Angular momentum redirection phase of vector beams in a non-planar geometry

Amy McWilliam, Claire Marie Cisowski, Robert Bennett, and Sonja Franke-ArnoldSchool of Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom

(Dated: September 22, 2021)

An electric field propagating along a non-planar path can acquire geometric phases. Previously,geometric phases have been linked to spin redirection and independently to spatial mode transfor-mation, resulting in the rotation of polarisation and intensity profiles, respectively. We investigatethe non-planar propagation of scalar and vector light fields and demonstrate that polarisation andintensity profiles rotate by the same angle. The geometric phase acquired is proportional to j = `+σ,where ` is the topological charge and σ is the helicity. Radial and azimuthally polarised beams withj = 0 are eigenmodes of the system and are not affected by the geometric path. The effects con-sidered here are relevant for systems relying on photonic spin Hall effects, polarisation and vectormicroscopy, as well as topological optics in communication systems.

I. INTRODUCTION

Throughout history, mirrors have been used for themost sacred and profane purposes, as well as for a multi-tude of scientific and technological purposes. The earliestdocumented mirrors have been constructed in Neolithictimes from polished obsidian [1], followed by metallic mir-rors, not unlike the ones used in this paper, developed inMesopotamia around 4000 BCE, Venetian molten glassmirrors from around 1400 and the modern day dielectricmirrors. Contrary to popular misconceptions, mirrors donot ‘swap left and right’ but rather ‘near and far’. In anoptical context we might say that the propagation direc-tion of a light beam perpendicular to the mirror surfaceis reversed.

Mirror reflection also affects light’s (spin and orbital)angular momentum, however in a subtly different way:the angular momentum component perpendicular to themirror surface is conserved, whereas the parallel com-ponent is reversed – effectively reversing the projectionof the angular momentum with respect to the propaga-tion direction. When taking light via multiple mirrorsalong a non-planar trajectory, these successive angularmomentum redirections add up, and the propagation ofthe electric field may be modelled as parallel transportalong the beam trajectory [2, 3], resulting in a rotationof both the polarisation and intensity profile.

Polarisation rotation may be understood by consider-ing the action of individual optical elements along thenon-planar beam path on the electric field vector [4–6].This has been confirmed using optical fibres curled intoa helix [7–9] or by using a succession of mirror reflections[10, 11]. Propagation along a non-planar trajectory alsocauses a rotation of the beam intensity profile [12–14],which can be seen from simple ray tracing.

In this paper, we investigate experimentally and the-oretically the rotation of intensity, polarisation and vec-tor field profiles, using the same experimental setup totransport a beam of light along a non-planar trajectory.We show systematically that the rotation angle dependssolely on the non-planarity of the beam path, and inter-pret it in terms of geometric phases. For polarisationrotations, these are the well-established spin-redirection

phases: Upon non-planar propagation, the right and lefthanded circularly polarised components of the light fieldacquire equal and opposite phases, leading to a rotationof the polarisation ellipse. Similarly, a spatial mode witha given orbital angular momentum (OAM) acquires aphase proportional to its topological charge [15], whichwe shall call orbital-redirection phase. We show the differ-ential phase shifts acquired by the various modal contri-butions result in a rotation of the overall intensity profile.

The geometric phase provides a unifying concept inphysics and specifically in optics [16, 17]. It describesa phase modulation that, unlike the dynamic phase, isindependent of the optical path length but results exclu-sively from the geometry of the optical trajectory [18].Such geometric phases play a crucial role in the pho-tonic spin Hall effect [19–21] and its scalar equivalent,the acoustic orbital angular momentum Hall effect [22],and spin-orbit transformations in general.

In our work we apply the concept of geometric phasesto the simplest possible experimental setup, comprisingnothing but a succession of metal mirrors in a non-planarconfiguration. For the first time, however, we study po-larisation and image rotation for a wide range of scalarand vector beams, including beams with inhomogeneousspatial polarisation distributions [23], which are non-separable in their spin and orbital degrees of freedom[24]. We systematically show that polarisation profilesand images are rotated by the same angle, confirmingthe concept of an angular redirection phase as the originof optical beam rotation.

