much smaller than unity, given that F 0 is at least as wide as

F 20 , and both are much wider than |µ|2. Also, the Wigner function for aquasi-homogeneous field takes the simple factorized form

W F QH(q, p) =

|K |2π

N /2

F 0(q)µ(p), (79)

where µ is the Fourier transform of µ. Note that µ is real, given thesecond relation in Eq. (77). Since µ is much narrower than F 0, µ is much

wider than F 0, so W F QHis significantly different from zero over a region of

phase space much larger than the minimum uncertainty cell size, |2K |− N .

5.5. Smoothness of the Wigner Function for Highly MixedStates

As discussed in Subsection 4.6, the Wigner function of a deterministicfunction can be highly oscillatory and include many negative regions.However, for a significantly mixed state involving a very large number of coherent modes (i.e., for ν 1), the Wigner function tends to be smoothand non-negative. While not a rigorous proof, we now give an argumentof why this is the case, based on the following two observations: first,the Wigner function of each coherent mode of a mixed state can oscillate




between positive and negative values, although overall it is “morepositive than negative” given the combination of Eqs. (46) and (66):

W f n (q, p) d N qd N p = 1. (80)

Second, from Eqs. (55) and (66), the following relation holds for n = n:

W f n (q, p)W f n (q, p) d N qd N p = |K |

2π

N f ∗n (q) f n (q) d N q

2

= 0. (81)

This relation implies that the positive and negative regions of the Wignerfunction of any mode cannot consistently coincide with those for anyother mode, although there must be a correlation or alignment of theseregions that guarantees that the overlap integral of the Wigner functionsvanishes. Given these two facts, the positively weighted superposition of the Wigner functions of a very large number of coherent modes is likelypositive at any given phase-space point. Also, the oscillations for eachmode average out following superposition, and the resulting Wigner

function tends to be smooth.To illustrate this argument, consider a Gaussian correlation of the form

F (q1, q2) = π 1/2

σ 2exp

−(q1 + q2)2

4σ 21

exp

−(q2 − q1)2

4σ 22

, (82)

where σ 1 ≥ σ 2. Let K = 1, for simplicity. The Wigner function for thiscorrelation can be found easily to be

W F (q, p) = exp− q2

σ 2

1

− σ 22 p2

. (83)

The Mercer expansion in Eq. (68) for this correlation can be found [53] tohave the eigenvalues of the form

n = 2π 1/2σ 1σ 2

σ 1 + σ 2

σ 1 − σ 2

σ 1 + σ 2

n

, (84)

as well as eigenfunctions that are scaled versions of the Hermite–Gaussfunctions in Eq. (61), that is

f n(q)

= π

σ 1σ 2

1/41

√ 2nn! H n q

√ σ 1σ 2 exp−

q2

2σ 1σ 2 . (85)

Therefore, the Wigner representation can be written as a weightedsuperposition of the Wigner functions of the modes of the form inEq. (71), where, according to Eqs. (63) and (51), the Wigner function of each mode is given by

W f n (q, p) = (−1)n Ln

2

q2

σ 1σ 2+ σ 1σ 2 p

2

exp

− q2

σ 1σ 2− σ 1σ 2 p

2

. (86)




Figure 14

n 10

q

p

q

p

(a) (b)

Frames from Media 12. (a) Wigner function of the nth coherent mode, given in

Eq. (86) for σ 1 = σ −12 = 4, i.e., ν = 1/4. (b) Partial sum up to n of these Wigner

functions, weighted by the factor in Eq. (84). The sum converges to the Wigner

function in Eq. (83). In both figures, the range shown for both q and p is [−8, 8].

Notice that the partial superpositions are free of negative regions.

The overall degree of purity can be found either from the substitution of Eq. (84) into Eq. (73) and the use of the geometric series formula, or fromthe substitution of Eq. (83) into Eq. (75). The result is simply

ν =

σ 2

σ 1, (87)

so σ 2 = σ 1 corresponds to a fully coherent Gaussian beam. This resultcan also be found by substituting Eq. (83) into Eq. (75). Notice that ν−2 =σ 1/σ 2 equals the ratio between the cross-sectional phase-space area of the Wigner function for F, given in Eq. (83), to that for a fully coherentGaussian function. That its, overall, the average relative variation of W F

in any phase-space direction is smaller by a factor of ν than that for theWigner function of a coherent Gaussian. Media 12 (Fig. 14) shows (a)the Wigner function for the nth mode of this expansion given in Eq. (86),and (b) the partial sum up to n of the superposition in Eq. (71). One canappreciate how the distribution becomes smoother as more modes areincluded.

It was argued in Subsection 4.6 that for “pure” or “coherent” systemsdescribed by a function f (q), the use of the Wigner function can beproblematic from a numerical point of view, due to the fast oscillationsand the doubling of the number of variables. However, for “mixed”states this is not the case. First, the correlation F depends on 2 N variables,as does the Wigner function. Second, as discussed earlier, the oscillationsof the Wigner function tend to wash out for significantly mixed states.








6. Other Phase-Space Representations

The Wigner function and the spectrogram are not the only represen-tations of a function in terms of both its original variables and theFourier conjugate ones. Extensive reviews of the many alternatives thathave been proposed are given in some of the references mentionedearlier [1,8,10–14,54]. Here, I just give a brief summary of some of thestandard representations and their properties.

6.1. Distributions Related to the Spectrogram

The similarity between Eqs. (55)and(39), together with the shift propertyin Eq. (54), means that the spectrogram is related to the Wigner functionaccording to

S f ,w(q, p) =

W f (q, p)W w(q − q, p − p) d N qd N p

= [W f ∗ W w](q, p), (88)

where the asterisk denotes convolution in both q and p. That is, thespectrogram is a “blurred” version of the Wigner function, where the“blur” is the Wigner function of the window. This relation also holdsfor the spectrogram and Wigner function of a mixed function F, since itholds for each of its coherent modes:

S F ,w(q, p) = [W F ∗ W w](q, p). (89)

One effect of this blurring is to smooth out the phase-space distribution,removing the fast oscillations and negative values sometimes presented by the Wigner function. In particular, it was mentioned earlier that forfunctions of the form f (q) = A(q) exp[iK (q)], where A varies slowlywithin the scale of the oscillations of the exponential, the Wigner functionhas a ridge structure around the manifold p = ∂/∂q, but also presentshigh oscillations within the concavities of this ridge. For these functions,such oscillations are not present in the spectrogram, which reduces to aridge localized around the manifold if an appropriate window functionis used, as will be shown later.

As mentioned earlier, the window function used in the definition of thespectrogram is sometimes dictated by the physics of the problem. Thisis the case, for example, of measurement processes that are described by an overlap integral like that in Eq. (55). Note that the definitionof the spectrogram in Eqs. (39) and (38) has this same form, wherethe measuring device is characterized by a wave function in the q

space corresponding to g(q) = {T (q, p)w}(q) ∝ w(q − q) exp(iK q · p).This “device” then makes a measurement of the process described by

f , focusing around the prescribed values q and p, but respecting thelimits established by the uncertainty relation. For example, considerthe measurement of a function of time through absorption by resonantvibration modes of the measuring device (e.g., our hearing); in thesimplest case where the mode can be modeled as a damped harmonicoscillator, the window function has the form w(t − t ) ∝ (t − t ) exp[γ (t −t )], where is the Heaviside step function and γ is the damping




coefficient. For a detailed treatment of the use of the spectrogram (andother distributions) to describe realistic measurements of stochastic timesignals, see Ref. [55].

In cases where the window function is not dictated by the physicalsituation, it can be chosen for mathematical convenience. For example,it is well known that the function that reaches the lower bound set bythe uncertainty relation is a Gaussian, whose Fourier transform is also aGaussian. Therefore, a Gaussian window minimizes the size of the blur

in phase space. When the window is chosen as a normalized Gaussian of width a, i.e.,

w(q) = π− N /4a− N /2 exp

− q2

2a2

, (90)

the spectrogram is referred to (particularly in the quantum-mechanicalliterature) as the Husimi function [56]:

H F ,a(q, p) = |K |

2π 3/2a

N F (q

1, q2) exp

−(q

1 − q)2 + (q2 − q)2

2a2

× exp[−iK (q2 − q1) · p] d N q1d N q2. (91)

Further, in systems where propagation or evolution is described byfractional Fourier transformation, as in the case of the quantumdescription of the optical field (see Subsection 10.1), the lowest-ordereigenstate of the system (the vacuum mode in quantum optics) is aGaussian centered at the origin and with a specific width, referred to hereas a0. When a Gaussian of this width is used as the window function, theHusimi function is referred to in the quantum optics literature as the Q function [57–59]:

QF (q, p) = H F ,a0(q, p). (92)

This distribution corresponds to the overlap of a quantum density matrixassociated with F and what is known as a coherent state. An associateddistribution also used in quantum optics is the Glauber–Sudarshan P function [60–62], defined mathematically such that its convolution withthe Wigner function of a Gaussian window of width a0 gives the Wignerfunction of F:

[W w ∗ PF ](q, p) = W F (q, p). (93)

It must be noted that, for pure states f and even for many mixedstates F, the P function is highly singular. It is therefore not oftenused for graphical representation over phase space of a function, but

instead as a theoretical tool for derivations involving the coherent moderepresentation of quantized fields.

6.2. The Kirkwood–Rihaczek and Margenau–Hill Distributions

Many other phase-space representations have been defined, includingthe Kirkwood–Rihaczek distribution [63,64], which for pure states is




given by

R f (q, p) = |K |

2π

N /2

exp(−iK q · p) ˜ f ∗(p) f (q). (94)

Unlike the other phase-space distributions described so far, R f is gener-ally complex. Its real part, sometimes referred to as the Margenau–Hilldistribution [65], is also used as a phase-space representation. Note

that, unlike the spectrogram (and its special cases, the Husimi andQ functions) and the P function, the Kirkwood–Rihaczek distribution(as well as its real part) satisfies marginal relations analogous to thosesatisfied by the Wigner function and given in Eqs. (43a) and (43b):

R f (q, p) d N p = | f (q)|2, (95a)

R f (q, p) d N q = |˜ f (p)|2. (95b)

However, the Kirkwood–Rihaczek distribution does not satisfy the

more general marginal property in Eq. (44). For mixed states, theKirkwood–Rihaczek distribution is given by

RF (q, p) = |K |

2π

N F (q, q) exp[iK (q − q) · p] d N q. (96)

6.3. The Ambiguity Function

Another commonly used representation that is defined in a phasespace is the ambiguity function [66–68]. However, this function is quitedifferent from those described earlier, since the phase space over which

it is defined is the Fourier conjugate of that for the Wigner function andother representations. The ambiguity function is the characteristic functionfor the Wigner function, i.e., its Fourier (or inverse Fourier) transformover all variables:

AF (q, p) = |K |

2π

N W F (q, p) exp[iK (q · p − q · p)] d N qd N p

= |K |

2π

N F

q − q

2, q + q

2

exp(−iK q · p) d N q. (97)

Instead of a marginal property, the ambiguity function of a pure statesatisfies the relations

A f (0, p) exp(iK q · p) d N p = | f (q)|2, (98a) A f (q, 0) exp(−iK q · p) d N q = |˜ f (p)|2, (98b)

and the analogous relations for mixed states. Note that, unless theWigner function is symmetric around two axes in each (qi, pi) subspace(following an appropriate scaling), the ambiguity function is complex.




The ambiguity function satisfies a phase-space mapping under linearcanonical transformations similar to that of the Wigner function. To seethis, consider the substitution of Eqs. (47) into the first expression inEq. (97):

A f (q, p) = |K |

2π

N W f (q, p) exp[iK (q · p − q · p)] d N qd N p

= |K

|2π N

W C(S) f (Aq + Bp, Cq + Dp)

× exp

iK (q, p)

O I

−I O

q

p

d N qd N p. (99)

By performing the change of variables (q, p)T = S(q, p)T whose Jacobian equals Det(S) = 1, this expression can be written as

A f (q, p) = |K |

2π

N W C(S) f

(q, p)

×expiK (q, p) O I

−I OS

−1 q

p d N qd N p

= |K |

2π

N W C(S) f

(q, p)

× exp

iK (q, p)S

T

O I

−I O

q

p

d N qd N p

= AC(S) f

(Aq + Bp, Cq + Dp), (100)

where Eq. (19) was used in the second step. On the other hand,phase-space shifts do not correspond to a rigid displacement of theambiguity function, but to multiplication by a linear phase factor.

6.4. The Cohen Class of Distributions

Except for the ambiguity function, all of the representations describedso far are members of the so-called Cohen class of phase-spacedistributions [54,69,70], which in terms of the ambiguity function can bedefined as

C F ,φ(q, p) = |K |

2π

N AF (q, p)φ (p, q)

×exp

[iK (q

·p

−q

·p)

]d N qd N p, (101)

where the kernel φ determines the representation. This kernel corre-sponds to the ratio of the characteristic function for the distribution inquestion to that for the Wigner function (i.e., the ambiguity function):

φ(p, q) = 1

AF (q, p)

|K |2π

N C F ,φ(q, p)

× exp[iK (q · p − q · p)] d N qd N p. (102)




The values of this kernel for several of the phase-space distributionsdescribed so far are given in Table 1. Note that, for all these distributions,the kernel is normalized so that φ(0, 0) = 1, which guarantees that theintegral of the distribution over all phase space is independent of thefunctional form of the kernel:

C F ,φ(q, p) dnqd N p =

F (q, q) d N q. (103)

Since all Cohen-class representations are bilinear, they all presentinterference contributions like those of the Wigner function. However,these effects can be suppressed, amplified, or relocated depending on thekernel, as can be seen from Eq. (101). Since the interference contributionsto the Wigner function are oscillatory, they contribute to the ambiguityfunction at regions away from the origin in the (q, p) space. If thekernel φ(q, p) decays away from the origin, as in the case of thespectrogram and its special cases, the Husimi and Q functions, then theseinterference terms are largely suppressed. For the P function, on the otherhand, for which the kernel grows extremely fast away from the origin,the interference terms are greatly enhanced, causing the distributionto be singular except in cases where AF decays sufficiently fast as to

overcome the kernel’s divergence. In the case of the Kirkwood–Rihaczekdistribution, the kernel is a phase factor with unit magnitude, so theinterference effects are of similar significance as those for the Wignerfunction, but shifted to a different region of phase space, as will bediscussed later.

Notice that, by using the convolution theorem, Eq. (101) states that allmembers of the Cohen class are given by the convolution over phasespace of the Wigner function with a blur function, i.e.,

C F ,φ(q, p) = [W f ∗ ](q, p), (104)

where the “blur” is proportional to the Fourier transform of the kernel:

(q, p) = |K |

2π

2 N φ (q, p) exp[iK (q · p − q · p)] d N qd N p,

(105a)

φ(q, p) =

(q, p) exp[iK (q · p − q · p)] d N qd N p. (105b)

Note from the second expression evaluated at the origin that the areaunder the blur is unity. In the case of the spectrogram and relateddistributions, the convolution in Eq. (104) is the one described in Eq. (89),so (q, p) = W w(q, p). However, in the case of the P function, is notwell defined. The blurs for the different Cohen-class distributions arealso presented in Table 1.

The inner-product property of the Wigner function in Eq. (55) is a specialcase of a more general relation for Cohen-class phase-space distributions,given by

C f ,φ1

(q, p)C g,φ2(q, p) d N qd N p =

|K |2π

N

g∗(q) f (q) d N q

2

,

(106a)




Table 1. Kernels and Blurs for Several Phase-Space Sistributions of the Cohen Class

Name Symbol Kernel φ(p, q) Blur (q, p)

Wigner W 1 δ(q)δ(p)

Spectrogram S

2π|K |

N Aw(q, p) W w(q, p)

Husimi H exp

− q2

4a2− K 2a2 p2

4

|K | N

π N exp

− q2

a2− p2

K 2a2

Q Q exp− q24a2

0

− K 2a20 p2

4 |K | N

π N exp− q2

a20

− p2

K 2a20

P P exp

q24a2

0

+ K 2a20 p24

?

Kirkwood–Rihaczek R exp(iK q · p) |K | N

(2π) N exp(iK q · p)

provided that

φ1(q, p)φ2(q, p) = 1. (106b)

That is, to calculate the inner product of two functions (and therefore todescribe measurement processes) in terms of their phase representations,dual pairs of representations must be used where the kernel of onedistribution is the inverse of the kernel of the other. Note that the Wignerfunction is therefore the only self-dual of these representations, whilethe P function is the dual of the Q function, and the Kirkwood–Rihaczekdistribution is the dual of its complex conjugate. Similarly, the momentsof these different representations correspond to average values of theoperators associated with the variables according to different orderingschemes, as discussed in detail in the reviews in Refs. [1,10,14].

