arX

iv:1

612.

0033

5v3

[ph

ysic

s.op

tics]

21

Oct

201

7

Single-shot measurement of the orbital-angular-momentum spectrum of light

Girish Kulkarni1, Rishabh Sahu1, Omar S. Magana-Loaiza2, Robert. W. Boyd2,3, and Anand K. Jha1∗1Department of Physics, Indian Institute of Technology, Kanpur, 208016, India

2The Institute of Optics, University of Rochester, Rochester, New York 14627, USA3Department of Physics, University of Ottawa, Ottawa, ON, K1N6N5, Canada

(Dated: July 20, 2018)

The existing methods for measuring the orbital-angular-momentum (OAM) spectrum suffer fromissues such as poor efficiency, strict interferometric stability requirements, and too much loss. Fur-thermore, most techniques inevitably discard part of the field and measure only a post-selectedportion of the true spectrum. Here, we propose and demonstrate a new interferometric techniquefor measuring the true OAM spectrum of optical fields in a single-shot manner. Our techniquedirectly encodes the OAM-spectrum information in the azimuthal intensity profile of the outputinterferogram. In the absence of noise, the spectrum can be fully decoded using a single acquisitionof the output interferogram, and, in the presence of noise, acquisition of two suitable interfero-grams is sufficient for the purpose. As an important application of our technique, we demonstratemeasurements of the angular Schmidt spectrum of the entangled photons produced by parametricdown-conversion and report a broad spectrum with the angular Schmidt number 82.1.

It was shown by Allen et al. that a photon in alight beam can have orbital angular momentum (OAM)values in the integer multiples of ~ [1]. This resulthas made OAM a very important degree of freedom forboth classical and quantum information protocols [2–11].This is because an information protocol requires a dis-crete basis, and the OAM degree of freedom providesa basis that is not only discrete but can also be high-dimensional [12–15]. This is in contrast to the polar-ization degree of freedom, which provides a discrete butonly a two-dimensional basis [2–4]. High-dimensionalquantum information protocols have many distinct ad-vantages in terms of security [16–18], transmission band-width [19, 20], gate implementations [21, 22], supersen-sitive measurements [23] and fundamental tests of quan-tum mechanics [24–27]. In the classical domain, the high-dimensional OAM-states can increase the system capac-ities and spectral efficiencies [9–11].

One of the major challenges faced in the implemen-tation of OAM-based high-dimensional protocols is theefficient detection of OAM spectrum, and it is currentlyan active research area [28–37]. There are several ap-proaches to measuring the OAM spectrum of a field. Onemain approach [28, 29] is to display a specific hologramonto a spatial light modulator (SLM) for a given inputOAM-mode and then measure the intensity at the firstdiffraction order using a single mode fiber. This way,by placing different holograms specific to different inputOAM-mode in a sequential manner, one is able to mea-sure the spectrum. However, this method is very ineffi-cient since the required number of measurements scaleswith the size of the input spectrum. Moreover, due to thenon-uniform fiber-coupling efficiencies of different inputOAM-modes [38], this method does not measure the trueOAM spectrum. The second approach relies on mea-

∗Electronic address: akjha9@gmail.com

suring the angular coherence function of the field andthen reconstructing the OAM-spectrum through an in-verse Fourier transform. One way to measure the angu-lar coherence function is by measuring the interferencevisibility in a Mach-Zehnder interferometer as a func-tion of the Dove-prism rotation angle [31, 32]. Althoughthis method does not have any coupling-efficiency issue,it still requires a series of measurements for obtainingthe angular coherence function. This necessarily requiresthat the interferometer be kept aligned for the entirerange of the rotation angles. A way to bypass the in-terferometric stability requirement is by measuring theangular coherence function [33, 34] using angular double-slits [39]. However, this method also requires a series ofmeasurements and since in this method only a very smallportion of the incident field is used for detection, it is notsuitable for very low-intensity fields such as the fields pro-duced by parametric down-conversion (PDC). The otherapproaches to measuring the OAM spectrum includetechniques based on rotational Doppler frequency shift[35, 36] and concatenated Mach-Zehnder interferometers[37]. However, due to several experimental challenges,these approaches [35–37] have so far been demonstratedonly for fields consisting of just a few modes. Thus theexisting methods for measuring the OAM spectrum in-formation suffer from either poor efficiency [28, 35] orstrict interferometric stability requirements [31, 32, 37]or too much loss [33, 34]. In addition, many techniques[28, 33, 34] inevitably discard part of the field and yieldonly a post-selected portion of the true spectrum.

In this article, we demonstrate a novel interferometrictechnique for measuring the true OAM spectrum in asingle-shot manner, that is, by acquiring only one imageof the output interferogram using a multi-pixel camera.Since our method is interferometric, the efficiency is veryhigh, and since it involves only single-shot measurements,the interferometric stability requirements are much lessstringent.

2

Results

Theory of single-shot spectrum measurement.