II. EXPERIMENTAL SETUP

We generate a non-planar beam path, as outlined inFig. 1 and inspired by a setup used in [25, 26], com-prising four mirror reflections. Mirrors M2, M3 and M4

define a (vertical) plane, and adjusting the position ofmirror M1 allows us to access a non-planar geometry.The beam path is arranged such that the input and out-put wavevectors k0 and k4 point in the same direction(chosen as along the z axis), which allows us to definerotations unambiguously. The rotation of intensity pat-

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terns is determined by analysis of images obtained via aCMOS camera, and the (spatially resolved) polarisationrotation by full Stokes tomography [27, 28].

Experimentally, we parameterize the degree of non-planarity by the angle

θNP = π/2− α, (1)

where α is twice the angle of incidence on mirror M1.Specifically, α is adjusted by shifting the position of mir-ror M1 and adjusting M2 accordingly. In order to min-imise unwanted optical activity we use metallic mirrors.

We generate beam profiles with arbitrary intensity andpolarisation structures using a digital micromirror device(DMD), following techniques outlined in [27, 28]. This al-lows us to investigate homogeneously polarised beams aswell as those with spatially varying polarisation profiles,simply by changing the multiplexed hologram pattern onthe DMD. While our setup generates vector beams assuperpositions of different spatial modes in the horizon-tal and vertical polarisation components, these may beconverted into a modal decomposition of circular polari-sations, allowing us to design arbitrary vector modes.

Throughout the paper we are using the coordinate sys-tem indicated in Fig. 1. A positive rotation angle θ thencorresponds to a clockwise rotation if defined in the direc-tion of beam propagation, which appears anticlockwisewhen observed on the camera.

FIG. 1. Experimental setup. A succession of mirror reflec-tions takes the beam along a curved non-planar trajectory. kn

labels the wave vector after the nth mirror (Mn) reflection.Inset: corresponding path of the wave vector on the k-sphere.

III. IMAGE ROTATION

Spatially resolved ray propagation, taking into accountthe action and alignment of the various optical compo-nents, reveals that any intensity profile exiting the ex-periment is rotated by the angle θ defined in (1), as il-

lustrated in Fig. 1 for the Hermite-Gaussian (HG) modeHG3,0.

Any mode rotation can be interpreted in terms of ge-ometric phases. If Ψ(r, φ) is the original mode expressedin cylindrical coordinates, then a rotation by θ will re-sult in the mode Ψ′(r, φ) = Ψ(r, φ + θ). Here we definea clockwise rotation around the propagation axis as pos-itive. Expressing this mode in Laguerre-Gaussian (LG)

basis, with LG`p = u`p(r) exp(i`φ), we can write the orig-

inal mode as

ψ(r, φ) =∑p,`

〈LG`p|ψ(r, φ)〉LG`

p, (2)

where 〈a|b〉 denotes the inner product which may be eval-uated as mode overlap. The rotated mode is then

ψ′(r, φ) =∑p,`

〈LG`p|ψ(r, φ+ θ)〉LG`

p

=∑p,`

e−i`θ 〈LG`p|Ψ(r, φ)〉LG`

p, (3)

where we have evaluated the inner product as

〈LG`p|Ψ(r, φ+ θ)〉 = 〈u`p(r)ei`φ|Ψ(r, φ+ θ)〉

= 〈u`p(r)ei`(φ−θ)|Ψ(r, φ)〉

= e−i`θ∑p,`

〈LG`p|Ψ(r, φ)〉 .

where the second line results from relabelling the angularvariables. We find that a rotation by θ is associated withan orbital-redirection phase of −`θ for every LG modeof topological charge `. This is of course a direct conse-quence of the fact that LG modes are eigenmodes of theangular momentum operator −i∂θ, and the fact that theangular momentum operator is the generating functionof rotations.

