Lecture 4:

Beams of radiation

In the previous lectures we covered the theory that describes monochromatic waves emitted bypoint sources, and then considered pulses of waves emitted by polychromatic point sources. Inthis lecture we shall drop the assumption that our source is point–like and consider monochromaticwaves produced by an extended source. In particular we shall consider waves that are highlycollimated. We shall consider beams of radiation (e.g. the kind of waves emitted by a laserpointer).

1 The paraxial approximation:

Here we are (again) trying to solve the Helmholtz equation in three dimensions in a homogeneousnon–dispersive medium

[∇2 + k02]φ=0 (1)

for the case of waves produced by an extended source. From lecture 2 we could write the solutionin terms of the three dimensional Green function G(x−x′), for some distributed source of wavesj(x)

φ(x)=

∫

G(x−x′)j(x′)d3x′

However we would then have to pick a different source distribution j(x) for each beam of radi-ation, and evaulating the three dimensional integral will in general be quite tricky. Let’s followa different approach, taking advantage of the fact that we are interested in collimated waves.

Figure 1. Schematic of a paraxial wave: the paraxial approximation assumes that the wave energy is

travelling almost entirely along one axis. If, for example this is the z-axis, then the wave is very close to

being eik0z, with some change to the cross section as we move along the z–axis. The paraxial Helmholtz

equation (3) govens this change in the wave with z.

1

We ignore the details of the source, and assume that our wave is travelling along the z− axis. (asin figure 1). Because the wave of interest is highly collimated, all its Fourier components will be

close to k =k0z. To express this we write the wave as φ(x, y, z)= ψ(x, y, z)eik0z where ψ(x, y, z)varies very slowly with the z coordinate (i.e. |∂ψ/∂z |≪k0ψ). The Helmholtz equation (1) thenbecomes

[∇2 + k02]φ= ∇‖

2ψ+∂2ψ

∂z2+ 2ik0

∂ψ

∂z=0 (2)

where ∇‖2 =

∂2

∂x2+

∂2

∂y2. Given that the z derivative of ψ is assumed to be small in comparison to

k0ψ, to leading order we can drop the second z derivative of ψ from (2) , leaving us with

−1

2∇‖

2ψ= ik0∂ψ

∂z(3)

which is the paraxial Helmholtz equation, and governs the motion of highly collimated waves. Youperhaps recognise (3). I chose the symbol ψ deliberately. Equation (3) is of exactly the sameform as the time dependent Schroedinger equation from quantum mechanics (see lecture 1), exceptthat the time variable has been replaced with the distance along the z − axis. This means thatall of the tricks people use to solve the time evolution of the Schroedinger equation can be carrieddirectly over to solve the propagation of a beam of radiation.

There is a fundamental and quite interesting reason that the paraxial approximation yieldssomething in the form of the time dependent Schroedinger equation : the Schroedinger equationis itself a paraxial approximation to a relativistic wave equation, but the paraxial approxim-ation is for propagation along the time axis rather than a spatial one. But before I get sidetrackedI want to show you the general solution to (3).

The general solution to (3) can be constructed using something called the propagator . Thepropagator K(x, y, z; x′, y ′, z ′) looks intimidating (lots of variables!) but it is just the solu-tion to (3) with the initial condition that at z=z ′ it is a delta function in x and y, centred at x′, y ′

K(x, y, z ′;x′, y ′, z ′)=K(x, z ′; x′, z ′)= δ(2)(x−x′) (4)

If we can find such a function then we can immediately write down the solution to (3) for anyinitial condition ψ= ψ0 at e.g. z=0

ψ(x, z) =

∫

d2x′K(x, z; x′, 0)ψ0(x′) (5)

The function (5) satisfies (3) because K does, and reduces to ψ0(x) at z=0 because of the initialcondition (4).