The Laguerre-Gaussian (LG) modes, represented asLGl

p(ρ, φ), are exact solutions of the paraxial Helmholtzequation. The OAM-mode index l measures the OAMof each photon in the units of ~, while the index p char-acterizes the radial variation in the intensity [1]. Thepartially coherent fields that we consider in this articleare the ones that can be represented as incoherent mix-tures of LG modes having different OAM-mode indices[33]. The electric field Ein(ρ, φ) corresponding to such afield can be written as

Ein(ρ, φ) =∑

l,p

AlpLGlp(ρ, φ) =

∑

l,p

AlpLGlp(ρ)e

ilφ, (1)

where Alp are stochastic variables. The correspondingcorrelation function W (ρ1, φ1; ρ2, φ2) is

W (ρ1, φ1; ρ2,φ2) ≡ 〈E∗

in(ρ1, φ1)Ein(ρ2, φ2)〉e=

∑

l,p,p′

αlpp′LG∗lp (ρ1, φ1)LG

lp′(ρ2, φ2). (2)

Here 〈· · · 〉e represents the ensemble average and〈A∗

lpAl′p′〉e = αlpp′δl,l′ , where δl,l′ is the Kronecker-deltafunction. When integrated over the radial coordinate, theabove correlation function yields the angular coherencefunction: W (φ1, φ2) ≡

∫

∞

0ρdρW (ρ, φ1; ρ, φ2), which, for

the above field, can be shown to be [33]

W (φ1, φ2) →W (∆φ) =1

2π

∞∑

l=−∞

Sle−il∆φ, (3)

where Sl =∑

p αlpp, ∆φ = φ1 − φ2, and where we have

used the identity∫

∞

0 ρLG∗lp (ρ)LG

lp′(ρ)dρ = δpp′/2π.

The quantity Sl is referred to as the OAM spectrumof the field. It is normalized such that

∑

l Sl = 1

and∫ π

−πW (φ1, φ1)dφ1 =

∫ π

−πW (φ2, φ2)dφ2 = 1. The

Fourier transform relation of Equation (3) is the angu-lar analog of the temporal Wiener-Khintchine theoremfor temporally-stationary fields (see Section 2.4 of [40]).Therefore, a measurement of W (∆φ) can yield the OAMspectrum of the input field through the inverse Fourierrelation

Sl =

∫ π

−π

W (∆φ)eil∆φd∆φ. (4)

Now, let us consider the situation shown in Fig. 1(a). Apartially coherent field of the type represented by Equa-tions (1) and (2) enters the Mach-Zehnder interferometerhaving an odd and an even number of mirrors in the twoarms [shown in Fig. 1(c) ]. As illustrated in Fig. 1(d),each reflection transforms the polar coordinate as ρ→ ρand the azimuthal coordinate as φ+φ0 → −φ+φ0 acrossthe reflection axis (RA). Here φ is the angle measuredfrom RA, and φ0 is the angular-separation between RAand the zero-phase axis of the incident mode (dashed-axis). The phase φ0 does not survive in intensity ex-pressions. So, without the loss of any generality, we take

φ0 = 0 for all incident modes. Therefore, for the input in-cident field Ein(ρ, φ) of Equation (1), the field Eout(ρ, φ)at the output port becomes

Eout(ρ, φ) =√

k1Ein(ρ,−φ)ei(ω0t1+β1)

+√

k2Ein(ρ, φ)ei(ω0t2+β2+γ). (5)

Here, t1 and t2 denote the travel-times in the two arms ofthe interferometer; ω0 is the central frequency of the field;β1 and β2 are the phases other than the dynamical phaseacquired in the two arms; γ is a stochastic phase which in-corporates the temporal coherence between the two arms;k1 and k2 are the scaling constants in the two arms, whichdepend on the splitting ratios of the beam splitters, etc.The azimuthal intensity Iout(φ) at the output port is de-fined as Iout(φ) ≡

∫

ρ〈E∗

out(ρ, φ)E∗

out(ρ, φ)〉edρ, and usingEquations (1)-(4), we can evaluate it to be

Iout(φ) =1

2π(k1 + k2) + γ

√

k1k2W (2φ)eiδ + c.c. (6)

Here, we have defined δ ≡ ω0(t2 − t1) + (β2 − β1), andγ = 〈eiγ〉 quantifies the degree of temporal coherence.The intensity expression in Equation (6) is very differ-ent from the output intensity expression one obtainsin a conventional Mach-Zehnder interferometer with aDove prism having either odd/odd or even/even num-ber of mirrors in the two interferometric arms [31, 32].In Equation (6), the output intensity and the angularcorrelation function both depend on the detection-planeazimuthal angle φ. As a result, the angular correlationfunction W (2φ) comes out encoded in the azimuthal in-tensity profile Iout(φ). In contrast, in the conventionalMach-Zehnder interferometers [31, 32], the output inten-sity has no azimuthal variation; one measures the an-gular correlation function by measuring the interference-visibility of the total output intensity as a function of theDove prism rotation angles. For a symmetric spectrum[Sl = S−l = (Sl + S−l)/2], we have using the formula inEquation 4

Sl =

∫ π

−π

W (2φ)ei2lφd(2φ)

=1

2γ cos δ√k1k2

∫ π

−π

[

Iout(φ) −k1 + k2

2π

]

cos (2lφ)dφ.

