Measurement of the Orbital Angular Momentum Spectrum of Fields with PartialAngular Coherence using Double Angular Slit Interference

Mehul Malik1, Sangeeta Murugkar2, Jonathan Leach2, and Robert W. Boyd1,2

1The Institute of Optics, University of Rochester, Rochester, New York 14627, USA and2Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, ON, K1N 6N5, Canada∗

(Dated: June 17, 2021)

We implement an interferometric method using two angular slits to measure the orbital angularmomentum (OAM) mode spectrum of a field with partial angular coherence. As the angular sep-aration of the slits changes, an interference pattern for a particular OAM mode is obtained. Thevisibility of this interference pattern as a function of angular separation is equivalent to the angularcorrelation function of the field. By Fourier transforming the angular correlation function obtainedfrom the double angular slit interference, we are able to calculate the OAM spectrum of the par-tially coherent field. This method has potential application for characterizing the OAM spectrumin high-dimensional quantum information protocols.

PACS numbers: 42.50.Ar, 42.50.Ex, 42.25.Kb

I. INTRODUCTION

Historically, the problem of determining the correla-tions between two space-time points in a wave field hasbeen addressed by the analysis of a two-beam interfer-ence experiment [1, 2]. The degree of spatial coherenceof a light source can be determined from the visibilityof fringes in the far-field, arising from the interferenceof light fields in a Young’s double slit experiment. Thenecessary condition for interference is that the slit sep-aration should be smaller than the transverse coherencelength of the field.

It is well known that angular position and its conju-gate variable, orbital angular momentum (OAM), formFourier transform pairs [3]. This is convincingly demon-strated by the experimental observation of a discretediffraction pattern in the OAM spectrum when light istransmitted through angular amplitude masks [4, 5]. Ex-tending this idea, one can construct an angular analog ofYoung’s double slit experiment with two angular slits [6].By measuring the visibility of the interference patternobtained from double angular slit interference, one cancalculate the degree of angular coherence of the source.For certain types of partially coherent sources, the OAMmode spectrum is simply a Fourier transform of the de-gree of angular coherence [6]. This provides us with a newway of measuring the OAM mode spectrum of partiallycoherent fields.

Interest in the measurement and characterization ofthe OAM mode spectrum has grown rapidly over the lastdecade. This is because OAM modes, by virtue of livingin a discrete, infinite-dimensional Hilbert space, are avery useful tool for quantum information science. En-tanglement in up to 12 dimensions has been shown usingOAM modes [7]. Efforts are under way to realize a quan-tum key distribution system using the OAM basis andthe conjugate basis of angular position [8, 9]. Recently, aterabit/sec free space communication system using OAM

modes was demonstrated [10]. The effects of turbulenceon quantum communication using OAM modes is cur-rently an active topic of research [11]. OAM modes arealso being used as a tool for alignment free quantum com-munication due to their rotational invariance [12]. It isclear that accurate methods of characterizing the OAMmode spectrum will be of great importance for such quan-tum communication systems employing OAM modes.

The simplest way of measuring the OAM mode spec-trum is by projection measurements of the constituentmodes using a forked hologram [13]. More novel meth-ods have been developed recently that involve interferingthe field with a rotated copy of itself in a Mach-Zehnderinterferometer [15, 16]. In this paper, we experimentallydemonstrate a new way of measuring the OAM modespectrum of fields with partial angular coherence thatdoes not require careful interferometric alignment or im-age rotation. Based on the theoretical work in Ref. [6],we first review the concepts of the degree of angular co-herence and double angular slit interference. Then, wedescribe our experimental implementation of double an-gular slit interference using a spatial light modulator. Fi-

∆φ

FIG. 1: (Color online) Partially coherent field propagatingthrough double angular slits with an angular separation of∆φ.

arX

iv:1

208.

