Seminar I – 1st year of Master study programme

Vectorial optical beams

Author: Jure Brence

Mentor: prof. dr. Irena Drevensek Olenik

Ljubljana, 2017

Abstract

Vectorial beams are a class of solutions to the vector wave equation that are defined by a spatiallyvarying direction of polarization. Cylindrical vector beams obey axial symmetry, such as radial andazimuthal polarization. One method of generating vector beams involves liquid crystal q-plates, basedon a micro-structured pattern of polymeric ribbons, providing a radial symmetry to the LC director.Vector beams are of interest because of their unique focusing properties, which allow a great amountof control over the electric field shape near the focus, including the potential for tighter focusingthan with normal beams. Such properties can be applied in fields such as optical trapping, imaging,laser machining and plasmon excitation.

Contents

1 Introduction 1

2 Mathematical Description 12.1 Gaussian beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Vectorial beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Generation 33.1 Liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Focusing Properties 5

5 Applications 85.1 Tight focusing and High-Resolution Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2 Three-Dimensional Focus Engineering and Laser Machining . . . . . . . . . . . . . . . . . 85.3 Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Conclusion 10

1 Introduction

Understanding and manipulating the polarization of light is crucial for many modern optical applications.The state of polarization of incident light affects the focusing of optical beams, the absorption andreflection of materials, propagation through guides and fibers, the angles of refraction in birefringentmaterials and more. The manipulation of polarization is crucial to applications such as metrology,microscopy, atomic physics, display techniques, optical data storage and communications [1].

Conventionally, polarization is thought of in terms of linear, circular and the general elliptical po-larization, which are spatially homogeneous states of polarization. In recent years there has been agrowing interest in the physics of light beams with a specific inhomogeneous arrangement of the state ofpolarization - vector beams [1].

This seminar aims to introduce and overview the field of cylindrical vectorial (CV) beams, a classof general vector beams that obey axial symmetry and have been the focus of the majority of recentresearch and applications.

2 Mathematical Description

To understand how we arrive at vectorial optical beams and how they differ from scalar beams, let usfirst derive the well known Hermite-Gauss and Laguerre-Gauss beam modes.

2.1 Gaussian beams

Scalar optical beams are one of the solutions to the scalar wave equation

∇2E − 1

c2∂2E

∂t2= 0.

The temporal dependence is assumed to be harmonic and in paraxial form, the general solution is assumedto have a dependence

E(r, t) = E0Ψ(r)eikz−iωt,

where the coordinate z is chosen to be along the optical axis, as well as the direction of propagation, and

k = 2π/λ is the wave number. After applying the slowly varying envelope approximation ∂2Ψ∂z2 k ∂Ψ

∂zwe obtain the paraxial wave equation

∇2⊥Ψ = −2ik

∂Ψ

∂z, (1)

1

where ∇2⊥ = ∂2

∂x2 + ∂2

∂y2 denotes the transverse components of the Laplace operator. Following the analogywith the 2D Schrodinger equation in quantum physics, the solution with the least spread in the z-directionis a Gaussian wave packet. In Cartesian coordinates, the solutions are called the Hermite-Gauss modes:

Ψ(x, y, z) = E0Hm[√

2x

w(z)]Hn[√

2y

w(z)]w0

w(z)e−iΦmn(z)ei

k2q(z)

r2 ,

where w(z) is the beam radius, w(z = 0) = w0 is the radius at the beam waist, q = z − iz0 the complex

phase with the beam parameter z0 =πw2

0

λ , Φmn(z) = (m + n + 1) arctan zz0

the Gouy phase shift andHm(x) the Hermite polynomial functions. Another well-known set of solutions, the Laguerre-Gaussmodes, is acquired by solving equation 1 in cylindrical coordinates, where Lp(x) represents the p-thLaguerre polynomial:

Ψ(r, ϕ, z) = E0

(√2

r

w(z)

)lLlp

[2

r2

w2(z)

]w0

w(z)e−iΦpl(z)ei

k2q(z)

r2e−ilϕ.

2.2 Vectorial beams

Hermite-Gauss and Laguerre-Gauss solutions describe the propagation of beams with spatially-homogeneouspolarization. To find vectorial solutions, we need to consider the full vector wave equation

∇×∇×E− k2E = 0.