To illustrate some of the distinctive features of these differentrepresentations, let us consider the function f (q) = exp(iKq3/12). Asmentioned in Subsection 4.6, it would be expected that the significantvalues of a phase-space distribution should be concentrated around thephase-space curve p = q2/4. Figure 15 shows (a) the Wigner function,

(b), (c) the Husimi function for two values of the width a, (d) themagnitude, and (e) the real part of the Kirkwood–Rihaczek distribution,all of which can be calculated analytically in terms of Airy functions. TheWigner function presents the characteristic interference contributionswithin the concavities of the curve. For the Husimi function, on the otherhand, these contributions are largely suppressed, except for some smalleffects near the vertex of the parabola in 15(c), because the width of the blur in q is not sufficiently narrow to resolve this vertex properly.Figures 15(d) and (e) illustrate the fact that it is the real part of theKirkwood–Rihaczek distribution (i.e., the Margenau–Hill distribution)and not its magnitude that holds some resemblance to the curve.However, even the real part of this distribution contains significant

oscillatory interference contributions over large regions of phase space.These regions correspond to the intersection of all values of q occupied

by f (q) and all values of p occupied by ˜ f (p). A second example isgiven in Media 14–16 (Fig. 16), which show the transformation of several of these distributions under fractional Fourier transformation fora sum of two Gaussians of the form f (q) = π 1/4{exp[−(q − 4)2/2] +exp[−(q + 4)2/2]}. Note that, of all the representations shown in thefigure, only the Wigner and the Q function rotate rigidly in phase spacewith θ . While the P function would also transform in this way, it is








Figure 15

(e)

q

(a)

p

p p

p p

q

q q

q

(c)(b)

(d)

(a) Wigner function, (b) Husimi function with a = 1, (c) Husimi function with

a = 2, (d) absolute value of the Kirkwood–Rihaczek function, and (e) real part

of the Kirkwood–Rihaczek function, for f (q) = exp(iq3/12) and K = 1. The

plot ranges are q ∈ [−8, 8] and p ∈ [−4, 12].

highly singular for this case and therefore not plotted. Notice also thecounterrotating interference contributions in the case of the real part of the Kirkwood–Rihaczek distribution.

Table 2 shows whether or not several of the Cohen-class distributions

transform as a simple mapping under different linear unitary transfor-mations. In the last column, the transformation G represents a generalunitary transformation that is not a combination of phase-space shiftsand linear canonical transformations.

Figures 15 and 16 illustrate the potentially very different appearancesof all the bilinear phase-space representations described so far for thecase of a pure function f (q). However, it is important to note thatthese differences tend to wash away in the case of mixed functions




Figure 16

p

q(a) (b)

(c) (d)

p

p p

q

q q

Frames from movies illustrating (a) Media 13 the Wigner function, (b) (Media

14) the Q function (i.e., the Husimi function with a = 1), (c) (Media 15)

the Husimi function with a = 2, and (d) (Media 16) the real part of the

Kirkwood–Rihaczek function, for the fractional Fourier transform of f (q) =π 1/4{exp[−(q − 4)2/2] + exp[−(q + 4)2/2]}, with σ = K = 1. The plot ranges

are q ∈ [−8, 8] and p ∈ [−8, 8]. In the animations, the degree of the fractional

Fourier transform varies from 0 to −π .

Table 2. Transformation Propertiesa of Several Cohen-Class Phase-Space

Distributionsb under Linear Unitary Transformationsc

Distribution F F θ S (b) H(τ ) V (τ ) C(S) T (q0, p0) G

W ×S × × × × × × ×

H × × × × × × ×Q × × × × ×P × × × × ×

R × × × × ×

a Here, indicates that the corresponding distribution transforms as a phase-space

mapping, while × indicates that it does not.b Wigner, spectrogram, Husimi, Q, P, and Kirkwood–Rihaczek.c Fourier transformations, fractional Fourier transformations, scalings, Fresnel transfor-

mations, chirpings, general linear canonical transformations, phase-space shifts, and moregeneral unitary transformations.

F (q1, q2): the more incoherent or mixed the function is, the more similarall these representations become. This is particularly easy to see in thequasi-homogeneous limit, where F satisfies the factorization in Eq. (76).















For such fields, the ambiguity function takes the form

AF QH(q, p) =

|K |2π

N /2

F 0(p)µ(q). (107)

This function is strongly localized around the origin of the phase space(q, p). From Eq. (101), the Cohen-class distributions can be written as

C F QH(q, p) = |K |

2π

3 N /2 F 0(p)µ(q)φ(p, q)

× exp[iK (q · p − q · p)] d N qd N p. (108)

If µ and F 0 are sufficiently sharply peaked, the approximation

F 0(p)µ(q)φ(p, q) ≈ F 0(p)µ(q)φ (0, 0) = F 0(p)µ(q) is valid. That is,the dependence on the kernel φ disappears, and the expression reducesto the result in Eq. (79):

C F QH(q, p) ≈

|K |2π

N /2

F 0(q)µ(p). (109)

Since F 0 and µ are sharply peaked functions, F 0 and µ are wide, slowlyvarying functions. Therefore, C F QH

(q, p) varies slowly in all phase-spacevariables, occupying a region of phase space much greater than theminimum uncertainty cell. To illustrate this, consider the correlationfunction in Eq. (82). The Wigner function [already calculated in Eq. (83)],as well as the Q, P, and Kirkwood–Rihaczek functions, can be found to be given by

W F (q, p) = exp

− q2

σ 21

− σ 22 p2

, (110a)

QF (q, p) = σ 1a0 (σ 21 + a2

0)(a20 + σ 22 )

exp

− q2

σ 21 + a20

− a20σ 22

a20 + σ 22

p2

,

(110b)

PF (q, p) = σ 1a0 (σ 21 − a2

0)(a20 − σ 22 )

exp

− q2

σ 21 − a20

− a20σ 22

a20 − σ 22

p2

,

(110c)

RF (q, p) = 1√ 1 + 4ν4

exp

1

1 + 4ν4

− q2

σ 21

− σ 22 p2 + 4iν4qp

,

(110d)

where in the last expression ν = √ σ 2/σ 1 is the overall degree of purity.

Notice that the P function is only well defined if a0 falls within the rangeσ 1 > a0 > σ 2. In the limit σ 1,2 → a0 (corresponding to a fully coherentGaussian function of width a0) the P function tends to a delta function.




On the other hand, in the highly incoherent case where σ 1 σ 2, theKirkwood–Rihaczek function converges towards the Wigner function, asdo also the P and Q functions if a0 is chosen such that σ 1 a0 σ 2.The convergence of all the Cohen-class representations occurs not onlyfor functions satisfying the factorization in Eq. (76), but for any functionF (q1, q2) that differs significantly from zero only for values of |q2 − q1|much smaller than the scale of variation of F (q, q).

As mentioned in Section 1, the main topic of this tutorial is the

conservation of the Wigner function along paths associated with asimple physical model. As discussed in Subsection 4.3, there are twogeneral classes of linear unitary transformation on a function f whoseeffect in phase space is a simple mapping (i.e., rearrangement of arguments) of the Wigner function: linear canonical transformations

C (S) and phase-space shifts T (q0, p0). The most general class of unitarytransformation that corresponds to a mapping of the Wigner functionis an arbitrary combination of these two types of transformations.Therefore, the conservation property of the standard Wigner function isrestricted to systems where propagation or evolution is described by acombination of phase-space shifts and linear canonical transformations.However, as will be seen in Section 11, it is possible to retain theconservation property for other physical systems by modifying thedefinition of the Wigner function.

7. Wigner-Based Modeling of Beam Propagation

In optics and quantum mechanics, one of the main properties of theWigner function is that, for a certain class of simple systems, itsspatial propagation or temporal evolution follows the same rules asthe simpler physical formalisms it mimics (i.e., geometrical optics orclassical mechanics). In the case of paraxial classical optical propagation,this class of systems includes, but is not limited to, the so-called ABCDsystems [71], which are characterized in both the ray and the wavemodels by a 4 × 4 symplectic matrix. In this section, we first givea brief review of the ray-optical and radiometric description of thistype of system, and then show that, by using the Wigner function, thepropagation of wave fields through these systems follows the same rules,namely, conservation along rays.

7.1. Paraxial ABCD Systems in the Ray Domain

At a plane of constant z, a ray is fully characterized by its intersection

with this plane and by its direction. The intersection’s position isspecified by its transverse coordinate, x = ( x, y), while directionis specified by the optical momentum two-vector, p = ( p x, p y),corresponding to the product of the local refractive index n(x, z) and thetransverse part of a unit vector in the ray’s direction (see Fig. 17). Theray is then completely characterized by the state vector(x, p), which givesthe coordinates of the ray in phase space. The paraxial approximationimplies that |p| = n sin θ ≈ n tan θ n, where θ is the angle between theray’s tangent vector and the z axis. For a simple class of optical systems,




Figure 17

x

y

z

n

0

p x

p y

θ

At a plane of constant z, a ray (red line) is specified by the coordinates x = ( x, y)

of its intersection with this plane, and by the optical momentum p = ( p x, p y)

given by the transverse components of a tangent vector (green arrow) whose

norm is the refractive index, n.

the propagation of a paraxial ray is described mathematically by a simplerelation involving linear combinations of x and p. These systems, knownas linear systems or ABCD systems [71], are fully characterized by a4 × 4 symplectic matrix S composed of the 2 × 2 matrices A, B, C,and D, so that the initial state vector of the ray, (x, p), correspondingto a plane intersecting the optical axis at z, transforms into the final statevector at z > z, (x, p), through multiplication by this matrix:

x

p

=

A B

C D

x

p

= S

x

p

. (111)

The fact that, in this context, S must be symplectic follows from asimple derivation of Eq. (111), based on the Hamiltonian formalism of ray optics [72–74]. The part of an optical system contained between theplanes at z and z can be described by Hamilton’s point characteristicV (x, z; x, z), which equals the optical length of an extremal path (i.e., aray) joining the points (x, z) and (x, z). (This description is completeprovided that there is one and only one extremal path joining thesepoints; otherwise, a different type of characteristic function must beused [74].) In the paraxial regime and for a simple class of systems,sufficiently accurate results are obtained from the quadratic expansion of the point characteristic. Further simplification results from assuming thatthe z axis is an axis of symmetry of the system, i.e., that the system hasmirror symmetry around two perpendicular planes intersecting at thisaxis. (This includes, as a special case, systems with rotational symmetryaround the z axis.) In this case, the quadratic expansion does not includelinear terms, and can be written, without loss of generality, in a formanalogous to Eq. (14):

V (x, z; x, z) ≈ V 0 + x · B−1Ax − 2x · B−1x + x · DB−1x

2, (112)




where B−1A and DB−1 are symmetric, i.e., they satisfy Eq. (15). We nowarrive at Eq. (111) by using Hamilton’s equations [74]:

p = −∂V

∂x = B−1(−Ax + x),

p = ∂V

∂x= −(B

−1)Tx + DB−1x,

(113)

where we used the symmetry of B−1A and DB−1. These two relations can

be solved for x and p: the first gives, straightforwardly, x = Ax + Bp, asexpected, while the second can be written, after eliminating x from it byusing the first relation, as p = Cx + Dp, where C is defined precisely byEq. (21), meaning that S is indeed symplectic.

To illustrate the type of system that can be modeled through a linearrelation involving a symplectic ABCD matrix, consider the followingthree simple examples:

(a) Propagation through a homogeneous transparent medium of indexn leaves the ray’s slope unchanged but changes the ray’s transverseposition in an amount proportional to both the slope (approximatelyequal to p/n in the paraxial approximation) and the propagation

distance z, so that the matrix is given by

SHM( z) =

I zI/n

O I

. (114)

Notice that this matrix is similar to the one in Eq. (28).

(b) Propagation through a perfect thin lens of focal distance f leaves thetransverse position unchanged, but redirects the rays in an amountproportional to x, so that the matrix takes the form in Eq. (29):

SLens( f ) = I O

−I/ f I . (115)

(c) Propagation by a distance z through a gradient index (GRIN)

medium with refractive index n(x) = n0

1 − n2

2|x|2 is given by the

matrix

SGRIN( z) =

I cos(n2 z) I sin(n2 z)/n2n0

−In2n0 sin(n2 z) I cos(n2 z)

. (116)

Like the matrix in Eq. (24), this matrix is, up to a scaling factor, arotation matrix by an angle θ = −n2 z.

Note that, indeed, all these matrices are symplectic, with unit

determinant. The fact that Det(S) must equal unity in this contextis a consequence of what is known as the conservation of etendueor Liouville’s theorem, which states that the phase-space volume(also known as etendue) of any bundle of rays must be conservedunder propagation. In differential form, this means that d xd yd p

xd p y =

d xd yd p xd p y, or equivalently,

∂(x, p)

∂(x, p)= |Det(S)| = 1. (117)






That is, ray-optical free propagation corresponds geometrically to ahorizontal linear shear in phase space. Similarly, propagation from zwithin a quadratic GRIN medium corresponds to a (scaled) phase-spacerotation by an angle −n2 z under ray-optical propagation, as seen fromthe substitution of Eq. (116) into Eq. (121):

L(x, p; z) = L{cos[n2( z − z)]x − sin[n2( z − z)]p/(n0n2),

×cos

[n2u( z

− z)

]p

+sin

[n2( z

− z)

]n0n2x

; z

}. (125)

7.3. Paraxial ABCD Systems in the Wave Domain

In the wave domain, the propagation of a monochromatic paraxial scalarfield from an initial plane at z, traveling exclusively toward larger valuesof z, can be described by a linear expression of the form

U (x, z) =

U (x, z)K(x, z; x, z) d2 x, (126)

where K(x, z

;x, z) is known as the propagator. An approximate form

for this propagator in terms of geometrical–optical quantities takes theform [77–79]:

K(x, z; x, z) = −ik

2π

n(x, z)n(x, z)

Det(MV )

× exp

iπ

2νM

exp[ikV (x, z; x, z)], (127)

where k is the free-space wavenumber, V (x, z; x, z) is Hamilton’s pointcharacteristic given by the optical path length of a ray joining the pointsspecified by the arguments, MV is the stability matrix

MV =

∂2V

∂ x ∂ x

∂ 2V

∂ x ∂ y

∂2V

∂ y ∂ x

∂ 2V

∂ y ∂ y

, (128)

and νM is the so-called Maslov index, which is an integer that accountsfor extra phases accumulated during the propagation of the field from zto z, due to the changes of sign of Det(MV ) that occur at caustics [i.e., atplaces where the rays emanating from (x, z) intersect].

The expression for the propagator in Eq. (127) was first proposedin optics by Walther [77,78]. An analogous propagator was proposed

earlier by Van Vleck [80] to describe the evolution of solutions toSchrodinger’s equation in terms of classical particle trajectories, whereinstead of V one uses the classical action. This result can be consideredas a form of the Huygens–Fresnel principle: the field at the initialplane can be decomposed into secondary sources, each emitting afield proportional to K(x, z; x, z) whose phase delay at the observationpoint is proportional to the optical distance from the secondary source,and whose optical intensity is proportional to the density of raysaround the observation point arriving from the secondary source point




[i.e., Det(MV )]. Note that the original expressions by Walther and VanVleck did not include the Maslov index correction, which was added byGutzwiller [81] within the quantum context.

For situations where the ABCD formalism is valid in the ray-opticalregime, sufficiently accurate results are obtained from using in Eq. (127)the quadratic expansion of the point characteristic in Eq. (112). It isstraightforward to see that, in this approximation, the propagated field isproportional to a linear canonical transformation as defined in Eqs. (17)

and (14) with K = k , applied to the initial field:

U (x, z) = exp

ikV 0 + iπ

2νM

C (S)U (x, z). (129)

That is, within the ABCD-system approximation, the same matrix S thatdescribes the propagation of rays through simple multiplication alsodescribes (up to a global phase factor) the propagation of waves througha linear canonical transformation. The expression in Eq. (129) is known asCollins’ formula [27]. For example, for propagation through free space,for which V 0 = z − z, νM = 0, and S is given in Eq. (114), this expressionreduces to the Fresnel propagation integral:

U (x, z) = −ik exp[ik ( z − z)]2π( z − z)

U (x, z) exp

−ik

|x − x|2

2( z − z)

d2 x. (130)

Similarly, propagation through a quadratic GRIN medium, for which S

has the form in Eq. (116), is proportional to a fractional Fourier transformof degree n2( z − z) [82].

7.4. Wave Propagation Through ABCD Systems in Terms ofthe Wigner Function

The Wigner function of the field U (x, z) over the transverse variablesgives a representation in terms of both transverse position x andtransverse momentum p:

W U (x, p; z) = k

2π

U ∗

x − x

2

U

x + x

2

exp(−ix · p) d xd y.

(131)

In many ways, this Wigner function behaves like the basic radiance fromthe theory of radiometry. First, due to the marginal property in Eq. (43a),the integral of the Wigner function over all directions gives the localoptical intensity:

I (x, z) = |U (x, z)|2 =

W U (x, p; z) d2 p. (132)

Second, since for paraxial fields traveling through ABCD systems wavepropagation is described by linear canonical transformations (up to aphase factor independent of the transverse coordinates), Eqs. (47) implythat the Wigner function is constant along rays in these systems, i.e.,

W U (x, p; z) = W U (x, p; z) = W U (Ax + Bp, Cx + Dp; z), (133)




or, in differential form,

∂W U

∂ z(x, p; z) =

∂W U

∂x(x, p; z),

∂W U

∂p(x, p; z)

∂ S

∂ z

x

p

. (134)

Note that Eqs. (132), (133), and (134) are identical to Eqs. (120), (121) and(122), with the Wigner function playing the role of the basic radiance.This implies, for example, that free propagation in the paraxial wave

domain corresponds to a horizontal linear phase-space shear like that inEq. (124) and illustrated in Fig. 7. Similarly, propagation across a thin lensimparting quadratic phase (i.e., a chirp) has the effect of a vertical linearshear on the Wigner function like that in Fig. 8, and propagation througha quadratic GRIN medium is equivalent to a phase-space rotation likethat in Eq. (125), illustrated in Fig. 5. The fact that propagation throughABCD systems corresponds to a simple linear mapping of the Wignerfunction was first noticed by Bastiaans [70,83–86]. Given this similarity between the properties of the radiance and the Wigner function, itis tempting to state that the Wigner function constitutes a rigorouswave-optical foundation for the theory of radiometry, at least within theparaxial regime. The one conceptual objection to this association, though,

is the possible negativity of the Wigner function for some bundles of rays.Under the radiometric interpretation, these rays would then be some sortof “anti-rays” that cancel some of the power transported by other raysthrough the system.

