Measuring the complex orbital angular momentum spectrum and spatial modedecomposition of structured light beams

Alessio D’Errico,1 Raffaele D’Amelio,1 Bruno Piccirillo,1 Filippo Cardano,1, ∗ and Lorenzo Marrucci1, 2

1Dipartimento di Fisica, Universita di Napoli Federico II,Complesso Universitario di Monte Sant’Angelo, Via Cintia, 80126 Napoli, Italy

2CNR-ISASI, Institute of Applied Science and Intelligent Systems, Via Campi Flegrei 34, Pozzuoli (NA), Italy

Light beams carrying orbital angular momentum are key resources in modern photonics. In manyapplications, the ability of measuring the complex spectrum of structured light beams in terms ofthese fundamental modes is crucial. Here we propose and experimentally validate a simple methodthat achieves this goal by digital analysis of the interference pattern formed by the light beamand a reference field. Our approach allows one to characterize the beam radial distribution also,hence retrieving the entire information contained in the optical field. Setup simplicity and reducednumber of measurements could make this approach practical and convenient for the characterizationof structured light fields.

In 1992 Allen et al. [1] showed that helical modes oflight – paraxial beams featuring an helical phase factoreimφ, where φ is the azimuthal angle around the beamaxis and m is an integer – carry a definite amount of or-bital angular momentum (OAM) along the propagationaxis, equal to m~ per photon [2]. Important realizationsof such optical modes are, for example, Laguerre-Gauss(LG) [1] and Hypergeometric Gaussian beams (HyGG)[3], which share the typical twisted wavefront but differ intheir radial profiles. Controlled superpositions of helicalmodes, possibly combined with orthogonal polarizationstates via spin-orbit interaction [4, 5], result in spatiallystructured beams that are proving useful for a broad setof photonic applications [6], such as for instance clas-sical and quantum optical communication [7–10], quan-tum information processing [11–13], and quantum sim-ulations [14, 15]. The ability to ascertain experimen-tally the OAM values associated with individual helicalmodes represents a fundamental requirement for all ap-plications based on twisted light. Hitherto, this has beendemonstrated by a variety of methods: exploiting doubleslit interference [16], diffraction through single apertures[17–20] or through arrays of pinholes [21], interferencewith a reference wave [22, 23], interferometers [24–26],OAM-dependent Doppler frequency shifts [27–29], phaseflattening and spatial mode projection using pitchforkholograms [30–32], q-plates [33, 34], spiral phase plates[35] and volume holograms [36], spatial sorting of helicalmodes by mapping OAM states into transverse momen-tum (i.e. propagation direction) [37, 38], quantum weakmeasurements [39].General structured fields are however not given by indi-vidual helical modes, but can always be obtained as suit-able superpositions of multiple helical modes. Accord-ingly, a full experimental characterization of these struc-tured fields can be based on measuring the complex coef-ficients (amplitude and phase) associated with each modeappearing in the superposition, for any given choice of the

∗ filippo.cardano2@unina.it

mode basis. In general, this is not a trivial task, but sev-eral methods for the reconstruction of the complex spec-trum associated with the OAM degree of freedom havebeen demonstrated thus far [24, 28, 29, 33, 35, 36, 40, 41],possibly including also the radial mode spectrum recon-struction [31, 32, 37–39, 42–44]. It is worth noting that,once these complex coefficients are known, the completespatial distribution of the electric field can be obtainedand important properties such as beam quality factorM2, beam width and wavefront are easily computed atany propagation distance [40, 45]. Inspired by previousworks [28, 29, 46] introducing Fourier analysis in thiscontext, here we present an approach to the measure-ment of light OAM spectrum and, more generally, tospatial mode decomposition of structured light that mayprove to be more practical than most alternatives. TheOAM complex spectrum information is contained in theintensity pattern resulting from the interference of thelight beam with a known reference field (such as a Gaus-sian beam), and can be hence easily extracted by a suit-able processing of the corresponding images recorded ona camera. First, Fourier transform with respect to theazimuthal angle leads to determining the complex coef-ficients associated with each OAM value, as a functionof the radial coordinate. Numerical integration over thelatter then allows one to use this information to deter-mine the OAM power spectrum and, eventually, to de-compose each OAM component in terms of radial modes,e.g. LG beams. Remarkably, the whole information as-sociated with the spatial mode decomposition, or withthe OAM power spectrum, is contained in a few images,whose number does not scale with the dimensionality ofthe set of detected helical modes. A unique series ofdata recorded for the characterization of a given field isused for obtaining the decomposition in any basis of spa-tial modes carrying OAM (LG, HyGG, Bessel,...), as thischoice comes into play only at the stage of image analysis.

