## Abstract

We propose to analyze the polarization and phase states of laser beams using a fixed set of non-polarizing phase elements. The experimental implementation of the proposed method is based on the use of multi-order phase-diffractive optical elements (DOEs). The presence or absence of intensity maxima (information bits) corresponding to the numbers of diffraction orders allows an identification code (a codeword) to be obtained. The resulting codeword makes it possible to uniquely determine the order of the vortex singularity and the order of the cylindrical polarization of the laser beam in various combinations based on simple relations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Cylindrical vector beams (CVBs) [1], including high-order CVBs, are of practical interest in such fields as optical communications with twisted light [2], amplitude and polarization modulation of focal distributions [3], optical manipulation [4,5], stellar coronagraphy [6,7], and polarization-based spatial filtering [8,9]. Some applications are based on the phenomenon of the so-called inverse energy flux [10], arising from the focusing of high-order CVBs. In this case, the integral inverse energy flux increases with the increasing order of a CVB. Thus, the range of applications of CVBs of various orders has recently expanded.

Optical vortex (OV) beams [11,12], which may have different polarization states providing an additional degree of freedom [13], are being actively used for multiplexing transmission channels too. In [2], Millione et al. used only the polarization degree of freedom. In this paper, we propose to combine both of these possibilities. In this connection, the prompt recognition of the generated polarization and phase states of laser beams is of particular importance. The possibility of multichannel (simultaneous) analysis of the vortex states of laser beams for uniform types of polarization was first demonstrated by Kotlyar et al. [14]. The proposed approach made it possible to detect the sign and order of the topological charge of the incident beam [15] and to measure even the fractional orbital angular momentum of the beam [16].

At the same time, the use of polarization as an additional degree of freedom complicates the task of signal demultiplexing. As a rule, the polarization state is identified using elements of polarization optics. This requires a change in the position of the polarization element during the analysis, which significantly slows down the study and reduces its efficiency. However, in some cases, for certain types of polarizations, one can use, for example, non-polarizing optics, including elements of singular optics. The idea of employing a single fork-shaped phase grating for detecting radially and azimuthally polarized laser beams (i.e., the first-order CVBs) was first proposed by Moreno et al. [17]. The experiments demonstrated a significant difference in the results for circular and non-uniform polarizations, but it was impossible to distinguish the radial polarization from the azimuthal one. In [18] B.S.B. Ram et al. proposed a method that can be efficiently used to distinguish the sign of the polarization singularity of CVBs. However, the patterns formed for recognition are difficult for their automatic processing, and the optical setup used is quite complex and contains polarization plates. The authors of the works [19–21] proposed interference methods based on a linear combination of orthogonally polarized vortex beams, making it possible to both generate and detect some cylindrical polarizations. However, such methods require not only diffraction gratings but also polarizing optics for detection. Most importantly, they do not allow polarization and phase combinations to be identified by their digital codes.

The use of vortex phase elements in combination with tight focusing and high-aperture diffraction axicons [22] makes it possible to distinguish between some types of polarizations (in particular, to distinguish between radial and azimuthal polarizations) and first-order and second-order OVs. Earlier we [22] also proposed to employ multi-order diffractive optical elements (DOEs) for complex analysis of laser beams. In this paper, in order to unambiguously identify the polarization state of a beam, we propose to simultaneously use a certain set of vortex filters of different orders in the paraxial regime, which is important for free-space optics communication. Thus, the polarization elements that require a change of position in the process of analysis (measurement) are replaced with a fixed set of non-polarizing phase elements. The capabilities of diffractive optics allow us to realize a set of phase elements in the form of a single multi-order DOE. The proposed method is based on the polarization and phase interaction of non-uniformly polarized laser beams with singular-phase structures. This interaction is more general than the spin–orbit interaction, and can be observed even in the paraxial regime. Moreover, such an interaction makes it possible to unambiguously determine the order of the vortex singularity and the order of the cylindrical polarization of the laser beam in arbitrary combinations.

## 2. Theory and simulation

Let us consider the interaction of OVs of various orders with different polarization states. For a vortex field with uniform polarization (linear, circular, or elliptical), we can write:

*r*,

*ϕ*) are the polar coordinates,

*m*is the order of the OV,

*A*(

*r*) is an arbitrary radius-dependent function, and (

*c*,

_{x}*c*)

_{y}^{T}is the vector of a uniform polarization state.