To illustrate these ideas, let us consider the simplest physical situation,corresponding to propagation through a region of free space surround-ing the plane z = 0. The substitution in Eq. (124) of z = 0 and with Lreplaced with W U in Eq. (132) gives the expression

I (x, z) =

W U (x − zp, p; 0) d2 p. (135)

That is, in free space, the intensity at any propagation distance is given by a vertical marginal projection of a horizontally sheared version of theinitial Wigner function. This is illustrated in Media 17 and Fig. 18 forthe case of a field that is independent of y, for which the phase space( x, p) is two-dimensional. In this figure, the initial field U 0( x) = U ( x, 0)

is chosen to be constant for | x| ≤ a/2 and zero otherwise. Within theparaxial regime, this field corresponds, for example, to a plane wavetraveling in the z direction that is diffracted a slit of width a. The Wignerfunction, which can be calculated analytically, is shown in Fig. 18(b).Its marginal in p does return the desired rectangular distribution. Asshown in Media 17, propagation in z is described by a horizontal linear

shearing of the Wigner function, whose marginal in p gives the intensityprofile at the corresponding propagation distance. This example showsthe importance of the negative values of the Wigner function in order toaccount for destructive interference. It also gives a simple visualizationof the fact that, at the propagation distance zpeak = 0.0545ka2, a peakin the axial intensity is achieved, associated with the vertical alignmentof some of the main positive regions of the Wigner function. This isprecisely the distance at which, for example, a pinhole camera with asquare pinhole of side a would achieve its best resolution. (For a circular









Figure 18

(a) (b)

(c)

(a), (b) Wigner function for a plane wave incident on a slit of width a, and

marginal projection in p, corresponding to the intensity (green line on top), for

(a) z = 0 and (b) z = zpeak , where the vertical alignment of positive regions leads

to a peak in the intensity on axis. The horizontal axis corresponds to x ∈ [−a, a],

while the vertical axis is p ∈ [−20/(ka), 20/(ka)]. The intensity distribution is

shown in (c) for x ∈ [−a, a] and z ∈ [0, 2 zpeak ]. The red line indicates z = zpeak .

In Media 17, z varies between 0 and 2 zpeak .

pinhole of diameter a, the numerical factor is slightly smaller, 0.044, asfound by Rayleigh.)

An alternative presentation of the same example is given in Fig. 19.Figure 19(a) shows the intensity distribution over the (scaled) physicalplane. Figure 19(b) shows a superposition of lines representing rays,which are either black or white depending on whether the Wignerfunction for the corresponding ray is negative or positive, and whose

opacity is proportional to the magnitude of the Wigner function,such that rays for which the Wigner function is close to zero arenearly invisible. These lines are superposed over a grey background(representing zero), so that the final shade of grey at each point dependson the number and weight of the positive and negative rays goingthrough it. Media 18 shows partial superpositions of these rays for thesake of illustration. It is seen there that, once a sufficient number of raysis included, the ray superposition in Fig. 19(b) reproduces the features of the wave pattern in Fig. 19(a).









Figure 19

x

z

0.5a

zpeak

–0.5a

0.5a

–0.5a

x

z zpeak

(a) (b)

(a) Intensity distribution over the ( x, z) plane for a plane wave diffracted by a

slit of width a at z = 0. Note that the z direction is significantly compressed

with respect to the x direction. (b) Superposition of a dense sampling of

lines representing rays, over a grey background. The lines are black or white

depending on whether the Wigner function for the corresponding ray is positiveor negative, and each line has an opacity proportional to the magnitude of the

Wigner function. Media 18 shows the gradual accumulation of the rays in this

figure.

It must be noted that ABCD systems are not the most general typeof system for which the rules of paraxial propagation for the Wignerfunction exactly mimic the radiometric model. As discussed in Section 4,the most general type of transformation that corresponds to a simplemapping of the Wigner function in phase space is a combination of linear

canonical transformations and phase-space shifts. This corresponds tosystems where Hamilton’s characteristic function can be approximated by a quadratic expansion like that in Eq. (112), but where this expansioncan include linear terms that account for asymmetries of the system.These linear terms lead to phase-space shifts. Lateral spatial shifts can be introduced, for example, by a tilted etalon, or by a deliberate lateralshift of the optical axis in order to model a misaligned optical element.Directional shifts can be introduced by, say, a transparent wedge, or bya deliberate small rotation of the optical axis in order to model a tiltedoptical element.

7.5. Phase-Space Interpretation of Measurement

As discussed in [45], an ideal measuring device can be thought of asa source that emits a characteristic field whose amplitude and phaseare such that it cancels as much of the incident field as possible. (Forexample, a point detector that measures the local intensity can be thoughtof as a point source emitting a spherical wave.) Let us consider theparaxial case. In general, let the field specific to some detector be given, ata plane z (after the detector), by the function g(x), times an undetermined






constant complex amplitude, A0. The power absorbed by the detector isthen the difference between the power of the incident field U and that of the field past the detector, U − A0g. This absorbed power is given by

D =

|U (x, z)|2d2 x −

|U (x, z) − A0g(x)|2d2 x

= A0 U ∗(x, z)g(x)d2 x + A∗

0 U (x, z)g∗(x)d2 x − A∗

0 A0, (136)

where, without loss of generality, we assumed |g(x)|2d2 x = 1. If the

detector is ideal, the amplitude A0 must be chosen so that the absorbedpower is maximal. By taking derivatives of Eq. (136) with respect to thereal and imaginary parts of A0 we find

A0 =

U (x, z)g∗(x)d2 x, (137)

which substituted back into Eq. (136) leads to the expression

D

= g∗(x)U (x, z)d2 x2

. (138)

That is, the measurement resulting from this device has a formproportional to the right-hand side of Eq. (55) with f = U . Therefore,according to that equation, the result of the measurement is proportionalto the overlap integral of the Wigner function of the field, W U , andthat of the detector, W g, at the corresponding plane. This form suggestsan intuitive ray-based interpretation for the measurement process: theWigner function of the detector indicates what rays can be accepted intothe detector and in what measure. The measurement is then equal to theintegral of the power of all rays from the incident field that is acceptedin the detector.

Of course, this interpretation should not be taken literally, since raysare not real, independent physical objects, but rather they are just amathematical construct. Further, the Wigner function of the field can benegative for some thin bundles of rays. However, due to the uncertaintyrelation, the Wigner function of the detector cannot be arbitrarily narrow,so that it cannot be focused on arbitrarily narrow bundles of rays of the field. In particular, it cannot focus on narrow bundles of rays overwhich the field’s Wigner function is purely negative; a sufficient amountof “positive rays” is always also accepted into the detector, so that theresult of the measurement is always non-negative. Something analogoushappens with possible negative regions of the Wigner function for themeasuring device.

7.6. Applications of the Wigner and Ambiguity Functions inthe Paraxial Regime

In the past three decades, a large number of articles have been publishedwhose topic is the application of the Wigner and ambiguity functions tothe study and visualization of optical systems within the paraxial regime.It is impossible to do justice to all of this body of work within the limited




space provided by this article, and I apologize in advance to the manyauthors whose interesting work is not mentioned or cited here. Sincethe main emphasis of this article is the use of the simple conservationproperties of the Wigner function, the applications singled out in thissection are ones that exploit these properties.

7.6a. Imaging Systems and the Optical Transfer Function

The expression for the intensity of a field propagating through free spacein terms of the Wigner function is given in Eq. (135). The correspondingexpression in terms of the ambiguity function is given by

I (x, z) =

AU (− zp, p; 0) exp(ik x · p) d2 p. (139)

Note that the Fourier transform in the transverse variables of both sidesof Eq. (139) gives

AU (− zp, p; 0) = k

2π 2

I (x, z) exp(−ik x · p) d2 x. (140)

In imaging systems, the intensity distribution at the image planecorresponding to a point source in object space is referred to as thepoint spread function (PSF). The Fourier transform of the PSF is knownas the optical transfer function (OTF), which characterizes the imagingcapabilities of the system in the case of spatially incoherent objects [87].As pointed out by Brenner et al. [88], for the field generated by apoint source in object space, the ambiguity function at “radial” planarsections defined by x = − zp corresponds to the OTF of the systemfor different amounts of defocus z, as can be seen from Eq. (140).Based on this realization, both the Wigner and ambiguity functions have been employed in the design of optical systems, e.g., to study axial

resolution [89] or to image transparent objects with partially coherentillumination [90].

7.6b. Propagation-Invariant Beams and Extended Depth of Field

There is a class of solutions of the paraxial wave equation whoseintensity profile is exactly invariant under propagation. These solutionsdo not correspond to realistic fields, since they require an infinite amountof optical power. However, they can be achieved approximately, leadingto fields whose transverse intensity profile is nearly invariant underpropagation within some range of distances. Perhaps the best known

such solution is that of Bessel beams [91], given by the expression

U (x, z) = u0 J n(kp0|x|) exp(inφq) exp

ikz

1 − p2

0

2

, (141)

where u0 is a constant amplitude factor, φq is the angle between x andthe x axis, J n is an nth order Bessel function of the first kind, n is thevortex charge of the beam, and p0 is a constant that sets the numericalaperture of the beam. In general, all propagation-invariant beams have




a transverse Fourier transform (with K = k ) that is restricted to a ring of radius p0, i.e.,

U 0(p) = u0δ(|p| − p0)

kp0 A(p), (142)

where A is an arbitrary function of p. The special case of Bessel beamscorresponds to A(p) = exp(inφ p), where φ p is the angle between p and

the p x axis. By using the definition of the Wigner function in terms of theFourier transform, one can easily find

W U (x, p; 0) = 2|u0|2

(2π p0)2

( p0 − |p|)|p|

p20 − |p|2

× A∗

p − p

2

A

p + p

2

exp(ik x · p)

(143a)

where is the Heaviside function and p(p) is a vector perpendicular top, defined as

p(p) = ( p y, − p x)

|p|

p20 − |p|2. (143b)

(The particular case of Bessel fields was found in [92].) This Wignerfunction is defined in a four-dimensional phase space, so it is difficult tovisualize its horizontal shearing due to free propagation. However, it isstraightforward to see that this Wigner function is invariant under suchshearing, that is, W U (x, p; z) = W U (x − zp, p; 0) = W U (x, p; 0), since theonly occurrence of x is in a dot product with p, which is perpendicularto p; so replacing x with x − zp has no effect. This invariance underhorizontal shearings, though, implies that the Wigner function extends

over all values of x, meaning that its integral is infinite. This is consistentwith the fact that propagation-invariant fields have infinite power.

There is an interesting second class of propagation-invariant field, whoseintensity profile shape is invariant under paraxial propagation, althoughit does not travel along a straight line but along a parabolic path. Thearchetypal beam of this type is the Airy beam [93]. Let us consider firstthe version of this beam for a space with only one transverse variable, x,in addition to the propagation distance, z. The formula for an Airy beamis then given by

U ( x, z)

=u0 exp(ikz)Ai x

a −z2

4k 2

a4 expi xz

2ka3

−z3

12k 3

a6 , (144)

where Ai is the Airy function of the first kind and a is a parameter thatregulates both the width of the lobes of the beam’s intensity as well asthe path these lobes follow. The intensity pattern of this beam, shown inFig. 20(a), is given by

I ( x, z) = |u0|2Ai2

x

a− z2

4k 2a4

. (145)




Figure 20

(a) (b)

(c)

(a), (b) Wigner function for an Airy beam, given by Eq. (147), over the

phase-space region x ∈ [−5a, 5a], p ∈ [−3/ka, 3/ka], for (a) z = 0 and (b)

z = 2.5ka2. In both cases, the marginal in p, equal to the spatial intensity

distribution, is shown as a green curve at the top. (c) Intensity profile for this Airy

beam, described by Eq. (145), over the same interval in x and for z ∈ [0, 5ka2].

The red line indicates z = 2.5ka2, corresponding to the Wigner function in part

(b). Media 19 shows the evolution of this Wigner function and its marginal under

propagation in z, where the value of z is indicated by the red line.

Therefore, the intensity profile is maintained under propagation in z upto a lateral displacement proportional to z2/(k 2a3). As in the case of theBessel beams, the Wigner function is easier to calculate from the field’stransverse Fourier transform at z = 0, given by a simple cubic phase:

U 0( p) = u0

a

k

2πexp

ik 3a3 p3

3

. (146)

Notice that this angular spectrum has constant magnitude, meaningthat the integral of its square modulus is infinite, as would be thepower required to generate a rigorous version of this beam. The Wignerfunction can be calculated analytically, giving a form analogous to thatshown in Fig. 15, but rotated by 90◦:

W U ( x, p; 0) = 41/3k |u0|2

2π a3Ai

41/3

k 2a2 p2 + x

a

. (147)






That is, the Wigner function has the shape of nested positive andnegative parabolic fringes of equal vertex curvature and (horizontal)axis of symmetry. While this Wigner function is not invariant underhorizontal shearing in an absolute sense, it is easy to see from Media 19(Fig. 20) that a horizontal shearing caused by propagating a distance

z is equivalent to a phase-space shift. By substituting x → x − zp

in Eq. (147), it is easy to show that this phase-space shift is given

by T

[ z2/(4k 2a4), z/(2k 2a3)

]; the spatial part of the shift is responsible

for the curvature of the beam’s path, while the directional part leadsto the phase factor in Eq. (144). By using the projection propertiesof Airy functions [94], one can show that the marginal in p of thisWigner function equals the intensity distribution in Eq. (145). Notethat the intriguing feature of Airy beams of following a curvedtrajectory becomes less mysterious under this phase-space picture: allrays composing this beam travel along straight lines, but the rays thatgive rise to a maximum at a given value of z are not the same ones asthose that give rise to the maximum at a different propagation distance.An analogous interpretation can be given in terms of caustics [93].A treatment that is similar to the one presented here was publishedrecently [95].

A three-dimensional version of the Airy beam can be composed asthe product of the field in Eq. (144) times a similar solution in termsof y [divided by u0 exp(ikz)], and the corresponding Wigner functionis proportional to the products of the Wigner functions in Eq. (147)with arguments ( x, p x) and ( y, p y), respectively. Approximations to these beams can be generated [96,97] by placing at the pupil of an opticalsystem a transparent mask whose phase has a cubic dependence [asin Eq. (146)] in both Cartesian directions. Coincidentally, this cubicphase mask was proposed by Dowski and Cathey [98] as a way toextend the depth of field of imaging systems. With such a cubic phase,the PSF corresponds approximately to the intensity of an Airy beam,which remains nearly invariant for a range of defocus distances. Further,

given this invariance and known profile, this PSF can be deconvolvedcomputationally, roughly independently of defocus. Of course, thereis a small deviation of the defocused PSFs with respect to the idealgeometrical location due to the curvature of the path followed by themaximum, but this deviation is negligibly small in imaging applications.Approximations to Bessel beams, particularly those of zeroth order, havealso been used to extend depth of focus [99–101]. These are achieved,for example, by the inclusion of either annular pupils or axicons withinthe imaging system. Other studies of extension of the depth of fieldof an imaging system based on phase-space distributions are givenin [102,103].

7.6c. The Talbot Effect

The ray-like propagation of the Wigner function also provides a simple,visual description of the Talbot effect. In this effect, fields that areperiodic over the initial transverse plane form images of themselvesat certain distances of propagation in free space, without the use of lenses or mirrors. Consider again, for simplicity, a two-dimensional field(independent of y). Let the field at the plane z = 0, U 0( x), be periodic with






period , i.e., U 0( x + ) = U 0( x) for any x. The Fourier transform of U 0is then a discrete sum of weighted Dirac deltas:

U 0( p) =

n

unδ

p − 2π n

k

, (148)

where un are the Fourier series coefficients of U 0( x), and the sum runsover all integers. [In practice, the initial field only includes a finite

number of periods, so the delta functions should in principle be replaced by tall and narrow finite functions. However, the idealized form inEq. (148) facilitates the analysis that follows.] As found by Testorf andco-workers [104–106], the Wigner function of such a field at the initialplane has the following form:

W U ( x, p; 0) = k

2π

n1

n2

u∗n1

un2δ

p − π

n1 + n2

k

× exp

i2π x

n2 − n1

. (149)

That is, the Wigner function is discrete (and singular) in p, differing from

zero only when p is an integer multiple of π/k , as shown in Fig. 21.(Again, for a realistic field, these delta functions must be replaced bynarrow finite distributions.) Notice that this spacing in p is half of that

between the delta functions in U 0( p) in Eq. (148), due to the presenceof half-point interference contributions in the Wigner function. At thediscrete set of values of p where it differs from zero, the Wigner functionhas different periodicity in x depending on whether p is an even or oddmultiple of π/k ; it can be seen from Eq. (149) that

W U

x +

2, n

π

k ; 0

= (−1)nW U

x, n

π

k ; 0

, (150)

for integer n. This can be appreciated in Fig. 21, where the values of theWigner function at even rows has periodicity /2, while for odd rows itoscillates between positive and negative values with period .

