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

TL;DR

This paper introduces a simple, experimentally validated method for measuring the complex orbital angular momentum spectrum and spatial mode decomposition of structured light beams using digital interference analysis, enabling comprehensive characterization.

Contribution

The authors present a novel, practical approach for fully characterizing structured light beams' orbital angular momentum and spatial modes through digital interference analysis.

Findings

Successfully measured complex OAM spectrum and spatial modes

Method reduces measurement complexity and number of measurements

Validated approach with experimental results

Abstract

Light beams carrying orbital angular momentum are key resources in modern photonics. In many applications, the ability of measuring the complex spectrum of structured light beams in terms of these fundamental modes is crucial. Here we propose and experimentally validate a simple method that achieves this goal by digital analysis of the interference pattern formed by the light beam and 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 reduced number of measurements could make this approach practical and convenient for the characterization of structured light fields.

Equations

$$E_s(r,\varphi ,z,t)=A_s(r,\varphi ,z)e^{-i(\omega t-kz)},$$

$$A_s(r,\varphi ,z)=\sum _{m=-\infty }^{\infty }{c}_m(r,z)e^{im\varphi },$$

$$c_m(r,z)=\frac{1}{2\pi }{\int }_0^{2\pi }d\varphi e^{-im\varphi }{A}_s(r,\varphi ,z).$$

$$P(m)=\frac{1}{S}{\int }_0^{\infty }drr|c_m(r,z){|}^2,$$

$$A(r,\varphi ,z)=\sum _{p=0}^{\infty }\sum _{m=-\infty }^{\infty }{b}_{p,m}{\text{LG}}_{p,m}(r,\varphi ,z),$$

$$b_{p,m}={\int }_0^{\infty }rdr{\text{LG}}_{p,m}^{\ast }(r,z)c_m(r,z),$$

$$I=$$

$${\tilde{I}}_{\alpha }=2\sum _m|A_{\text{ref}}||c_m|\cos [m\varphi +\alpha +{\beta }_m],$$

$${\tilde{I}}_0-i{\tilde{I}}_{\pi /2}=2\sum _m|A_{\text{ref}}||c_m|e^{i[m\varphi +{\beta }_m]}$$

$$c_m(r,z)=\frac{1}{4\pi A_{\text{ref}}^{\ast }(r,z)}{\int }_0^{2\pi }d\varphi ({\tilde{I}}_0-i{\tilde{I}}_{\pi /2})e^{-im\varphi },$$

$$\begin{split}LG_{p,m}(\rho,\zeta,\phi)=&\sqrt{\frac{2^{|m|+1}p!}{\pi(p+|m|)!(1+\zeta^{2})}}\Biggr{(}\frac{\rho}{\sqrt{1+\zeta^{2}}}\Biggr{)}^{|m|}\times\\ &e^{-\frac{\rho^{2}}{1+\zeta^{2}}}L_{p}^{|m|}(2\rho^{2}/(1+\zeta^{2}))\times\\ &e^{i\frac{\rho^{2}}{\zeta+1/\zeta}}e^{im\phi-i(2p+|m|+1)\arctan(\zeta)},\end{split}$$

$$\begin{split}\hbox{HyGG}_{p,m}(\rho,\zeta,\phi)=&\sqrt{\frac{2^{1+|m|+p}}{\pi\Gamma(1+|m|+p)}}\frac{\Gamma(1+|m|+p/2)}{\Gamma(1+|m|)}\times\\ &i^{|m|+1}\zeta^{p/2}(\zeta+i)^{-(1+|m|+p/2)}\times\\ &\rho^{|m|}e^{-i\rho^{2}/(\zeta+i)+im\phi}\times\\ &{}_{1}F_{1}\biggr{(}-p/2;|m|+1;\rho^{2}/(\zeta(\zeta+i))\biggr{)},\end{split}$$

$$\begin{split}\hat{Q}(\delta)=&\cos\biggr{(}\frac{\delta}{2}\biggr{)}\begin{pmatrix}1&0\\ 0&1\end{pmatrix}+\\ &i\sin\biggr{(}\frac{\delta}{2}\biggr{)}\begin{pmatrix}0&e^{-im\phi}\\ e^{im\phi}&0\end{pmatrix},\end{split}$$

