Non-integer OAM beam shifts of

Hermite-Laguerre-Gaussian beams

A.M. Nugrowati∗, J.P. Woerdman

Huygens Laboratory, Leiden UniversityP.O. Box 9504, 2300 RA Leiden, The Netherlands

Abstract

We have studied the effect of non-integer Orbital Angular Momentum (OAM)on OAM enhanced beam shifts, for in-plane (Goos-Hanchen) and out-of-plane(Imbert-Fedorov) shifts, using Hermite-Laguerre-Gaussian beams. Contraryto naive expectation we find, theoretically and experimentally, that the non-integer OAM beam shifts do not interpolate linearly between the integerOAM beam shifts.

Keywords: beam shifts, optical angular momentum, external reflection

1. Introduction

Beams having a finite transverse extent are known to be deflected differ-ently from the geometric prediction, upon interaction with interfaces [1–4].Recently, an extensive research has been carried out addressing how this in-teraction is affected by the Orbital Angular Momentum (OAM) of the beam,which so far has been focussed on the integer OAM case. Specifically, OAMcouples to the Goos-Hanchen (GH) shift [1] that is a longitudinal shift andin-plane with the incoming light, and the Imbert-Fedorov (IF) shift [2, 3]that is transversal and out-of-plane of incidence. Each of these two shiftscan be separated into (i) the spatial type of shift (∆GH, ∆IF) that is inde-pendent of beam focussing, and (ii) the angular type of shift (ΘGH, ΘIF) thatis enhanced by beam propagation upon focussing [5, 6]. The angular shift isclosely related to the Spin Hall Effect of Light [7–9]. In this Communication

∗Corresponding authorEmail address: nugrowati@physics.leidenuniv.nl (A.M. Nugrowati)

Preprint submitted to Optics Communication February 28, 2018

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our interest is in partial (e.g. external) reflection of OAM beams. In thiscase, it has been reported [5, 6, 10–13] that the shifts are linearly enhancedby the (integer) OAM content of the beam. For completeness we note that inthe case of total (internal) reflection the beam shifts are purely spatial andare not affected by the OAM value of the beam [11, 14].

Figure 1: Schematic of angular Goos-Hanchen (ΘGH) and Imbert-Fedorov (ΘIF) shiftsnormalized to the wavenumber k0 and Rayleigh length zR (Λ = k0zR). We deal withpartial external reflection, i.e. refractive index n1<n2.

In OAM beam generation one uses typically Laguerre-Gaussian (LG)beams since their (integer) azimuthal mode index ` is directly proportional tothe OAM value [15]. Such integer OAM beams have a cylindrically symmet-ric beam profile. Our interest is in beam shifts of non-integer OAM beams;such beams have anisotropic beam profiles. Several techniques have been in-troduced to generate non-integer OAM beams [16–19], mainly motivated bytheir potential in carrying higher density classical and quantum information.A light beam carrying a non-integer OAM value can be made in such a way[11, 19, 20] that the (anisotropic) beam profile is preserved upon propagation,apart from the overall scaling and a quadratic phase factor. This is crucialfor their use in beam shift experiments since it eliminates the dependenceon the position of the reflecting (or transmitting) interface. In this Commu-nication, we report our study of beam shifts of a structurally propagationinvariant beam carrying non-integer OAM, upon external reflection on a di-electric surface. We find, theoretically and experimentally, that the angulartype of beam shifts (shown in Fig. 1) has a strong nonlinear dependence onthe non-integer OAM value, that is different from the linear dependence inthe case of integer OAM [10, 12, 13].

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2. Theory

We start with an input Hermite-Gaussian (HGn,m) beam having the modeindices n,m entering a ‘π/2-mode converter’ [20]. By varying the orientationangle α of its transverse profile with respect to those of the converter (seeFig. 2 below), the output beam is a propagation invariant Hermite-Laguerre-Gaussian (HLGn,m (x, y|α)) beam carrying an arbitrary non-integer OAMvalue; i.e. ` = (m− n) sin 2α [19]. At α = Nπ/2 and α = (2N + 1)π/4 whereN is an integer, the output beams are HGn,m and LGp,` modes, respectively,with ` = (m− n) and p = min(m,n), p > 0 the radial mode index.