(7)

So, if the precise values of k1, k2, γ and δ are known thena single-shot measurement of the output interferogramIout(φ) yields the angular coherence function W (2φ) andthereby the OAM spectrum Sl. Here, by “a single-shot measurement” we mean recording one image of theoutput interferogram using a multi-pixel camera. Therecording may involve collecting several photons per pixelfor a fixed exposure time of the camera.

Theory of two-shot noise-insensitive spectrum

measurement. Although it is in principle possible to

3

FIG. 1: Describing the proposed experimental technique and its working priciple. (a) Schematic of the setup forsynthesizing partially coherent field of the type represented by Equations (1) and (2) with known OAM spectra. (b) Schematicof the setup for producing the partially coherent fields through parametric down-conversion. (c) Schematic of the Mach-Zehnderinterferometer. (d) Describing how a mirror reflection changes the azimuthal phase of an LG mode. An incident beam with the

azimuthal phase profile eil(φ+φ0) transforms into a beam having the azimuthal phase profile e−il(φ−φ0), where φ0 is the anglebetween the reflection axis (RA) and the zero-phase axis (dashed axis) of the incident mode. (e) Illustrating the interferenceeffect produced by the interferometer when the incident field is an LGl

p=0(ρ, φ) mode with l = 4. At the output, we effectively

have the interference of an eilφ mode with an e−ilφ mode, and we obtain the output interference intensity in the form of a petalpattern with the number of petals being 2|l| = 8. SLM: spatial light modulator; SF: spatial filter; DM: dichroic mirror; BBO:type-I beta barium borate crystal; T: translation stage.

measure the OAM spectrum in a single-shot manner asdiscussed above, it is practically extremely difficult to doso because of the requirement of a very precise knowl-edge of k1, k2, γ, and δ. Moreover, obtaining a spec-trum in this manner is susceptible to noise in the mea-sured Iout(φ), which results in errors in the measuredspectrum. We now show that it is possible to elimi-nate this noise completely while also relinquishing theneed for a precise knowledge of k1, k2, γ, and δ, justby acquiring one additional output interferogram. Wepresent our analysis for a symmetric spectrum, that is,for [Sl = S−l = (Sl + S−l)/2]. (See Methods section forthe non-symmetric case). Let us assume that the exper-imentally measured output azimuthal intensity Iδout(φ)at δ contains some noise Iδn(φ) in addition to the signalIout(φ), that is,

Iδout(φ) = Iδn(φ) +1

2π(k1 + k2) + 2γ

√

k1k2W (2φ) cos δ.

Now, suppose that we have two interferograms, Iδcout(φ)

and Iδdout(φ), measured at δ = δc and δ = δd, respectively.

The difference in the intensities ∆Iout(φ) = Iδcout(φ) −Iδdout(φ) of the two interferograms is then given by

∆Iout(φ) = ∆In(φ) + 2γ√

k1k2(cos δc − cos δd)W (2φ),

where ∆In(φ) = Iδcn (φ) − Iδdn (φ) is the difference in thenoise intensities. Multiplying each side of the above

equation by ei2lφ, using the formula in Equation 4,and defining the measured OAM spectrum as Sl ≡∫ π

−π ∆Iout(φ)ei2lφd(2φ) =

∫ π

−π ∆Iout(φ)ei2lφdφ, we get

Sl =

∫ π

−π

∆In(φ)ei2lφdφ+ 2γ

√

k1k2(cos δc − cos δd)Sl. (8)

We see that in situations in which there is no shot-to-shot variation in noise, that is, ∆In(φ) = 0, the mea-sured OAM-spectrum Sl is same as the true input OAM-spectrum Sl up to a scaling constant. One can thus ob-tain the normalized OAM-spectrum in a two-shot mannerwithout having to know the exact values of k1, k2, γ, δcor δd. Nevertheless, in order to get a better experimentalsignal-to-noise ratio, it would be desirable to have γ ≈ 1,k1 ≈ k2 ≈ 0.5, δc ≈ 0, and δd ≈ π. Now, in situation inwhich ∆In(φ) 6= 0, it is clear from Equation (8) that themeasured spectrum will have extra contributions. How-ever, since we do not expect very rapid azimuthal varia-tions in ∆In(φ), the extra contributions should be moreprominent for modes around l = 0 and should die downfor large-l modes.

Measuring lab-synthesized OAM spectra. We nowreport the experimental demonstrations of our techniquefor laboratory-synthesized, symmetric OAM spectra. Asshown in Fig. 1(a), a He-Ne laser is spatially filteredand made incident onto a Holoeye Pluto SLM. The

4

FIG. 2: Experimental results obtained with the lab-synthesized fields. (a) and (b) Measured output interferogramsand the corresponding azimuthal intensities for input LGl

p=0(ρ, φ) modes with l = 1, 4, and 16 for δc ≈ 2mπ and δd ≈ (2m+1)π,respectively, where m is an integer. (c) Measured output interferograms, the azimuthal intensities, and the measured spectrumfor the synthesized input field with a Gaussian OAM-spectrum. (d) Measured output interferograms, the azimuthal intensities,and the measured spectrum for the synthesized input field with a Rectangular OAM-spectrum. In all the above azimuthalintensity plots, the red lines are the experimental plots and the black lines are the theoretical fits.