We now find an expression for the propagator K. First we notice that

ψk = exp

[

i

(

k · (x−x′)− 1

2k0k2(z − z ′)

)]

(6)

is a solution to (3)

−1

2∇‖

2ψk =1

2k2ψk = ik0

∂ψk

∂z=

1

2k2ψk

2

Summing (6) over all k we can then construct a function that both satisfies the paraxial Helmholtzequation, and is a delta function δ(2)(x−x′) when z= z ′

K(x, z; x′, z ′)=

∫

d2k

(2π)2exp

[

i

(

k · (x−x′)− k2

2k0(1− iη)(z − z ′)

)]

(7)

(remember that the Fourier representation of the two dimensional delta function is δ(2)(x−x′)=(2π)−2

∫

d2k exp[ik · (x−x′))]). I have assumed that z >z ′ and added in the small quantity η in

order to make the integral converge (as usual the limit η→ 0 will be taken at the end).

Using the formula for the Gaussian integral that we saw in lecture 3∫

−∞

∞exp(−ax2 + bx)dx =

π/a√

exp(b2/4a) for both the integrals over kx and ky we obtain from (7)

K(x, z; x′, z ′)=

∫

dkx

2πexp

[

ikx(x− x′)− kx2

2k0(i+ η)(z − z ′)

]

×∫

dky

2πexp

[

iky(y− y ′)− ky2

2k0(i+ η)(z − z ′)

]

=k0

2πi(z − z ′)exp

[

ik0(x−x′)2

2(z − z ′)

]

(8)

we have therefore found the propagator for a beam moving through a homogeneous medium, whichwe shall use in the remainder of this lecture:

K(x, z; x′, z ′) =k0

2πi(z − z ′)exp

[

ik0(x−x′)2

2(z − z ′)

]

(9)

Before we move on it is worth briefly thinking about the question, ‘what does the paraxial approx-

imation neglect, and when can we use it?’. Firstly, it clearly only works for a wide beam, wherethe power flow is almost entirely along one axis. Secondly, because the beam is constrained tohave Fourier components only very close to k=k0z, it ignores reflection which would require a verylarge change of wave–vector, taking k0z→−k0z. In any real experiment, there will be reflectionsof the beam as soon as you try to do anything with it (e.g. put it through a lens) and the paraxialapproximation will not account for this attenuation of the beam.

1.1 A digression, the paraxial approximation across physics:

1.1.1 The paraxial approximation and the path integral:

The propagator K that we’ve defined above is an object that is commonly used to solve the timedependent Schrodinger equation. For a particle of massm moving in two dimensions it is given by

K(x, t; x′, t′)=m

2πi~ (t− t′)exp

(

im

2~(t− t′)(x−x′)2

)

(10)

where t − t′ is the length of time since the quantum wave was a delta function distribution (att= t′). Clearly, once the time interval t− t′ has been replaced with the distance along the z axisz − z ′, and m/~ with k0 then we have exactly (9).

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In quantum mechanics there is a general way to compute the propagator (10) in any system,known as the path integral (see R. P. Feynman and A. R. Hibbs, “Quantum Mechanics and Path

Integrals” Dover (2010)), where the propagator is written as

K(x, t; x′, t′)=

∫

Dx ei

~S[x(t)]

(11)

In this equation Dx indicates an integration over all possible trajectories x(t) that start at x′ attime t′ and finish at x at time t. The quantity S[x(t)] is the classical action (the integral of theLagrangian over time), which takes a different value depending on the trajectory x(t).

In practice one performs the ‘path integral’ in (11) through discretizing the path x(t) into a list ofvalues x′,xǫ,x2ε, ...,x for the times t′, t′+ ǫ, t′+2ǫ, ..., t and then writing the sum over all paths asa product of integrals over all possible intermediate positions

∫

Dx=N∫

d2xǫ

∫

d2x2ǫ..., where N is

a normalization constant. At the end of the calculation the time step ǫ is taken to zero. When Sis quadratic in the position and velocity variables this infinite number of integrals is just a series ofGaussian integrals, which can be done analytically using the same formula we used in the previoussection.