The field defined by Eq. (1) has a set of orthogonal states in accordance with the order of the vortex phase singularity *m*, which can be uniquely detected using multi-order DOEs [14]. A OV with *p*th-order cylindrical polarization (radial, azimuthal, or mixed) has the form of

*ϕ*

_{0}is the phase shift (

*ϕ*

_{0}= 0 corresponds to the radial polarization, and

*ϕ*

_{0}= π corresponds to the azimuthal polarization). Obviously, the field

**F**

*(*

_{p,m}*r*,

*ϕ*) has two degrees of freedom, i.e., the order of the vortex singularity

*m*and the order of the cylindrical polarization

*p*.

Let us investigate the possibility of determining the parameters *m* and *p* using multi-order DOEs matched with a set of optical vortices and having the following transmission function:

*Q*+ 1 is the number of generated diffraction orders;

*q*is the number of the diffraction order, which in this case coincides with the topological charge of the OV; and

*α*,

_{q}*β*are the spatial carrier frequencies of the corresponding diffraction orders. Note that the DOE defined by Eq. (3) can be made binary if the spatial frequencies for the complex-conjugate vortex orders are chosen to be equal to

_{q}*α*= -

_{-q}*α*,

_{q}*β*= -

_{-q}*β*[23].

_{q}Let the multi-order DOE be supplemented with a spherical lens and illuminated by a laser beam defined by Eq. (2). The distribution in the focal plane is described by the Fourier transform:

*u*-

*u*,

_{q}*v*-

*v*) are the offset delta functions, and

_{q}*C*are the decomposition coefficients:

_{q}It follows from Eq. (5) that, in contrast to the uniform polarization described by Eq. (1), a cylindrically polarized field leads to the formation of not one, but two, correlation peaks, corresponding to the sum and difference of orders of the vortex singularity and polarization:

*ϕ*

_{0}, i.e., it will not be possible to classify the type of polarization (cylindrical, radial, or azimuthal), but the order

*p*can be determined.

The presence of correlation peaks in orders *q _{1}* and

*q*makes it easy to determine both the order of the vortex singularity m and the polarization order

_{2}*p:*

Figure 1 shows the profile of a binary DOE designed by Eq. (3) with *Q* = 6 for analyzing the polarization and phase states of vortex cylindrically polarized beams defined by Eq. (2),as well as a scheme of the correspondence of diffraction orders in the focal plane to OV orders *q*. In the considered case, the zero order (*q* = 0) is duplicated in order to displace it from the center, since the central part can be illuminated by radiation that has passed beyond the DOE region and does not carry any information. The simulation results for various orders of polarization *p* and the vortex singularity *m* are shown in Fig. 2. When *p* = 0 (the first row of Fig. 2), the polarization corresponds to a linear one (cos(*ϕ*_{0}), sin(*ϕ*_{0}))^{T} with an angle of inclination, *ϕ*_{0}, to the *x* axis. In this case, one correlation peak is formed (with the exception of zero, which is duplicated), corresponding to the OV order *q* = *m*. In the absence of vortex phase singularity, i.e., at *m* = 0 (the first column of Fig. 2), a pair of complex conjugate orders is formed, whose modulus corresponds to the polarization order |*q*| = *p*. In other cases, more complex combinations of two correlation peaks are formed, whose positions in diffraction orders are completely determined by Eq. (6).

As can be seen from the above simulation results, all states are uniquely determined if the DOE contains a sufficient number of diffraction orders (information channels of the analyzer). The presence of correlation peaks in the diffraction orders and the use of Eq. (7) allow one to determine the order of the vortex singularity *m* and the order of polarization *p*.

Since we consider only positive values of *p*, then *q*_{1} ≥ *q*_{2}. The number of orders analyzed by DOE is determined by the number of channels *Q*. The maximum value of the determined *p* and |*m*| is equal to *Q*/2. Thus, the DOE designed by Eq. (3) provides recognition of 2*Q* phase states with uniform polarization and *Q*^{2}/2 polarization and phase states with cylindrical polarization. The gain is *Q*/4 times higher. Obviously, with increasing numbers of states in question, one will require an increase in the number of channel analyzers, *Q*. Note that in [24], we successfully used a DOE with *Q* = 12 to decompose the incident field in OVs, and in [25], a DOE with *Q* = 32 was used.