3178

v2 [

phys

ics.

optic

s] 1

6 N

ov 2

012

2

I(Δφ, ℓ , φcoh≈ 2π) I(Δφ, ℓ , φcoh= 0.5) I(Δφ, ℓ , φcoh= 0.25)O

AM

+24

0+1

2

Δφ (rads)π−π 0

Δφ (rads)π−π 0

OA

M+2

40

+12

Δφ (rads)π−π 0

OA

M+2

40

+12

b)a) c)

FIG. 2: (Color online) Simulation results for angular interference patterns for OAM mode projections from ` = 0 to ` = 24for three values of coherence angle (a) φcoh ≈ 2π rads (b) φcoh = 0.5 rad and (c) φcoh = 0.25 rad. The red rectangular boxesindicate the value of ` used in our experiment. The z-axis shows normalized intensity from 0 to 1, indicated by grayscale values.

nally, we calculate the OAM mode spectrum from thevisibility of the measured angular interference pattern.

II. THEORY

The cross-spectral-density function is used to quan-tify field correlations between two points in the space-frequency domain. The non-negative definiteness of thisfunction, in conjunction with the Mercer theorem, im-ply that it is a Hilbert-Schmidt kernel and that it has acoherent-mode representation of the form [1]

W (r1, r2) =∑n

αnψ∗n(r1)ψn(r2) (1)

The properties of W (r1, r2) ensure that all the coeffi-cients αn are real and non-negative, i.e. αn ≥ 0, and thatat least one αn is non-zero. This further implies that forany partially coherent field, there exists at least one basisin which the cross-spectral-density function can be repre-sented as the superposition of modes that are completelycoherent in the space-frequency domain.

For fields in the Laguerre-Gauss (LG) basis, one candefine a cross-spectral-density function in the angle-frequency domain by integrating over the radial dimen-sion [6]. The cross-spectral-density function then repre-sents the correlation between the fields at angular posi-tions φ1 and φ2 and can be expressed as a superpositionof OAM modes

W (φ1, φ2) =

∞∑`=−∞

C`e−i`∆φ (2)

where ∆φ = φ1 − φ2. If there is only one term in theabove expansion, the field can be considered to be com-pletely coherent in the angle-frequency domain. If thereis more than one term in the expansion, the field can becharacterized as being partially coherent. For such fields,

one can introduce the angular analog to coherence length,the coherence angle, φcoh. The coherence angle quanti-fies the angular separation over which two field points aremutually coherent.

These concepts are best illustrated by the case of an-gular interference from a double angular slit (Fig. 1).When a partially coherent field is transmitted through amask containing two angular slits at angular positions φ1

and φ2 with unit transmission, an interference pattern isobtained with intensity in an OAM mode ` given by [6]

I`(∆φ) =1

π

∞∑`′=−∞

C`′ [1 + λ(∆φ)cos(`∆φ)]. (3)

Here, λ(∆φ) is the degree of angular coherence and isobtained by normalizing the cross spectral density func-tion:

λ(∆φ) =W (φ1, φ2)

[W (φ1, φ1)]1/2[W (φ2, φ2)]1/2

=W (φ1, φ2)∞∑

`′=−∞

C`′

(4)

where ∆φ = φ1 − φ2. Eq. 3 is the angular interfer-ence law, since it quantifies the interference between thefields coming from two separate angular positions. This isanalogous to the general interference law for stationaryoptical fields [1]. Eq. 4 is analogous to the law relat-ing the degree of spatial coherence to the cross-spectraldensity function in the space-frequency domain (Eq. 1).Also, one can see that λ(∆φ) is equal to the visibility ofthe interference pattern, which is a function of the angu-lar slit separation. The width of λ(∆φ) is a measure ofthe coherence angle (φcoh) over which the field remainscoherent. For a completely coherent field, the degree ofangular coherence λ(∆φ) is equal to unity for all valuesof ∆φ, and φcoh approaches a value of 2π.