Solutions with azimuthal symmetry are described by the general form

E(r, z) = U(r, z)eikz−iωteϕ, (2)

which, when inserted into the vector wave equation within a slowly varying envelope approximation givesthe differential equation

1

r

∂

∂r

(r∂U

∂r

)− U

r2+ 2ik

∂U

∂z= 0.

If we compare this equation with the analogous differential equation for the cylindrical scalar beam,

1

r

∂

∂r

(r∂Ψ

∂r

)+

1

r2

∂2Ψ

∂ϕ2+ 2ik

∂Ψ

∂z= 0,

we can clearly see a difference in the second term. The solution to this differential equation is anazimuthally polarized electric field with a complementary radially polarized magnetic field. Togetherthey form an azimuthally polarized cylindrical vector beam. If we assume a magnetic field with a formanalogous to 2, we arrive at a azimuthally polarized magnetic field and a radially polarized electric field,forming a radially polarized cylindrical vector beam.

A commonly used approximation of the CV beams near the beam-waist takes the amplitude profileof the LG01 mode [1].

E(r, z) = Are−r2/w2

eα, α = r,Φ.

It can also be shown that it is possible to express CV beams as a superposition of orthogonally polarizedHermite-Gauss modes [2]:

Er = HG10ex +HG01ey, (3)

EΦ = HG01ex +HG10ey,

which is a useful property for generating CV beams. Linear combinations of radial and azimuthalpolarizations are usually referred to as generalized cylindrical vector beams.

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Figure 1: Amplitude profile and electric field polarization for different modes: a) x-polarized fundamentalGaussian mode, b) x-polarized HG10, c) x-polarized HG01, d) y-polarized HG01, e) y-polarized HG10,f) x-polarized LG01, g) radial polarization h) azimuthal polarization, i) generalized CV beam. Adaptedfrom [1].

3 Generation

In the past decade, a number of very different methods of generating CV beams has been developed. Theyare broadly classified into two groups. The first are active generation methods, which require an activegain medium and typically take place within a laser cavity. Intracavity devices utilize axial birefringenceor axial dichroism to induce discrimination in favor of the axially symmetric modes [1]. Interferometricmethods make use of the relations in equation 3. One such approach is the Sagnac interferometer thatproduces CV beams as a superposition of HG01 and HG10 modes [3].

The second group are passive generation methods, which use specially designed optical elements toconvert linear or circular polarization into radial or azimuthal polarization. The simplest example ofsuch devices is a radial analyzer, an element that transmits only the radial component of the incomingpolarization [4]. Circularly polarized light can be expressed in terms of radial and azimuthal polarization

Ein/E0 = ex + i ey = (cosφ er − sinφ eφ) + i (sinφ er + cosφ eφ) = eiφ(er + i eφ).

The light passing through a radial analyzer that absorbs the eφ polarization will then take the form

Eout/E0 = eiφer.

A simple radial analyzer evidently produces a radially polarized beam, but with a spiral phase factor,introducing a helical shape to the beam. A true CV beam can be obtained by correcting the spiralphase factor with a spiral phase element with opposite helicity, which is a commercially available opticalelement.

3.1 Liquid crystals

Liquid crystals (LC) are widely used in various applications that involve manipulating the polarizationof light. LC are optically anisotropic and belong in the category of uniaxial materials. The difference

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Figure 2: a) Surface rubbing of a twisted nematic LC polarization converter, b) the resulting orientationof the LC molecules in the cell seen from above. Reproduced from [5].

Figure 3: a) The radial structure of polymer ribbons, b) transmission figure of the q-plate, placed betweena crossed polarized and analyzer, c) a sketch of the LC alignment induced by the ribbon pattern. Adaptedfrom [6].

between the ordinary and extraordinary index of refraction is typically around 0.2. The optical axiscoincides with the preferential molecule orientation, called the director. It is possible to influence thedirection of the director with suitable treatment of surfaces or by application of an external electric field.

As such, liquid crystals are also the basis for several methods of generating CV beams. A twistednematic liquid crystal may be sandwiched between linearly and circularly rubbed plates (figure 2a),continually rotating from the initial rubbing direction to the circularly distributed direction on thesecond plate. Incoming light, linearly polarized along the rubbing direction on the first plate followsthe rotation of the molecules, creating the CV polarization on the other side. Incoming light, polarizedparallel or perpendicular to the rubbing on the first plate will result in azimuthal or radial outgoingpolarization, respectively [5].