The point here is that the path integral is often taken to be something peculiar to the strangeworld of quantum mechanics, where the particle is somehow (due to quantum weirdness) samplingevery possible classical trajectory between a source and a detector. However, because the paraxialHelmholtz equation (3) is of exactly the same form as the time dependent Schroedinger equation,we can equally use the same path integral (11) to calculate the evolution of a collimated beamof radiation. But now we cannot retreat into quantum weirdness (e.g. our path integral couldbe telling us about the sound coming out of a large loud speaker). Instead we must say—moreconcretely—that the wave propagates through all of space and that we must sum the phases forall possible propagation paths in order to construct the wave as a function of position.

1.1.2 The Schroedinger equation is a paraxial approximation in time:

The connection between the time dependent Schroedinger equation and the paraxial Helmholtzequation is no accident: the Schroedinger equation is itself a paraxial approximation to a dif-ferent wave equation. To see this, let’s imagine trying to construct a relativistic analogue of thetime–dependent Schrodinger equation in the same way we did in lecture 1. We start from therelativistic equation for the energy and momentum of a particle in terms of its rest mass m0

E2/c2− p2 =m02c2 (12)

Performing the usual trick (see lecture 1) used to turn classical mechanics into quantum mechanics

(E→ i~∂

∂tand p→−i~∇) this becomes the following equation for a wave φ

[

∇2− 1

c2∂2

∂t2− m0

2c2

~2

]

φ= 0 (13)

This equation is known as the Klein–Gordon equation and is satisfied by all relativistic quantumparticles (although it is only in the case of spin zero particles that ψ is a scalar). We can alwayswrite the energy E=~ω of a particle as the rest energy m0c

2 plus an extra part due to e.g. motion,or the interaction with a potential. However for slowly moving particles in weak potentials, thetime dependence is almost entirely due to the rest energy. To connect this with our treatment ofbeams of radiation, we might say that the wave–function is a beam that is almost entirely directedalong the time axis.

4

We therefore write our wave–function φ as φ(x, t) = e−i

m0c2

~tψ(x, t) where ψ is only weakly

dependent on time |∂ψ/∂t|≪m0c2~−1ψ. Substituting this into the Klein–Gordon equation (13)

we obtain something very much like (2)

[

∇2ψ− 1

c2∂2ψ

∂t2+

2im0

~

∂ψ

∂t

]

= 0

and again, to leading order, the second time derivative of ψ can be neglected, being much smallerthan all the other terms. This leaves

− ~2

2m0∇

2ψ= i~∂ψ

∂t

which is the time dependent Schroedinger equation. As promised, we have shown that thetime dependent Schroedinger equation is a paraxial approximation (in time) to the relativisticKlein–Gordon equation.

1.2 The Gaussian beam:

After our digression, let’s return to the problem of beam propagation. We’ll now apply ourpropagator to a simple kind of beam to see how it evolves in space. We assume an intensity profilefor the beam in the z = 0 plane that is a Gaussian, with a maximum at the centre of the x − y

coordinate system:

ψ(x, z= 0) = ψ0(x) = e−

x2

2R2. (14)

Using (5), and the expression for the propagator (9) we find the evolution of the beam along thez axis can be written as

ψ(x, z) =

∫

d2x′K(x, z; x′, 0)e−

x′2

2R2

=k0

2πiz

∫

d2x′exp

[

ik0(x2 + x′2− 2x · x′)

2z

]

e−

x′2

2R2

=k0

2πizexp

[

ik0x2

2z

]∫

d2x′exp

[

−ik0x · x′

z− 1

2

(

1

R2− ik0

z

)

x′2

]

Applying the Gaussian integral formula that is given above, we find after a few manipulations, theresult of this integral to be