Redirection-phases are said to be geometric becausethey depend on the geometry of the path formed on therelevant sphere of angular momentum directions. Forparaxial beams, the (spin/orbital) angular momentumvectors are aligned with the k vectors, but the componentof S and L perpendicular to the mirror does not changeorientation after every mirror reflection [29]. With thisin mind, the k redirection sphere, shown as the inset inFig. 1, can be translated into an angular momentum redi-rection sphere, shown in Fig. 2. For image rotations con-sidered here, θ is equal to the solid angle Ω, correspondingto the blue curve on the orbital redirection sphere uponpropagation, in Fig. 2.

Any transverse spatial mode of mode order N can beexpressed as superposition ofN+1 modes from a differentmode family with the same mode order. For HG modesHGn,m and LG modes LG`

p the mode number is given byN = n+m = 2p+ |`| respectively.

This allows us to illustrate the orbital-redirectionphase for simple examples: The first order mode HG1,0

3

FIG. 2. Angular momentum redirection sphere for J = L+S,illustrated for the case where J is aligned with the initial wavevector.

rotated by an angle θ in the clockwise direction (withbeam propagation), may be expressed as

HG′1,0 =1√2

(LG+10 e−iθ + LG−1

0 eiθ)

= cos θ HG1,0 + sin θ HG0,1, (4)

in the LG basis and the HG basis respectively.Similarly, the second order mode HG2,0 mode rotated

by θ can be written as:

HG′2,0 =1

2(LG+2

0 e−i2θ −√

2LG01 + LG−2

0 e+i2θ)

= cos2(θ)HG2,0 +sin(2θ)√

2HG1,1 + sin2(θ)HG0,2.

(5)

Measurements of various HG intensity profiles beforeand after non-planar propagation are shown in Fig. 3 fora fixed angle α.

FIG. 3. Image rotation: Experimental images (from the cam-era’s perspective) of the original (top) and rotated (bottomrow) mode profiles for a fixed non-planar trajectory withθNP = (21.4 ± 0.5), indicated as yellow dashed line.

In order to confirm the relationship between non-planarity and image rotation experimentally, we realisenon-planar trajectories, with α set to various angles be-tween 15 and 70, and identify the image rotation for all14 HGn,m modes, with a mode number N = n+m ≤ 4.We first low-pass Fourier filter the camera images to re-move artefacts due to diffraction, and then convert themto polar plots. We obtain the rotation angle from the an-gular offset between the rotated and input beam profiles.

As expected, the rotation angle does not depend on thespecific transverse input mode. For each geometric con-figuration, determined by angle α, we therefore evaluatethe rotation θ by averaging over the individual rotationangles found for our 14 test modes. Fig. 4 shows the ro-tation angles θ of the intensity distribution for variousincidence angles α, confirming that θ = θNP = π/2 − α,i.e. the image rotation angle is directly given by the de-gree of non-planarity of the beam trajectory.

FIG. 4. Image and polarisation rotation for non-planar tra-jectories: Experimental measurements and theoretical predic-tion of the relationship between non-planarity set by α andthe intensity rotation θ. The lower plot shows the differencebetween the measured θ and the theoretically expected θNP.The error in α is estimated to be smaller than 0.5.

IV. POLARISATION ROTATION

Using the same experimental setup we confirm thatpropagation along non-planar trajectories rotates notonly images, but also the axis of the polarisation ellipse

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characterizing the electric field. We study the rotation ofhomogeneously polarised beams, independent on the spa-tial state of the mode, here illustrated for fundamentalGaussian beams.

The propagation of the light along a non-planar tra-jectory can be modelled using Jones vector formalism[30], where the action of each mirror on the polarisationvector is described by a Jones matrix that arises fromsimple ray optics. As we are working in a 3D geome-try, the electric field is described by 3D Jones vectors

E = Ehh+Ev v+Ez z, where h, v and z are unit vectorsin the horizontal, vertical and propagation direction.