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RESULTS AND DISCUSSION

Description of the technique. In the following, welimit our attention to the case of scalar optics, as theextension to the full vector field is simply obtained byapplying the same analysis to two orthogonal polariza-tion components. Considering cylindrical coordinates(r, φ, z), the electric field amplitude associated with amonochromatic paraxial beam propagating along the zdirection is given by:

Es(r, φ, z, t) = As(r, φ, z)e−i(ωt−k z), (1)

where ω is the optical frequency and k is the wave num-ber. We refer to Es as the signal field, to distinguish itfrom the reference beam that will be introduced later on.The information concerning the spatial distribution ofthe field is contained in the complex envelope As(r, φ, z).Being periodic with respect to the azimuthal coordinateφ, such complex function can be expanded into a sum offundamental helical modes eimφ, carrying m~ OAM perphoton along the z axis [2]:

As(r, φ, z) =

∞∑m=−∞

cm(r, z)eimφ, (2)

where coefficients cm are defined in terms of the angularFourier transform

cm(r, z) =1

2π

∫ 2π

0

dφ e−imφAs(r, φ, z). (3)

The probability P (m) that a photon is found in the m-order OAM state is obtained from the coefficients cmby integrating their squared modulus along the radialcoordinate:

P (m) =1

S

∫ ∞0

dr r |cm(r, z)|2, (4)

where S =∑m

∫∞0dr r |cm(r, z)|2 is the beam power at

any transverse plane. The quantity P (m) is also referredto as the OAM power spectrum, or spiral spectrum ofthe beam, and does not depend on the longitudinal co-ordinate z, because of OAM conservation during prop-agation. A complete analysis of the field in terms oftransverse spatial modes is obtained by replacing eimφ inEq. 2 with a complete set of modes having a well definedradial dependance, e.g. LG modes:

A(r, φ, z) =

∞∑p=0

∞∑m=−∞

bp,m LGp,m(r, φ, z), (5)

where LGp,m(r, φ, z) is the complete LG mode of integerindices p and m (with p ≥ 0), as explicitly defined in theMethods. The link between coefficients cm and bp,m isthen given by:

bp,m =

∫ ∞0

r dr LG∗p,m(r, z) cm(r, z), (6)

where we introduced the radial LG amplitudesLGp,m(r, z) = LGp,m(r, φ, z)e−imφ, for which the φ de-pendence is removed.

The procedure we present here allows one to measurethe complex quantities cm(r), or equivalently the coef-ficients bp,m. We achieve this goal by letting the sig-nal optical field interfere with a reference wave Eref =Aref(r, φ, z)e

−i(ωt−kz), having the same polarization, fre-quency, wavelength and optical axis of the beam underinvestigation, and whose spatial distribution is known.The simplest choice for this reference is a Gaussian beam.At any plane transverse to the propagation direction, theintensity pattern I formed by the superposition of signaland reference beams is (we omit the functional depen-dance on the spatial coordinates)

I =Is + Iref + Iα. (7)

Here Is and Iref are the intensities corresponding to thesole signal and reference fields, respectively, while the

term Iα = 2 Re(eiαAsA∗ref) corresponds to their interfer-

ence modulation pattern, α being a controllable opticalphase between the two. The interference modulation pat-tern can be experimentally singled out by taking threeimages, namely I, Iref (blocking the signal beam) and Is(blocking the reference beam), and then calculating the

difference Iα = I − Iref − Is.The interference modulation pattern is linked to the