ψ(x, z)=

R2

R2 +iz

k0

exp

− x2

2(

R2 +iz

k0

)

(15)

To extract a bit more physics from this we’ll split it into amplitude and phase contributions. Aftera few manipulations (15) can be written as

ψ(x, z) =1

1 + ζ2

√

exp

(

− x2

2R2(1 + ζ2)

)

1− iζ

1+ iζ

√

exp

(

iζx2

2R2(1+ ζ2)

)

=w(0)

w(ζ)exp

(

− x2

2w(ζ)2

)

exp

(

i

[

ζx2

2w(ζ)2+ arg(1− iζ)

])

(16)

5

where the dimensionless coordinate ζ=z/k0R2 has been introduced, and the beam width has been

defined as

w(ζ)=R 1 + ζ2√

From the second line of (16) we can get a better understanding for how the beam depends on thebeam axis coordinate z. The narrowest part of the beam (the waist) is at z=0, where the width

(full width half maximum) is 2 2 log(2)√

R, with R a number of our choice (14). As we move away

in either direction from this narrowest point, the beam retains its Gaussian envelope but spreadsout at a rate determined by the dimensionless variable ζ. The points z = ±k0R

2 are known asthe Rayleigh range, and are where the width of the beam has increased by a factor of 2

√. The

narrower the waist of the beam, the more divergent it is and hence the smaller the Rayleigh range.

The phase of the beam contains two contributions. The first exp(iζ x2/2w(ζ)2) is a quadraticphase variation over the cross section of the beam that indicates the flow of power out from thebeam centre as it spreads. The second phase factor, exp(i arg(1 − iζ)) is known as the Gouy

phase. This is a phase shift generally associated with wave fields as they change from converging(in this case towards x = 0), to diverging. The Gouy phase changes by π as we move from farbehind to far in front of the beam waist. Note that even the propagator (9) has a Gouy phase, butit occurs over an infinitesimal distance: as z passes through z ′ the prefactor of z− z ′ changes signand the wave acquires an additional phase of π. For some plots and animations of the Gaussianbeam as a function of z see the corresponding Jupyter notebook.

Before moving on to the next section we should reiterate something about the paraxial approxima-tion. Although it appears that we can choose any beam waist size R we like, this is only possiblewithin the limits of the paraxial approximation. As R is made ever smaller towards 1 / k0 theparaxial approximation starts to become unrealistic, and we obtain the false impression that thebeam waist can be made arbitrarily small compared to the wave–length. In fact the beam waistcan only be reduced to a minimal value, referred to as the diffraction limit .

1.3 The Airy beam—a constantly bending beam:

In 1979 Berry and Balazs pointed out something very interesting about the motion of certainshapes of wave–packet evolving according to the time dependent Schroedinger equation (American

Journal of Physics 47, 264 (1979)). They found that a wave–packet shaped as a particular kindof special function known as the Airy function (see accompanying Jupyter notebook), acceleratesat a constant rate even though it is in free space and subject to zero force (in principle withoutlimit, reaching an arbitrary high velocity).

Although this is completely counter–intuitive, it does not obviously violate the physics weknow. The wave–packet is an interference pattern between its Fourier components, and the‘motion’ is the shift in the interference pattern as the relative phases of the Fourier compon-ents vary in time. As we saw in lecture 3, interference features in pulses can move in apparentlyunphysical ways (e.g. faster than the speed of light in vacuum). We need to be careful withthe variables we choose to attach physical meaning to.