$$r_{\text{max}}$$

$$r_{\text{min}}$$

$$r_1$$

$$\sqrt{\frac{2p+m-2-[1+4(p-1)(p+m-1){]}^{1/2}}{2}},$$

$$r_p$$

$$\sqrt{\frac{2p+m-2+[1+4(p-1)(p+m-1){]}^{1/2}}{2}},$$

$${\tilde{r}}_p$$

$$\left\{\begin{array}[]{ll}r_{\text{min}}<r_{1}&\text{(i)}\\ r_{p}<r_{\text{max}}&\text{(ii)}\\ \Lambda>2d&\text{(iii)}\end{array}\right.$$

$$p<\left(\frac{N^2{d}^2}{4w^2(z)}-m-1\right)/2$$

Full text

In 1992 Allen et al. Allen et al. (1992) showed that helical modes of light – paraxial beams featuring an helical phase factor $e^{im\phi}$, where $\phi$ is the azimuthal angle around the beam axis and $m$ is an integer – carry a definite amount of orbital angular momentum (OAM) along the propagation axis, equal to $m\hbar$ per photon Yao and Padgett (2011). Important realizations of such optical modes are, for example, Laguerre-Gauss (LG) Allen et al. (1992) and Hypergeometric Gaussian beams (HyGG) Karimi et al. (2007), which share the typical twisted wavefront but differ in their radial profiles. Controlled superpositions of helical modes, possibly combined with orthogonal polarization states via spin-orbit interaction Bliokh et al. (2015); Cardano and Marrucci (2015), result in spatially structured beams that are proving useful for a broad set of photonic applications Rubinsztein-Dunlop et al. (2017), such as for instance classical and quantum optical communication Vallone et al. (2014); Willner et al. (2015); Mirhosseini et al. (2015); Bouchard et al. (2017), quantum information processing Wang et al. (2015); Malik et al. (2016); Hiesmayr et al. (2016), and quantum simulations Cardano et al. (2016, 2017). The ability to ascertain experimentally the OAM values associated with individual helical modes represents a fundamental requirement for all applications based on twisted light. Hitherto, this has been demonstrated by a variety of methods: exploiting double slit interference Sztul and Alfano (2006), diffraction through single apertures Hickmann et al. (2010); Ferreira et al. (2011); Mourka et al. (2011); Mazilu et al. (2012) or through arrays of pinholes Berkhout and Beijersbergen (2008), interference with a reference wave Harris et al. (1994); Padgett et al. (1996), interferometers Leach et al. (2002); Slussarenko et al. (2010); Lavery et al. (2011), OAM-dependent Doppler frequency shifts Courtial et al. (1998); Vasnetsov et al. (2003); Zhou et al. (2016), phase flattening and spatial mode projection using pitchfork holograms Mair et al. (2001); Kaiser et al. (2009); Schulze et al. (2013), q-plates Karimi et al. (2009a, 2012), spiral phase plates Bierdz et al. (2013) and volume holograms Gruneisen et al. (2011), spatial sorting of helical modes by mapping OAM states into transverse momentum (i.e. propagation direction) Berkhout et al. (2010); Mirhosseini et al. (2013), quantum weak measurements Malik et al. (2014).

General structured fields are however not given by individual helical modes, but can always be obtained as suitable superpositions of multiple helical modes. Accordingly, a full experimental characterization of these structured fields can be based on measuring the complex coefficients (amplitude and phase) associated with each mode appearing in the superposition, for any given choice of the mode basis. In general, this is not a trivial task, but several methods for the reconstruction of the complex spectrum associated with the OAM degree of freedom have been demonstrated thus far Leach et al. (2002); Karimi et al. (2009a); Bierdz et al. (2013); Gruneisen et al. (2011); Vasnetsov et al. (2003); Zhou et al. (2016); Forbes et al. (2016); Zhao et al. (2017), possibly including also the radial mode spectrum reconstruction Mirhosseini et al. (2013); Berkhout et al. (2010); Malik et al. (2014); Schulze et al. (2013); Kaiser et al. (2009); Berkhout et al. (2011); Mirhosseini et al. (2016); Fickler et al. (2017). It is worth noting that, once these complex coefficients are known, the complete spatial distribution of the electric field can be obtained and important properties such as beam quality factor $M^{2}$, beam width and wavefront are easily computed at any propagation distance Schulze et al. (2012); Forbes et al. (2016). Inspired by previous works Abouraddy et al. (2011); Vasnetsov et al. (2003); Zhou et al. (2016) introducing Fourier analysis in this context, here we present an approach to the measurement of light OAM spectrum and, more generally, to spatial mode decomposition of structured light that may prove to be more practical than most alternatives. The OAM complex spectrum information is contained in the intensity pattern resulting from the interference of the light beam with a known reference field (such as a Gaussian beam), and can be hence easily extracted by a suitable processing of the corresponding images recorded on a camera. First, Fourier transform with respect to the azimuthal angle leads to determining the complex coefficients associated with each OAM value, as a function of the radial coordinate. Numerical integration over the latter then allows one to use this information to determine the OAM power spectrum and, eventually, to decompose each OAM component in terms of radial modes, e.g. LG beams. Remarkably, the whole information associated with the spatial mode decomposition, or with the OAM power spectrum, is contained in a few images, whose number does not scale with the dimensionality of the set of detected helical modes. A unique series of data recorded for the characterization of a given field is used for obtaining the decomposition in any basis of spatial modes carrying OAM (LG, HyGG, Bessel,…), as this choice comes into play only at the stage of image analysis.