Given its periodicity in x and discreteness in p, one can see that theWigner funcion is also periodic under propagation in z, due to the linearshear relation

W U ( x, p; z) = W U ( x − pz, p; 0). (151)

The right-hand side of this relation equals W U ( x, p; 0) when pz is aninteger multiple of . Since the Wigner function vanishes unless p is aninteger multiple of π/k , then the Wigner function returns to its original

configuration when z = zn = nk 2

/π (the Talbot distances) where nis an integer. This is illustrated by Media 20 (Fig. 21). As discussed inRefs. [104,106], this phase-space picture also helps visualize the so-calledfractional Talbot effect. Consider, for example, the case when z = z1/2 =k 2/2π . Given the phase-space shearing, all even rows of the Wignerfunction, corresponding to p = 2π n/k , shift by an integer numberof periods, returning to their state at z = 0. On the other hand, theodd rows, corresponding to p = (2n + 1)π/k , shift by a half-integernumber of periods. Given the periodicity properties in Eq. (150), the






Figure 21

x

p

Λ

x

z

Λ

(b)

(a)

2π/k Λ

(a) Intensity profile of a section of a field that is periodic in x with period ,

for z from the initial plane z = 0 to the first Talbot image, at z1. (b) Wigner

function for this field. Note that the Wigner function is discrete and delta-like in

p. The green line on top shows the marginal projection in p, corresponding to

the intensity. Media 20 shows the change of these quantities under propagation

in z between 0 and z1 (the first Talbot image). Note that at z = z1/2 the intensity

is a perfect image of the initial intensity except for a lateral shift of /2.

resulting Wigner function is identical to the initial one except for a globalshift in x of /2. This explains why, at this propagation distance, theintensity pattern is identical to that at z = 0 except for a half-periodshift. Extensions of self-imaging to polychromatic fields have also beenstudied based on the Wigner function [107].

Note that this phase-space picture of the Talbot effect has several featuresin common with that of the propagation-invariant beams discussedin Subsection 7.6b. In both cases, the Wigner function after horizontalshearing is identical to the initial one (or a shifted version of it). The

only difference is that, in this case, this reproduction of the initial statehappens only at a discrete set of propagation distances. Nevertheless,this similarity helps explain other common features, like the one referredto as ”self-healing.” Consider the case where a section of the initialfield (a Bessel or Airy beam, or a periodic field) is perturbed somehow,e.g., through blocking by an obstacle. After some propagation, thisperturbation washes away, so that the beam returns to its originalunperturbed shape. As in the case of Bessel and Airy beams, localfeatures of the field after propagation should not be thought of as beingthe same physical entity as the corresponding features of the initial field,even if they are identical in shape. Such misinterpretation can lead tounphysical conclusions, like the violation of special-relativistic causality.

7.6d. Sampling and Holography

The Wigner function also gives a useful visualization of why the well-known sampling theorem can be relaxed for chirped signals [108,109],e.g., for paraxial beams that have propagated away from theirwaist plane. Shannon’s sampling theorem states that the maximumsampling spacing over the transverse variable x that allows a complete






reconstruction of the field is inversely proportional to the supportof the Fourier transform of the field at the initial plane. Paraxialfree propagation does not change this support, given the relation

U (p, z) = U 0(p) exp[ikz(1 − |p|2/2)]. One could therefore think thatthe maximum sampling spacing remains constant under propagation.Gori [110] showed that, for a field that is strictly limited to an interval,say, at z = 0, the maximum transverse sampling spacing needed to fullyrecover the field increases linearly with z. Similarly, for fields whose

Fourier transform U 0(p) is (at least approximately) restricted to a region|p| < pM, the sampling frequency needed to approximately reproducethe field can be relaxed under propagation. A simple interpretation of this relaxation in terms of the Wigner function follows a derivation thatis mathematically similar to that mentioned earlier for the Talbot effect, but with x and p exchanged. Consider sampling a two-dimensional fieldat a discrete number of equidistant points over the initial line ( z = 0),i.e., U s( x) =

nU ( x, z)δ( x − n), where is the sampling distance.The Wigner function of this sampled field can be found to be given by [108,109]

W U s ( x, p) =1

n

n (−1)

nn

δ

x − n

2

W U x, p − n

π

k ; z

= 1

n

n

δ

x − n

2

W U

x, p − n 2π

k ; z

+ (−1)nW U

x, p − (2n + 1)

π

k ; z

. (152)

That is, the Wigner function of the sampled field equals a series of spatially sampled replicas (with half the sampling spacing) of the Wignerfunction of the original field, stacked on top of each other in the p

direction at a separation of π/k , which is half the separation betweenthe corresponding replicas of the Fourier transform. This is becausethe replicas corresponding to odd n are due to half-point interferenceeffects. Given the alternating sign in front of them, these intermediatereplicas can be cancelled through a suitable projection, so they are notconsidered in the schematic analysis that follows. Figure 22(a) showsschematically the replicas corresponding to even n, separated vertically by 2π/k . To be able to reconstruct the original field, this separationmust be sufficient so that the different replicas do not overlap. If thefield is not chirped, this condition reduces to 2π/k ≥ 2 pM, where 2 pM

is the support of the angular spectrum. This relation can be written as ≤ π/kpM, which is the standard sampling theorem. If the field ischirped, e.g., due to propagation away from its waist plane, its Wignerfunction is sheared, and the replicas can be stacked more densely withoutoverlapping significantly, relaxing the minimum value of .

The revised sampling theorem described earlier has been applied to dig-ital holography [111,112]. Similarly, the Wigner function has been used toexplain the principles of regular [113–116] and volume holography [117].Other recent applications of the Wigner function include descriptionsof the moire effect [118] and the detection of subsurface targets closelylocated beneath a randomly rough surface [119].




Figure 22

x

p

pM

x

p

pM

x

p

pM

(a) (b) (c)

(a) Representation of three stacked sampled replicas of the Wigner function,

whose spacing in p, given by 2π/k , must be greater than the support in p for

each replica, 2 pM, so they do not overlap. (b) After propagation in p, the Wigner

function is sheared, and so is each replica following sampling, resulting in more

effective separation between the replicas. (c) The sheared replicas can be packed

more densely without leading to significant overlapping.

7.7. Partially Coherent Fields

The fact that paraxial propagation through ABCD systems correspondsto a simple remapping of the Wigner function makes this formalisma useful numerical tool in the case of partially coherent fields. Inthe frequency domain, a partially coherent field is described by itscross-spectral density, defined as the correlation of the field at two points:

J (x1, z1; x2, z2) = U ∗(x1, z1)U (x2, z2). (153)

(Note that the cross-spectral density is often denoted by the letter W , but we denote it here by J instead to prevent confusion with the Wignerfunction.) At any plane z1 = z2 = z, this function depends on fourvariables (x1 and x2). At the same plane, the Wigner function alsodepends on four variables (x and p):

W J (x, p; z) = k

2π

J

x − x

2, z; x + x

2, z

exp(−ik x · p) d2 x, (154)

so the number of variables is not doubled. Further, in general, theless coherent a field is (i.e., the smaller its overall degree of purity ν),the smoother its Wigner function tends to become; so its sampling

over phase space can be relaxed. Given this, if, say, the intensityof a field needs to be computed at a large number of points N ,traditional calculations based on propagation integrals would requirethe computation of a quadruple integral of an oscillatory integrand foreach point. The computation time would then be proportional to D4 N ,where D is the number of samples required in each variable for theintegration. On the other hand, if the Wigner function is used, thereis an initial computational investment in calculating this function atm×m×m×m sample points over the four-dimensional phase space, each




requiring the evaluation of a double oscillatory integral that involves M 2

operations. Once this calculation is completed and its results stored inmemory, propagation corresponds to a simple remapping of the Wignerfunction, and the evaluation of the intensity at any point involves onlyintegration over the directional variables, i.e., m2 operations. Therefore,the total computation time is proportional to m4 M 2 + m2 N , versus D4 N

for the traditional approach. For fairly incoherent fields, M is comparablewith D, and both are significantly larger than m, so the Wigner-based

computation becomes advantageous when N is sufficiently large [120].In both the traditional and Wigner approaches, computational savingscan be achieved by using fast Fourier transform algorithms or asymptoticintegral estimations, whenever appropriate. For partially coherent fieldsinvolving only a reduced number of coherent modes, however, it might be more efficient to use a modal decomposition and propagate eachmode independently.

While the Wigner function gives direct access to one-point fieldproperties such as the optical intensity, its ray-like propagationproperties have also been exploited for the propagation of two-pointquantities like the cross-spectral density, through the inversion of Eq. (154) [121,122].

7.8. Wigner Functions that Account for Polarization in theParaxial Regime

Wigner functions have also been proposed to model polarization effectsin the paraxial regime. These functions take the form of a 2 × 2matrix [123,124], corresponding to the substitution of the 2 × 2cross-spectral density matrix into Eq. (154):

WJ(x, p; z) = k

2π Jx − x

2, z; x + x

2, z exp(−ik x · p) d2 x, (155)

where {J(x1, z1; x2, z2)}i, j = U ∗i (x1, z1)U j(x2, z2), for i, j being either xor y. Alternatively, Luis [125,126] defined a Stokes–Wigner function thattakes the form of a 4-vector whose elements are a generalization of theStokes parameters, calculated as the trace of the product of WJ and eachof the four Pauli matrices

WJ(x, p; z) = 1

2

[WJ(x, p; z)] xx + [WJ(x, p; z)] yy

[WJ(x, p; z)] xx − [WJ(x, p; z)] yy

[WJ(x, p; z)] xy + [WJ(x, p; z)] yx

i[WJ(x, p; z)] xy − i[WJ(x, p; z)] yx

. (156)

The marginal in p of this Wigner function gives the four Stokesparameters at the corresponding point.

8. The Use of the Wigner Function for MoreComplicated Systems and/or Nonparaxial Fields

The simple propagation properties of the Wigner function are limitedto paraxial systems in which rays evolve according to linear relations.




For more complicated systems involving optical elements introducingphase delays whose dependence on the transverse position are morecomplicated than quadratic (e.g., lenses with aberrations), propagationdoes not correspond to a simple rearrangement of the arguments of theWigner function; it must be described instead either through an integralpropagation kernel [70] involving all phase-space variables, or by a seriesof differential operators [10,14,41] acting on the Wigner function. Thisis also the case for diffraction by planar obstacles, where one can use

an integral relation to model diffraction [127,128]. In the case of fairlyspatially incoherent fields, a radiometric-like model can be used, wherethe primary effect of the obstacle is to block the rays incident on it(i.e., the value of the Wigner function for these rays is set to zero), whilethose missing the obstacle continue to propagate (i.e., the value of theWigner function for these rays remains unchanged). Additionally, theedges of the obstacle behave as secondary sources emitting rays [129],whose Wigner function is related to that of the rays incident on the edges.Approximate paraxial propagation models based on the Wigner functionhave also been proposed for the case of turbulent media [130–133], aswell as for nonlinear Kerr media [134–138].

Even for propagation in free space (or a homogeneous transparentmedium), the Wigner function is not conserved along rectilinear pathsfor fields that are not paraxial. To show this, let us use the angularspectrum representation of a field traveling in free space:

U (x, z) = k

2π

U 0(p) exp{ik [x · p + z p z(p)]} d2 p, (157)

where U 0(p) is the Fourier transform over the transverse position x of

U 0(x) = U (x, 0), and p z(p) =

1 − |p|2. Notice that, even though thefield is nonparaxial, we still choose to treat z as a propagation parameter,separate from the other two spatial coordinates. Let us assume that

U 0(p) = 0 for |p| > 1, so that the field contains no evanescentcomponents. The Wigner function for this field at any plane of constant

z can be written in terms of either U or U 0:

W U (x, p; z) = k

2π

U ∗

x − x

2, z

U

x + x

2, z

exp(−ik x · p) d2 x

= k

2π

U ∗0

p − p

2

U 0

p + p

2

exp(ik x · p)

× exp

ik z

p z

p + p

2

− p z

p − p

2

d2 p. (158)

In the paraxial regime, where U 0(p) = 0 only for |p| 1, one can use theapproximation p z(p) ≈ 1−|p|2/2. Under this approximation, the quantityin square brackets in the last term of Eq. (158) reduces to p · p, resultingin the fact that W U (x, p; z) = W U (x − zp, p; 0), as expected. However, beyond the paraxial regime, the propagation of the Wigner function isnot in general described by a simple rearrangement of arguments. Thispropagation is ruled not by a simple first-order differential equationlike Eq. (134) but instead by a high-order differential equation [139]. As




mentioned in the next subsection, this difficulty is alleviated only in thecase of significantly spatially incoherent fields.

8.1. Generalized Radiometry for Nonparaxial Fields

The first use of the Wigner function in classical optics was within the

context of nonparaxial partially coherent fields. The objective of suchwork was to give a wave-optical foundation to the heuristic theoryof radiative transfer [140,141] and in particular to the radiometricformalism [6]. These efforts concentrated on deriving a wave-basedradiance. This was first done, independently, by Dolin in 1964 [4] and by Walther in 1968 [5]. In particular, Walther proposed the use of thestandard Wigner function over the transverse variables, and introduced

an extra obliquity factor p z(p) =

1 − |p|2 to account for nonparaxialityof the field:

LW(x, p; z) = p z(p)W J (x, p; z), (159)

where W J is the Wigner function for the cross-spectral density, definedin Eq. (154). Several other generalized radiances, as these wave-baseddefinitions of the radiance are referred to in the literature, have beenproposed. Many articles on this topic are collected in the volume edited by Friberg [142], and useful summaries on the subject are given inthe articles by Wolf [143] and Friberg [144]. Many of these generalizedradiances result from the use of other members of the Cohen class [145].For example, Walther proposed a second generalized radiance thatturns out to be the Kirkwood–Rihaczek distribution times an obliquityfactor [146].

There are two objections to the wave-based definition of the radiancein Eq. (159): it can be negative, and it is not rigorously conserved along

rays beyond the paraxial regime. However, it is easy to show that, in thelimiting case of significantly incoherent fields of the quasi-homogeneoustype, these two problems disappear [145,147–148]. Consider a fieldwhose cross-spectral density at a plane z = 0 has the form

J (x1, 0; x2, 0) = I 0

x1 + x2

2

µ0(x2 − x1), (160)

where I 0 is a non-negative function that varies slowly in comparison withµ0, which differs significantly from zero only within a region around theorigin of its argument. One can write the radiance in Eq. (159) in the formof Eq. (158) as

LW(x, p; z) = k

2π p z(p)

˜ J 0

p − p

2, p + p

2

exp(ik x · p)

× exp

ik z

p z

p + p

2

− p z

p − p

2

d2 p, (161)

where ˜ J 0(p1, p2) = U ∗0 (p1)U 0(p2). For the quasi-homogeneous initial

field in Eq. (160), this expression becomes




LW(x, p; z) = k

2π p z(p)µ0(p)

˜ I 0(p) exp(ik x · p)

× exp

ik z

p z

p + p

2

− p z

p − p

2

d2 p. (162)

Given the slow variation of I 0(x), ˜ I 0(p) differs from zero only forsmall values of |p|. Therefore, the quantity within square brackets inthe second line of Eq. (162) can be approximated by its second-order

expansion in p. It is easy to see that even-order terms in this expansioncancel, so only the linear term survives, leading to

LW(x, p; z) ≈ k

2π p z(p)µ0(p)

˜ I 0(p) exp

ik

x + z

∂ p z

∂p(p)

· p

d2 p

= k

2π p z(p) µ0(p) I 0

x − z

p

p z(p)

= LW

x − z

p

p z(p), p; 0

. (163)

That is, LW is approximately non-negative and conserved along rays,even those forming large angles with the z axis. This relation holdsnot only for the factorizable quasi-homogeneous fields in Eq. (160) but

also for any field where the directional correlation ˜ J 0(p1, p2) differssignificantly from zero only for separations of its two arguments thatare much smaller than unity.

As discussed in Subsection 6.4, any definition of the generalized radiance based on a standard Cohen-class distribution would lead to the sameresult in this limit (including Walther’s second generalized radiancedefinition [146]). Light generated by thermal sources typically presentscoherent widths that are small compared with the scale of variation of theintensity, so any Cohen-class generalized radiance can be used to justifythe radiometric model for the case of thermal light.

9. Short Pulses and Propagation in LinearDispersion Media

Another area of optics where phase-space distributions have beenapplied is the characterization of pulses, and the description of theirpropagation through dispersive media [149–153]. Most work on thistopic has been limited to a one-dimensional treatment, useful forstudying highly collimated fields traveling through homogeneous mediaor propagation modes in waveguides, particularly optical fibers, wherethe approximately invariant transverse structure of the field can befactorized from the more interesting longitudinal/temporal behavior.The vast majority of this work is based on the use of the phasespace composed of time and frequency [149,151,154,155], often referredto as the chronocyclic phase space. However, some authors considerinstead the phase space composed of the propagation distance andwavenumber [150,152,153].

As discussed in this section, within some approximations, the prop-agation of pulses through certain optical elements turns out to be




mathematically analogous to paraxial beam propagation through ABCDsystems [156,157]. Useful reviews of this analogy and its applications tothe characterization of ultrashort pulses is given in the book chapter andreview article by Dorrer and Walmsley [154,155]. For simplicity, most of the treatment below considers single pulses, described by a componentof the electric field, E ( z, t ), where z is the spatial propagation variable.However, for the characterization of pulsed lasers used for ultrafastapplications, it is convenient to use a statistical formalism analogous

to that of the theory of partial coherence. Here, each pulse in the trainis treated as a member of an ensemble. Since the different pulses aregenerally not identical, the pulse train is characterized by the correlationfunction [158,159]

( z1, z2; t 1, t 2) = E ∗( z1, t 1) E ( z2, t 2), (164)

where the angular brackets denote ensemble averaging and the two timearguments are defined within a range of the size of the pulse temporalseparation.