Liquid crystals are also used in a different approach, where instead of spatially varying polarizationrotation, the axis of retardation varies. We will take a closer look into one such method that has beendeveloped in cooperation between LC researchers from Ljubljana and Tianjin in China [6].

A LC cell with the director aligned in the plane perpendicular to the direction of light propagationexhibits birefringence. Light, polarized parallel to the director, propagates with a different velocitythan the perpendicular polarization and consequently the cell creates a phase shift θ between the twopolarizations. Applying voltage to transparent electrodes allows one to control the phase shift and makethe cell act as, for example, a quarter- or half- waveplate. A quarter waveplate turns circular polarizationinto linear polarization, oriented of 45o with respect to the director. If the direction of the director isradially distributed, as shown on figure 3c, the outgoing beam will be a cylindrical vectorial beam. Inthe Jones formalism, a radially aligned LC cell, known as a q-plate, can be described by a matrix [7]:

Mθ(φ) =

[cos θ2 + i sin θ

2 cos 2φ i sin θ2 sin 2φ

i sin θ2 sin 2φ cos θ2 − i sin θ

2 cos 2φ

], (4)

where φ denotes the azimuthal angle and θ the phase shift that is dependent on the applied voltage. An

incident circularly polarized beam, described by the Jones vector 1√2

[1−i

]results in outgoing polarization

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Figure 4: States of polarization, phase fronts and generated beam profiles detected with/without theanalyzer for four different optical retardations: a) θ = 2π , b) θ = 3π/2 , (c) θ = π and d) θ = π/2. Theorientation of black arrows in dash circles represents the transmission axis of the analyzer [6].

of the form

E =E0√

2

(cos

θ

2

[1−i

]+ i sin

θ

2e−i2φ

[1i

]), (5)

expressed as a superposition of scalar beams with left-handed and right-handed circular polarization.For a phase shift θ = π

2 or θ = 3π2 , when the LC cell functions as a quarter wave plate, the expression

simplifies into

θ =π

2: E ∝ e−iφ

[cos (π4 − φ)− sin (π4 − φ)

]∝ e−iφ(er − eφ), (6)

θ =3π

2: E ∝ e−iφ

[sin (π4 − φ)cos (π4 − φ)

]∝ e−iφ(er + eφ).

We see that the radial LC cell produces a sum or difference of the radial and azimuthal polarizations -a generalized cylindrical vector beam. Unfortunately, a spiral phase factor is also produced, but can beeliminated by the use of a spiral phase element. Image 4 shows an analysis of the polarization, producedfrom the incident fundamental Gaussian beam by such a q-plate.

The radial LC alignment pattern is achieved by a process of two-photon polymerization-based directlaser writing (TPP-DLW) that creates sparse out-of-plane polymeric ribbons. A nematic liquid crystal,filled into the prepared cell, aligns itself in the direction of the ribbons. This principle allows the creationof not only a radial alignment pattern (q = 1), but also defects with arbitrary topological charges. LCdefects with q = 1/4 or q = 3/4 can not naturally appear in liquid crystals and could be used to producevectorial beams with exotic polarizations and high potential for interesting physics or applications [6].

4 Focusing Properties

Cylindrical vector beams are interesting primarily because of their unique focusing properties. Incidentradial polarization results in a dominantly longitudinal field near the axis. The transversal and lon-gitudinal components are spatially separated close to the focal point. These properties allow focusing

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Figure 5: The geometry of the focusing problem. An incident CV beam is focused by an objectivelens with focal length f . Points on the lens plane are described by (r, ϕ,−f), points on the sphericalwavefront with radius f by (f sin θ cosϕ, f sin θ sinϕ, f cos θ) and points of interest near the focus by(ρ, φ, z). Adapted from [1].

into a tighter spot than with homogeneously polarized beams. Different focal behavior for radial andazimuthal polarization, combined with relatively easy conversion between the two, opens possibilities forprecise three-dimensional shaping of the focal field. We will derive the focal field and its properties usingnumerical methods, based on the diffraction theory of Richards and Wolf [9, 10]. The method expressesthe vector components of the focal field with a diffraction integral over a spherical wavefront, producedby a convex lens [1, 8].