As we have seen above, the time dependent Schroedinger equation takes the same form as theparaxial Helmholtz equation with time playing the role of the beam axis, z. Therefore if weconstruct a beam with a cross section that is uniform along the y-axis, but is an Airy functionalong the x-axis then the beam should ‘accelerate’ with increasing z, i.e. it should bend. We’llnow show this using the propagator and the integral representation of the Airy function

Ai(ax) =1

π

∫

0

∞

cos

(

t3

3+ axt

)

dt

=1

2π

∫

−∞

∞

eit

3

3 eiaxtdt

(17)

6

(if special functions scare you then you can just think of them as shorthands for commonly foundbut difficult integrals). We are free to choose the constant a, which has dimensions of inverselength. This constant fixes the scale of the interference features in the beam. Assuming an initialbeam that is uniform along the y-axis, but an Airy function along the x–axis

ψ(x, z= 0)= ψ0(x)=Ai(ax) =1

2π

∫

−∞

∞

eit

3

3 eiaxtdk

and integrating this against the propagator (9), we obtain

ψ(x, z) =

∫

d2x′K(x, z; x′, 0)Ai(ax′)

=k0

2πizexp

(

ik0x2

2z

)∫

dy ′ e−

(

η−ik0

2z

)

y ′2 1

2π

∫

−∞

∞

eit

3

3

∫

dx′e−

(

η−ik0

2z

)

x′2

ei(

at−k0x

z

)

x′

dt

=exp

(

ik0x2

2z

)

1

2π

∫

−∞

∞

eit

3

3 exp

−iz

(

at− k0x

z

)

2

2k0

dt

(18)

where I again used the Gaussian integration formula (as you may have noticed, this is a godsendin paraxial optics) to evaluate the integrals over x′ and y ′. As an integral, (18) is not very useful(we don’t get any immediate insight into what the formula is telling us). So let’s do a bit moreshuffling, first expanding the integrand

ψ(x, z)=1

2π

∫

−∞

∞

eit

3

3 exp

−iz

(

a2 t2− 2atk0x

z

)

2k0

dt (19)

and then ‘completing the cube’ (in the formula below α and β are aribtrary constants)

αt3 + βt2 =α

(

t+β

3α

)

3

− t

(

β2

3α

)

−α

(

β

3α

)

3

(20)

Applying (20) to (19), our integral starts to look very much like the integral representation of theAiry function (17) we started with

ψ(x, z)=1

2πe

i

3ζ3

∫

−∞

∞

ei

3(t−ζ)3

eit[ax−ζ2]dk (21)

where I have introduced the dimensionless variable ζ = za2 / 2k0. Changing the integrationvariable in (18) to τ = t− ζ we can write the evolution of the beam along the z axis again in termsof the Airy function

ψ(x, z) =e−

2i

3ζ3

eiζax 1

2π

∫

−∞

∞

ei

3τ3

eiτ [ax−ζ2]dk

=e−

2i

3ζ3

eiζaxAi(ax− ζ2)

(22)

We have therefore found exactly the behaviour we anticipated from Berry and Balazs’ result: a fixedpoint x=x0 in a beam shaped as an Airy function at z=0 moves as ax=ax0+ ζ2, which representsa uniform bending of the beam. The rate at which the beam bends is set by the quantity ζ:for a smaller value of a, the Airy function at z = 0 is more spread out, and the beam bends lessrapidly. For some plots and animations of the Airy beam as a function of z see the correspondingJupyter notebook. Again, we must remember that this is all within the paraxial approximation,which becomes invalid once the beam propagation deviates too much from the z–axis.

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2 Application—the Poisson Arago spot:

In the early 1800s there were no Maxwell equations, and it was still not firmly decided whether lightshould be considered as a wave or a particle. While Fresnel developed a scalar wave theory, whichhe believed explained many optical phenomena, those such as Poisson retained the Newtonianview that light should be considered as a collection of particles. In an attempt to refute Fresnel’stheory, Poisson deduced that—according to the wave theory of light—there ought to be a brightspot in the centre of the shadow behind an opaque spherical or circular object. This seemed socounter intuitive to him that he put it forward as evidence against Fresnel’s theory. However,Dominique Arago took up this prediction and very quickly experimentally verified it. This wasa significant success for the wave theory of light, and the bright spot as became known as thePoisson–Arago spot .