Within the linear domain and neglecting the effects of absorption, adispersive medium or system accepts monochromatic solutions of the

form exp[i(kz − ωt )]. The temporal frequency ω and wavenumber (orspatial frequency) k are related through the dispersion relation

k (ω) = ωn(ω)

c, (165)

where c is the speed of light in vacuum. The refractive index n is assumedto be purely real, although in reality this is not possible (except infree space) due to the constraints imposed by causality and expressedmathematically through the Kramers–Kronig relations. However, formany cases of practical interest, absorption is negligible over the rangeof frequencies occupied by the field, so its effects can be neglected. The

plot of ω versus k is known as the dispersion curve.A pulse traveling in a dispersive medium of this type can be expressedas a superposition of monochromatic components:

E ( z, t ) = 1√ 2π

U 0(ω) exp{i[k (ω) z − ωt ]} dω, (166)

where U 0(ω) is the spectral amplitude, given by the temporal Fouriertransform (with K = −1) of E at z = 0:

U 0(ω) = 1√ 2π E (0, t ) exp(iωt ) dt . (167)

In the case in which the pulses occupy a limited frequency region wherethe dispersion curve can be approximated by a segment of a parabola,oriented either vertically or horizontally in the ω versus k plane, thepropagation/evolution of the pulse is mathematically analogous tothe free-space propagation of a paraxial beam. These two parabolicapproximations lead to similar but nonequivalent treatments, as we nowdiscuss.




9.1. Wigner Function of Time versus Frequency

Consider the Taylor expansion of k up to second order around the centralfrequency of the pulse, ω0, i.e.,

k (ω) ≈ k 0 + k 1(ω − ω0) + k 2(ω − ω0)2

2, (168)

where k i = k (i)(ω0). The pulse’s propagation can be modeled in terms of the chronocyclic Wigner function, defined as

W E , z(t , ω; z) = 1

2π

E ∗

z, t − t

2

E

z, t + t

2

exp(iωt ) dt . (169)

[For trains of ultrashort pulses, the field product in this definition can bereplaced by ( z, z; t − t /2, t + t /2), where is the correlation defined inEq. (164).] From the substitution of Eq. (166) into Eq. (170), it is easy toshow that, if the approximation in Eq. (168) is valid, the Wigner functionpropagates as

W E , z(t , ω; z) = W E , z[t − k (ω) z, ω; 0], (170)

where, given Eq. (168), k (ω) = k 2ω + k 1 − k 2ω0 is a linear functionof frequency. That is, propagation in z is described within thisapproximation by a simple linear shearing of the Wigner function overthe phase space (t , ω). One way to understand this follows from writingEq. (166) in the following form:

E ( z, t ) = exp[i(k 0 z − ω0t )] ¯ E z( z, τ ) , (171)

where the factored exponential accounts for the carrier phase, τ = t −k 1 z = t − z/vg is the retarded group time, with vg = 1/k 1 being the group

velocity at ω0, and

¯ E z( z, τ ) = 1√ 2π

U 0(ω0 + ω) exp

i

−ωτ + k 2 z

2ω2

dω, (172)

where the change of variable ω = ω − ω0 was used. It is easy to see that¯ E z( z, τ ) satisfies a Schrodinger-type equation analogous to the paraxialwave equation that describes beam diffraction in free space:

i∂ ¯ E z

∂ z( z, τ ) = −k 2

2

∂ 2 ¯ E z

∂τ 2( z, τ ) . (173)

The solution of this equation has a form analogous to the Fresnel paraxialpropagation formula Eq. (130), i.e.,

¯ E z( z, τ ) = exp(iπ/4)√ 2π k 2 z

¯ E z( z, τ ) exp

−i

(τ − τ )2

2k 2 z

dτ

= C [S z( z)] ¯ E z(0, τ ) , (174)

where z retains its role as a propagation parameter and τ is analogous tothe transverse position variable. As the last step in this equation states,




propagation in z corresponds to a linear canonical transformation in τ

with a matrix analogous to that in Eq. (114), i.e.,

S z( z) =

1 k 2 z

0 1

, (175)

i.e., to a linear shearing of the (τ,ω − ω0) phase space, mathematicallyanalogous to the free-space-propagation shearing in Eq. (124). The effect

of the factored carrier phase is a phase-space shift.As discussed in Refs. [154,155], devices such as quadratic temporal phasemodulators introduce chirp factors and hence play a role analogous tolenses; i.e., they correspond to linear canonical transformations withmatrices analogous to that in Eq. (115). The combination of dispersivemedia and quadratic temporal phase modulators then allows thecomposition of general “temporal ABCD systems” that can performtransformations such as imaging, magnification, and Fourier andfractional Fourier transformation.

9.2. Wigner Function of Position versus Wavenumber

Consider instead the parametrization of ω in terms of k , and assume that,over the range of spatial frequencies occupied by the pulse, a quadraticTaylor expansion of the frequency around k 0 (the central wavenumber)is sufficient:

ω(k ) ≈ ω0 + ω1(k − k 0) + ω2(k − k 0)2

2. (176)

In this case, it is more convenient to consider the Wigner function over z

of the field:

W E ,t ( z, k ; t ) =1

2π

E ∗

z −z

2 , t

E

z +z

2 , t

exp(−ik z) d z, (177)

which can be shown to satisfy the evolution relation

W E ,t ( z, k ; t ) = W E ,t [ z − ω(k )t , k ; 0], (178)

provided the approximation in Eq. (176) is valid. Evolution in t thencorresponds to a linear shearing in the phase space ( z, k ), since ω(k ) =ω2k + ω1 − ω2k 0 is linear in k . Again, this is because the field can be written as a solution to a Schrodinger-like equation, although ondifferent variables. Let us again factor out the carrier phase and expressthe remaining function in terms of regular and retarded times,

E ( z, t ) = exp[i(k 0 z − ω0t )] ¯ E t (τ, t ). (179)

After changing variables to k = k − k 0 and noticing that vg = ω1, we findthe integral expression

¯ E t (τ, t ) = 1√ 2π

U 0[ω(k 0 + k )]ω(k ) exp

i−k vgτ − ω2t

2k 2

dk ,

(180)




which clearly satisfies the evolution equation

i∂ ¯ E t

∂ t (τ, t ) = − ω2

2v2g

∂ 2 ¯ E t

∂τ 2(τ, t ). (181)

Evolution in t then corresponds to a Fresnel-type linear canonicaltransformation in τ of the form

¯ E t (τ, t ) = C [St (t )] ¯ E t (τ, 0), (182)corresponding to the matrix

St (t ) =

1 ω2v−2g t

0 1

. (183)

This matrix is also analogous to that for free propagation of a beam, givenin Eq. (114), although in this case the propagation parameter is time, notdistance.

10. Phase-Space Tomography

Phase-space distributions like the Wigner and ambiguity functionshave been used for the recovery of the coherence properties of mixedstates from measurements of the marginal projections through a setof techniques known as phase-space tomography, first proposed byBertrand and Bertrand [35]. This approach works particularly wellfor systems where phase space is two-dimensional and for whichpropagation or evolution corresponds either to a Fresnel-type integral(i.e., to a linear shearing of phase space) or to a fractional Fouriertransform (i.e., to a rotation of phase space). The idea is that, if sufficient marginal projections are obtained through measurements atdifferent stages, the Wigner function can be estimated through standardtomographic methods like the inverse Radon transform. Once theWigner function is known, the correlation function F (q1, q2) can berecovered. Much of the pioneering work on this type of technique camefrom the same research group [36–38], who applied it to three differentphysical contexts, as discussed in what follows.

10.1. Phase-Space Tomography in Quantum Optics

The first experimental implementation of phase-space tomography was by Smithey et al. [38] (following work by Vogel and Risken [160]), whoused this technique to recover of the quantum state of a light field. Tolet us discuss this technique, I give a very brief description of the useof the Wigner function in quantum optics. To make the presentation assimilar as possible to that in the rest of this article, the standard quantumnotation of operators and density matrices is avoided. For more standardtreatments of the topic, the reader can consult the books by Mandel andWolf [50], Gerry and Knight [161], Leonhardt [39], and Schleich [162].

In classical optics, the electric field is the “wave function” of interest, andits variables are position and/or time. Consider a Cartesian component




of the electric field at a fixed point for a monochromatic mode within acavity. In the classical picture, this component can be written as

E (t ) = E 0 exp(−iωt ), (184)

where ω is the frequency of oscillation. Note that we are using thecomplex (or analytic signal) representation of the field. Let us define thenormalized quadratures

q(t ) = α[ E (t )], p(t ) = α[ E (t )], (185)

where α is a constant (inversely proportional to the square root of hω)that guarantees that q2+ p2 gives, instead of the intensity, the total energyof the mode divided by hω, i.e., the number of photons in the mode.Classically, q and p can be fully determined simultaneously, so they can be represented by a single point in the classical phase space (q, p), asshown in Fig. 23(a). The angular coordinate of this point in this planecorresponds to the phase of the field, while the radial one is proportionalto the amplitude, and therefore related to the intensity or number of photons. As time advances, this point rotates around the origin alonga circular path, with frequency ω. (Note that this phase-space motion is

mathematically analogous, although entirely different physically, to thatfor a ray propagating paraxially in a quadratic GRIN medium, where qis the transverse position, p is the transverse optical momentum, and n2 z

replaces ωt .)

In the quantum-optical description of the field, on the other hand, thequadrature q is not a simple function of time, but the variable of a wavefunction (q; t ). Further, q and p cannot be prescribed simultaneously,since they are Fourier conjugate variables (with K = N = 1). However, by using a phase-space representation like the Wigner function, wecan represent the state of the field at any given time as a function of both q and p, shown in Fig. 23(b). That is, at a fixed time, instead of having a delta-like localized distribution, the field is represented by

a distribution W (q, p; t ) of “possible quadrature pairs” or “possibleamplitudes and phases” whose extent in phase space is limited by theuncertainty relation: the narrower the distribution is in q, the wider it isin p, and vice versa. Since we have, instead of a single point, an extendeddistribution, the field does not have well-defined amplitude and phase:the smaller the uncertainty in the phase (related to the angular spread of the distribution with respect to the origin) of the field is, the larger theuncertainty in the number of photons (related to the radial spread), andvice versa.

As mentioned earlier, temporal evolution in the classical picturecorresponds to uniform rigid rotation around the origin. The same istrue in the quantum picture when using a phase-space representationlike the Wigner, Q, or P functions, since temporal evolution correspondsto fractional Fourier transformation in q of (q; 0), with scaling σ = 1and degree θ = ωt . Recall from Subsection 5.2 that fractional Fouriertransformation corresponds to a rigid clockwise rotation for these threerepresentations.

Let us consider the Wigner function. Since Gaussian wave functionsachieve the maximum possible joint localization in q and p, the Wignerrepresentation of a quantum field that is closest to the phase-space




Figure 23

(a) (b) (c)

p

q

p

q

p

q

Representation, in the phase space composed of the two quadratures of the

electric field, of (a) a classical monochromatic field, (b) a quantum coherent

state, and (c) a quantum squeezed state. Note that the state in (c) is squeezed

such that it has a more confined phase spread (corresponding to the angular

spread of the distribution) than the coherent state in (b). Media 21 shows the

evolution of these states and their marginals in p over one optical cycle.

representation of a classical field (a single point in phase space)corresponds to a Gaussian wave function, shifted to the desiredphase-space point. This Wigner function is itself Gaussian in both qand p. If this Gaussian has circular cross-section, then the state iscalled a coherent state, meaning that its width in q and p does notchange under temporal evolution. If the phase-space Gaussian is insteadelongated in one direction, then the state is referred to as a squeezedstate, and its marginal distributions will not only oscillate harmonically but also “breathe,” expanding and contracting at a frequency 2ω astime progresses. If the elongation is in the radial phase-space direction[as shown in Fig. 23(c)], then the phase of the field (related to theangular spread subtended by the distribution from the origin) is betterdetermined than for a coherent state of the same centroid, at the cost of

the number of photons being more uncertain.

By using a series of balanced homodyne measurements, one canestimate the marginal in p of the Wigner function of the quantum statecorresponding to different stages of its evolution. These projections atdifferent times are equivalent to projections in different directions of the Wigner function at a fixed time, from which the Wigner functionis inferred by use of the inverse-Radon transformation. An extensivereview of this method is given in the book by Leonhardt [39].

10.2. Phase-Space Tomography of Classical Fields

Phase-space tomography has also been applied for the spatial ortemporal characterization of classical fields. In the temporal domain, ithas been used for characterizing trains of ultrashort pulses by estimatingthe chronocyclic Wigner function [37,163]. Within the context of thespatial characterization of stationary fields, the first proposition of using phase-space tomography to recover the cross-spectral densityof a paraxial optical field is due to Nugent [164], and the firstexperimental implementation of this approach was by McAlister






et al. [36]. This approach has been used for the characterization of x-raysources [165,166] whose main variation is in one transverse coordinate.Tu and Tamura [167,168] realized that the same method can be cast inmuch simpler terms if, instead of the Wigner function, one employsthe ambiguity function. The key to this approach is evident from theanalogue of Eq. (140) for fields that depend only on one transversevariable, i.e.,

A J (− zp, p; 0) =k

2π

I ( x, z) exp(−ik xp) d x. (186)

This equation states that the ambiguity function at a radial slice of thephase space ( x, p) with slope − z−1 is given by the Fourier transform of the intensity at the prescribed line of constant z. Therefore, by samplingthe intensity at many planes, both before and after the focal plane ( z = 0),one can estimate the ambiguity function at sufficient radial slices to allowaccurate reconstruction through interpolation. The cross-spectral densityis then calculated as

J ( x1, 0; x2, 0) = A J ( x2 − x1, p; 0) expik x1 + x2

2 p

d p. (187)

This approach can be used not only for recovering the coherence frommeasured intensities, but also for the design of holographic elementsin order to achieve desired intensity patterns (or PSFs for imagingsystems) at different propagation distances, as proposed by Horstmeyeret al. [169,170].

Hazak [171] and Gori et al. [172] realized that, in the more general casewhere the transverse cross-spectral density depends on both x and y, theinformation contained in the field’s intensity over a volume (a functionof three variables) is not sufficient for recovering the initial cross-spectraldensity (a function of four variables). Within the approach of Tu andTamura, this problem can be appreciated from the fact that Eq. (140)gives access to the ambiguity function only within a three-dimensional“slice” in which x is parallel to p. To address this problem, Raymeret al. [173] proposed introducing a fourth independent parameter in theform of the varying focal distance of a cylindrical lens inserted before themeasurement of the intensity over a volume. The effect of such lensesis to give access to marginal projections of the Wigner function thatcorrespond to the square modulus of different anamorphic fractionalFourier transforms. A system of this type has been used for beams wherethe field is separable in the two transverse coordinates [174]. Recently,Rojas et al. [175] performed simultaneous tomographic measurementsover the phase spaces of position–direction and time–frequency.

11. Generalized Wigner Functions: Achieving ExactConservation

As discussed in the previous sections, the Wigner function allows adescription of the propagation or evolution of optical wave fields interms of a distribution that is conserved along the paths associatedwith simpler physical models. The condition for this description is




that wave propagation in the system corresponds to a combination of linear canonical transformations and phase-space shifts. However, evenin a situation as simple as free-space propagation beyond the paraxialregime, this conservation law is no longer satisfied exactly. In this section,we discuss modifications to the definition of the Wigner function forwhich the property of conservation along paths under propagationis preserved. The two cases that we concentrate on are nonparaxialfield propagation and pulse propagation through general dispersive

transparent media.

11.1. General Procedure for Constructing Conserved WignerFunctions

We now give a simple prescription for the definition of generalizationsof the Wigner function that present the property of describing propaga-tion/evolution exactly as a rearrangement of the arguments [176,177],which is valid whenever the solutions to the wave equations in questioncan be expressed as superpositions of exponentials of imaginary linearcombinations of the spatial or spatiotemporal variables. We concentrate

on two situations:i. Nonparaxial propagation in free-space of a field without evanescentcomponents. For the sake of illustration, let the field be monochromatic,fully coherent, scalar, and limited to two dimensions, so that it satisfiesthe Helmholtz equation:

∇ 2U ( x, z) + k 2U ( x, z) = 0. (188)

This field can be expressed as a weighted superposition of plane wavesof the form

U ( x, z)

= k

2π π

−π

A(θ ) exp[ik ( x sin θ

+ z cos θ )

]dθ , (189)

where A is the complex amplitude of the plane-wave componenttraveling in a direction at an angle θ with respect to the z axis.

ii. One-dimensional propagation of pulses through dispersive transparentmedia. According to Eq. (166), pulses traveling through dispersive mediacan be written as

E ( z, t ) = 1√ 2π

U 0(ω) exp{i[k (ω) z − ωt ]} dω. (190)

Notice that, in both these situations, the field of interest is of the form

E (r) =

K

2π

A(η) exp[iK r · g(η)] dη, (191)

where, for the nonparaxial field in Eq. (189), r = ( x, z), K = k , E = U ,η = θ , A = A, and g = (sin θ, cos θ ), while for the pulse in Eq. (190),r = ( z, t ), K = 1, E = E , η = ω, A = U 0, and g = [k (ω), −ω]. Theexpression in Eq. (191) looks like a restricted Fourier synthesis, in thesense that the frequency vector g is restricted to a curve (or manifold, for




higher dimensions). For the case of nonparaxial propagation, this curveis the unit circle (or the unit sphere in three dimensions) of plane-wavedirections, while for pulse propagation it is the dispersion curve k versusω. Another physical situation, not discussed here, whose solutions can bewritten in the form in Eq. (191) is that of free-space relativistic quantummechanics, where the manifold described by g is a hyperboloid in theenergy–momentum 4D space [177].