As shown in Appendix B, the action of propagationalong the non-planar trajectory is described by the Jonesmatrix,

P =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

(6)

where, θ = θNP = π/2 − α. This is, of course, a generalrotation matrix, showing that the mirror system rotatesan input polarisation state by an angle θ about the zaxis.

Similarly to image rotation, a polarisation rotation canbe explained in terms of geometric phases, in this case be-tween the circular polarisation components. Any parax-ial linearly polarized beam of light, propagating along thez direction, may be written as:

E = E0(r, φ)(

cos θ0h+ sin θ0v), (7)

where E0(r, φ) describes the transverse spatial mode, and0 ≤ θ0 ≤ 2π is the angle of the polarisation direction tothe horizontal. Applying the rotation matrix (6) resultsin the rotated mode

E′ = E0(r, φ)(

cos(θ0 + θ)h+ sin(θ0 + θ)v). (8)

One may easily verify that the initial and rotated field,expressed instead in terms of the circular polarisation

basis r = (h− iv)/√

2 and l = (h+ iv)/√

2, become

E = (E0(r, φ)/√

2)(

eiθ0 r + e−iθ0 l), (9)

E′ = (E0(r, φ)/√

2)(

ei(θ0+θ)r + e−i(θ0+θ) l). (10)

We can interpret the rotation as the right and left handpolarisation component, carrying an angular momentumof −~ and +~ per photon, acquire a phase factor ofexp(iθ) and exp(−iθ), respectively. This result also holdsfor general elliptical beams, but for simplicity we haveomitted the lengthy calculation.

A rotation by θ is associated with a spin-redirectionphase of −σθ for every circularly polarised modes of he-licity σ, leading to polarisation rotation, and an orbital-redirection phase of −`θ for every spatial mode compo-nent with an OAM of `~. For both cases, the rotation

FIG. 5. Polarisation rotation: (a) Experimental observationof polarisation rotation of a horizontally polarised Gaussianbeam (left), for different non-planar geometries (b) Polari-sation profile of a homogeneously polarised HG1,1 mode be-fore and after rotation. The polarisation colour scheme usedthroughout is shown as inset, with intensity represented asopacity.

angle θ is equal to the solid angle Ω formed by the pathtraced on the appropriate (spin or orbital) angular mo-mentum redirection sphere shown in Fig. 2.

We confirm this experimentally, by identifying the po-larisation rotation of an initially horizontally polarisedbeam after it is passed along a non-planar trajectorywith various angles of θNP = π/2 − α. A selection ofthese measurements are shown in Fig. 5.

Polarisation rotation of homogeneously polarised lightis independent on the spatial mode, as illustrated inFig. 5(b) on the example of a horizontally polarised HG1,1

mode. It is evident that both the intensity and polarisa-tion profiles have rotated by the same angle.

Our spatially resolved measurements show that, asexpected, the polarisation remains homogeneous acrossthe beam profile. We determine the rotation angle θby averaging over the spatially resolved Stokes measure-ments before and after propagation, with the error inθ given by the standard deviation. A plot of measuredexperimental angle, α against measured polarisation ro-tation, θ, is shown in Fig. 4, confirming that the polari-sation rotation is equal to the non-planarity parameter,θ = θNP = π/2− α.