LGlp=0(ρ, φ) modes are generated using the method by

Arrizon et al. [42] and then made incident into the Mach-Zehnder interferometer shown in Fig. 1(c). The measuredinterferogram and the corresponding azimuthal intensityIδout(φ) for a few LGl

p=0(ρ, φ) modes for δc ≈ 2mπ, andδd ≈ (2m+ 1)π, where m is an integer, are presented inFig. 2(a) and Fig. 2(b), respectively. A very good matchbetween the theory and experiment indicates that theLGl

p=0(ρ, φ) modes produced in our experiments are of

very high quality. By controlling the strengths of the syn-thesized LGl

p=0(ρ, φ) modes for l ranging from l = −20to l = 20, we synthesize two separate fields: one with arectangular spectrum and the other one with a Gaussianspectrum. The representative interferogram correspond-ing to a particular field as input is obtained by addingindividual interferograms for l ranging from l = −20 tol = +20. Two such representative interferograms, onewith δ = δc and other one with δ = δd are recorded

5

FIG. 3: Experimental results obtained with fields produced by parametric down-conversion. (a) and (b) Measuredoutput interferogram and the azimuthal intensity for δ = δc and δ = δd, respectively. (c) The normalized measured spectrum Sl

as computed using Equation (8), and the normalized theoretical spectrum as calculated using the formalism of Ref. [41], for oursetup parameters, namely, a type-I collinear down-conversion with a 2-mm thick BBO crystal and a 0.85-mm beam-waist pumplaser. The theoretical spectrum has no fitting parameters. The angular Schmidt number K = 1/(

∑lS2l ) for the measured

spectrum is evaluated to be 82.1.

for each field. Figures 2(c) and 2(d) show the measuredoutput interferograms, the corresponding azimuthal in-tensities, and the measured spectrum Sl computed usingEquation (8) for the synthesized Gaussian and Rectan-gular OAM spectrum, respectively. We find a very goodmatch between the synthesized spectra and the measuredspectra. There is some mismatch in the measured spectrafor low-l modes. We attribute this to SLM imperfections,various wave-front aberrations, and the non-zero shot-to-shot noise variation ∆In(φ).

Measuring angular Schmidt spectrum of entan-

gled states. The state of the two-photon field producedby parametric down-conversion (PDC) has the followingSchmidt-decomposed form when the detection system issensitive only to the OAM-mode index [33]:

|ψ2〉 =∞∑

l=−∞

√

Sl|l〉s| − l〉i. (9)

Here s and i stand for signal and idler photons, respec-tively, |l〉 represents a mode with OAM-mode index l,and Sl is referred to as the angular Schmidt spectrum.The angular Schmidt spectrum quantifies the dimension-ality and the entanglement of the state in the OAM basis[32, 41, 43]. There are a variety of techniques for measur-ing the angular Schmidt spectrum [28, 32, 33, 44, 45]. Inthe context of spatial entanglement, there has even been atheoretical proposal [46] and its subsequent experimentalimplementation [47] for measuring the spatial Schmidt

spectrum in a single-shot manner using coincidence de-tection. However, all the above mentioned work, includ-ing the single-shot work in the spatial domain [46, 47],are sensitive to noise and require either a very preciseknowledge of the experimental parameters, such as beamsplitting ratio, or a very stable interferometer. In con-trast, as an important experimental application of ourtechnique, we now report an experimental measurementof the angular Schmidt spectrum of the PDC photonsthat not only is a single-shot, noise-insensitive techniquebut also does not require coincidence detection.As derived in Ref. [33], the angular coherence function

Ws(φ1, φ2) corresponding to the individual signal or idlerphoton has the following form:

Ws(φ1, φ2) →Ws(∆φ) =1

2π

∞∑

l=−∞

Sle−il∆φ, (10)

where Sl =∑

p αlpp is the OAM spectrum of individ-

ual photons. Comparing Equations (9) and (10), we findthat the OAM spectrum of individual photons is sameas the angular Schmidt spectrum of the entangled state.Therefore, it is clear that one can measure the angularSchmidt spectrum in a single-shot manner by measuringthe OAM-spectrum of individual photons in a single-shotmanner. As depicted in Fig. 1(b), entangled photons areproduced by PDC with collinear type-I phase matching.The photons are collected by a lens arrangement whosecollection angle is larger than the emission cone-angle ofthe crystal. This ensures that no part of the produced

6

field is discarded from the measurement and thus that thetrue spectrum is measured. This field is then made inci-dent into the Mach-Zehnder interferometer of Fig. 1(b).For a type-I collinear down-conversion with a 2-mm thickBBO crystal and a 0.85-mm beam-waist pump laser, themeasured output interferograms and the correspondingazimuthal intensities for two values of δ have been shownin Fig. 3(a) and Fig. 3(b). Fig. 3(c) shows the normalizedmeasured spectrum as computed using Equation (8) andthe normalized theoretical spectrum as calculated usingthe formalism of Ref. [41]. The near-perfect match with-out the use of any fitting parameter shows that we haveindeed measured the true theoretical angular Schmidtspectrum of PDC photons. There is some mismatchfor low-l values, which we attribute to the very smallbut finite shot-to-shot noise variation ∆In(φ). The an-gular Schmidt number computed as K = 1/(

∑

l S2l ) is

K = 82.1, which, to the best of our knowledge, is thehighest-ever reported angular Schmidt number so far.