To derive the beam shifts for non-integer OAM beams, we follow thegeneral expression of a HLG mode given by Abramochkin and Volostnikov[19],

HLGn,m(x, y|α) = e−x2−y2

n+m∑k=0

ik cosn−k α sinm−k α

P(n−k,m−k)k (− cos 2α)Hn+m−k(

√2x)Hk(

√2y), (1)

with P the Jacobi and H the Hermite polynomials. We further use theexpression given by Aiello [21] for the observable GH and IF shifts (〈x〉 and〈y〉, respectively) of general beams. The contribution of the angular and thespatial shifts of the fundamental TEM00 mode, denoted by the superscript”0”, to the total shifts is given by[

〈x〉〈y〉

]=

[∆0

GH

∆0IF

]+

[a11 a12a21 a22

] [Θ0

GH

Θ0IF

], (2)

where the dimensionless shifts of the TEM00 mode are

∆0GH = w⊥Im

(∂ ln r⊥∂θ

)+ w‖Im

(∂ ln r‖∂θ

), (3a)

−Θ0GH = w⊥Re

(∂ ln r⊥∂θ

)+ w‖Re

(∂ ln r‖∂θ

), (3b)

∆0IF = −

a‖a⊥ cot θ

R2‖a

2‖ +R2

⊥a2⊥[(

R2‖ +R2

⊥)

sin η + 2R‖R⊥ sin(η − ϕ‖ + ϕ⊥

)],

(3c)

Θ0IF = −

a‖a⊥ cot θ

R2‖a

2‖ +R2

⊥a2⊥

[(R2‖ +R2

⊥)

cos η], (3d)

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with w‖/⊥ = R2‖/⊥a

2‖/⊥/(R

2‖a

2‖ +R2

⊥a2⊥), r‖/⊥ = R‖/⊥ exp

(iϕ‖/⊥)

)is the Fres-

nel reflection coefficient at incident angle θ, a‖/⊥ the electric field components,and η their phase difference. The matrix elements in Eq.(2) are expressed as

aij =

2 Im

∫f ∗(βi

∂

∂βj

)fdxdy∫

|f |2dxdy, (4)

with f the general beam expression and i, j the indices of the matrix elements(β1 = x, β2 = y). This matrix has been derived by Aiello for integer OAMbeam shifts [21].

For completeness, we would like to point out that the matrix ai,j is thesame as the 4-by-4 matrix for the dimensionless OAM beam shifts in Meranoet al. [11] with the GH shift k0 〈x〉 = ∆`

GH + (z/zR) Θ`GH and the IF shift

k0 〈y〉 = ∆`IF + (z/zR) Θ`

IF, having the OAM content `, the wavenumber k0,and the Rayleigh length zR. Note the z dependency of the angular shifts [5].

We introduce now a simple method to derive non-integer OAM beamshifts: we decompose the HLG mode into two integer OAM beams withopposite signs (i.e. two LG modes with ±` signs). For clarity we restrictourselves to the beam shifts of the HLG1,0(x, y|α) mode but our approachis generally true for arbitrary HLGn,m. By varying α, we realize any non-integer OAM value between −1 6 ` 6 1. The decomposition of the modeis

HLG1,0(x, y|α) =cosα + sinα√

2LG0,1(x, y)

+cosα− sinα√

2LG0,−1(x, y), (5)

where LG0,±1(x, y) = e−x2−y2(x ± iy)±1L±10 (2x2 + 2y2) and L the Laguerre

polynomial. Finally, substituting Eq.(5) in Eq.(4) yields the matrix elements

a11 = (1 + |m− n|) (2 + cos 2α) /2, (6a)

a12 = −|m− n| sin 2α, (6b)

a21 = |m− n| sin 2α, (6c)

a22 = (1 + |m− n|) (2− cos 2α) /2, (6d)

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of our non-integer OAM enhanced beam shifts.The antidiagonal matrix elements (a12 and a21) corresponding to the spa-

tial beam shifts, behave similarly to the integer case [11, 12] in the sense thatthey are linearly dependent on the non-integer OAM value (m − n) sin 2α.However, for the diagonal matrix elements (a11 and a22), corresponding tothe angular shifts (ΘGH and ΘIF), the results for the non-integer case areessentially different from the integer case. Apart from the contribution of|m− n| to the shift, that is similar to the integer case [11], there is an extrafactor (2± cos 2α) /2 showing the beam shift dependency on the angle α.This leads to a nonlinear interpolation between the integer OAM shifts.