V. ROTATION OF VECTOR BEAMS

Finally, we study the rotation of beams of light present-ing an inhomogeneous polarisation distribution. Theseare non-separable in spatial and polarisation degreesof freedom. General vector beams can be written asψ = ψl(r, ϕ)l + ψr(r, ϕ)r, but here we restrict ourselves,without loss of generality, to beams of the form

ψ = LG`1p1 l + e−iϕLG`2

p2 r. (11)

As discussed in the previous sections, the action ofthe non-planar propagation results in orbital redirection

5

FIG. 6. Polarisation plots of experimentally measured vector beams before (top row) and after (bottom row) the system, for

a fixed rotation angle of θ = (21.4 ± 0.5). Beams in (a, b) contain only linear polarisation and are of the form LG±10 l +

exp (−iϕ)LG±10 r, for ϕ = (0, π). The first column in (c) is a Poincare beam and the second column shows a beam containing

polarisations along a great circle on the Poincare sphere. Smaller inserts show the corresponding theory plots.

phase of exp(−i`φ) for mode with OAM of `~, and a spinredirection phase of exp(−iσφ) for a mode with a spin ofσ~ along the propagation direction. The total geomet-ric phase is hence proportional to the total total angularmomentum number j = ` + σ. Applied to our vectorbeam Eq.(11), this leads to a simultaneous rotation ofthe intensity pattern and the polarisation by θ:

ψ′ = e−i(`1+1)θLG`1p1 l + eiϕe−i(`2−1)θLG`2

p2 r. (12)

In particular, if j = 0, the original beam will be recoveredafter rotation, meaning that the output beam will be in-distinguishable from the input, both in polarisation andintensity. As σ can only take values±1, this only happensfor LG modes with l1 = −1 and l2 = 1. These beamsremain unchanged and can be considered the eigenmodesof the system and therefore, can be thought of as con-served quantities. This makes intuitive sense, as beamswith these properties are rotationally symmetric, bothin their intensity and polarisation profile, making themrotation invariant. For the simple case of p1,2 = 0, weobtain radial polarisation for ϕ = 0, and azimuthal po-larisation for ϕ = π, as shown in Fig. 6(a). However,for all other values of j, the angular redirection phasewill cause a rotation of the beam. Fig. 6(b, c), showsthe rotation of a selection of general vectorial beams forwhich j 6= 0, including counter-rotating vector beams, forwhich `1 = 1 and `2 = −1, these beams correspond tothe mirror images of radially and azimuthally polarizedbeams.

VI. CONCLUSION

The orbital and spin angular momentum refer to fun-damentally different properties of light: The former is ex-

trinsic and arises from twisted phasefronts of a light beamor photon wavefunction, the latter is intrinsic, relates tothe vector nature of electromagnetic fields or more specif-ically their circular polarisation. It is therefore surprisingthat taking a light beam along a non-planar trajectoryaffects both spin components in the same way, by addingan angular momentum redirection phase of exp(−ijθNP),where j = σ + ` is the combined spin and angular mo-mentum number of the light, and θNP parametrizes thenon-planarity of the trajectory.

We have experimentally and theoretically looked at therotation of intensity, polarisation and vector field profiles,that arises from propagation along a closed non-planartrajectory. This rotation has been interpreted in termsof both spin and orbital redirection phases, for the polari-sation and intensity, respectively. We have demonstratedthat the intensity and polarisation rotate in the same di-rection by the same angle, which is dependent on the solidangle enclosed by the path on the associated redirectionsphere. By looking at the rotation of vector modes, withspatially varying polarisation, we have shown that thegeometric phase acquired is proportional to the total an-gular momentum number of the beam, j = `+ σ. Whenj = 0, the total geometric phase is null, and the polarisa-tion profile appears unrotated by our system. This is thecase for radial and azimuthally polarised modes, whichcan be considered as eigenmodes of the system.

Due to their robustness against aplanarity, radial andazimuthal vector modes constitute ideal candidates foroptical communication based on optical fibres, inherentlysubject to twisting and bending constraints [12], and formicroscopy applications, where these modes can also betightly focused [31]. Our findings are also relevant fortwisted optical cavities and mirror-based resonators [32–34] and for quantum metrology [35]. The redirection

6

phases evidenced in this work can be linked to opticalHall effects, as both can be attributed to the existenceof Berry potentials [15], and also constitute elegant ad-ditions to the landscape of geometric phases of light, sofar dominated by their planar counterparts [36, 37].