Discussion

In conclusion, we have proposed and demonstrated asingle-shot technique for measuring the angular coher-ence function and thereby the OAM spectrum of lightfields that can be represented as mixtures of modes withdifferent OAM values per photon. Our technique in-volves a Mach-Zehnder interferometer with the centralfeature of having an odd and an even number of mir-rors in the two interferometric arms, and it is very ro-bust to noise and does not require precise characteri-zation of setup parameters, such as beam splitting ra-tios, degree of temporal coherence, etc. This techniquealso does not involve any inherent post-selection of thefield to be measured and thus measures the true OAMspectrum of a field. As an important application of thistechnique, we have reported the measurement of a veryhigh-dimensional OAM-entangled states in a single shotmanner, without requiring coincidence detection. Forsuch high-dimensional states, our technique improves thetime required for measuring the OAM spectrum by ordersof magnitude. This can have important implications interms of improving the signal-to-noise ratio and reducingthe interferometric stability requirements for both classi-cal and quantum communication protocols that are basedon using OAM of photons.

The technique presented in this article is for fields thatcan be represented as incoherent mixtures of modes car-rying different OAM-mode indices. However, in manyimportant applications, such as OAM-based multiplexingin communication protocols [9–11], one uses fields thatare coherent superpositions of OAM-carrying modes. Forsuch fields, we believe that the generalizations of the re-construction techniques [48, 49] used for complex-valuedobjects could be a possible way of getting the state in-formation in a single-shot manner. Moreover, in recentyears, finding efficient ways for measuring a partially co-herent field is becoming an important research pursuit[50], and we believe that, at least in the OAM degree

of freedom, the generalized versions of the existing tech-niques for coherent fields in combination with our tech-nique presented in this article might pave the way to-wards a full quantum state tomography in a single-shotor a few-shots manner.

Methods

Details of the experiment with lab-synthesized

fields. In this experiment, the LGlp=0(ρ, φ) modes were

generated by an SLM using the method by Arrizon et al.

[42]. These modes were made sequentially incident intothe interferometer and the corresponding output inter-ferograms were imaged using an Andor iXon Ultra EM-CCD camera having 512×512 pixels. For each individualLGl

p=0(ρ, φ) mode the camera was exposed for about 0.4seconds. The sequential acquisition was automated to en-sure that δ is the same for all the modes. The azimuthalintensity Iδout(φ) plots were obtained by first precisely po-sitioning a very narrow angular region-of-interest (ROI)at angle φ in the interferogram image and then integrat-ing the intensity within the ROI up to a radius that issufficiently large. To reduce pixelation-related noise, theinterferograms were scaled up in size by a factor of fourusing a bicubic interpolation method. In order to ensureminimal shot-to-shot noise variation, the interferometerwas covered after the required alignment with a box andthe measurements were performed only after it had sta-bilized in terms of ambient fluctuations.

Details of the experiment measuring angular

Schmidt spectrum. As depicted in Fig. 1(b), a 405 nmultraviolet pump laser with a beam radius 0.85 mm andhaving a Gaussian transverse mode profile was made in-cident on a 2-mm thick beta barium borate (BBO) crys-tal. The crystal was phase-matched for collinear type-I PDC. The pump power of 100 mW ensured that wewere working within the weak down-conversion limit, inwhich the probability of producing a four-photon stateis negligibly small compared to that of producing a two-photon state. The residual pump photons after the crys-tal were discarded by means of a dichroic mirror (DM).The down-converted photons were passed through an in-terference filter of spectral width 10 nm centered at 810nm and then made incident into the Mach-Zehnder inter-ferometer of Fig. 1(b). The output of the interferometerwas recorded using an Andor iXon Ultra EMCCD cam-era having 512×512 pixels with the acquisition time of13 seconds. The azimuthal intensity Iδout(φ) plots wereobtained by first precisely positioning a very narrow an-gular region-of-interest (ROI) at angle φ in the interfero-gram image and then integrating the intensity within theROI up to a radius that is sufficiently large. To reducepixelation-related noise, the interferograms are scaled upin size by a factor of eight using a bicubic interpolationmethod.

We note that since we are using collinear down-conversion, the individual signal and idler photons haveequal probability of arriving at a given output port of

7

the interferometer. As a result, what is recorded by thecamera at a given output port is the sum of the inter-ferograms produced by the signal and idler fields at thatoutput port. However, since the individual signal andidler fields have the same OAM spectrum, the azimuthalprofile of the sum interferogram is same as that of the in-dividual interferograms produced by either the signal orthe idler field. One assumption that we have made here isthat the probability of simultaneous arrivals of the signaland idler photons at the same EMCCD-camera pixel isnegligibly small. This assumption seems perfectly validgiven that the EMCCD camera has 512×512 pixels, andas shown in Fig. 3, the output interferograms occupymore than half of the EMCCD camera pixels.