Results and Discussion

Description of the technique. In the following, we limit our attention to the case of scalar optics, as the extension to the full vector field is simply obtained by applying the same analysis to two orthogonal polarization components. Considering cylindrical coordinates $(r,\phi,z)$, the electric field amplitude associated with a monochromatic paraxial beam propagating along the $z$ direction is given by:

[TABLE]

where $\omega$ is the optical frequency and $k$ is the wave number. We refer to $E_{s}$ as the signal field, to distinguish it from the reference beam that will be introduced later on. The information concerning the spatial distribution of the field is contained in the complex envelope $A_{s}(r,\phi,z)$. Being periodic with respect to the azimuthal coordinate $\phi$, such complex function can be expanded into a sum of fundamental helical modes $e^{im\phi}$, carrying $m\hbar$ OAM per photon along the $z$ axis Yao and Padgett (2011):

[TABLE]

where coefficients $c_{m}$ are defined in terms of the angular Fourier transform

[TABLE]

The probability $P(m)$ that a photon is found in the $m$-order OAM state is obtained from the coefficients $c_{m}$ by integrating their squared modulus along the radial coordinate:

[TABLE]

where $S=\sum_{m}\int_{0}^{\infty}dr\,r\,|c_{m}(r,z)|^{2}$ is the beam power at any transverse plane. The quantity $P(m)$ is also referred to as the OAM power spectrum, or spiral spectrum of the beam, and does not depend on the longitudinal coordinate $z$, because of OAM conservation during propagation. A complete analysis of the field in terms of transverse spatial modes is obtained by replacing $e^{im\phi}$ in Eq. 2 with a complete set of modes having a well defined radial dependance, e.g. LG modes:

[TABLE]

where $\text{LG}_{p,m}(r,\phi,z)$ is the complete LG mode of integer indices $p$ and $m$ (with $p\geq 0$), as explicitly defined in the Methods. The link between coefficients $c_{m}$ and $b_{p,m}$ is then given by:

[TABLE]

where we introduced the radial LG amplitudes $\text{LG}_{p,m}(r,z)=\text{LG}_{p,m}(r,\phi,z)e^{-im\phi}$, for which the $\phi$ dependence is removed.

The procedure we present here allows one to measure the complex quantities $c_{m}(r)$, or equivalently the coefficients $b_{p,m}$. We achieve this goal by letting the signal optical field interfere with a reference wave $E_{\text{ref}}=A_{\text{ref}}(r,\phi,z)e^{-i(\omega t-kz)}$, having the same polarization, frequency, wavelength and optical axis of the beam under investigation, and whose spatial distribution is known. The simplest choice for this reference is a Gaussian beam. At any plane transverse to the propagation direction, the intensity pattern $I$ formed by the superposition of signal and reference beams is (we omit the functional dependance on the spatial coordinates)

[TABLE]

Here $I_{s}$ and $I_{\text{ref}}$ are the intensities corresponding to the sole signal and reference fields, respectively, while the term $\widetilde{I}_{\alpha}=2\,\text{Re}{(e^{i\alpha}A_{s}\,A_{\text{ref}}^{*})}$ corresponds to their interference modulation pattern, $\alpha$ being a controllable optical phase between the two. The interference modulation pattern can be experimentally singled out by taking three images, namely $I$, $I_{\text{ref}}$ (blocking the signal beam) and $I_{s}$ (blocking the reference beam), and then calculating the difference $\widetilde{I}_{\alpha}=I-I_{\text{ref}}-I_{s}$.