Appendix A: Basis rotation

We here give an explicit derivation of image rotationin terms of a basis change to the LG mode basis, link-ing the rotation angle θ and the orbital redirection phase.We start with the general case and then give the transfor-mation matrices for specific examples of some HG modesused in the experiment.

An HG mode rotated by an angle θ, denoted |HG′〉, canbe written as a superposition of fundamental HG modesby first expressing the mode in the Laguerre-Gaussian(LG) basis, applying a rotation matrix RLG(θ), then re-turning to the HG basis [38]:

|HG′〉 = BHG←LG · RLG(θ) · BLG←HG |HG〉 (A1)

where RLG(θ) = diag(e−i`1θ, e−i`2θ, .., e−i`N+1θ) impartsan orbital-redirection phase, e−i`θ, to the LG componentof topological charge `. Since LG and HG modes bothform their own complete and orthonormal basis, one canbe converted into a superposition of the other via:

|HGn,m〉 =∑`,p

〈LG`p|HGn,m〉 |LG`

p〉 (A2)

|LG`p〉 =

∑n,m

〈HGn,m|LGlp〉 |HGn,m〉 (A3)

where the overlaps 〈LGlp|HGn,m〉 and 〈HGn,m|LGl

p〉 yieldthe elements of the mode conversion matrices. Whilecomplicated general expressions for these overlaps existin the literature, it is simplest to calculate these directlyfrom the overlap of the respective modes using:

〈a|b〉 =

∫∫(ua(x, y, 0))∗ub(x, y, 0)dxdy (A4)

where u are the amplitudes of the beams.For first order modes, the explicit forms of the matrices

describing the mode transformations are

BN=1HG←LG =

1√2

(1 1i −i

)(A5)

RN=1LG (θ) =

(e−iθ 0

0 e+iθ

)(A6)

BN=1LG←HG =

1√2

(1 −i1 i

)(A7)

For second order modes, the transformations are:

BN=2HG←LG =

12 − 1√

212

i√2

0 − i√2

− 12 −

1√2− 1

2

(A8)

RN=2LG (θ) =

e−i2θ 0 00 1 00 0 e+i2θ

(A9)

BN=2LG←HG =

12 − i√

2− 1

2

− 1√2

0 − 1√2

12

i√2− 1

2

(A10)

Appendix B: Polarisation rotation

For this, we need to consider a system of three orthog-onal vectors corresponding to each mirror, the incidentwavevector, k, and two orthogonal polarisation compo-nents, one which lies in the plane of incidence of the mir-ror, p, and one which lies perpendicular to it, s. In thisappendix we apply the three-dimensional polarisation raytracing approach discussed in [39] to our system. The ac-tion of an optical element q on a polarisation vector eqis;

eq = Pq · eq−1 (B1)

where Pq is some 3 × 3 matrix to be found. This equa-tion is written in global coordinates. It is equivalent toanother equation written in local coordinates, namely;

wq = Jq · wq−1 (B2)

Any general polarisation vector can be written in terms ofa pair of orthogonal basis states a and b. Then the globalrelation (B1) becomes the following pair of statements;

e′a = Pq · ea e′b = Pq · eb (B3)

which represent six equations in total. The matrix Pqhas nine elements, so we need three more equations touniquely define it. We choose the matrix form of the lawof reflection (in global coordinates)

kq = Pq · kq−1 (B4)

furnishing us with nine equations in nine unknowns,which can be solved for the elements of Pq. The prop-erties of each optical element are specified by its Jonesmatrix, which works on local Jones vectors. We there-fore need to transform from global to local coordinates,let the optical element act, then transform back to globalcoordinates. All of this should be contained within Pq,so it will have the form;

Pq = OG←L · Jq ·OL←G (B5)

7

where OG←L transforms from local to global and OL←Gtransforms from global to local.