The incoming beam will be a generalized cylindrical vector beam, described as

E(r, ϕ) = E0P (r)(cosϕ0er + sinϕ0eϕ),

where the relative angle of the polarization is φ0 and P (r) is the field strength distribution in the plane.The geometry of the problem is illustrated on figure 5. The lens transforms the planar wavefront intoa spherical one that converges into the focal point. We need to find how the planar field strengthdistribution transforms onto a spherical front: P (r) → P (θ). One way to derive the transformationdemands the conservation of power

(E0P (r))22πrdr = (E0P (θ))22πf2 sin θdθ

P 2(θ) = P 2(r)r

f2 sin θ

dr

dθ.

The ideal parabolic lens has a simple relation between θ and r, as illustrated on figure 5:

r/f = tan θ ≈ sin θ,

dr = f cos θdθ.

For a lens of this type the transformed field strength distribution becomes

P (θ) = P (r)√

cos θ. (7)

The refraction of the lens also transforms the polarization vectors:

e′r = cos θer + sin θez,

e′ϕ = eϕ.

In 1959, Richards and Wolf developed an approach for computing electric and magnetic fields near focusfor high aperture lenses [9, 10]. The method is widely used to date in analyzing the vector nature of

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focused beams. The integral representation can be interpreted in terms of the Huygens-Fresnel principle,but with secondary plane waves instead of spherical waves. The Richards-Wolf method is a diffractionintegral over the vector field on a spherical wavefront ω, with radius equal to the focal length of the lens:

E(r) =−ik2π

∫∫Ω

a(θ, ϕ)eik(s·r)dΩ. (8)

Here a(θ, ϕ) denotes the field strength factor

a(θ, ϕ) = E0fP (θ)(cosϕ0e′r + sinφ0e

′ϕ). (9)

The complex exponential factor in equation 8 contains the scalar product of r = (ρ, φ, z), the position vec-tor of a point in the focal plane in cylindrical coordinates, and of unit vector s = (sin θ cosϕ, sin θ sinϕ, cos θ),that points toward a point on the integration surface. The scalar product can be expressed as

s · r = z cos θ + r sin θ cos (ϕ− φ). (10)

After inserting equations 9 and 10 into the Richards-Wolf integral, we obtain the form

E(ρ, φ, z) = E0−ikf

2π

θmax∫0

dθ

2π∫0

P (θ)(cosϕ0e′r + sinϕ0e

′ϕ)eik(z cos θ+ρ cos (ϕ−φ)) sin θdθdϕ. (11)

The integration limit θmax denotes the maximum angle θ at which the lens collects and focuses light. Theappropriate numerical aperture is NA = tan (θmax). It is desirable to rewrite this integral in cylindriccoordinates of the focal plane, by using the identities

eρ = cosφ ex + sinφ ey (12)

eφ = − sinφ ex + cosφ ey. (13)

The diffraction integral can already be simplified by completing the integration over ϕ by making use ofthe identity

2π∫0

cos (nφ)eikr sin θ cosφdφ = 2πinJn(kr sin θ),

where Jn(x) denotes the n-th Bessel function. The final analytical expression for the focal field of theform E(ρ, φ, z) = Eρ eρ + Ez ez + Eφ eφ is

Eρ(ρ, φ, z) = E0kf cosφ0

θmax∫0

P (θ) sin θ cos θJ1(kρ sin θ)eikz cos θdθ, (14)

Ez(ρ, φ, z) = iE0kf cosφ0

θmax∫0

P (θ) sin2 θJ0(kρ sin θ)eikz cos θdθ,

Eφ(ρ, φ, z) = E0kf sinφ0

θmax∫0

P (θ) sin θJ1(kρ sin θ)eikz cos θdθ.

Although these integrals must be calculated numerically, a quick examination of the field componentsreveals some interesting properties. All three expressions are independent of the azimuthal angle φ,therefore obeying cylindrical symmetry. The longitudinal component Ez has the factor i, meaningit oscillates with a phase shift of 90o in respect to the radial and azimuthal component. Secondly,both transversal components have the Bessel function J1(kρ sin θ) in the integral, while the longitudinalcomponent has J0(kρ sin θ). Of all Bessel functions Jn, only the first, n = 0 has a nonzero value forρ → 0. In other words, the longitudinal component is dominant close to the optical axis. A radiallypolarized incident beam (ϕ0 = 0) will close to the focal point have a longitudinal component near the axisand a donut-shaped radial component further from the axis. An incident azimuthal beam (ϕ0 = π/2)will only have the donut-shaped azimuthal component. Figure 6 shows a numerical calculation of thefield distribution using equations 14.