The goal is to define a generalized Wigner function W E (r, ξ ) for a field

of the form in Eq. (191) that is conserved under propagation/evolutionand whose marginal projection gives a desired bilinear property, e.g., theoptical intensity. Here, ξ is a suitable variable, e.g., the angle θ withrespect to the z axis of a ray passing through r in the case of nonparaxialfields, or the uniform velocity v of a light particle at position z and time tin the case of dispersive propagation. Mathematically, the conservationcondition can be written as a first-order differential equation of the form

u(ξ ) · ∂W E

∂r(r, ξ ) = 0, (192)

where u(θ ) = (sin θ, cos θ ) in the case of nonparaxial fields [so that

Eq. (192) indicates conservation along rays], and u(v) = (v, 1) fordispersive pulse propagation [so that Eq. (192) implies the conservationof W E along uniform velocity trajectories]. The marginal relation, on theother hand, is given by

I (r) = W E (r, ξ ) dξ. (193)

For the sake of illustration, we require that the marginal projection yieldsthe intensity. However, other local bilinear physical quantities like theelectric, magnetic, or total energy densities, the Poynting vector, or thecoherency matrix can be considered instead.

In order to find W E , the first step is to substitute Eq. (191) into thedefinition of the intensity (or the bilinear quantity of interest), leadingto

I (r) = |E (r)|2

= K

2π

A∗(η1)A(η2) exp{iK r · [g(η2) − g(η1)]} dη zdη2. (194)

For partially coherent fields and pulse trains, the product of Aand its complex conjugate must be replaced by the correspondingcorrelation. By using the change of variables η1,2(ξ,α) and introducingthe appropriate Jacobian, this expression can be written in the form of

Eq. (193), where

W E (r, ξ ) = K

2π

A∗[η1(ξ,α)]A[η2(ξ,α)]∂(η1, η2)

∂(ξ,α)

× exp(iK r · {g[η2(ξ,α)] − g[η1(ξ,α)]}) dα. (195)

Since the only dependence on r is within the exponent, it is easy tosee that this definition of the generalized Wigner function satisfiesthe conservation requirement in Eq. (192) as long as the following




Figure 24

g1

g2

Illustration of the geometric interpretation of the change of variables needed

for the construction of a Wigner function that is conserved under propagation

or evolution. The values of η1 and η2 correspond to two intersections of the

curves with a straight line normal to u(ξ ), and α controls the displacement of

this straight line.

relation holds:

u(ξ ) · {g[η2(ξ,α)] − g[η1(ξ,α)]} = 0. (196)

This relation dictates the geometry of the change of variables needed forthe definition of the generalized Wigner function, as shown in Fig. 24.Consider the curve spanned by g(η) for all values of η. Then, in order toevaluate W E at a given value of ξ , the integration in α must parametrizeall pairs of points g(η1) and g(η2) that are joined by a line segmentnormal to u(ξ ). That is, η1 and η2 must be chosen as corresponding toall pairs of points along the curve spanned by g that are intersectionswith straight lines whose slopes are normal to u(ξ ). In other words, ξ

regulates the slope of the line containingg(η

1)

andg(η

2)

while, for eachξ , the displacement of this line is regulated by α.

In the case of dispersion, the curve can present inflection points,in which a case there might be more than two intersections with astraight line. If this is the case, all possible pairs must be accountedfor by the parametrization. On the other hand, for higher-dimensionalproblems where g spans not a curve but a surface or some higher-ordermanifold, the condition in Eq. (196) is not sufficient to find a uniquechange of variables, i.e., a unique definition of the generalized Wignerfunction with the desired properties. As discussed below, some otherphysically motivated criterion must be imposed in this case to removethis ambiguity.

11.2. Nonparaxial Field Propagation

For two-dimensional nonparaxial free propagation, g(θ ) = (sin θ, cos θ )

spans a unit circle. Note that, in this case u(θ ) = g(θ ), so Eq. (196) takesthe form

u(θ ) · {u[θ 2(θ,α)] − u[θ 1(θ,α)]} = 0. (197)




Figure 25

(b)

g1

g2

θ

α /2

θ1

θ2

u

u1

u2

φ

g1

g2

g3

u(θ)

(a)

(a) Illustration of the change of variables when the curve is the unit circle, for

which α can be chosen as the angular separation of the two points, which varies

from 0 to π , and θ indicates the direction of the bisector. (b) For nonparaxial

fields in three dimensions, the manifold spanned by g is a unit sphere. The

change of variables here is from u1 and u2 (two points over this sphere) to u, the

direction of the bisector of the two points, α, the angular separation between the

points, and φ, which controls the orientation of the line joining the two points

for fixed u and α.

One solution to this constraint corresponds to the simple change of variables θ 1,2(θ,α) = θ ∓ α/2, as shown in Fig. 25(a), for −π ≤ α ≤ π .The Jacobian of this change of variables equals unity, so the generalizedWigner function can be written as [178–181]

W U ( x, z, θ ) = k

2π

π

−π

A∗

θ − α

2

A

θ + α

2

× exp2i( x cos θ − z sin θ ) sinα

2 dα. (198)

This Wigner function satisfies exactly the marginal relation and is exactlyconstant along straight paths in the direction specified by θ . It has been shown [45] that it also satisfies an inner-product overlap propertyanalogous to that in Eq. (55).

Analogous generalized Wigner functions can be defined in thethree-dimensional case [182,183], where the manifold spanned by g isthe surface of a unit sphere. Let this sphere be parametrized directly by the unit vector u [that is, g(u) = u]. The change of variables isslightly more complicated in this case, since it is necessary to includeone more integration variable in the definition of the Wigner function.The condition in Eq. (196) now becomes

u · [u2(u, α , φ ) − u1(u, α , φ )] = 0, (199)

where φ is the extra integration variable needed for the change of variables. That is, u must be constrained to the plane normal to u2 −u1. For the sake of symmetry and in order to remove the ambiguitymentioned earlier, let us also choose u to be coplanar with u1 and u2.This implies that u must bisect u1 and u2, as shown in Fig. 25(b). As inthe 2D case, we choose α as the angle between u1 and u2. The resulting




generalized Wigner function then has the form

W U (r, u) =

k

2π

2 π

0

2π

0

A∗ [u1(u, α , φ )] A [u2(u, α , φ )]

× exp {i [u2(u, α , φ ) − u1(u, α , φ )] · r} dφ sin αdα, (200)

for r

=( x, y, z) and

u1,2(u, α , φ ) = u cosα

2∓ [w1(u) cos φ + w2(u) sin φ] sin

α

2, (201)

where w1(u) and w2(u) are two unit vectors perpendicular to u and toeach other.

Wigner functions of this type have been used to describe thepropagation not only of the intensity of scalar fields but also of theelectric and magnetic energy densities and Poynting vector [184,185],polarization [185,186], and cross-spectral density [187] of nonparaxialelectromagnetic fields. In the case of polarization, the Wigner functions

are 3 × 3 matrices [analogous to the 2 × 2 matrix Wigner functionin Eq. (155)] whose elements correspond to correlations of differentCartesian components of the field. Further, upon propagation acrossor reflection from interfaces between homogeneous media, thesegeneralized Wigner functions transform following approximately aradiometric model: to leading order, they follow ray-optical laws of refraction and reflection and are multiplied by a Fresnel transmissivity orreflectivity, with corrections taking the form of derivatives of the leadingterm, which are less significant for more smoothly varying, less coherentfields [188,189]. Generalized Wigner functions have also been derived tomodel propagation through homogeneous anisotropic (e.g., birefringentand/or chiral) transparent media [190]. For these media, g spans not

one but two manifolds, one associated with each eigenpolarizationof the medium. In general, these manifolds are ellipsoids, given thedependence of the refractive index on the propagation direction.

The generalized Wigner functions in Eqs. (198) and (200) are valid forfields that include components propagating in any direction. However,in the case of nonparaxial fields composed exclusively of plane wavestraveling towards larger values of z (i.e., within the forward hemisphereof directions), these Wigner functions can be written in a form that makesthe description of propagation formally equivalent to that for paraxialfields [191]. To see this, consider the generalized Wigner function fortwo dimensions in Eq. (198). The field is assumed to be composed of

only forward-propagating plane waves; i.e., the integral in Eq. (189)extends only from −π/2 to π/2. Therefore, the Wigner function inEq. (198) vanishes for cos θ < 0. Now perform a change in the directionalparameter, so that instead of the angle θ , we specify direction by the slopeτ = tan θ . After also including a Jacobian factor |∂(θ)/∂(τ)|, we find theexpression

W U ( x, τ ; z) = W U ( x, z, arctan τ )

1 + τ 2. (202)




Figure 26

(a) (b)

(c)

(a), (b) Nonparaxial Wigner function in terms of x and τ for a truncated

cylindrical wave with numerical aperture of 1/√

2 (half-angle of π/4), at (a)

z = 0 and (b) z = 10/k . In both cases, the marginal in τ , plotted at the

top, equals the intensity profile at the corresponding propagation distance. (c)

Intensity profile over the region x ∈ [−10/k , 10/k ], z ∈ [0, 20/k ]. The red line

indicates z = 10/k , corresponding to the Wigner function in (b). See Media 22.

The propagation and marginal properties can now be written in the sameform as in the paraxial regime:

W U ( x, τ ; z) = W U ( x − zτ, τ ; 0), (203) W U ( x, τ ; z) dτ = I ( x, z). (204)

That is, propagation corresponds to a horizontal linear shear in the

phase space ( x, τ ), and the intensity is given by the marginal projectionover τ . This is illustrated in Fig. 26 for a truncated cylindrical wave,corresponding to A(θ ) being constant for |θ | ≤ π/4 and zero otherwise.The alternative representation in Eq. (202) allows, among other things,the definition of a generalized ambiguity function for nonparaxial fields,given by the Fourier transform over x and the inverse Fourier transformover τ of W U ( x, τ ; z). Given the full analogy with the paraxial regime,the standard phase-space tomography techniques used in the paraxial2D case can be extended to the nonparaxial regime [191].






The generalized Wigner functions in Eqs. (198) and (200) are formallyequivalent to wave-based definitions of the radiance proposed byOvchinnikov and Tatarskii [192] and Pedersen et al. [193–195]. Thederivation followed by those authors, however, was quite different: first,the Wigner function over all three spatial variables is calculated, yieldinga distribution over a six-dimensional phase space. This distributionturns out to be confined to the interior of a sphere in the Fouriervariables, and to diverge at the surface of this sphere. Then, a radial

projection of this distribution over the 3D Fourier variable is performed,with an appropriate weight factor, to eliminate the redundant variablecorresponding to the magnitude of the direction vector. The resultingdistribution presents the desired properties of conservation alongrectilinear rays, as well as a directional marginal related to the field’sflux. However, to compute this distribution, one must first know thefield in all space. A very different derivation for what is essentiallythe same distribution (up to an obliquity factor) was also proposed by Littlejohn and Winston [196]. Wolf et al. [178] also proposed aFourier-subspace radial projection for the case of two spatial dimensions, but by expressing the field as a weighted superposition of plane waves,they found the expression in Eq. (198). This result was then extended

to fields in 3D space and with any state of coherence, leading to afamily of generalized Wigner functions [179,182,183] whose marginalsyield different local properties like the intensity, flux, and energy density.Sheppard and Larkin [180,181] found a more direct derivation in thetwo-dimensional case through a change of the angular variables tocentroid and difference, consistent with the construction discussed here.The generalized Wigner functions in both two and three dimensionscan also be calculated in terms of a series of derivatives acting on thestandard Wigner function [197].

11.3. Pulse Propagation Through Transparent Dispersive

Media

In the case of propagation through dispersive media, the generalizedWigner function behaves as a statistical–mechanical distribution of independent “light particles” traveling at different velocities v, since itis defined to obey the propagation and marginal properties [176,177]

W E ( z, t , v) = W E ( z − vt , 0, v) = W E

0, t − z

v, v

, (205)

W E ( z, t , v) = I ( z, t ). (206)

Of course, the precise definition of the generalized Wigner functiondepends on the dispersion relation, according to the constraint inEq. (196). Since in this case u(v) = (v, 1) and g(ω) = [k (ω), −ω], thisconstraint takes the form

v[k (ω2) − k (ω1)] − (ω2 − ω1) = 0. (207)




The solution of this constraint and the resulting Wigner functions areshown in what follows for two simple examples.

11.3a. Lorentz Model Dispersion

Consider first a nonabsorbing approximation to the Lorentz model of dispersion for a medium presenting a single resonance at ωR, given by

k (ω) = ω

c

1 − β

ω − ωR

, (208)

where β is a positive constant related to the plasma frequency of themedium. As shown in Ref. [176], the change of variables that results fromimposing Eq. (207) can be written as

ω1,2(v, α) = ωR −

βωRv

c − vexp

±α

2

. (209)

It is easy to verify that this change of variables does satisfy the constraintin Eq. (207) for k given in Eq. (208). The corresponding Wigner functionthen takes the form

W E ( z, t , v) = cβωR

4π(c − v)2

U ∗0

ωR −

βωRv

c − vexp

α

2

× U 0

ωR −

βωRv

c − vexp

−α

2

× exp

2i

βωRv

c − v

z − vt

vsinh

α

2

dα. (210)

This Wigner function is shown in Media 25 (Fig. 27) for the case wherethe pulse’s spectral amplitude U 0(ω) is constant within the range ω ∈[0.5ωR, 0.85ωR] and vanishes outside of it, for a medium where β =0.4ωR. The marginal projection of this distribution in v, equal to thespatial intensity profile, is also shown.

11.3b. Waveguide Dispersion

As a second example, consider the case of a perfect metallic hollowwaveguide, whose dispersion is given by

k =

ω2 − ω2co

c, (211)

for ω < ωco, where ωco is the cutoff frequency. A convenient change of variables that satisfies Eq. (207) for this dispersion relation is given by

ω1,2(v, α) = ωco cosh

α(v) ∓ α

2

, (212)






Figure 27

0

10cω R

k

(a)

(b) (c)

v

z

v

z

0.5 ω R 0.85 ω

R

ω R

(a) Dispersion curve in Eq. (208) with β = 0.4ωR corresponding to a transparent

Lorentz-model medium with one resonance. The green rectangle indicates the

pulse’s spectral amplitude, U 0(ω). (b), (c) Generalized Wigner function in

Eq. (210) for this pulse, in terms of z ∈ [0.2ct − 20ωR/c, 0.2ct + 20ωR/c] and

v ∈ [0, 0.4c], for (b) t = 0 and (c) t = 100/ωR. The marginal in v, plotted at the

top, equals the intensity profile at the corresponding propagation time. Media 25

shows the horizontal shearing around the line v = 0.2c of the Wigner function

and the evolution of the spatial intensity profile for t between 0 and 100/ωR.

Notice that we chose to shear around v = 0.2c instead of around v = 0 in order

to move with the pulse, i.e., to factor out the pulse’s spatial displacement.

where α(v) = arctanh(v/c) is the relativistic velocity parameter. Thegeneralized Wigner function then takes the form

W E ( z, t , v) = cω2co

2π(c2 − v2)

2α

−2α

U ∗0

ωco cosh

α − α

2

× U 0

ωco cosh

α + α

2

sinh

α − α

2

sinh

α + α

2

× exp

2i

ωco( z − vt )√ c2 − v2

sinhα

2

dα. (213)

The evolution of this generalized Wigner function and the correspondingintensity profile is illustrated in Media 24 in Fig. 28 for a pulse witha spectrum constrained to the range ω ∈ [1.1ωco, 1.5ωco], within whichU 0(ω) is constant.

In the examples we just presented, we plotted the Wigner function overthe phase space of z versus v for different propagation times t. However,there are other ways to present these Wigner functions. For example, if the temporal intensity profiles are of more interest than the spatial ones,








Figure 28

0

c

k

(a)

(b) (c)

1.2 ω co

ω co1.1 1.5 ω coω co

v v

z z

ω

MEDIA 24 (a) Dispersion curve for a hollow metallic waveguide, given in

Eq. (211). The green rectangle indicates the pulse’s spectral amplitude, U 0(ω).