OAM mode decomposition by the following expression:

Iα = 2∑m

|Aref||cm| cos [mφ+ α+ βm], (8)

where βm(r, z) = Arg[cm(r, z)]−Arg[Aref(r, z)]. By com-bining two interference patterns obtained with α = 0 andα = π/2 one then gets:

I0 − i Iπ/2 = 2∑m

|Aref||cm| ei[mφ+βm] (9)

Finally, Fourier analysis with respect to the azimuthalcoordinate allows one to determine the coefficients cm(r):

cm(r, z) =1

4πA∗ref(r, z)

∫ 2π

0

dφ(I0 − i Iπ/2)e−imφ, (10)

which contains all the information associated with thespatial distribution of the electric field.

The method just described is required for a full modaldecomposition and requires taking a total of four images(that is I with α = 0 and α = π/2, plus Iref and Is),maintaining also a good interferometric stability betweenthem. However, for applications requiring the measure-ment of the OAM power spectrum only, that is ignoringthe radial structure of the field, and for which the OAMspectrum is bound from below (that is, there is a mini-mum OAM value) there is a simplified procedure that iseven easier and more robust (the case for which the spec-trum is limited from above can be treated equivalently).

3

This usually requires to have the signal beam pass firstthrough a spiral optical phase element, described by thetransfer factor eiMφ (this can be achieved with a q-plateor a spiral phase plate with the appropriate topologicalcharge). If M is higher than the absolute value of theminimum negative OAM component of the signal beam,the spiral spectrum of the beam after this optical com-ponent will contain only modes associated with positiveOAM values. Then, one can extract the associated prob-

abilities P (m) by Fourier analysis of I0 only (see Eq. 8),

with no need of measuring also Iπ/2, thus reducing thenumber of required images to three. We discuss this indetail in the final part of the paper, showing also thatsuch procedure is intrinsically less sensitive to noise cor-responding to beam imperfections.

FIG. 1. Sketch of the experimental apparatus. a) A He-Nelaser beam passes through a polarizer (P) and is spatiallycleaned and collimated by means of an objective (Ob), a pin-hole (ph) and a lens (L). A half-wave plate (HWP) and apolarizing beam splitter (PBS) are used in order to split thebeam into the signal and reference arms, whose relative in-tensities can be controlled by HWP rotation. Fields resultingfrom a complex superposition of multiple helical modes wereobtained by using q-plates and quarter-wave plates (QWPs),as shown in panels b-c. After preparing the signal field, weplace a further sequence of a QWP and a q-plate in case weneed to shift the entire OAM spectrum. The reference fieldis a TEM0,0 Gaussian mode. In the upper arm of the inter-ferometer, by orienting the QWP at 0 or 90◦ with respect tothe beam polarization we can introduce a α = 0 or π/2 phasedelay between the signal and the reference field, respectively.The two beams are superimposed at the exit of a beam split-ter (BS) and filtered through a polarizer, so that they sharethe same polarization state. The emerging intensity patternis recorded on a CCD camera (with resolution 576 × 668).b) A QWP oriented at 45◦ or 0, followed by q-plate withq = 4 and δ = π or δ = π/2, is used for the generation of alight beam containing a single mode (m = 8) or three modes(m = −8, 0, 8), respectively. c) two q-plates with q = 1 andq = 1/2 are aligned to generate spectra with m ∈ [−3, 3]. d)A set of more complex distributions was obtained by displac-ing laterally the centre of a q-plate (q = 1 and δ = π) withrespect to the axis of the impinging Gaussian beam.