Theory of non-symmetric OAM-spectrum mea-

surement. This section presents our analysis for a non-symmetric spectrum, that is, when Sl = S−l conditionis not necessarily met. Just as in the case of symmet-ric spectrum, let us assume that the measured azimuthalintensity Iδout(φ) at the output contains the noise termIδn(φ) in addition to the signal Iout(φ). Thus

Iδout(φ) = Iδn(φ) + Iout(φ)

= Iδn(φ) +k1 + k2

2π+ γ

√

k1k2[W (2φ)e−iδ + c.c.]. (11)

Now, suppose we have two interferogram measured attwo different values of δ, say at δc and δd. The difference∆Iout(φ) in the intensities of the two interferogram isthen given by

∆Iout(φ) = Iδcout(φ) − Iδdout(φ)

= ∆In(φ) + γ√

k1k2[W (2φ)e−iδc +W ∗(2φ)eiδc

−W (2φ)e−iδd −W ∗(2φ)eiδd ], (12)

where ∆In(φ) = Iδcn (φ) − Iδdn (φ) is the difference in thenoise intensities. Unlike in the case of symmetric spec-trum, ∆Iout(φ) is not proportional to the angular coher-ence function W (2φ). Multiplying each side of Equation(12) by ei2lφ and using the angular Wiener-Khintchinerelation Sl =

∫ π

−πW (2φ)ei2lφd(2φ), we obtain

∫ π

−π

∆Iout(φ)ei2lφd(2φ) =

∫ π

−π

∆In(φ)(φ)ei2lφd(2φ)

+ γ√

k1k2[Sle−iδc + S−le

iδc − Sle−iδd − S−le

iδd ]. (13)

Now, multiplying each side of Equation (12) by e−i2lφ

and using the angular Wiener-Khintchine relation Sl =∫ π

−πW (2φ)ei2lφd(2φ), we obtain

∫ π

−π

∆Iout(φ)e−i2lφd(2φ) =

∫ π

−π

∆In(φ)(φ)e−i2lφd(2φ)

+ γ√

k1k2[S−le−iδc + Sle

iδc − S−le−iδd − Sle

iδd ]. (14)

Adding Equations (13) and (14), we get

∫ π

−π

∆Iout(φ) cos(2lφ)d(2φ) =

∫ π

−π

∆In(φ) cos(2lφ)d(2φ)

+ γ√

k1k2(Sl + S−l)(cos δc − cos δd). (15)

Subtracting Equation (14) from Equation (13), we get

∫ π

−π

∆Iout(φ) sin(2lφ)d(2φ) =

∫ π

−π

∆In(φ) sin(2lφ)d(2φ)

− γ√

k1k2(Sl − S−l)(sin δc − sin δd). (16)

Now the question is how should one define the spectrumso that the defined spectrum becomes proportional to thetrue spectrum. Upon inspection we find that for the non-symmetric case it is not possible to define the spectrumthe way we did it in the case of symmetric spectrum.Nevertheless, in the special situation in which δc + δd =π/2, it is possible to define the measured spectrum justlike we did it in the symmetric case. Let us considerthe situation when δc = θ and δd = π/2 − θ such thatδc + δd = π/2. Equations (15) and (16) for this situationcan be written as∫ π

−π

∆Iout(φ) cos(2lφ)d(2φ) =

∫ π

−π

∆In(φ) cos(2lφ)d(2φ)

+ γ√

k1k2(Sl + S−l)(cos θ − sin θ). (17)

and∫ π

−π

∆Iout(φ) sin(2lφ)d(2φ) =

∫ π

−π

∆In(φ) sin(2lφ)d(2φ)

+ γ√

k1k2(Sl − S−l)(cos θ − sin θ). (18)

Adding Equations (17) and (18), we get

∫ π

−π

∆Iout(φ) [cos(2lφ) + sin(2lφ)] d(2φ)

=

∫ π

−π

∆In(φ) [cos(2lφ) + sin(2lφ)] d(2φ)

+ 2γ√

k1k2(cos θ − sin θ)Sl. (19)

So, now if we define the measured spectrum Sl to be

Sl ≡∫ π

−π

∆Iout(φ) [cos(2lφ) + sin(2lφ)] d(2φ)

=

∫ π/2

−π/2

2∆Iout(φ) [cos(2lφ) + sin(2lφ)] dφ

=

∫ π

−π

∆Iout(φ) [cos(2lφ) + sin(2lφ)] dφ, (20)

we get,

Sl =

∫ π

−π

∆In(φ) [cos(2lφ) + sin(2lφ)] dφ

+ 2γ√

k1k2(cos θ − sin θ)Sl. (21)

8

In situations in which the noise neither has any explicitfunctional dependence on δ nor has any shot-to-shot vari-ation, we have ∆In(φ) = 0. Thus the defined spectrumSl becomes proportional to the true spectrum Sl. We seethat just as in the case of symmetric spectrum, one doesnot have to know the exact values of k1, k2, γ and θ. Theonly thing different in this case is that one has to takethe two shots such δc + δd = π/2.