The interference modulation pattern is linked to the OAM mode decomposition by the following expression:

[TABLE]

where $\beta_{m}(r,z)=\text{Arg}[c_{m}(r,z)]-\text{Arg}[A_{\text{ref}}(r,z)]$. By combining two interference patterns obtained with $\alpha=0$ and $\alpha=\pi/2$ one then gets:

[TABLE]

Finally, Fourier analysis with respect to the azimuthal coordinate allows one to determine the coefficients $c_{m}(r)$:

[TABLE]

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

The method just described is required for a full modal decomposition and requires taking a total of four images (that is $I$ with $\alpha=0$ and $\alpha=\pi/2$, plus $I_{\text{ref}}$ and $I_{s}$), maintaining also a good interferometric stability between them. However, for applications requiring the measurement of the OAM power spectrum only, that is ignoring the radial structure of the field, and for which the OAM spectrum is bound from below (that is, there is a minimum OAM value) there is a simplified procedure that is even easier and more robust (the case for which the spectrum is limited from above can be treated equivalently). This usually requires to have the signal beam pass first through a spiral optical phase element, described by the transfer factor $e^{iM\phi}$ (this can be achieved with a $q$-plate or a spiral phase plate with the appropriate topological charge). If $M$ is higher than the absolute value of the minimum negative OAM component of the signal beam, the spiral spectrum of the beam after this optical component will contain only modes associated with positive OAM values. Then, one can extract the associated probabilities $P(m)$ by Fourier analysis of $\widetilde{I}_{0}$ only (see Eq. 8), with no need of measuring also $\widetilde{I}_{\pi/2}$, thus reducing the number of required images to three. We discuss this in detail in the final part of the paper, showing also that such procedure is intrinsically less sensitive to noise corresponding to beam imperfections.

Experimental results. We demonstrate the validity of our technique by determining the OAM spectrum and the radial profile of the associated helical modes for a set of structured light fields. The setup is shown in Fig.1 and described in detail in the figure caption. Here, structured light containing multiple OAM components is generated by means of $q$-plates, consisting essentially in a thin layer of liquid crystals whose local optic axes are arranged in a singular pattern, characterized by a topological charge $q$ Marrucci et al. (2006, 2011). The way such device modifies the spatial properties of a light beam is described in detail in the Methods section.

In Figs. 2a-c and 2d-f we report the results of our first experiment, consisting in the measurement of both amplitude and phase of coefficients $c_{m}(r)$ of optical fields having one ($m=8$) and three ($m=\{-8,0,8\}$) different helical modes, respectively, accompanied by the associated OAM power spectrum (see Eq. 4). We generate such structured light by shining a $q$-plate with $q=4$ with a left-circularly (horizontally) polarized Gaussian beam and setting the plate optical retardation $\delta$ to the value $\pi$ ($\pi/2$), respectively (see the Methods). Our data nicely follow the results from our simulations, with some minor deviations that are due to imperfections in the preparation of the structured fields. In particular, in panel a, the small peaks centered around $m=-8$ are related to the possible ellipticity of the polarization of the Gaussian beam impinging on the $q$-plate, while a small contribution at $m=0$ corresponds to the tiny fraction of the input beam that has not been converted by the $q$-plate. In panels b and e the radial profiles used for our simulations are those corresponding to the Hypergeometric-Gaussian modes Karimi et al. (2007), the helical modes that are expected to describe the optical field at the exit of the $q$-plate Karimi et al. (2009b) (see the Methods for details). Error bars shown in our plots are those associated with the variability in selecting the correct center $r=0$ in the experimental images, which is identified as one of the main source of uncertainties in the spectral results. They are estimated as three times the standard deviation of the data computed after repeating our analysis with the coordinate origin set in one of 25 pixels that surround our optimal choice. Other possible systematic errors, such as for example slight misalignments between the signal and reference fields, are not considered here.

Data reported in Fig. 2 prove our ability to measure the complex radial distribution of the field associated with individual helical modes in a superposition. For each of these, we can use our results to obtain a decomposition in terms of a complete set of modes. For a demonstration of this concept, we consider the field obtained when a left-circularly-polarized beam passes through a $q$-plate with $q=4$ and $\delta=\pi$. The latter contains only a mode with $m=8$, as shown in Fig. 2a-c. By evaluating the integrals reported in Eq. 6 we determine the coefficients $b_{p,8}$ of a LG decomposition. For our analysis, we use LG beams with an optimal waist parameter $\widetilde{w}_{0}$ (different from the one of the impinging Gaussian beam), defined so that the probability of the lowest radial index $p=0$ is maximal Vallone (2017). In Fig. 3 we plot squared modulus and phase of the coefficients $b_{p,8}$ determined experimentally, matching nicely the results obtained from numerical simulations.