(b), (c) Generalized Wigner function in Eq. (213) for this pulse, in terms of

z ∈ [0.6ct − 40ωco/c, 0.6ct + 40ωco/c] and v ∈ [0.4c, 0.8c], for (b) t = 0

and (c) t = 200/ωco. The marginal in v, plotted at the top, equals the intensity

profile at the corresponding propagation time. Media 24 shows the horizontal

shearing around the line v = 0.6c of the Wigner function and the evolution of

the spatial intensity profile for t between 0 and 200/ωco. Notice that shearing

around v = 0.6c instead of around v = 0 factors out the spatial displacement of

the pulse.

it might be more useful to use a variant of these Wigner functions interms of t and a “slowness” variable s = 1/v, as was done in Ref. [198].This way, propagation in z corresponds to a horizontal linear shearing inthe phase space (t , s). Alternatively, if it were preferable to express theseWigner functions in the chronocyclic phase space, one can substitutev(ω) = 1/k (ω) and include a Jacobian factor for normalization, so thatthe conserved chronocyclic Wigner function is given by

¯W E ( z, t , ω)

=k (ω)

[k (ω)]2W E z, t ,

1

k (ω) , (214)

such that W E ( z, t , ω) dω = I ( z, t ). Finally, it must be noted that versions

of other members of the Cohen class of phase-space representations thatare also conserved exactly under general transparent dispersive mediahave also been proposed [199–201].






11.4. Reduction to the Standard Wigner Function forParabolic Manifolds

Notice that, for the simple case in which g spans a parabola with anylocation and orientation, the generalized Wigner function reduces to aform of the standard Wigner function. For a general parabolic curve, thisvector can be parametrized as

g(η) = a + bη + cη2, (215)

where a, b, and c are three constant vectors. After making the change of variables η1,2(ξ,α) = η(ξ) ∓ α/2, which has a Jacobian equal to |η(ξ )|,Eq. (196) can be written as

αu(ξ ) · [b + η(ξ)c] = 0, (216)

so that η(ξ) = −b · u(ξ)/c · u(ξ ). The generalized Wigner function is thengiven by

W E (r, ξ ) = |η(ξ )| K

2π A∗

η(ξ) − α

2 A η(ξ) + α

2 × exp {iK αr · [b + η(ξ)c]} dα

= |η(ξ )| W E {η(ξ), r · [b + η(ξ)c]}, (217)

where W E is the standard Wigner function. A similar proof can beperformed for higher dimensionalities.

The reduction of the generalized Wigner function to the standardone for parabolic curves or manifolds explains the usefulness of thestandard Wigner function in the case of paraxial free propagation andnarrow-bandwidth pulses. In these cases, only a small segment of themanifold spanned by g is of interest, which can then be accuratelyapproximated by a parabola: for paraxial optics we can approximate g =(sin θ , cos θ ) ≈ (θ, 1 − θ

2

/2), while for narrow-band pulse propagationwe can use either of the quadratic approximations in Eq. (168) or (176).The same is true in the case of quantum mechanics where, following theKlein–Gordon or Dirac equations, the manifold for free evolution of aparticle is a (two-sheeted) hyperboloid of “radius” equal to the particle’smass in the energy–momentum space. In the nonrelativistic limit, onlythe region near the (positive energy) vertex of this hyperboloid is of importance, so it can be approximated by a paraboloid. The standardWigner function is therefore useful in the nonrelativistic limit.

12. Concluding Remarks

An overview of the use of the Wigner function in optics was presented,with an emphasis on its application to model field propagation. Asdiscussed briefly in Section 6, there are many other phase-spacedistributions that are commonly used in physics and engineering, eachof them with a unique set of properties that make it well suited for somepurposes. However, there are two properties that, together, confer theWigner function a particular connection to physical intuition. The first isthat the Wigner function for an optical field after spatial propagation or




temporal evolution corresponds to a simple mapping of arguments of theWigner function for the initial field. This property is valid in cases wherepropagation or evolution correspond mathematically to an arbitrarycombination of linear canonical transformations and phase-space shifts.Such cases include propagation of paraxial stationary fields throughoptical systems that are linear in the ray domain, propagation of time-dependent fields through transparent media with linear dispersion,and the temporal evolution of quantum states of light. This mapping

of the Wigner function can be interpreted as conservation along pathsassociated with a simpler physical model, like ray optics, free particledynamics, or classical field evolution. The second key property is that,at any point of propagation over such systems, the marginal projectionof the Wigner function over the Fourier variables is associated withan observable, physically meaningful local property of the field. Giventhese two properties, the Wigner function’s behavior mimics that of the phase-space distributions used in the simpler physical models forthese situations (e.g., the basic radiance from the theory of radiometry).However, the Wigner function incorporates the fundamental limitationsimposed by the space–bandwidth product theorem, and this causes it to be potentially negative within some regions of phase space, and to have

strict restrictions in its functional form.By modifying the definition of the Wigner function according to asimple geometrically motivated prescription, the exact satisfaction of its two key properties can be extended at least for certain physicalsituations that are not described by linear canonical transformationsand phase-space shifts. These situations include all cases where thewave equation accepts a complete set of solutions corresponding topurely imaginary exponentials of linear combinations of the variables.This is the case of nonparaxial field propagation (if evanescent wavesare excluded) and of pulse propagation through homogeneous mediawith arbitrary dispersion (if absorption is neglected). This extension brings to mind the following question: what is the most general

class of system and field for which a generalized Wigner functioncan be defined that satisfies exactly both the marginal relation to theintensity and the property of conservation along paths? This generalquestion was not addressed in this article. Let us consider the caseof propagation in general transparent inhomogeneous linear media.There are inhomogeneous refractive index distributions that allow arigorous description of the propagation of intensity as a conservedweight distribution for rays, even beyond the paraxial regime. One trivialexample of such case is that of inhomogeneous and anisotropic mediathat are mappings of homogeneous media through what is known astransformation optics [202], since the Wigner function would then bethe result of the corresponding mapping of the Wigner function for a

homogeneous medium. On the other hand, there are other distributionsfor which this type of representation is certainly impossible. Take the caseof a gradient index waveguide that is symmetric with respect to the z axis but that is anharmonic, i.e., that present nonlinear mode dispersion. Inthe ray picture, propagation in z would correspond to a swirling of phasespace, since different rays oscillate around the axis with different periods.In the wave domain, on the other hand, fields that are concentratedspatially and directionally far from the axis of the waveguide are knownto return to their initial state after some propagation distance, due to




the phenomenon known as a wave packet revival [203]. Therefore, thereseems to be an irreconcilable difference in the behavior of the rays andthe waves.

In many cases, phase-space distributions like the Wigner function arevaluable more for conceptual reasons than for computational ones.They provide an intuitive framework that leads to useful ideas andinterpretations, even if the methods resulting from them can then beexpressed in a more direct way that bypasses the phase-space picture.

Computationally, the increased number of variables and complicatedfunctional form of the Wigner function or its generalizations discussedhere can be important obstacles towards numerical implementation,particularly in the case of coherent fields or pure states. For partiallycoherent fields or mixed states, on the other hand, the value of Wignerfunctions can be not only conceptual but also computational, becauseusing standard propagation formulas can become computationallyprohibitive in this case. If used properly, the Wigner function canallow significant computational savings in these cases, as discussed inRef. [120].

This article is dedicated to the memory of Luis Edgar Vicent (1972–2011),

friend, colleague and coauthor of Ref. [120].

References and Notes

1. N. L. Balazs and B. K. Jennings, “Wigner’s function and other distribution

functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).

2. E. P. Wigner, “On the quantum correction for thermodynamic equilib-

rium,” Phys. Rev. 40, 749–759 (1932).

3. J. Ville, “Theorie et applications de la notion de signal analytique,” Cables

Transm. 2A, 6174 (1948).

4. L. S. Dolin, “Beam description of weakly inhomogeneous wave fields,”

Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).5. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259

(1968).

6. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley,

1983), pp. 13–27.

7. P. Moon and D. E. Spencer, The Photic Field (MIT Press, 1981).

8. L. Cohen, Time–Frequency Analysis (Prentice Hall, 1995).

9. W. Mecklenbrauker and F. Hlawatsch, The Wigner Distribution: Theory

and Applications in Signal Processing (Elsevier, 1997).

10. H. W. Lee, “Theory and application of the quantum phase-space distribu-

tion functions,” Phys. Rep. 259, 147–211 (1995).

11. G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting

operators and general phase-space methods in quantum mechanics. I.

Mapping theorems and ordering of functions of noncommuting operators,”

Phys. Rev. D 2, 2161–2186 (1970).


operators and general phase-space methods in quantum mechanics. II.

Quantum mechanics in phase space,” Phys. Rev. D 2, 2187–2205 (1970).


operators and general phase-space methods in quantum mechanics. III.




A generalized Wick theorem and multitime mapping,” Phys. Rev. D 2,

2206–2225 (1970).

14. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution

functions in physics: Fundamentals,” 106 121–167 1984.

15. M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, ed., Phase-Space

Optics: Fundamentals and Applications (McGraw-Hill, 2009).

16. A. Torre, Linear Ray and Wave Optics in Phase Space: Bridging Ray and

Wave Optics via the Wigner Phase-Space Picture (Elsevier, 2005).

17. D. Dragoman, “The Wigner distribution function in optics and optoelec-tronics,” in Progress in Optics XXXVII , E. Wolf, ed. (Elsevier, 1997),

pp. 1–56.

18. J. T. Sheridan, W. T. Rhodes, and B. M. Hennelly, “Wigner Optics,” Proc.

SPIE 5827, 627–638 (2005).

19. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill,

1962).

20. G. Folland and A. Sitaram, “The uncertainty principle: a mathematical

survey,” J. Fourier Anal. Appl. 3, 207–238 (1997).

21. E. U. Condon, “Immersion of the Fourier transform in a continuous group

of functional transformations,” Proc. Natl. Acad. Sci. 23, 158–163 (1937).

22. V. Namias, “The fractional order Fourier transform and its application toquantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).

23. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, “Fractional transfor-

mations in optics,” in Progress in Optics XXXVIII , E. Wolf, ed. (1998),

pp. 263–242.

24. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier

Transform with Applications in Optics and Signal Processing (John Wiley

& Sons, 2001).

25. A. Erdelyi, Asymptotic Expansions (Dover, 1956).

26. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and

H. M. Ozaktaz, “Anamorphic fractional Fourier transform: optical imple-

mentation and applications,” Appl. Opt. 34, 7451–7456 (1995).

27. S. A. Collins, “Lens-system diffraction integral written in terms of matrixoptics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).

28. M. Moshinsky and C. Quesne, “Linear canonical transformations and their

unitary representations,” J. Math. Phys. 12(8), 1772–1783 (1971).

29. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum

Press, 1979), Ch. 9, 10.

30. T. Alieva and M. J. Bastiaans, “Properties of the linear canonical integral

transformation,” J. Opt. Soc. Am. A 24, 3658–3665 (2007).

31. D. Mustard, “The fractional Fourier transform and the Wigner distribu-

tion,” J. Aust. Math. Soc. B-Appl. Math. 38, 209–219 (1996) Published

earlier as Applied Mathematics Preprint AM89/6 School of Mathematics,

UNSW, Sydney, Australia (1989).

32. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional

Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).

33. A. W. Lohmann and B. H. Soffer, “Relationships between the

Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11,

1798–1801 (1994).

34. See Chapter 4 by G. Saavedra and W. Furlan in Ref. [15], pp. 107–164.

35. J. Bertrand and P. Bertrand, “A tomographic approach to Wigner’s func-

tion,” Found. Phys. 17, 397–405 (1987).




36. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer,

“Optical phase retrieval by phase space tomography and fractional-order

Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).

37. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic

tomography for measuring the amplitude and phase structure of optical

pulses,” Opt. Lett. 18, 2041–2043 (1993).

38. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of

the Wigner distribution and the density matrix of a light mode using optical

homodyne tomography: application to squeezed states and the vacuum,”Phys. Rev. Lett. 70, 1244–1247 (1993).

39. U. Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press,

1997).

40. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Vol. 1,

(Wiley, 1977), pp. 214–227.

41. J. E. Moyal, “Quantum mechanics as a statistical theory,” Math. Proc.

Camb. Phil. Soc. 45, 99–124 (1949).

42. R. G. Littlejohn and R. Winston, “Generalized radiance and measure-

ment,” J. Opt. Soc. Am. A 12, 2736–2743 (1995).

43. D. Dragoman, “Wigner-distribution-function representation of the cou-

pling coefficient,” Appl. Opt. 34, 6758–6763 (1995).44. A. Wax and J. E. Thomas, “Optical heterodyne imaging and Wigner phase

space distributions,” Opt. Lett. 21, 1427–1429 (1996).

45. M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am.

A 17, 1256–1264 (2000).

46. M. V. Berry, “Semi-classical mechanics in phase space: a study of Wigners

function,” Philos. Trans. R. Soc. Lond. 287, 237–271 (1977).

47. M. V. Berry and N. L. Balazs, “Evolution of semiclassical quantum states

in phase space,” J. Phys. A Math. Phys. 12, 625–642 (1979).

48. M. V. Berry, “Quantum scars of classical closed orbits in phase space,”

Proc. R. Soc. Lond. Ser. A 423, 219–231 (1989).

49. M. A. Alonso and G. W. Forbes, “Phase-space distributions forhigh-frequency fields,” J. Opt. Soc. Am. A 17, 2288–2300 (2000).

50. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cam-

bridge University Press, 1995), Sec. 4.7.

51. M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt.

Soc. Am. 73, 251–255 (1983).

52. A. Starikov, “Effective number of degrees of freedom of partially coherent

sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).

53. See Ref. [50], p. 261.

54. L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77,

941–981 (1989).

55. J. H. Eberly and K. Wodkiewicz, “The time-dependent physical spectrum

of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977).

56. K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.

Math. Soc. Jpn. 22, 264–314 (1940).

57. Y. Kano, “A new phase-space distribution function in the statistical theory

of the electromagnetic field,” J. Math. Phys. 6, 1913–1915 (1965).

58. C. L. Mehta and E. C. G. Sudarshan, “Relation between Quantum

and Semiclassical Description of Optical Coherence,” Phys. Rev. 138,

B274–B280 (1965).




59. R. J. Glauber, “Optical coherence and photon statistics,” in Quantum

Optics and Electronics, C. Dewitt, A. Blandin, and C. Cohen-Tannoudji,

ed. (Gordon and Breach, 1965), pp. 65.

60. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys.

Rev. 131, 2766–2788 (1963).

61. J. R. Klauder, “Continuous representation theory. I. Postulates of continu-

ous representation theory,” J. Math. Phys. 4, 1055–1058 (1963).

62. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechan-

ical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279(1963).

63. J. G. Kirkwood, “Quantum statistics of almost classical ensembles,” Phys.

Rev. 44, 31–37 (1933).

64. A. W. Rihaczek, “Signal energy distribution in time and frequency,” IEEE

Trans. Info. Theory 14, 369–374 (1968).

65. H. Margenau and R. N. Hill, “Correlation between measurements in

quantum theory,” Prog. Theoret. Phys. 26, 722–738 (1961).

66. P. M. Woodward, Probability and Information Theory with Applications

to Radar (Pergamon, 1953).

67. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64,

779–788 (1974).

68. L.-P. Guigay, Ambiguity Function in Optical Imaging, Chapter 2 of Ref.[15], pp. 45–62.

69. L. Cohen, “Generalized phase-space distributions,” J. Math. Phys. 7,

781–786 (1966).

70. M. J. Bastiaans, Wigner Distribution in Optics, Chapter 1 of Ref. [15], pp.

1–44.

71. R. K. Luneburg, Mathematical Theory of Optics (University of California

Press, 1966), pp. 246–257.

72. See Ref. [71], pp. 103–110.

73. M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University

Press, 1999), pp. 142–144.

74. H. A. Buchdahl, Hamiltonian Optics (Dover, 1993), pp. 7–12.

75. See Ref. [6], pp. 13–25.

76. See Ref. [6], pp. 75–80.

77. A. Walther, “Lenses, wave optics and eikonal functions,” J. Opt. Soc. Am.

59, 1325–1333 (1969).

78. A. Walther, The Ray and Wave Theory of Lenses (Cambridge University

Press, 1995), pp. 169–187.

79. M. A. Alonso and G. W. Forbes, “Semigeometrical estimation of Green’s

functions and wave propagators in optics,” J. Opt. Soc. Am. A 14,

1076–1086 (1997).

80. J. H. Van Vleck, “The correspondence principle in the statistical interpre-

tation of quantum mechanics,” Proc. Natl. Acad. Sci. USA 14, 178–188

(1928).81. M. C. Gutzwiller, “Periodic orbits and classical quantization conditions,”

J. Math. Phys. 12, 343–358 (1971).

82. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their

optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).

83. M. J. Bastiaans, “Wigner distribution function applied to optical signals

and systems,” Opt. Commun. 25, 26–30 (1978).

84. M. J. Bastiaans, “Transport equations for the Wigner distribution func-

tion,” J. Mod. Opt. 26, 1265–1272 (1979).




85. M. J. Bastiaans, “The Wigner distribution function and Hamilton’s char-

acteristics of a geometric–optical system,” Opt. Commun. 30, 321–326

(1979).

86. M. J. Bastiaans, “Wigner distribution function and its application to

first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).

87. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988),

pp. 101–136.

88. K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castaneda, “The ambiguity

function as a polar display of the OTF,” Opt. Commun. 44, 323–326(1983).

89. S. B. Oh and G. Barbastathis, “Axial imaging necessitates loss of lat-

eral shift invariance: proof with the Wigner analysis,” Appl. Opt. 48,

5881–5888 (2009).

90. S. B. Mehta and C. J. R. Sheppard, “Using the phase-space imager to

analyze partially coherent imaging systems: bright-field, phase contrast,

differential interference contrast, differential phase contrast, and spiral

phase contrast,” J. Mod. Opt. 57, 718–739 (2010).

91. J. E. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys.

Rev. Lett. 58, 1499–1501 (1987).