3

It has been shown in Ref. [6] that for partially coher-ent fields with a broad Gaussian distribution in `, theangular correlation function is simply the Fourier trans-form of the OAM mode distribution. More specifically,for a Gaussian OAM mode distribution with a width σ,

C(`) =1√2πσ

exp

(−`2

2σ2

). (5)

For such a distribution of OAM, the degree of angularcoherence is given by

λ(∆φ) = exp

(−σ2∆φ2

2

). (6)

By measuring the visibility of the interference patternas a function of slit separation ∆φ, we can construct thedegree of coherence λ(∆φ) of the field. The width of thevisibility function, 1/σ, gives us the coherence angle ofthe field, φcoh. From this, we can calculate σ, which isthe width of the OAM mode spectrum of the partially co-herent field, and reconstruct the OAM mode distributionfrom Eq. (5).

Simulation results for angular interference patterns forOAM mode projections from ` = 0 to ` = 24 are plottedin Fig. 2 (a)-(c) for three different values of φcoh. It isclear from Eq. 6 that the degree of coherence λ(∆φ) isindependent of the OAM mode number used for projec-tion. Consequently, the width of the visibility envelope,i.e. the coherence angle (φcoh), is also independent of theOAM mode number used for projection. This is clearlyillustrated in the simulated results in Fig. 2. The inverserelationship between the coherence angle (φcoh) and theOAM mode distribution width (σ) is also apparent. Thesimulated interference pattern obtained by projecting inOAM mode ` = 12 is outlined in red. In the next section,we discuss experimental results for this particular case.

III. EXPERIMENT AND RESULTS

We implemented a double angular slit experiment us-ing a setup shown in Fig. 3(a). Light from a laser diodeat 650nm is coupled into a single mode fiber (SMF) witha 10x microscope objective. The Gaussian single modeoutput from the SMF is collimated and is incident on thefront surface of a spatial light modulator (SLM). It is asimple task to use an SLM to generate a coherent super-position of OAM modes; however, for this experiment,we require partially coherent Gaussian distributions ofOAM modes with varying degrees of coherence.

To generate these distributions, we generate a coher-ent superposition of OAM modes with a single hologram.The range of OAM modes that we generate at this stagehas a width σ, but each of the modes in the superposition

Laser, λ = 650 nm

f= 500mm

X10

X10

X10

Photodetector

SLM

Phase Intensity Hologram

x =

0

2π

(a)

(b)

FIG. 3: (Color online) (a) Schematic of the experimentalsetup (b) Example of a phase hologram impressed on the SLM

is given a random phase. We then generate a differentsuperposition of OAM modes using a new hologram inwhich each of the modes is given a new random phase.Changing the holograms in this way ensures that thereis no phase relationship between the modes generated bysubsequent holograms. This procedure is repeated overa thousand times such that the resulting time-averagedbeam that reflects off the SLM is a partially coherentsuperposition of OAM modes with a coherence angleφcoh ≈ 1/σ.

One should note that at this stage, we lack precisecontrol over the degree of coherence of the partially co-herent OAM mode distribution that is generated. Webelieve that a significant contributing factor to this ef-fect is an `-dependent efficiency when using an SLM togenerate modes that carry OAM. As SLMs are pixellateddevices, and one requires an SLM to impart an azimuthalphase shift of 2π` to generate a particular OAM mode,there is decreasing efficiency for generating modes withincreasing OAM number. This can be understood as an`-dependent Nyquist limit when generating OAM modeswith a pixellated device. For our experiment, the re-duction in efficiency for large values of OAM results ingenerating a partially coherent mode distribution witha narrower width (σ) and larger coherence angle (φcoh)than intended. Calibration of this effect would improvethe accuracy with which we can create mode distributionswith a given degree of coherence. However, the strengthof our proposed technique is that we have an accuratemeasure of the final OAM mode distribution width andcoherence without relying on the `-dependent efficiency

4

I(Δφ, ℓ = 12, φcoh≈ 2π) I(Δφ, ℓ = 12, φcoh= 0.71) I(Δφ, ℓ = 12, φcoh= 0.29)b)a) c)

-π π-π/2 π/20 −π π−π/2 π/20

Inte

nsity

0.5

1

0

Inte

nsity

0.5

1

0

Inte

nsity

0.5

1

0

Data

VisibilityFit

-π π-π/2 π/20Δφ (rads)Δφ (rads)Δφ (rads)