3. Experiment

αwire

open cavityHe-Ne laser HG1,0

π/2 - mode convertermode matching

lens

collimatinglens

polarizer

quadrant detector

variable

lens

waveform generator

lock-in amplifier

θi

2θi

mirror

HWP

x

y

retarderprism

HLG1,0(x,y|α)

Figure 2: Set-up for measuring non-integer OAM beams. Inset: a typical HLG beam withthe intensity profile cross sections along the detector transverse axes.

To demonstrate the nonlinear interpolation of the angular beam shiftsfor non-integer OAM beams at partial external reflection, we use the set-upshown in Fig. 2. A HLG1,0 beam is generated by modifying the orientationangle α of the incoming HG1,0 mode with respect to the ‘π/2-mode converter’.

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The converter consists of a mode matching lens and a pair of cylindrical lensesseparated at the appropriate distance [15]. The generated beam is collimatedand by using a half waveplate and a polarizer we tune the intensity and theinput polarization, respectively. This beam is reflected from the hypothenusaair-glass interface of a glass prism (n2 = 1.51). To measure the angular beamshift, we use a quadrant detector and obtain the relative beam displacementbetween two orthogonal directions of the incoming polarization states. Ourgenerated HLG beams have the same symmetry beam axes (see inset ofFig. 2) as those of a standard quadrant detector, which greatly simplifies thebeam shift measurement [20]. To maximize the observable shift, we switchthe polarization state of the beam between 0o/90o for ΘGH and −45o/45o forΘIF, by means of a variable retarder. We further focus the beam to reachthe condition for purely angular beam shifts to occur, i.e. z � zR [21]. Theangle of incidence at the air-glass interface is fixed at θ = 45o and we readthe displacement signal of the reflected beam by using a lock-in amplifier.

For completeness, we would like to point out that a quadrant detectorfurnishes the median instead of the centroid of the beam intensity distribu-tion [11, 21]. This leads to an additional factor in the matrix elements aijfor beam profiles that does not have the fundamental Gaussian distribution.We avoid this ’problem’ when using a quadrant detector, by choosing HLGbeams that have a Gaussian cross section (see the inset of Fig. 2) along themeasured shifts [20].

Our result, shown in Fig. 3, clearly demonstrates the nonlinear depen-dence of angular beam shifts on the non-integer OAM value. The data fol-lows the theoretical prediction given in Eqs.(6a) and (6d). Due to the chosenHLG beams orientations, as discussed in the previous paragraph, we obtainthe same curve for both ΘGH and ΘIF. It is straightforward to generalize ourtwo-component decomposition method in Eq.(5) to any arbitrary non-integerOAM values carried by any higher order HLG beams. This is illustrated inFig. 4, which shows the nonlinear interpolation of the beam shifts betweenhigher-order integer OAM values.

4. Conclusion

In conclusion, we have reported theoretically and experimentally, the an-gular GH and IF shifts of HLG beams carrying non-integer OAM values.HLG beams are chosen to fulfill the requirement for meaningful beam shifts,

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−1 −0.5 0 0.5 1

1

2

Matrix elements: a11 and a22

OAM value: (m-n) sin 2α

ΘO

AM / Θ

TEM00

theorylinear prediction

IF (a22)

ΘGH (a11)

Θ

Figure 3: Result of the angular shifts for HLG1,0|α carrying non-integer OAM value -1<`<,normalized to that for TEM00.

−10 −5 0 5 100

2

4

6

8

10

12

(m−n) sin 2α

HLG0,1

linear predictionnon-integer OAM

ΘO

AM / Θ

TEM

00

HLG0,2

HLG0,3

HLG0,10

Matrix elements: a11 and a22

Figure 4: Calculated angular shifts of higher order non-integer OAM beams.

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in which the beam should be structurally stable upon propagation, apartfrom the overall scaling due to diffraction.

Previous work has reported that the beam shifts in the external reflec-tion case are linearly dependent on the integer OAM value [12]; it wouldseem natural that for non-integer OAM case, one should interpolate linearlybetween the integer OAM beam shifts. We have found, however, that theinterpolation of beam shifts to non-integer OAM values by using HLG beamsis highly nonlinear. Not only the OAM value, but also the OAM density [22]affect the overall beam shifts; the latter is more pronounced for non-integerOAM beams and the nonlinearity depends on the chosen mode profile.

Acknowledgments

This work is supported by the EU within FET Open-FP7 ICT as part ofSTREP Program 255914 Phorbitech. We acknowledge the fruitful discussionswith Andrea Aiello, Sumant Oemrawsingh, and Steven Habraken.

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