## Abstract

The chapter provides a brief overview of shaping and measuring techniques of the vortex spectra (squared amplitudes and initial phases of vortex modes) including radial indices. The main physical mechanisms causing the formation of laser beams with a complex vortex composition, in particular, in biological media, are indicated, and the need for a digital analysis of vortex spectra is substantiated. It is the analysis of vortex spectra that allows us to find the orbital angular momentum and informational entropy (Shannon’s entropy) of perturbed laser beams in real time. In the main part of the chapter, we consider in detail a new approach for measuring vortex spectra without cuts and gluing of the wavefront, based on digital analyzing high-order intensity moments of complex beams and sorting the vortex beam in computer memory sells. It is shown that certain types of weak local inhomogeneities cause a vortex avalanche causing a sharp dips and bursts of the orbital angular momentum spectra and quick ups and downs of the informational entropy. An important object of analysis is also the vortex spectra of beams scattered by simple opaque obstacles such as a hole, a disk, and a sector aperture.

### Keywords

- optical vortex
- moment’s intensity
- orbital angular momentum
- medical optics

## 1. Introduction

As is well known, optical vortices [1] accompany light scattering processes due to both simple and complex medium inhomogeneities (see, e.g., [2] and references therein). Scattered light carries a huge array of information, both on the composition and structure of scattering (diffraction) centers, and on the structure of the initial light beam. At the same time, optical information can be read off both by analyzing the frequency spectrum [3, 4, 5] and the spectrum of optical vortices [6, 7]. An important aspect of this problem is the study of biomedical objects [8, 9], for example, the composition of the blood or the reflection of a vortex beam from the skin surface [10] for express diagnostics of skin diseases. The fact is that a vortex beam scattered by the skin is transformed into a speckle-like structure resembling one that occurs when light passes through a medium with weak turbulence [11, 12]. At the same time, the speckle structure is formed by the skeleton of optical vortex array [13, 14, 15]. In turn, the analysis of such a complex vortex structure is conveniently carried out on the basis of vortex fractal techniques [16] (see also [17] and references therein). There are a variety of approaches for the fractal vortex models of laser beams scattered by biological tissues [18, 19] based on the light scattering by nonspherical particles [20] that involves the representation of the wave field in terms of Legendre polynomials. However, a real optical experiment for measuring the vortex spectrum requires the use of particular approaches to the orthogonal basis for the scattered beam expansion in terms of special functions, which can differ significantly from the corresponding theoretical models.

The problem of using the properties of optical vortices in various areas of science and technology requires the development of reliable but simple techniques for measuring the spectrum of optical vortices in complex beams scattered by various objects. Therefore, the attention of many researchers is drawn to measuring the orbital angular momentum (OAM) of vortex beams [7, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] that is directly related to the spectrum of optical vortices. Really, the complex amplitude

where

The basis of this expression is an implicit connection between the topological charge

At the same time, the measurement of the fractional OAM does not give complete information about the combined beam content. As can be seen from expression (2), the same value of the OAM can correspond to a different composition of vortex modes with squared amplitudes

In this chapter, we consider in detail the technique of measuring the vortex spectrum based on the analysis of high-order intensity moments that excludes cuts and gluing of the beam without losses of information on initial mode phases. Unlike the method of holographic gratings that transforms a combined beam into vortex modes with different propagation directions, just as a prism converts “white” light into a spatial frequency spectrum, we will try to demonstrate how a perturbed singular beam can be sorted into vortex modes sited in computer memory cells and then reproduce the beam main characteristics: OAM, information entropy, and initial topological charge in real time. Moreover, knowing the digital spectrum, the beam can be recovered again, and by adjusting the spectral vortex amplitudes we can improve the structure of the transmitted field.

## 2. Preliminary remarks

As far as we know, the first theoretical and experimental studies of optical vortex arrays refer to 1991, when the authors of [41] succeeded in reproducing holographically individual letters and words due to ordering optical vortex array in typical phase skeleton on the base of the technique that had been developed back in the early 1980s [29]. However, only after the article [42] by Berry did the studies of the vortex array properties become widespread. As a result, it was shown that the diffracted beam turns into a combined beam containing a large number of optical vortices with integer topological charges. These are sometimes called the beams with fractional topological charges. Using Eq. (2), we can verify that beams with a fractional topological charge also have fractional OAM

If the Bessel beam is represented in the form of a conical beam of plane waves with fractional phase bypath

where

The result of the plotting is represented by the curve 1 in Figure 1(a). The OAM oscillates at large values of the topological charge

The OAM oscillations disappear (see Figure 1(c), curve 2). As the authors of [45] revealed, a gradual increase in OAM is observed only at small values of the fractional topological charge. In fact, we are dealing with different beams, although the basis for their shaping is the same physical process. As we will see later, the choice of the normalization of vortex modes endows the combined beam with special properties.