Experimental results. We demonstrate the validityof our technique by determining the OAM spectrum and

FIG. 2. Experimental reconstruction of light OAM spectrum.We report the experimental characterization of optical fieldscontaining one (a-b-c) and three (d-e-f) helical modes, gener-ated using a q-plate with q = 4 and δ = π or π/2, respectively.Panels a and d show the OAM distributions in the two cases.Error bars are calculated as three times the standard error.Panels b,c and e,f show the measured amplitude and phaseprofiles of the non-vanishing helical modes that are presentin the beam, where blu, red and green coloured points areassociated with modes with m = 0, 8,−8, respectively. Theseresults are compared with theoretical simulations, representedas continuous curves with the same color scheme adopted forthe experimental results. For each value of m, we plot nor-malized coefficients cm = cm/Sm, where Sm is the total powerassociated with the helical mode. As expected from theory, afraction of the beam is left in the fundamental Gaussian state,while an equal amount of light is converted into helical modeswith m = ±8, both having the radial profile of a HyGG−8,8

mode. Simulated profiles of Gaussian and HyGG modes cor-respond to w0 = 1.45 mm and z = 30 cm, the latter beingthe distance between the q-plate and the camera. Error barsare smaller than experimental points.

the radial profile of the associated helical modes for a setof structured light fields. The setup is shown in Fig.1 anddescribed in detail in the figure caption. Here, structuredlight containing multiple OAM components is generatedby means of q-plates, consisting essentially in a thin layerof liquid crystals whose local optic axes are arranged in asingular pattern, characterized by a topological charge q[47, 48]. The way such device modifies the spatial proper-ties of a light beam is described in detail in the Methodssection.

In Figs. 2a-c and 2d-f we report the results of our firstexperiment, consisting in the measurement of both am-plitude and phase of coefficients cm(r) of optical fieldshaving one (m = 8) and three (m = {−8, 0, 8}) differ-

4

ent helical modes, respectively, accompanied by the as-sociated OAM power spectrum (see Eq. 4). We generatesuch structured light by shining a q-plate with q = 4 witha left-circularly (horizontally) polarized Gaussian beamand setting the plate optical retardation δ to the value π(π/2), respectively (see the Methods). Our data nicelyfollow the results from our simulations, with some minordeviations that are due to imperfections in the prepara-tion of the structured fields. In particular, in panel a,the small peaks centered around m = −8 are related tothe possible ellipticity of the polarization of the Gaussianbeam impinging on the q-plate, while a small contribu-tion at m = 0 corresponds to the tiny fraction of the in-put beam that has not been converted by the q-plate. Inpanels b and e the radial profiles used for our simulationsare those corresponding to the Hypergeometric-Gaussianmodes [3], the helical modes that are expected to de-scribe the optical field at the exit of the q-plate [49] (seethe Methods for details). Error bars shown in our plotsare those associated with the variability in selecting thecorrect center r = 0 in the experimental images, which isidentified as one of the main source of uncertainties in thespectral results. They are estimated as three times thestandard deviation of the data computed after repeatingour analysis with the coordinate origin set in one of 25pixels that surround our optimal choice. Other possiblesystematic errors, such as for example slight misalign-ments between the signal and reference fields, are notconsidered here.

Data reported in Fig. 2 prove our ability to measure thecomplex radial distribution of the field associated withindividual helical modes in a superposition. For each ofthese, we can use our results to obtain a decompositionin terms of a complete set of modes. For a demonstra-tion of this concept, we consider the field obtained whena left-circularly-polarized beam passes through a q-platewith q = 4 and δ = π. The latter contains only a modewith m = 8, as shown in Fig. 2a-c. By evaluating theintegrals reported in Eq. 6 we determine the coefficientsbp,8 of a LG decomposition. For our analysis, we useLG beams with an optimal waist parameter w0 (differentfrom the one of the impinging Gaussian beam), definedso that the probability of the lowest radial index p = 0is maximal [50]. In Fig. 3 we plot squared modulus andphase of the coefficients bp,8 determined experimentally,matching nicely the results obtained from numerical sim-ulations.