In addition to Eq. (7), the polarization and phase state of the analyzed beam can be identified using the configuration of a codeword (see Fig. 3) made up of information bits at the centers of the corresponding diffraction orders (presence and absence of a correlation peak are denoted by 1 and 0, respectively). Examples of codewords for some situations are shown in Fig. 3.

## 3. Experiments

Figure 4 shows the optical scheme used in the experiment. The output of a solid-state laser was spatially filtered and expanded using a system consisting of a pinhole *PH* (a hole diameter of 40 μm) and a spherical lens *L _{1}* (

*f1*= 250 mm). Then, the extended laser beam passed through a HOLOEYE LC 2012 spatial light modulator

*SLM*operating in transmissive mode (1024 × 768 pixels, 36 μm pixel pitch), which was used to generate a vortex beam with a given topological charge in the + 1 diffraction order. A combination of lenses

_{1}*L*(

_{2}*f*= 150 mm),

_{2}*L*(

_{3}*f*= 150 mm) and diaphragm

_{3}*D*was utilized for spatial filtering of the vortex beam produced by the first modulator. A polarizer

*P*was used to isolate the linear

*x*-component from the generated beam. A HOLOEYE PLUTO VIS spatial light modulator

*SLM*(1920 × 1080 pixels, 8 μm pixel pitch) operating in reflective mode was used to implement the phase mask of a multi-order analyzing DOE demultiplexing the beam. With a beam splitter

_{2}*BS*and a mirror

*M*, the modulated laser beam reflected from the second modulator was directed to a lens

*L*(

_{4}*f*= 500 mm), which focused the laser beam on a ToupCam UCMOS08000KPB

_{4}*CMOS*image sensor (resolution 3264 × 2448 pixels, 1.67 μm pixel pitch). To convert a linearly polarized laser beam into cylindrical polarized beams of the 1st and 2nd order, we used a polarization converter

*C*, which represented in each case a corresponding S-waveplate. Figure 5 shows the results of the detection of laser beams with a different topological charge and cylindrical polarization order, obtained experimentally with the proposed scheme. The results in Fig. 5 are in agreement with the simulation results shown in Fig. 2.

## 4. Conclusions

We have theoretically justified and experimentally implemented an optical scheme for analyzing the polarization and phase states of laser beams using a non-polarizing phase element in the form of a multi-order DOE with 2*Q* channels. The developed system allows one to uniquely determine the order of the vortex singularity and the order of the cylindrical polarization of the beam in various combinations by a unique digital code. The proposed approach provides recognition of 2*Q* phase states with uniform polarization and *Q*^{2}/2 polarization and phase states with cylindrical polarization. In the latter case, the gain increases by a factor of *Q*/4.

The experimental implementation of the method is based on the use of two spatial light modulators (the first one forms a complex amplitude of the beam, and the second one produces an analyzing multichannel DOE with 12 channels) and S-waveplates of different orders generating CVBs.

In the experiment, we have investigated beams with all possible combinations of OVs and cylindrical polarizations up to the second order inclusive, taking into account their sign. All the results are consistent with the theory and show that polarization and phase states of the beams can be uniquely detected in accordance with the number of DOE channels. The proposed optical system can be used to detect vortex beams with different polarization states, which is an additional degree of freedom in coding information in communication systems. Note that we cannot to distinguish the radial polarization from the azimuthal one, however in in a number of applications, including optical communications, they are not used as independent states since there is a strong interaction between them during propagation [26].

## Funding

Russian Foundation for Basic Research (18-29-20045-mk); RF Ministry of Science and Higher Education (007-GZ/Ch3363/26).

## Acknowledgments

This work was financially supported by the Russian Foundation for Basic Research (18-29-20045-mk) in part of numerical calculations and experimental results and by the Ministry of Science and Higher Education (007-GZ/Ch3363/26) in part of theoretical results.

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