## 3. Theoretical background of the digital vortex sorting and experimental results

### 3.1 Nondegenerate case

For a simplest model for our studies, we choose a scalar field of wave in the form of superposition

where

Our goal is to analyze the distribution of the wave field intensity

where the intensity distribution is written as

The right-hand sides of the linear Eq. (9) are easy to calculate, and the left-hand sides are directly measured experimentally using the Eq. (7) that gives the desired values of the squared amplitudes

As intensity of array of the beams depends only on the difference of phase between two pairs of modes, in computation of the initial phases we will assume that one of beams phase is given (let us say

where

A key element of the experimental setup in Figure 2(a) was a spatial light modulator SLM (Thorlabs EXULUS-4K1), which converted the fundamental

Before proceeding to the current measurements, it was necessary to adjust the experimental setup. For this, a number of calibration measurements were carried out. The digitization of the intensity distribution

### 3.2 Degenerate case and vortex avalanche

We consider the case when the initial LG beam with a zero radial index

where

It is important to notice the intensity moments

where

Let us consider new normalized coordinates so that the beam axes are directed by an angle

where

It seems that for solving this problem it is reasonable to choose intensity moment functions in the form of Hermite polynomials Eq. (14) as

In particular, the terms

Thus, after the Fourier transform the distribution of intensity

The equations for squared amplitudes

where

The system of Eq. (18) enables us to find the values of the squared amplitudes via the measured values of the intensity moments

For measurements, the experimental setup shown in Figure 2(a) was used. However, in contrast to the non-degenerate case, the main experimental data were measured in the second arm, provided that the detection plane was located at the double focus region of the cylindrical lens, while the first arm was used for general adjustment [39].

We will implement the above technique for a new type of combined beams with conspicuous dips and bursts in the OAM spectrum similar to that shown in Figure 1(a) and (b). Let us write the complex amplitude of the combined vortex beam in the form

where

Figure 1(b) and (c) displays the case of even vortices beams in the curves of

As the perturbation grows the area of the hologram defect increases (Figure 3(b)). Small variations in the holographic grating structure lead to a cardinal reconstruction of the vortex spectrum (Figure 4(a)). Weak perturbations

For large perturbations

Absolutely other situation arises when the holographic grating with

Further increasing the parameter

## 4. OAM, informational entropy and topological charge of truncated vortex beams

A special case is represented by natural changes of the vortex spectrum due to external influences on the laser beam (diffraction by opaque obstacles, external interference in the beam, etc.). The simplest opaque obstacles are the sector, circular and annular apertures where we stop our attention using a simple technique of the vortex digital sorting.

### 4.1 Sector aperture

Let us consider propagation a scalar beam of Laguerre-Gaussian

where

where the beam amplitudes are

where

The complex amplitude (22) can describe the spatial evolution of the perturbed beam if we replace

#### 4.1.1 The vortex spectrum

We consider vortex beams spectra for topological charge that most clearly reflect properties of the sector perturbation. Computer simulation and experimental data of typical vortex spectra is shown in Figure 7. We revealed a clear maximum of vortex mode intensity

#### 4.1.2 The orbital angular momentum

The OAM per photon of a complex perturbed beam can be calculated in accordance with Eq. (23). The mode amplitudes are given by the normalized field

The OAM is practically unchanged a wide range of angles and remains almost equal to the initial OAM, despite the rapid increase in the number of vortex states (see Figure 8(a)). After the second spectral maximum is formed in the negative region of topological charges Figure 7, there is a sharp decrease of the OAM. The OAM is equal to zero already at the angle

#### 4.1.3 Informational entropy (Shannon’s entropy)

The normalized squared amplitude

The Shannon entropy (24) characterizes the amount of uncertainty (randomness) that arises when a perturbation acts on a vortex beam. For example, in the case of the same amplitudes of the mode beams, the Shannon entropy of

In Figure 8(b) presents the dependence of the Shannon Entropy

#### 4.1.4 The topological charge

According to Berry [42], the topological charge of the vortex array is defined as the difference of the fluxes of vortex trajectories through the beam cross section taking into account vortex directions and “weights,” and is calculated as