Data availability. The authors declare that the maindata supporting the findings of this study are availablewithin the article. Extra data are available from the cor-responding author upon request.

Acknowledgements

We acknowledge financial support through an initiationgrant no. IITK /PHY /20130008 from Indian Instituteof Technology (IIT) Kanpur, India and through the re-

search grant no. EMR/2015/001931 from the Scienceand Engineering Research Board (SERB), Departmentof Science and Technology, Government of India.

Author contributions

G.K. and A.K.J. proposed the idea, worked out the the-ory, and performed the experiments. R.S. contributed to-wards producing LG beams using the method by Arrizonet al. [42] and helped with numerical studies. O.S.M.L.and R.W.B. contributed to the theoretical analysis of thework. A.K.J. supervised the overall work and all authorscontributed in preparing the manuscript.

Competing financial interests: The authors declareno competing financial interests.

References

[1] Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. &Woerdman, J. P. Orbital angular momentum of lightand the transformation of laguerre-gaussian laser modes.Phys. Rev. A 45, 8185–8189 (1992).

[2] Bennett, C. H. & Brassard, G. Proceedings of ieee in-ternational conference on computers, systems, and sig-nal processing, bangalore, india,(new york). Proceed-ings of IEEE International Conference on Computers,Systems, and Signal Processing, Bangalore, India,(NewYork) (1984).

[3] Bennett, C. H. Quantum cryptography using any twononorthogonal states. Phys. Rev. Lett. 68, 3121–3124(1992).

[4] Ekert, A. K. Quantum cryptography based on bell’s the-orem. Phys. Rev. Lett. 67, 661–663 (1991).

[5] Duan, L.-M., Lukin, M., Cirac, J. I. & Zoller, P. Long-distance quantum communication with atomic ensemblesand linear optics. Nature 414, 413–418 (2001).

[6] Marcikic, I., De Riedmatten, H., Tittel, W., Zbinden,H. & Gisin, N. Long-distance teleportation of qubits attelecommunication wavelengths. Nature 421, 509–513(2003).

[7] Ladd, T. D. et al. Quantum computers. Nature 464,45–53 (2010).

[8] Kimble, H. J. The quantum internet. Nature 453, 1023–1030 (2008).

[9] Wang, J. et al. Terabit free-space data transmission em-ploying orbital angular momentum multiplexing. NatPhoton 6, 488–496 (2012).

[10] Bozinovic, N. et al. Terabit-scale orbital angular momen-tum mode division multiplexing in fibers. Science 340,1545–1548 (2013).

[11] Yan, Y. et al. High-capacity millimetre-wave commu-nications with orbital angular momentum multiplexing.Nature Communications 5, 4876 (2014).

[12] Barnett, S. M. & Pegg, D. T. Quantum theory of rotationangles. Phys. Rev. A 41, 3427–3435 (1990).

[13] Yao, E., Franke-Arnold, S., Courtial, J., Barnett, S. &Padgett, M. Fourier relationship between angular posi-tion and optical orbital angular momentum. Opt. Express

14, 9071–9076 (2006).[14] Jha, A. K. et al. Fourier relationship between the angle

and angular momentum of entangled photons. Phys. Rev.A 78, 043810 (2008).

[15] Jack, B., Padgett, M. J. & Franke-Arnold, S. Angulardiffraction. New Journal of Physics 10, 103013 (2008).

[16] Karimipour, V., Bahraminasab, A. & Bagherinezhad, S.Quantum key distribution for d -level systems with gen-eralized bell states. Phys. Rev. A 65, 052331 (2002).

[17] Cerf, N. J., Bourennane, M., Karlsson, A. & Gisin, N. Se-curity of quantum key distribution using d -level systems.Phys. Rev. Lett. 88, 127902 (2002).

[18] Nikolopoulos, G. M., Ranade, K. S. & Alber, G. Er-ror tolerance of two-basis quantum-key-distribution pro-tocols using qudits and two-way classical communication.Phys. Rev. A 73, 032325 (2006).

[19] Fujiwara, M., Takeoka, M., Mizuno, J. & Sasaki, M. Ex-ceeding the classical capacity limit in a quantum opticalchannel. Phys. Rev. Lett. 90, 167906 (2003).

[20] Cortese, J. Holevo-schumacher-westmoreland channel ca-pacity for a class of qudit unital channels. Phys. Rev. A69, 022302 (2004).

[21] Ralph, T. C., Resch, K. J. & Gilchrist, A. Efficient toffoligates using qudits. Phys. Rev. A 75, 022313 (2007).

[22] Lanyon, B. P. et al. Simplifying quantum logic usinghigher-dimensional hilbert spaces. Nat Phys 5, 134–140(2009).

[23] Jha, A. K., Agarwal, G. S. & Boyd, R. W. Supersensitivemeasurement of angular displacements using entangledphotons. Phys. Rev. A 83, 053829 (2011).