Shifting the OAM power spectrum. As mentionedabove, shifting the OAM spectrum of the signal field maybe used to simplify its measurement, when reconstruct-ing the radial profile is not needed. In our case, we letthe signal field pass through a q-plate with q = M/2and δ = π, after preparing it in a state of left-circularpolarization. If M is large enough, i.e. higher than thesmallest OAM component of the signal field, we havethat c(m) 6= 0 only if m > 0. This allows in turn us-ing Eq. 8 to determine the OAM spectrum, instead ofEq. 9 that requires the measurement of Iπ/2 also. At

FIG. 3. Complete spatial mode decomposition in terms of LGbeams. We consider the light beam emerging from a q-platewith (q = 4, δ = π), described by a HyGG−8,8 mode [49].We evaluate the overlap integral between the radial envelopec8(r) measured in our experiment at z = 30 cm and LGp,8

modes at the same value of z and characterized by the opti-mal beam waist w0 = w0/9 [50], where w0 is the input beamwaist. In panels a) and b) we plot the squared modulus andthe phase of the resulting coefficients (blue markers), respec-tively, showing a good agreement with the values obtainedfrom numerical simulations (red markers). The phase corre-sponding to b1,8 is absent in the plot since the correspondingcoefficient amplitude is vanishing.

the same time, this approach is less sensitive to possi-ble noise due to beam imperfections, typically associatedwith small spatial frequencies and affecting lowest-orderhelical modes, as reported also in Refs. [28, 29]. Let usnote that once the beam passes through an optical el-ement adding the azimuthal phase eiMφ, thanks to theconservation of OAM during free propagation, the asso-ciated power OAM spectrum is only shifted by M units,that is P (m)→ P (m+M). The radial distribution of in-dividual helical modes, on the other hand, is altered dur-ing propagation, that is cm(r, z) 6→ cm+M (r, z). For thisreason, this alternative procedure proves convenient onlywhen determining the OAM probability distribution butcannot be applied to the reconstruction of the full modaldecomposition. In Fig. 4, we report the measured powerspectrum of different fields containing helical modes withm ∈ [−3, 3] (see the Methods and the figure caption forfurther details on the generation of such complex fields),as determined by shifting the OAM spectrum by M = 8by means of a q-plate with q = 4 and δ = π.

OAM spectra for displaced q-plates. As a finaltest, we used our technique for characterizing more com-plex optical fields, such as those emerging from a q-platewhose central singularity is displaced with respect to theinput Gaussian beam axis (Fig.1c). In Fig. 5 we reportthe OAM probability distributions obtained when trans-lating a q-plate (q = 1, δ = π) in a direction that isparallel to the optical table, with steps of ∆x = 0.125mm. Our data are in excellent agreement with resultsobtained from numerical simulations. In the same figurewe show part of the associated total intensity patterns I0(see Eq. 7) recorded on the camera. In addition, for eachconfiguration we show in Fig. 5 that the first and secondorder moments of the probability distributions are char-acterized by Gaussian profiles 〈m〉 = 2q exp(−2x2

0/w20)

5

and 〈m2〉 = (2q)2 exp(−2x20/w

20) [51]. Fitting our data

so that they follow the expected Gaussian distributions(red curves) we obtain wfit

0 = 1.36 ± 0.04 mm from 〈m〉(panel g), and wfit

0 = 1.39 ± 0.06 mm from 〈m2〉 (panelh), which are close to the expected value w0 = 1.45±0.18mm.

Range of detectable helical modes. Finite size ofthe detector area and the camera resolution impose nat-ural limitations to our approach, that cannot be used tocharacterize helical modes with arbitrary values of m andradial profiles. Starting from these considerations, in theMethods we describe how to evaluate the bandwidth ofdetectable LG modes in terms of sensor area and reso-lution, and provide an explicit example for our specificconfiguration (associated data are reported in Fig. 6).