Berry showed [42] that the vortex beam TC has always integer values when the spiral phase plate is perturbed equal to the integer value of the unperturbed plate step. We investigated the changes of the TC under the beam sector perturbations [52] and perturbation of the holographic grating [47]. Note that all further calculations are based on the requirement that the perturbation does not introduce changes in the mode phases. We performed a series of computer TC estimations of the perturbed beam (20) for various initial TC. The following restrictions were used. As the spectra of vortices in Figure 7 show, the squared mode amplitudes quickly tend to zero as their TC increases. Therefore, we can restrict ourselves to a finite mode numbers

### 4.2 Circular and annular apertures

The problem of the birth and annihilation of phase singularities has been considered as far back as at the beginning of the last century in connection with the peculiarities of light diffraction at the edges of the half-plane or lenses and micro-objectives of telescopes and microscopes (see [53] and references therein). As a rule, the discussion came down to the technique of suppressing the corresponding aberrations. In this section, we focus on the digital vortex sorting after beam diffraction by the circular and annular apertures addressing the Shannon entropy problem of the diffracted combined vortex beams.

Note also that recently, special attention has been paid to studies on increasing the information capacity of optical channels due to *a single phase screen* [55]. In this section, we will focus on the digital sorting of LG modes with various radial indices.

We consider the perturbation of the vortex

If the axis of the annular aperture coincides with the beam axis then the perturbation excites only LG beams with the same TC (

The LG vortex modes amplitudes restricted by the ring of thickness

where

Recall that under the action of axial perturbation, vortex modes with new topological charges do not appear in the perturbed beam. Therefore, the digital sorting of vortex modes in a perturbed beam can be carried out in accordance with Eq. (15) for the nondegenerate case (see Section 3.1). In this case, as a function of moments *p* and *k* enables us to obtain a closed system of linear equations for the squared amplitudes

The experimental results of measuring the vortex spectra are shown in Figure 9, where the average values of the squared amplitudes are plotted along the ordinate axis. A characteristic feature of the dependences

Another interesting feature of the axial aperture action is manifested under the combined vortex beam perturbation consisting of two vortex beams with the same values but different TC signs (

## 5. Conclusions

We examined the technique of digital sorting of vortex modes that makes it possible to measure in real time the vortex spectrum (squared amplitudes and initial phases) including radial indices, OAM, and informational entropy of perturbed singular beams. The considered approach is based on the measurement of intensity moments of higher orders and a digital solution of a linear equations system that eliminates the cuts and gluing of the beam wavefront without losing information on the modes initial phases. Moreover, the digital vortex spectrum also enables us to restore the initial combined beam and, correcting parameters of the spectral modes, to improve its characteristics.

The digital approach has been tested on vortex beams free of wave defects perturbed both by local defects of holographic gratings responsible for the beam generation and by the sectorial, circular and annular aperture. We revealed that a local perturbation of the holographic grating near the central forked defect causes bursts and dips in the OAM spectrum. The depth and height of the spectral dips and bursts are controlled by the parameters of the holographic grating and can vary over a wide range. The perturbation inserted by the sector aperture is regulated by the sector angle. Over a wide range of sector angles, the beam OAM remains almost unchanged. However, when the sector angle is relatively large, so that most of the light flux is cut off by the aperture, the optical uncertainty principle begins to act, and the OAM sharply decreases to almost zero that is accompanied by a rapid growth of the Shannon entropy. At the same time, the beam topological charge remains unchanged for any sectorial perturbations. The axial perturbation via a circular and annular aperture does not change either the OAM or topological charge. However, a wide range of Laguerre-Gauss modes with the same topological charges but different radial indices leads to a rapid increase in information entropy as the pupil of the circular aperture or the ring thickness of the annular aperture decreases. This allows not only to estimate the noise level in the optical information transmission line, but also to record external interference in the information flow. We also note that the digital sorting of optical vortices in a perturbed light flux opens up broad prospects for its employment for medical express-diagnostics of skin diseases, since, for example, this allows us to detect slight changes in the vortex spectrum of a laser beam scattered by inflamed or dehydrated skin areas.

## Acknowledgments

The authors are grateful to E. Abramochkin (Samara Branch of the Lebedev Physical Institute, Russian Academy of Sciences, Samara, Russia) for a useful discussion of the mathematical approach. The reported study was funded by RFBR according to the research project № 19-29-01233.