Shifting the OAM power spectrum. As mentioned above, shifting the OAM spectrum of the signal field may be used to simplify its measurement, when reconstructing the radial profile is not needed. In our case, we let the signal field pass through a $q$-plate with $q=M/2$ and $\delta=\pi$, after preparing it in a state of left-circular polarization. If $M$ is large enough, i.e. higher than the smallest OAM component of the signal field, we have that $c(m)\neq 0$ only if $m>0$. This allows in turn using Eq. 8 to determine the OAM spectrum, instead of Eq. 9 that requires the measurement of $I_{\pi/2}$ also. At the same time, this approach is less sensitive to possible noise due to beam imperfections, typically associated with small spatial frequencies and affecting lowest-order helical modes, as reported also in Refs. Vasnetsov et al. (2003); Zhou et al. (2016). Let us note that once the beam passes through an optical element adding the azimuthal phase $e^{iM\phi}$, thanks to the conservation of OAM during free propagation, the associated power OAM spectrum is only shifted by $M$ units, that is $P(m)\rightarrow P(m+M)$. The radial distribution of individual helical modes, on the other hand, is altered during propagation, that is $c_{m}(r,z)\not\rightarrow c_{m+M}(r,z)$. For this reason, this alternative procedure proves convenient only when determining the OAM probability distribution but cannot be applied to the reconstruction of the full modal decomposition. In Fig. 4, we report the measured power spectrum of different fields containing helical modes with $m\in[-3,3]$ (see the Methods and the figure caption for further details on the generation of such complex fields), as determined by shifting the OAM spectrum by $M=8$ by means of a $q$-plate with $q=4$ and $\delta=\pi$.

OAM spectra for displaced $q$-plates. As a final test, we used our technique for characterizing more complex optical fields, such as those emerging from a $q$-plate whose central singularity is displaced with respect to the input Gaussian beam axis (Fig.1c). In Fig. 5 we report the OAM probability distributions obtained when translating a $q$-plate ($q=1$, $\delta=\pi$) in a direction that is parallel to the optical table, with steps of $\Delta x=0.125$ mm. Our data are in excellent agreement with results obtained from numerical simulations. In the same figure we show part of the associated total intensity patterns $I_{0}$ (see Eq. 7) recorded on the camera. In addition, for each configuration we show in Fig. 5 that the first and second order moments of the probability distributions are characterized by Gaussian profiles $\langle m\rangle=2q\exp(-2x_{0}^{2}/w_{0}^{2})$ and $\langle m^{2}\rangle=(2q)^{2}\exp(-2x_{0}^{2}/w_{0}^{2})$ Piccirillo et al. (2015). Fitting our data so that they follow the expected Gaussian distributions (red curves) we obtain $w_{0}^{\text{fit}}=1.36\pm 0.04$ mm from $\langle m\rangle$ (panel g), and $w_{0}^{\text{fit}}=1.39\pm 0.06$ mm from $\langle m^{2}\rangle$ (panel h), which are close to the expected value $w_{0}=1.45\pm 0.18$ mm.

Range of detectable helical modes. Finite size of the detector area and the camera resolution impose natural limitations to our approach, that cannot be used to characterize helical modes with arbitrary values of $m$ and radial profiles. Starting from these considerations, in the Methods we describe how to evaluate the bandwidth of detectable LG modes in terms of sensor area and resolution, and provide an explicit example for our specific configuration (associated data are reported in Fig. 6).

Conclusions

In this study we introduced a new technique for measuring the orbital angular momentum spectrum of a laser beam, accompanied by its complete spatial mode decomposition in terms of an arbitrary set of modes that carry a definite amount of OAM, such as LG beams or others. Based on the azimuthal Fourier analysis of the interference pattern formed by the signal and the reference field, relying on only a few measurements this approach allows one to readily extract the information contained in both the radial and azimuthal degrees of freedom of a structured light beam. The most general method requires taking four images, including the intensity patterns of the signal beam, the reference beam and two interference patterns between them. Information on the modal decomposition of the signal field is then retrieved using a simple dedicated software. Since the spatial mode decomposition is obtained during this post-processing procedure, the same set of images can be used to decompose a beam in different sets of spatial modes. As demonstrated here, the experimental implementation of our approach requires a simple interferometric scheme and minimal equipment. Hence, it may be readily extended to current experiments dealing with the characterization of spatial properties and OAM decomposition of structured light.