CONCLUSIONS

In this study we introduced a new technique for measur-ing the orbital angular momentum spectrum of a laserbeam, accompanied by its complete spatial mode decom-position in terms of an arbitrary set of modes that carrya definite amount of OAM, such as LG beams or others.Based on the azimuthal Fourier analysis of the interfer-ence pattern formed by the signal and the reference field,relying on only a few measurements this approach allowsone to readily extract the information contained in boththe radial and azimuthal degrees of freedom of a struc-tured light beam. The most general method requirestaking four images, including the intensity patterns ofthe signal beam, the reference beam and two interfer-ence patterns between them. Information on the modaldecomposition of the signal field is then retrieved usinga simple dedicated software. Since the spatial mode de-composition is obtained during this post-processing pro-cedure, the same set of images can be used to decomposea beam in different sets of spatial modes. As demon-strated here, the experimental implementation of our ap-

FIG. 4. Measure of shifted OAM power spectrum. OAMprobability distributions are measured for two different opti-cal fields, obtained when shining a sequence of two q-plateswith q1 = 1 and q2 = 1/2. A further q-plate with q = 4 shiftsthe final spectrum by M = 8 units. a) OAM spectrum for thecase δ1 = π and δ2 = π. b) The same data are reported fora different field, obtained when δ1 = π and δ2 = π/2. Errorbars represent the standard error multiplied by three.

FIG. 5. OAM spectrum for a shifted q-plate. We measure theOAM power spectrum at the exit of a q-plate (q = 1, δ = π)shifted with respect to the axis of the impinging Gaussianbeam, which is left-circularly polarized. The overall spectrumis shifted by M = 8 units since we are using a further q-platewith q = 4 and δ = π. However, we plot the original OAMdistribution associated with the signal field. a-f) Experimen-tal (green) and simulated (red) OAM power spectra when thelateral shift is equal to a∆x, with a = 1, 3, 6, 9, 12, 15 and ∆x=0,125 mm, respectively. Error bars represent the standarderror multiplied by three. g-i) Examples of the experimentalintensity pattern I0 used for determining the power spectrareported in panels a,c,e. The number of azimuthal fringes re-veals that the OAM spectrum has been shifted. j- k) Firstand second moment (〈m〉 and 〈m2〉) measured as a functionof the lateral displacement. Error bars are not visible becausesmaller than the experimental points.

proach requires a simple interferometric scheme and min-imal equipment. Hence, it may be readily extended tocurrent experiments dealing with the characterization ofspatial properties and OAM decomposition of structuredlight.

METHODS

A. Spatial modes carrying OAM

Using adimensional cylindrical coordinates ρ = r/w0 andζ = z/zR, where w0 is the waist radius of the Gaus-sian envelope and zR the Rayleygh range, respectively,Laguerre-Gaussian LGp,m modes have the well known

6

FIG. 6. Detectable LG modes. Different colors, as reportedin the legend, indicate whether a specific LGp,m, in a trans-verse plane z = 30 cm and with a beam waist w0 = 0.16 mm[w(z = 30 cm)→ 0.4 mm] can be resolved in our setup. Theseparameters correspond to the ones used for the complete spa-tial decomposition of the HyGG beam generated by a q-plate(q = 4, δ = π) in terms of LG beams.

expression:

LGp,m(ρ, ζ, φ) =

√2|m|+1p!

π(p+ |m|)!(1 + ζ2)

(ρ√

1 + ζ2

)|m|×

e− ρ2

1+ζ2 L|m|p (2ρ2/(1 + ζ2))×

eiρ2

ζ+1/ζ eimφ−i(2p+|m|+1) arctan(ζ),

(11)

where L|m|p (x) is the generalized Laguerre polynomial, p

is a positive integer and m is the azimuthal index asso-ciated with the OAM.

When a Gaussian beam passes through an optical el-ement that impinges on it a phase factor eimφ, the out-going field is described by a Hypergeometric-Gaussianmode HyGGp,m [3, 49] with p = −|m|:

HyGGp,m(ρ, ζ, φ) =

√21+|m|+p

πΓ(1 + |m|+ p)

Γ(1 + |m|+ p/2)

Γ(1 + |m|)×

i|m|+1ζp/2(ζ + i)−(1+|m|+p/2)×

ρ|m|e−iρ2/(ζ+i)+imφ×

1F1

(− p/2; |m|+ 1; ρ2/(ζ(ζ + i))

),

(12)

where Γ(z) is the Euler Gamma function and 1F1(a; b; z)is the confluent Hypergeometric function.