FIG. 4: (Color online) Normalized intensity for OAM mode projection `=12, of partially coherent light transmitted throughtwo angular apertures, as a function of the angular separation ∆φ for three OAM mode distributions with different degrees ofangular coherence, characterized by a coherence angle (a) φcoh ≈ 2π rads (b) φcoh = 0.71 rad and (c) φcoh = 0.29 rad. Themeasured interference patterns as a function of angular slit separation ∆φ (dashed black line) are plotted alongside a fit (solidred line) obtained from theory. A fit to the visibility envelope is also plotted in (b) and (c) (long-dashed green line).

of an SLM.For every partially coherent Gaussian distribution of

OAM modes that we generate, the same SLM is thenused to (i) generate a forked phase hologram to projectthe field into an OAM mode with angular momentum `,and (ii) generate an amplitude mask of two angular slitswith 15◦ widths, with one slit fixed at 0◦ and the otherrotating from −π to π radians. One should note that thesame SLM is used to generate the partially coherent fieldsand carry out the angular interference experiment. Anexample of the phase and amplitude pattern impressedon the SLM for the angular interference experiment isshown in Fig. 3(b). The light from the SLM is imagedonto the aperture of a SMF connected to a photodetector.

The theory presented in the previous section assumesinfinitesimal angular slits. Since this is not possible toimplement in the lab, we use slits with a finite width of15◦. Analogous to the linear case, this results in a finitespread of OAM modes. In our experiment, we chose tomeasure the interference pattern given by Eq. (3) byprojecting the field into a mode with ` = 12, which ispresent in the envelope of OAM modes diffracted fromour finite width slits. The intensity at the photodetectoras a function of the angular separation ∆φ between theangular slits is shown in Fig. 4 (a)-(c). For a completelycoherent field (φcoh ≈ 2π), there is only a single OAMmode (` = 0) in the generated field and the degree ofcoherence λ(∆φ) ≈ 1. The forked hologram with ` = 12projects the field into an OAM mode with ` = −12. Eq.(3) then simplifies to the form

I(∆φ) =1

π[1 + cos(12∆φ)]. (7)

From this, we can see that the interference pattern thusobtained will have 12 fringes. This can be interpreted asthere being 12 angular positions of the second angular slitas it rotates from −π to π radians, for which the fieldstransmitted through the two angular slits are exactly in

phase. This gives rise to the 12 fringes seen in the inten-sity plot in Fig. 4(a) (plotted with a dashed black line).A fit to these interference fringes based on the theoryin section II is plotted with a solid red line. Since thefield used in Fig. 4(a) has only one OAM mode and iscompletely coherent in angular position, the visibility en-velope of the fringe pattern is close to one for all values ofangular separation. In Fig. 4(b), the degree of coherenceis reduced and there are additional OAM modes in theinput Gaussian distribution. One can see that the visi-bility envelope is narrower, which leads to only 5 fringesbeing detected in the interference pattern. This is dueto the fact that the angular width over which the fieldremains mutually coherent, or the coherence angle φcoh,is reduced. It can be seen from Fig. 4 (a)-(c) that asthe width of the visibility envelope given by the coher-ence angle decreases further, fewer interference fringesare visible as ∆φ goes from −π to π radians.

When the input OAM mode distribution has a broadgaussian shape, we can plot the visibility envelope of theinterference pattern using Eq. (6). This condition is heldfor the interference patterns seen in In Figs. 4 (b) and(c). We plot a fit to the visibility envelope (with a long-dashed green line) of the angular interference pattern forthese two cases. The width of the visibility curve givesus the value of the coherence angle, φcoh. The measuredvalue of φcoh in Figs. 4 (b) and (c) is 0.71 and 0.29 ra-dian respectively. From Eqs. (5) and (6), we see that thewidth of the OAM mode distribution is equal to the in-verse of the coherence angle. Using this, we calculate themeasured OAM mode distribution width σm = 1.4 and3.4. The measured OAM mode distributions for thesetwo cases are plotted in Fig. 5.