92. W. P. Schleich, J. P. Dahl, and S. Varro, “Wigner function for a free particlein two dimensions: a tale of interference,” Opt. Commun. 283, 786–789

(2010).

93. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.

47, 264–267 (1979).

94. M. V. Berry and F. J. Wright, “Phase-space projection identities for

diffraction catastrophes,” J. Phys. A. Math. Gen. 13, 149–160 (1980).

95. R.-P. Chen, H.-P. Zheng, and C.-Q. Dai, “Wigner distribution function of

an Airy beam,” J. Opt. Soc. Am. A 28, 1307–1311 (2011).

96. G. Siviloglou and D. Christodoulides, “Accelerating finite energy Airy

beams,” Opt. Lett. 32, 979–981 (2007).

97. G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observa-

tion of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901-1-213901-4(2007).

98. E. R. Dowski and W. T. Cathey, “Extended depth of field through

wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).

99. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt.

Soc. Am. 50, 749 (1960).

100. C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, “Optical microscopy with

extended depth of field,” Proc. R. Soc. Lond. A 387, 171–186 (1983).

101. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, “Bessel annular

apodizers: imaging characteristics,” Appl. Opt. 26, 2770–2772 (1987).

102. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, “Ambiguity

function as a design tool for high focal depth,” Appl. Opt. 27, 790–795

(1988).

103. Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Lu, “Analysis of optical systems

with extended depth of field using the Wigner distribution function,” Appl.

Opt. 45, 8586–8595 (2006).

104. M. Testorf and J. Ojeda-Castaneda, “Fractional Talbot effect: analysis in

phase space,” J. Opt. Soc. Am. A 13, 119–125 (1996).

105. M. Testorf, “Designing Talbot array illuminators with phase-space optics,”

J. Opt. Soc. Am. A 23, 187–192 (2006).




106. M. E. Testorf, Self-imaging in Phase Space, Chapter 9 of Ref. [15], pp.

279–307.

107. J. Lancis, E. E. Sicre, A. Pons, and G. Saavedra, “Achromatic white-light

self-imaging phenomenon: an approach using the Wigner distribution

function,” J. Mod. Opt. 42, 425–434 (1995).

108. A. Stern and B. Javidi, “Sampling in the light of Wigner distribution,” J.

Opt. Soc. Am. A 21, 360–366 (2004); errata, 21, 1602–1612 (2004).

109. B. M. Hennelly, J. J. Healy, and J. T. Sheridan, Sampling and Phase Space,

Chapter 10 of Ref. [15], pp. 309–336.110. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39,

293–297 (1981).

111. A. Stern and B. Javidi, “Improved-resolution digital holography using the

generalized sampling theorem for locally band-limited fields,” J. Opt. Soc.

Am. A 23, 1227–1235 (2006).

112. A. Stern and B. Javidi, “Space-bandwidth conditions for efficient

phase-shifting digital holographic microscopy,” J. Opt. Soc. Am. A 25,

736–741 (2008).

113. K. B. Wolf and A. L. Rivera, “Holographic information in the Wigner

function,” Opt. Commun. 144, 36–42 (1997).

114. A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castaneda, “Holography andthe Wigner function,” in The Art and Science of Holography: A Tribute to

Emmett Leith and Yuri Denisyuk , H. J. Caulfield, ed. (SPIE Press, 2004),

pp. 127–144.

115. A. W. Lohmann, “The space–bandwidth product, applied to spatial fil-

tering and to holography,” in Selected Papers on Phase-Space Optics,

M. E. Testorf, J. Ojeda-Castaneda, and A. W. Lohmann, ed. (SPIE Press,

2006), pp. 11–32.

116. M. Testorf and A. W. Lohmann, “Holography in phase space,” Appl. Opt.

47, A70–A77 (2008).

117. S. B. Oh and G. Barbastathis, “Wigner distribution function of volume

holograms,” Opt. Lett. 34, 2584–2586 (2009).

118. M. Testorf, “Analysis of the moire effect by use of the Wigner distributionfunction,” J. Opt. Soc. Am. A 17, 2536–2542 (2000).

119. N. Morelle, M. E. Testorf, N. Thirion, and M. Saillard, “Electromagnetic

probing for target detection: rejection of surface clutter based on the

Wigner distribution,” J. Opt. Soc. Am. A 26, 1178–1186 (2009).

120. L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for

the propagation of partially coherent fields,” Opt. Commun. 207, 101–112

(2002).

121. E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19,

2024–2026 (1994).

122. A. T. Friberg and S. Yu. Popov, “Radiometric description of intensity

and coherence in generalized holographic axicon images,” Appl. Opt. 35,

3039–3046 (1996).

123. K. Duan and B. Lu, “Wigner-distribution-function matrix and its applica-

tion to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B

22, 1585–1593 (2005).

124. R. Castaneda and J. Garcia-Sucerquia, “Electromagnetic spatial coherence

wavelets,” J. Opt. Soc. Am. A 23, 81–90 (2006).

125. A. Luis, “Spatial-angular Mueller matrices,” Opt. Commun. 263, 141–146

(2006).




126. A. Luis, “Ray picture of polarization and coherence in a Young interfer-

ometer,” J. Opt. Soc. Am. A 23, 2855–2860 (2006).

127. R. Castaneda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of

diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273,

8–20 (2007).

128. R. Castaneda and J. Carrasquilla, “Spatial coherence wavelets and

phase-space representation of diffraction,” Appl. Opt. 47, E76–E87

(2008).

129. M. A. Alonso, “Diffraction of paraxial partially coherent fields by planar

obstacles in the Wigner representation,” J. Opt. Soc. Am. A 26, 1588–1597

(2009).

130. A. Wax and J. E. Thomas, “Measurement of smoothed Wigner phase-space

distributions for small-angle scattering in a turbid medium,” J. Opt. Soc.

Am. A 15, 1896–1908 (1998).

131. H. T. Yura, L. Thrane, and P. E. Andersen, “Closed-form solution for

the Wigner phase-space distribution function for diffuse reflection and

small-angle scattering in a random medium,” J. Opt. Soc. Am. A 17,

2464–2474 (2000).

132. C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical

coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811(2000).

133. A. C. Fannjiang, “White-noise and geometrical optics limits of

Wigner–Moyal equation for wave beams in turbulent media,” Commun.

Math. Phys. 254, 289–322 (2005).

134. D. Dragoman, “Wigner distribution function in nonlinear optics,” Appl.

Opt. 35, 4142–4146 (1996).

135. D. Dragoman, J. P. Meunier, and M. Dragoman, “Beam-propagation

method based on the Wigner transform: a new formulation,” Opt. Lett.

22, 1050–1052 (1997).

136. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical

theory for incoherent light propagation in nonlinear media,” Phys. Rev. E65, 035602R (2002).

137. M. Lisak, L. Helczynski, and D. Anderson, “Relation between different

formalisms describing partially incoherent wave propagation in nonlinear

optical media,” Opt. Commun 220, 321–323 (2003).

138. H. Gao, L. Tian, B. Zhang, and G. Barbastathis, “Iterative nonlinear

beam propagation using Hamiltonian ray tracing and Wigner distribution

function,” Opt. Lett. 35, 4148–4150 (2010).

139. S. Abe and J. T. Sheridan, “Wigner optics in the metaxial regime,” Optik

114, 139–141.

140. Yu. A. Kravtsov and L. A. Apresyan, “Radiative transfer: new aspects of

the old theory,” in Progress in Optics, Vol. XXXVI, E. Wolf, ed. (North

Holland, 1996), pp. 179–244.

141. L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and

Wave Aspects (Gordon and Breach, 1996).

142. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry Mile-

stone Series, Vol. MS69 (SPIE Optical Engineering Press, 1993).

143. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).

144. A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936

(1982).




145. G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space

representations of the cross-spectral density operator,” Opt. Commun. 62,

67–72 (1987).

146. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623

(1973).

147. A. T. Friberg, “On the generalized radiance associated with radiation from

a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).

148. J. T. Foley and E. Wolf, “Radiometry as a short-wavelength limit of

statistical wave theory with globally incoherent sources,” Opt. Commun.55, 236–241 (1985).

149. J. Azana, “Time–frequency (Wigner) analysis of linear and nonlinear pulse

propagation in optical fibers,” EURASIP J. Appl. Signal Process. 2005,

1554–1565 (2005).

150. P. Loughlin and L. Cohen, “A Wigner approximation method for wave

propagation,” J. Acoust. Soc. Am. 118, 1268–1271 (2005).

151. J. Ojeda-Castaneda, J. Lancis, C. M. Gomez-Sarabia, V. Torres-Company,

and P. Andres, “Ambiguity function analysis of pulse train propagation:

applications to temporal Lau filtering,” J. Opt. Soc. Am. A 24, 2268–2273

(2007).

152. P. Loughlin and L. Cohen, “Approximate wave function from approximatenon-representable Wigner distributions,” J. Mod. Opt. 55, 3379–3387

(2008).

153. L. Cohen, P. Loughlin, and G. Okopal, “Exact and approximate moments

of a propagating pulse,” J. Mod. Opt. 55, 3349–3358 (2008).

154. C. Dorrer and I. A. Walmsley, Phase space in ultrafast optics, Chapter 11

of Ref. [15], pp. 337–383.

155. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromag-

netic pulses,” Adv. Opt. Phot. 1, 308–437 (2009).

156. B. H. Kolner, “Space–time duality and the theory of temporal imaging,”

IEEE J. Quantum Electron. 30, 1951–1963 (1994).

157. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed.

(Wiley, 2007), pp. 188.158. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral

coherence properties of temporally modulated stationary light sources,”

Opt. Express 11, 1894–1899 (2003).

159. B. J. Davis, “Observable coherence theory for statistically periodic fields,”

Phys. Rev. A 76, 043843 (2007).

160. K. Vogel and H. Risken, “Determination of quasiprobability distributions

in terms of probability distributions for the rotated quadrature phase,”

Phys. Rev. A 40, 2847–2849 (1989).

161. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge

University Press, 2005), pp. 56–71.

162. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).

163. C. Dorrer and I. Kang, “Complete temporal characterization of short

optical pulses by simplified chronocyclic tomography,” Opt. Lett. 28,

1481–1483 (2003).

164. K. A. Nugent, “Wave field determination using three-dimensional intensity

information,” Phys. Rev. Lett. 68, 2261–2264 (1992).

165. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and

I. McNulty, “X-ray imaging: a generalized approach using phase-space

tomography,” J. Opt. Soc. Am. A 22, 1691–1700 (2005).




166. C. Q. Tran, A. P. Mancuso, B. B. Dhal, K. A. Nugent, A. G. Peele, Z. Cai,

and D. Paterson, “phase space reconstruction of focused x-ray fields,”

J. Opt. Soc. Am. A 23, 1779–1786 (2006).

167. J. Tu and S. Tamura, “Wave field determination using tomography of the

ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).

168. J. Tu and S. Tamura, “Analytic relation for recovering the mutual intensity

by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206

(1998).

169. R. Horstmeyer, S. B. Oh, and R. Raskar, “Iterative aperture mask designin phase space using a rank constraint,” Opt. Express 18, 22545–22555

(2010).

170. R. Horstmeyer, S. B. Oh, O. Gupta, and R. Raskar, “Partially coherent

ambiguity functions for depth-variant point spread function design,” pre-

sentation during PIERS, Marrakesh, March, 2011.

171. G. Hazak, “Comment on ‘Wave field determination using three-

dimensional intensity information’,” Phys. Rev. Lett. 69, 2874–2874

(1992).

172. F. Gori, M. Santarsiero, and G. Guattari, “Coherence and the spatial

distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).

173. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field recon-

struction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140(1994).

174. A. Camara, T. Alieva, J. A. Rodrigo, and M. L. Calvo, “Phase space

tomography reconstruction of the Wigner distribution for optical beams

separable in Cartesian coordinates,” J. Opt. Soc. Am. A 26, 1301–1306

(2009).

175. P. Rojas, R. Blaser, Y. M. Sua, and K. F. Lee, “Optical phase-space-time-

frequency tomography,” Opt. Express 19, 7480–7490 (2011).

176. J. C. Petruccelli and M. A. Alonso, “Phase space distributions tailored for

dispersive media,” J. Opt. Soc. Am. A 27, 1194–1201 (2010).

177. J. C. Petruccelli,Generalized Wigner Functions, Ph.D. Thesis, University

of Rochester (Rochester, NY, 2010).

178. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for

Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).

179. M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields

in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).

180. C. J. R. Sheppard and K. G. Larkin, “Wigner function for nonparaxial

wave fields,” J. Opt. Soc. Am. A 18, 2486–2490 (2001).

181. C. J. R. Sheppard and K. G. Larkin, “Wigner function for highly conver-

gent three-dimensional wave fields,” Opt. Lett. 26, 968–970 (2001).

182. M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields

in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).

183. M. A. Alonso, “Radiometry and wide-angle wave fields. III. Partial

coherence,” J. Opt. Soc. Am. A. 18, 2502–2511 (2001).184. M. A. Alonso, “Exact description of free electromagnetic wave fields in

terms of rays,” Opt. Express 11, 3128–3135 (2003).

185. M. A. Alonso, “Wigner functions for nonparaxial, arbitrarily polarized

electromagnetic wave fields in free-space,” J. Opt. Soc. Am. A. 21,

2233–2243 (2004).

186. J. C. Petruccelli, N, J. Moore, and M. A. Alonso, “Two methods for

modeling the propagation of the coherence and polarization properties of

nonparaxial fields,” Opt. Commun. 283, 4457–4466 (2010).




187. J. C. Petruccelli and M. A. Alonso, “Ray-based propagation of the

cross-spectral density,” J. Opt. Soc. Am. A 25, 1395–1405 (2008).

188. J. C. Petruccelli and M. A. Alonso, “Propagation of partially coherent

fields through planar dielectric boundaries using angle-impact Wigner

functions I. Two dimensions,” J. Opt. Soc. Am. A 24, 2590–2603 (2007).

189. J. C. Petruccelli and M. A. Alonso, “Propagation of nonparaxial partially

coherent fields across interfaces using generalized radiometry,” J. Opt.

Soc. Am. A 26, 2012–2022 (2009).190. J. C. Petruccelli and M. A. Alonso, “Generalized radiometry model for the

propagation of light within anisotropic and chiral media,” J. Opt. Soc. Am.

A 28, 791–800 (2011).

191. S. Cho and M. A. Alonso, “Ambiguity function and phase-space tomogra-

phy for nonparaxial fields,” J. Opt. Soc. Am. A 28, 897–902 (2011).

192. G. I. Ovchinnikov and V. I. Tatarskii, “On the problem of the relationship

between coherence theory and the radiation-transfer equation,” Radiophys.

Quantum Electron. 15, 1087–1089 (1972).

193. H. M. Pedersen, “Exact geometrical description of free space radiative

energy transfer for scalar wavefields,” in Coherence and Quantum Optics

VI , J. H. Eberly, L. Mandel, and E. Wolf, ed. (Plenum, 1990), pp. 883–887.

194. H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J.

Opt. Soc. Am. A 8, 176–185 (1991); errata, 8, 1518 (1991).

195. H. M. Pedersen, “Geometric theory of fields radiated from three-

dimensional, quasi-homogeneous sources,” J. Opt. Soc. Am. A 9,

1626–1632 (1992).

196. R. G. Littlejohn and R. Winston, “Corrections to classical radiometry,” J.

Opt. Soc. Am. A 10, 2024–2037 (1993).

197. S. Cho, J. C. Petruccelli, and M. A. Alonso, “Wigner functions for paraxial

and nonparaxial fields,” J. Mod. Opt. 56, 1843–1852 (2009).

198. M. A. Alonso, T. Setala, and A. T. Friberg, “Optimal pulses for arbitrary

dispersive media,” J. Eur. Opt. Soc. R.P. 6, 1100 (2011).

199. A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels,

“Quadratic time–frequency representations with scale covariance and

generalized time-shift covariance: A unified framework for the affine,

hyperbolic, and power classes,” Digital Signal Processing 8, 3–48 (1998).

200. F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels,

“The power classes-quadratic time–frequency representations with scale

covariance and dispersive time-shift covariance,” IEEE Trans. Signal

Process. 47, 3067–3083 (1999).

201. A. Papandreou-Suppappola, R. L. Murray, B.-G. Iem, and G.F. Boudreaux-Bartels, “Group delay shift covariant quadratic

time–frequency representations,” IEEE Trans. Signal Process. 49,

2549–2564 (2001).

202. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic

fields,” Science 312, 1780–1782 (2006).

203. R. W. Robinett, “Quantum wave packet revivals,” Phys. Rep. 392, 1–119

(2004).




Miguel A. Alonso received the degree of Engineer

in Physics in 1990 from the Universidad Autonoma

Metropolitana in Mexico City and a Ph.D. in Optics

in 1996 from The Institute of Optics, University of

Rochester. After postdoctoral work at Macquarie Uni-

versity in Sydney, Australia, and a faculty position at

the National Autonomous University of Mexico (UNAM)

in Cuernavaca, Mexico, he joined the faculty of The

Institute of Optics in 2003, where he is currently an Associate Professor. Hismain research interest is in finding new mathematical models to describe the

propagation of waves through a variety of systems. He is a Deputy Editor of

Optics Express, Chair of Spotlight on Optics, and a fellow of the Optical Society

of America.

Documents

Wigner Functions in Optics- Describing Beams as Ray Bundles and Pulses as Particl Ensembles