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Figure 6: Numerically calculated spatial distribution of field strength of a) longitudinal component, b)radial component, c) total field strength for a radially polarized incident beam, focused by a NA = 0.8lens. The azimuthal component is zero for this case. For an azimuthally polarized incident beam thelongitudinal and radial components do not exist and the azimuthal component is depicted on figure d).Adapted from [1].

5 Applications

Cylindrical vector beams have intriguing focusing properties when focused with a high-NA lens, allowinga variety of applications in optical imaging and manipulation. There has been promise of applicationsmuch broader than the exploitation of focusing properties. Research has been done on topics such asterahertz vector beams and waveguides, propagation of CVB in nonlinear materials, effects of turbulentatmosphere on the propagation of CVB, surface plasmon resonance excitations and studies in singularoptics regarding connections between angular momentum and spin [1].

5.1 Tight focusing and High-Resolution Imaging

A radially polarized vector beam focuses into a smaller spot than a beam with linear polarization. This isdue to the strong and narrow longitudinal component dominating the focal field. Figure 7a compares thecalculated field strength profile for a radial CV beam with a scalar Gaussian beam. The CV beam clearlycreates a tighter focus. Studies have revealed a spot size of 0.161λ2 for radial polarization, comparedwith 0.26λ2 for linear polarization [11]. This property has potential applications in a large number ofproblems that require small focal sports. As an example, it can enhance high-resolution imaging innumerous variations of microscopy [1].

5.2 Three-Dimensional Focus Engineering and Laser Machining

The longitudinal and transversal components are spatially separated near the focus for an incidentradially polarized beam. Since the relative angle of polarization ϕ0 can be continuously adjusted by adouble λ/2 wave-plate, it is possible to achieve specific shapes of the focal field. One interesting shapeis the flattop profile, achieved at ϕ = 24o for an objective lens with NA = 0.8 [1]. An example flattopprofile is shown on figure 7. Such as profile offers an improvement in the quality of laser machiningand cutting. With conventional methods, the focusing setup of a strong laser beam is a question oftrade-off between a small focal spot and the depth of focus.A flattop profile provides a better ratio of

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Figure 7: Left: numerically calculated focal shape of radial and axial field components for a radiallypolarized CV beam, focused by a high-NA lens, compared with the result for a linearly polarized beamby under the same conditions. Right: 2D plot of the time-averaged Poynting vector < Sz > in the r–zplane for highly focused radial polarization. Adapted from [1].

Bottom: example of a 3D flattop focus generated with CV beam.

cutting depth to cutting width. In many cases, the absorption efficiency for metals strongly dependson the polarization. It has been found that radial polarization offers advantages over linear or circularpolarization for better absorption in laser machining [12].

5.3 Optical trapping

Strong focused laser beams are the main tool of the trade in the field of optical trapping. Due to electricdipole interactions an optical beam exhibits three different forces on an irradiated particle: scatteringforce FS ∝< Sz >, absorption force FA ∝< Sz > and gradient force FG ∝ ∇|E|2. The gradient forcepulls particles toward the focal point and is proportional to the gradient of the field strength. The forcesdue to scattering and absorption have a destabilizing effect on the trap and are proportional to < Sz >,the time averaged axial component of the Poynting vector S = 1

µ0E × B, that represents the rate of

directional energy flux density of the beam. Stable optical trapping is difficult for metallic particlesdue to a strong scattering force that destabilizes the trap. For radially polarized CV beams, the focallongitudinal component is 90o out of phase with the transversal component and the axial component ofthe Poynting vector vanishes near the optical axis, as depicted on figure 7b. Furthermore, the tighterfocusing of CV beams provides a higher gradient force, resulting in a stronger and more precise opticaltrap [13,14].

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6 Conclusion

Cylindrical vector beams are optical beams with spatially inhomogeneous polarization that obeys axialsymmetry. Various methods of generation have been devised, from intracavity setups that allow themanufacture of vectorial lasers, to passive free-space devices that convert a Gaussian beam into a CVbeam. Many solutions are readily purchased from major optical equipment companies. Although meth-ods of manipulation of CV beams have not been discussed in the seminar, devices have been developedthat maintain polarization symmetry, while performing reflection, retardation or polarization rotationthat allows conversion different generalized CV beam states of polarization. With applications rangingfrom improvements in imaging, laser cutting, optical trapping and optical communications, as well asresearch possibilities in singular optics, the field of cylindrical optical beams is sure to experience rapidgrowth and visibility in the future.

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