5

Ampl

itude

Ampl

itude

0.30

0.00

0.10

0.20

ℓ8-8 -6 -4 -2 0 2 4 6

0.12

0.00

0.02

0.04

0.06

0.08

0.10

ℓ8-8 -6 -4 -2 0 2 4 6

σm = 1.4

σm = 3.4

b)

a)

FIG. 5: (Color online) Calculated OAM mode spectrum forthe two measured distribution widths of (a) σm = 1.4 and (b)σm = 3.4. The normalized amplitude is plotted as a functionof OAM mode number `.

IV. CONCLUSIONS

We have demonstrated experimental results to sup-port the theoretical analysis of angular coherence pro-posed in Ref. [6] for a partially coherent light source.We employed an SLM and a classical light source to cre-ate a partially coherent Gaussian distribution of OAMmodes. When this beam was passed through a doubleangular-slit, a characteristic interference pattern consist-ing of fringes inside a visibility envelope was obtained fora specific OAM mode projection. The width of this vis-ibility envelope served as a direct measure of the degreeof angular coherence of the source, and allowed us to re-construct the input OAM mode distribution to a goodapproximation. This was consistent with our simulationresults which were modeled on the theory of partial an-gular coherence put forth in Ref. [6]. We envision thatsuch measurements will be useful for quantum informa-tion protocols that rely on the high-dimensionality of the

OAM state space [8, 17]. As these protocols are usuallyimplemented in free-space communication systems [9], itis important to characterize the effects of turbulence onthe angular coherence of such a system. Additionally, ithas been shown in Ref. [6] that such a measurement couldbe carried out on either the signal or idler arm in orderto obtain the angular coherence properties of the entan-gled two-photon field produced by spontaneous paramet-ric down conversion [13, 14]. We expect that angularinterference from a double angular-slit holds promise forexploring many interesting quantum phenomena in thehigh-dimensional space of orbital angular momentum.

This work was supported by the Canada ExcellenceResearch Chairs (CERC) Program, the Natural Sciencesand Engineering Research Council of Canada (NSERC),and the DARPA InPho Program.

∗ Electronic address: memalik@optics.rochester.edu[1] L. Mandel, E. Wolf: Optical Coherence and Quantum

Optics: Cambridge University Press (1995)[2] P. B. Dixon, G. Howland, M. Malik, D. J. Starling, R. W.

Boyd, J. C. Howell: Physical Review A 82 (2010) 023801[3] E. Yao, S. Franke-Arnold, J. Courtial, S. Barnett,

M. Padgett: Optics Express 14 (2006) 9071[4] B. Jack, M. Padgett, S. Franke-Arnold: New Journal Of

Physics 10 (2008) 103013[5] A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S.

Buller, S. M. Barnett, S. Franke-Arnold, M. J. Padgett:Physical Review A 78 (2008) 043810

[6] A. K. Jha, G. S. Agarwal, R. W. Boyd: Physical ReviewA 84 (2011) 063847

[7] A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, E. An-dersson: Nature Physics 7 (2011) 677

[8] R. W. Boyd, A. Jha, M. Malik, M. O’Sullivan, B. Roden-burg, D. J. Gauthier: Proceedings of SPIE 7948 (2011)79480L

[9] M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini,J. Leach, M. P. J. Lavery, M. J. Padgett, R. W. Boyd:Optics Express 20 (2012) 13195

[10] J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan,H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. E.Willner: Nature Photonics 6 (2012) 488

[11] B.-J. Pors, C. H. Monken, E. R. Eliel, J. P. Woerdman:Optics Express 19 (2011) 6671

[12] V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S.Slussarenko, L. Marrucci, F. Sciarrino Nature Commu-nications 3:961 (2012)

[13] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger: Nature 412(2001) 313

[14] J. P. Torres, A. Alexandrescu, L. Torner: Physical ReviewA 68 (2003) 050301(R)

[15] R. Zambrini, S. M. Barnett: Physical Review Letters 96(2006) 113901

[16] H. D. L. Pires, J. Woudenberg, M. P. van Exter: OpticsLetters 35 (2010) 889

[17] S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs,A. Zeilinger: New Journal Of Physics 8 (2006) 75