Methods

**A. Spatial modes carrying OAM

** Using adimensional cylindrical coordinates $\rho=r/w_{0}$ and $\zeta=z/z_{R}$, where $w_{0}$ is the waist radius of the Gaussian envelope and $z_{R}$ the Rayleygh range, respectively, Laguerre-Gaussian $LG_{p,m}$ modes have the well known expression:

[TABLE]

where $L_{p}^{|m|}(x)$ is the generalized Laguerre polynomial, $p$ is a positive integer and $m$ is the azimuthal index associated with the OAM.

When a Gaussian beam passes through an optical element that impinges on it a phase factor $e^{im\phi}$, the outgoing field is described by a Hypergeometric-Gaussian mode $\text{HyGG}_{p,m}$ Karimi et al. (2007, 2009b) with $p=-|m|$:

[TABLE]

where $\Gamma(z)$ is the Euler Gamma function and ${}_{1}F_{1}(a;b;z)$ is the confluent Hypergeometric function.

**B. Generating structured light using $q$-plates

** A $q$-plate is formed by a thin layer of liquid crystals; the angle $\alpha$ describing the orientation of the optic axis of such molecules is a linear function of the azimuthal angle, that is $\alpha(\phi)=\alpha_{0}+q\phi$ ($q$ is the topological charge). In our experiments we set $\alpha_{0}=0$. In this case, the action of the $q$-plate is described by the following Jones matrix (in the basis of circular polarizations):

[TABLE]

where $m=2q$ and $\delta$ is the plate optical retardation, controllable by applying an external electric field Piccirillo et al. (2010). It is worth noting that the second term of Eq. 13 introduces the azimuthal dependance associated with the OAM degree of freedom. When a left or right circularly polarized Gaussian beam passes through a $q$-plate with $\delta=\pi$, positioned at the waist of the beam, the output beam is given by HyGG*-|m|,±m*, respectively Karimi et al. (2009b). In our experiment we generated superpositions of several OAM modes using single or cascaded $q$-plates, characterized by specific values of $q$ and $\delta$ that are reported in the figure captions.

**C. Limitations on the set of detectable spatial modes.

** We briefly discussed in the main text that the finite size of the detector area and the finite dimension of sensor pixels impose certain restrictions on the features of the helical modes that can be resolved in our setup. Let us consider the simple case wherein we want to decompose the signal field in terms of LGp,m modes, and we want to evaluate the $p,m$-bandwidth of detectable modes. We consider only the case $m>0$, since only the absolute value $|m|$ is relevant to our discussion. Consider a camera with $N\times N$ pixels, with pixel dimensions $d\times d$ (in our setup $N=576$ and $d=9$ $\mu$m). We define the following quantities:

[TABLE]

Here $r_{\text{max}}$ is the maximum radius available on the sensor; $r_{\text{min}}$ is the minimum radial distance where azimuthal oscillation associated with the OAM content of the LGp,m mode can be detected, before facing aliasing issues; $r_{1}$ is a lower bound for the first root of the Laguerre polynomials contained in the expression of LG modes; similarly, $r_{p}$ is the upper bound for the $p-$th root, while $\tilde{r}_{p}$, with $r_{p}<\tilde{r}_{p}$, delimits the oscillatory region of the Laguerre polynomials Ismail and Li (1992); Gatteschi (2002). Interestingly, the spatial region $r_{1}<r<\tilde{r}_{p}$ well approximates the area containing all the power associated with the mode. At the same time, the quantity $\Lambda=(r_{p}-r_{1})/p$ well describes the average distance between consecutive nodes of the LG mode, 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:

[TABLE]

Indeed, we are requiring that (i) the field is vanishing below the azimuthal aliasing threshold given by $r_{\text{min}}$, that (ii) all the power associated with the mode is contained in the sensor area and (iii) that the field radial oscillations have a spatial period such that at least two pixels are contained in a single period, respectively (radial aliasing limit). It is easy to check that in our configuration, where the beam waist is $w(z)=0.4$ mm, conditions (i) and (iii) are always satisfied for the values of $\{p,m\}$ that are solution of (ii), i.e. the limiting factor is only the dimension of the sensor area. By solving such inequality, we get the relation

[TABLE]

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

Funding Information

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