Figure 2. ThePoisson–Arago spot is the tiny green dot in the centre of the dark circle in themiddle. Taken

from Robert Vanderbei’s webpage, where he observes Poisson’s spot using only a Green laser pointer and a

telescope. We can estimate the parameters for our theory from the picture he gives of his living room. The

green laser pointer will have wavelength 532nm→ k0 = 2π/λ = 1.19× 107m−1. It looks as though z∼ 2m

and (there is a small disc on the front of the telescope) R∼0.01m. We thus have ζ = z/k0R2∼0.002. To

see a sharp spot we should therefore consider the limit of small ζ.

Let us try to demonstrate the existence of this bright spot within the paraxial formalism we havedeveloped. Assuming our object is illuminated from negative z, we take our initial field as thatin the plane of the opaque object: equal to unity everywhere except on the rear surface (which hasradius R):

ψ(x, z= 0)= ψ0(x) =

0 r <R

1 r >R

Using polar coordinates and applying the propagator (9) we then find that the wave beyond theobject (z > 0) is given by

ψ(x, z > 0)=k0

2πiz

∫

0

2π

dθ ′∫

R

∞

r ′ dr ′ exp

[

ik0(r2 + r ′2− 2rr ′ cos(θ− θ ′))

2z

]

e−ηr ′2

(23)

8

where η again serves to make the integral converge, and we take the limit η→0 at the end. Aftera little re–arranging we can take a factor out of the integral (23)

ψ(x, z > 0)=k0

2πizexp

(

ik0r2

2z

)∫

0

2π

dθ ′∫

R

∞

r ′ dr ′ exp

[

−(

η− ik0

2z

)

r ′2− ik0r cos(θ− θ ′)

zr ′

]

Using the expansion of a plane wave in terms of Bessel functions given at the end of the referencematerial in lecture 1 (exp(ix cos(θ)) =

∑

lilJl(x)e

inθ) we can perform the integral over θ, leavingonly the Bessel function J0

ψ(x, z > 0)=k0

izexp

(

ik0r2

2z

)∫

R

∞

r ′ dr ′ J0

(

k0rr′

z

)

e−

(

η−ik0

2z

)

r ′2

To evaluate the integral within the above expression we consider the limit where z / k0R2 ≪ 1,

i.e. the ratio of the distance from the disc to the radius of the disc, divided by k0R is verysmall. Provided r/R≪ 1, in this limit the oscillating exponential averages everything out exceptclose to r ′ = R. We may thus remove the Bessel function from the integral, approximating itsvalue to be J0(k0rR/z)

ψ(x, z > 0) ∼k0

izexp

(

ik0r2

2z

)

J0

(

k0rR

z

)∫

R

∞

r ′ dr ′ e−

(

η−ik0

2z

)

r ′2

[r/R≪ 1, z/k0R2≪ 1]

∼exp

(

ik0r2

2z

)

exp

(

ik0R2

2z

)

J0

(

k0rR

z

)

where I took the limit η→ 0. There is a Jupyter notebook that goes with this lecture, where Inumerically show that this approximation is okay. Introducing the dimensionless coordinatesζ = z/k0R

2 and ρ= r/R this becomes

ψ(x, z > 0)∼ exp

(

iρ2

2ζ

)

exp

(

i

2ζ

)

J0

(

ρ

ζ

)

(24)

Since the zeroth order Bessel function is equal to 1 at the origin, we reach the conclusion thatthere is a bright spot within the shadow region behind a circular (or spherical) object. The widthof this spot is determined by the value of ζ. Very close to the back of the object ζ≪ 1 the spotis very small, spreading as we increase z. The reason this bright spot occurs is because the waveilluminating the edge of the object has the same phase all along the edge. The waves emitedbehind the object from this illuminated edge also arrive at x = 0 in phase, giving rise to a brightspot.

9