B. Generating structured light using q-plates

A q-plate is formed by a thin layer of liquid crystals; theangle α describing the orientation of the optic axis of suchmolecules is a linear function of the azimuthal angle, thatis α(φ) = α0 + qφ (q is the topological charge). In ourexperiments we set α0 = 0. In this case, the action of

the q-plate is described by the following Jones matrix (inthe basis of circular polarizations):

Q(δ) = cos

(δ

2

)(1 00 1

)+

i sin

(δ

2

)(0 e−imφ

eimφ 0

),

(13)

where m = 2q and δ is the plate optical retardation,controllable by applying an external electric field [52]. Itis worth noting that the second term of Eq. 13 introducesthe azimuthal dependance associated with the OAMdegree of freedom. When a left or right circularlypolarized Gaussian beam passes through a q-plate withδ = π, positioned at the waist of the beam, the outputbeam is given by HyGG−|m|,±m, respectively [49]. In ourexperiment we generated superpositions of several OAMmodes using single or cascaded q-plates, characterizedby specific values of q and δ that are reported in thefigure captions.

C. Limitations on the set of detectable spatialmodes.

We briefly discussed in the main text that the finite sizeof the detector area and the finite dimension of sensorpixels impose certain restrictions on the features of thehelical modes that can be resolved in our setup. Let usconsider the simple case wherein we want to decomposethe signal field in terms of LGp,m modes, and we wantto evaluate the p,m-bandwidth of detectable modes. Weconsider only the case m > 0, since only the absolutevalue |m| is relevant to our discussion. Consider a camerawith N × N pixels, with pixel dimensions d × d (in oursetup N = 576 and d = 9 µm). We define the followingquantities:

rmax = dN/2, (14)

rmin = md/π, (15)

r1 = w(z)×√2p+m− 2− [1 + 4(p− 1)(p+m− 1)]1/2

2, (16)

rp = w(z)×√2p+m− 2 + [1 + 4(p− 1)(p+m− 1)]1/2

2, (17)

rp = w(z)√

2p+m+ 1. (18)

Here rmax is the maximum radius available on the sensor;rmin is the minimum radial distance where azimuthal os-cillation associated with the OAM content of the LGp,m

mode can be detected, before facing aliasing issues; r1

is a lower bound for the first root of the Laguerre poly-nomials contained in the expression of LG modes; sim-ilarly, rp is the upper bound for the p−th root, whilerp, with rp < rp, delimits the oscillatory region of theLaguerre polynomials [53, 54]. Interestingly, the spatial

7

region r1 < r < rp well approximates the area contain-ing all the power associated with the mode. At the sametime, the quantity Λ = (rp − r1)/p well describes theaverage distance between consecutive nodes of the LGmode, defining the periodicity of their radial oscillations.A given LGp,m mode is then “detectable” (or properly“resolvable”) if all the following conditions are satisfied: rmin < r1 (i)

rp < rmax (ii)Λ > 2d (iii)

(19)

Indeed, we are requiring that (i) the field is vanishing be-low the azimuthal aliasing threshold given by rmin, that(ii) all the power associated with the mode is contained inthe sensor area and (iii) that the field radial oscillationshave a spatial period such that at least two pixels arecontained in a single period, respectively (radial aliasinglimit). It is easy to check that in our configuration, wherethe beam waist is w(z) = 0.4 mm, conditions (i) and (iii)are always satisfied for the values of {p,m} that are solu-tion of (ii), i.e. the limiting factor is only the dimensionof the sensor area. By solving such inequality, we get therelation

p <

(N2d2

4w2(z)−m− 1

)/2 (20)

In Fig. 6 we plot a colormap for a rapid visualization ofdetectable modes. If we apply this analysis to the case ofFig. 3, in which a beam with m = 8 is studied, we obtainthat only radial modes with p < 16 can be detected. Ingeneral, for smaller values of w(z) the determination ofdetectable LG mode is more complex and requires thecomplete resolution of the system of inequalities systemgiven in (19).

FUNDING INFORMATION

This work was supported by the European ResearchCouncil (ERC), under grant no.694683 (PHOSPhOR).

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