We will first consider the global to local transforma-tion. Our orthogonal coordinate system before the op-tical element is sq,pq,kq−1, which we need to projectonto the local z direction for that element. This kind ofchange of basis is obtained by a matrix whose rows arethe coordinate vectors of the new basis vectors (local,sq,pq,kq−1) in the old basis (global, x,y, z), so;

OL←G =

sx,q sy,q sz,q−1

px,q py,q pz,q−1

kx,q ky,q kz,q−1

(B6)

To transform from local to global coordinates, we takethe inverse of this but replace the basis vectors with thoseafter the element.

O−1G←L = OTG←L =

s′x,q p′x,q kx,qs′y,q p′y,q ky,qs′z,q p′z,q kz,q

(B7)

where the first equality follows from the fact thatthis is an orthogonal matrix, since the basis vectorssq,pq,kq−1 are orthogonal. This now means the ma-trix Pq shown in Eq. (B1) is fully determined if the Jonesmatrix of the particular optical element and the vectorssq,pq,kq−1 are known. These can all be expressed en-tirely in terms of kq and kq−1;

sq =kq−1 × kq|kq−1 × kq|

s′q = sq

pq = kq−1 × sq p′q = kq × s′q (B8)

The power of this method lies in the fact that Q opticalelements can be cascaded by multiplying their Pq matri-ces to find an overall matrix P ;

P = PQ · PQ−1 · . . . · Pq · . . . · P2 · P1 (B9)

Therefore to describe a system all we need are the Jonesmatrices of each element and the k vectors, which in turnwill be determined by the position of each mirror.

As shown in Fig. 1 in the main text, we have four mir-rors and five k vectors. The Jones matrix for reflectionat an ideal mirror is;

Jq =

1 0 00 −1 00 0 1

(B10)

and the five k vectors are;

k0 =

001

k1 =

sinα0

− cosα

k2 =

010

k3 =

100

k4 =

001

= k0 (B11)

Using these in Eqs. (B5)-(B10), the combined matrix Pqfor this system turns out to be;

P = P4 · P3 · P2 · P1 =

sinα − cosα 0cosα sinα 0

0 0 1

(B12)

Defining θ = π/2 − α as in the main text, we end upwith;

P = P4 · P3 · P2 · P1 =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

(B13)

which is Eq. (6) in the main text. This represents an an-ticlockwise rotation of points in the xy plane around thez axis. Since the beam is travelling in the positive z di-rection, this represents a clockwise rotation when viewedalong the propagation axis of the beam.

Appendix C: Expressions for LG and HG modes

The Laguerre-Gauss modes are expressed in cylindrical

coordinates r =√x2 + y2, φ = arctan(y/x), and are

given by;

LG`p =

C`pw

[r√

2

w

]|`|L|`|p

(2r2

w2

)e− r2

w2 +i

(k r2z

2(z2R

+z2)+`φ+Φ

)

(C1)

where C`p =√

2p!π(|`|+p)! , Φ = −(2p + |`| + 1)χ with

χ = arctan(z/zR) is the Gouy phase, L|`|p is an associ-

ated Laguerre polynomial, w = w0(1 + (z/zR)2)1/2 is the

beam radius for waist w0 and zR = πw20/λ is the Rayleigh

range. The Hermite-Gauss modes are given by

HGnm = HGn(x)HGm(y) (C2)

where

HGn(x) =Cn√wHn

(√2x

w

)e− x2

w2 + ikx2z

2(z2+z2R

)−i(n+ 1

2 )χ

(C3)

where Cn =(

2π

)1/4√ 12nn! and Hn is a Hermite polyno-

mial of order n. These can then be used in the overlapintegral (A4) to generate the elements of the matricesshown in Appendix A, for example

〈LG10|HG20〉 =

∫ ∞−∞

∫ ∞−∞

(LG1

0

)∗HG20dxdy = − 1√

2(C4)

which is the entry in the first row, second column of thematrix shown in A8, and appears as the coefficient of thesecond term in Eq. (5).

8

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