[24] Kaszlikowski, D., Gnacinski, P., Zukowski, M., Mik-laszewski, W. & Zeilinger, A. Violations of local realismby two entangled N -dimensional systems are strongerthan for two qubits. Phys. Rev. Lett. 85, 4418–4421(2000).

[25] Collins, D., Gisin, N., Linden, N., Massar, S. & Popescu,S. Bell inequalities for arbitrarily high-dimensional sys-tems. Phys. Rev. Lett. 88, 040404 (2002).

[26] Vertesi, T., Pironio, S. & Brunner, N. Closing the de-tection loophole in bell experiments using qudits. Phys.

9

Rev. Lett. 104, 060401 (2010).[27] Leach, J. et al. Violation of a bell inequality in

two-dimensional orbital angular momentum state-spaces.Opt. Express 17, 8287–8293 (2009).

[28] Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entangle-ment of the orbital angular momentum states of photons.Nature 412, 313–316 (2001).

[29] Heckenberg, N. R., McDuff, R., Smith, C. P. & White,A. G. Generation of optical phase singularities bycomputer-generated holograms. Opt. Lett. 17, 221–223(1992).

[30] Zambrini, R. & Barnett, S. M. Quasi-intrinsic angularmomentum and the measurement of its spectrum. Phys.Rev. Lett. 96, 113901 (2006).

[31] Pires, H. D. L., Woudenberg, J. & van Exter, M. P. Mea-surement of the orbital angular momentum spectrum ofpartially coherent beams. Opt. Lett. 35, 889–891 (2010).

[32] Di Lorenzo Pires, H., Florijn, H. C. B. & van Exter,M. P. Measurement of the spiral spectrum of entangledtwo-photon states. Phys. Rev. Lett. 104, 020505 (2010).

[33] Jha, A. K., Agarwal, G. S. & Boyd, R. W. Partial angularcoherence and the angular schmidt spectrum of entangledtwo-photon fields. Phys. Rev. A 84, 063847 (2011).

[34] Malik, M., Murugkar, S., Leach, J. & Boyd, R. W. Mea-surement of the orbital-angular-momentum spectrumof fields with partial angular coherence using double-angular-slit interference. Phys. Rev. A 86, 063806 (2012).

[35] Vasnetsov, M., Torres, J., Petrov, D. & Torner, L. Ob-servation of the orbital angular momentum spectrum ofa light beam. Optics letters 28, 2285–2287 (2003).

[36] Zhou, H. et al. Orbital angular momentum complex spec-trum analyzer for vortex light based on the rotationaldoppler effect. Light: Science & Applications 6, e16251(2017).

[37] Leach, J., Padgett, M. J., Barnett, S. M., Franke-Arnold,S. & Courtial, J. Measuring the orbital angular momen-tum of a single photon. Phys. Rev. Lett. 88, 257901(2002).

[38] Qassim, H. et al. Limitations to the determination of alaguerre–gauss spectrum via projective, phase-flattening

measurement. JOSA B 31, A20–A23 (2014).[39] Jha, A. K. et al. Angular two-photon interference and

angular two-qubit states. Phys. Rev. Lett. 104, 010501(2010).

[40] Mandel, L. & Wolf, E. Optical coherence and quantumoptics (Cambridge university press, 1995).

[41] Torres, J. P., Alexandrescu, A. & Torner, L. Quantumspiral bandwidth of entangled two-photon states. Phys.Rev. A 68, 050301 (2003).

[42] Arrizon, V., Ruiz, U., Carrada, R. & Gonzalez, L. A.Pixelated phase computer holograms for the accurate en-coding of scalar complex fields. J. Opt. Soc. Am. A 24,3500–3507 (2007).

[43] Law, C. K. & Eberly, J. H. Analysis and interpretation ofhigh transverse entanglement in optical parametric downconversion. Phys. Rev. Lett. 92, 127903 (2004).

[44] Peeters, W. H., Verstegen, E. J. K. & van Exter, M. P.Orbital angular momentum analysis of high-dimensionalentanglement. Phys. Rev. A 76, 042302 (2007).

[45] Giovannini, D. et al. Determining the dimensionality ofbipartite orbital-angular-momentum entanglement usingmulti-sector phase masks. New Journal of Physics 14,073046 (2012).

[46] Chan, K. W., Torres, J. P. & Eberly, J. H. Transverseentanglement migration in hilbert space. Phys. Rev. A75, 050101 (2007).

[47] Just, F., Cavanna, A., Chekhova, M. V. & Leuchs, G.Transverse entanglement of biphotons. New Journal ofPhysics 15, 083015 (2013).

[48] Fienup, J. R. Reconstruction of a complex-valued objectfrom the modulus of its fourier transform using a supportconstraint. JOSA A 4, 118–123 (1987).

[49] Gaur, C., Mohan, B. & Khare, K. Sparsity-assisted solu-tion to the twin image problem in phase retrieval. JOSAA 32, 1922–1927 (2015).

[50] Waller, L., Situ, G. & Fleischer, J. W. Phase-space mea-surement and coherence synthesis of optical beams. Na-ture Photonics 6, 474–479 (2012).