Digital Sorting of Optical Vortices in Perturbed Singular Beams

3.1 Nondegenerate case

For a simplest model for our studies, we choose a scalar field of wave in the form of superposition LGnmof the lowest order with the Nmonochromatic beams, where m>0and n=0are the azimuthal and radial indices, respectively, so that all the vortex modes in superposition have the same waist radius w0at z=0. Let us consider the wave field in the waist plane z=0in the form [25].

where Nm=2−m−1m!πstands for the normalization factor, Cmis the mode amplitude, βmis the initial mode phase. We consider the nondegenerate case of the field representationm≥0orm≤0due to the axial symmetry; combined beams with different topological signs ±mare indistinguishable in intensity moments.

Our goal is to analyze the distribution of the wave field intensity ℑrφz=0in such a way as to express the squared amplitudes and the initial phases of the vortex modes in terms of physically measured quantities in the region of the beam waist. For this, we make use of the intensity moments approach [35] in the form

where the intensity distribution is written as

Mp,qrφstands for the intensity moments function, p,q=0,1,2,…. Now the problem is to choose a combination of intensity moments Jp,qin such a way as to exclude all terms of the last two sums and leave only the first one in Eq. (7). The equations for the squared amplitudes and the initial phases are separated if the moment function is written as Mprφ=rp. Then the equations for the squared amplitudes take the form

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 Сm2.

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 β0=0). Now it is important to select the combination of moments of intensity Jp,qthat so either the second or the third sum in Eq. (2) is considered in the calculation. It is natural to calculate not all differences of phase βm,m′, but only β0,m. However, we did not find such combinations Jp,qthat could filter out only terms with β0,m. Generally speaking, to calculate the phase difference βm,m′it is required to find the Mequations for variables Xm,m′=CmCm′cosβmm′and Ym,m′=CmCm′sinβm,m′in Eq. (2) the second and third sum, number of which is equal to the number of 2-combinations of Nelements. It turns out that the number of equations can be significantly decreased if using the moments J2p+1,1for Xm,m′variables and J1,2p+1for Ym.m′variables. The system of linear equations for the phase difference are written in the form

where n1=m−2p−k±3,n2=m−2p−k±1, Mm,n1,2=∫0∞rm+n1,2+1G2dr. The number of linear equations in each system Eqs. (10) and (11) is K=3N−3,N≥6. It is noteworthy that Eqs. (10) and (11) contain only the terms with an odd difference of indices, including β0,m, so that a finite solution enables us to obtain all phases of the partial beams in the form tanβm,m′=Ym,m′/Xm,m′.

A key element of the experimental setup in Figure 2(a) was a spatial light modulator SLM (Thorlabs EXULUS-4K1), which converted the fundamental TEM00mode of the He-Ne laser (wavelength λ=0.6328mcmand power 1 mW) into a composite beam with mode amplitudes Cmand initial phases βm. To make this possible, the laser beam was additionally filtered by the system FF. The beam splitter BS formed two working arms of the experimental setup. The beam in the first arm projected by a spherical lens onto the input pupil of the CCD1 camera (Thorlabs DCC1645) is subjected to the image computer processing. The result of the computer processing is a digitization of the beam intensity distribution and calculations of high-order intensity moments Jpq. Additional computer software made it possible to compose a system of linear Eqs. (9)–(11) and calculate both the mode amplitudes Cmand their initial phases βmin real time. The second arm was in use for the control measurement of the OAM of the composite beam. For this purpose, the beam was focused by a cylindrical lens CL onto the input pupil of the second CCD2 camera (Thorlabs DCC1645) then the second-order intensity moment Jxywas measured and the average OAM per photon was calculated in accordance with the technique described in detail in the paper [26].

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 ℑxyrequired normalization of the transverse coordinates x,yin units of the Gaussian beam waist at the plane of the CCD camera. To measure the initial beam waist radius w0, the second-order intensity moments J12were used in accordance with the beam quality measurement method [46]. In order to ensure calibration measurements, a special computer program was developed that assigned random values of the amplitudes and initial phases of the vortex modes in Eq. (1) due to the random-number generator. Then, an appropriate diffraction grating was formed on the liquid crystal element of the SLM modulator and the calibration beam was restored. A typical intensity distribution of the calibration beam is shown in the callout of Figure 2(c). The corresponding spectra of the squared amplitudes Сm2and initial phases βmare presented in Figure 2(b) and (c). Red and blue colors in Figure 2(a) and (b) present experimental and theoretical data, respectively. Note that the OAM measured in the second arm was ℓz=4.2and the first arm calculated according to Eq. (2) was ℓz=4.5, while the theory gives ℓz=4.9. The measurement error for N=10beams did not exceed 3–4% for amplitudes and 5–6% for phases. The most interesting effect is observed in the region of OAM oscillations in Figure 1(a) with a characteristic perturbation of the holographic grating in Figure 2(c). This effect is accompanied by a sharp restructuring of the vortex spectrum. For small perturbations of the singular beam, additional vortex modes arise only near the initial value of the topological charge m=M(see [47]). However, in general case of the hologram perturbation, vortex modes with opposite signs of topological charges appear. Such a vortex spectrum reconstruction cannot be detected using the above approach. It is required to expand the measurement technique.

3.2 Degenerate case and vortex avalanche

We consider the case when the initial LG beam with a zero radial index n=0,m≠0is subjected to local perturbation at the central region of the holographic grating. As a result of the perturbation, vortex modes with various types of topological charges appear in the beam. The method of the beam expansion in a series over the vortex modes depends on the type of the grating perturbation. We suppose (and then prove) that the chosen perturbation does not excite LG modes with radial indices n=0, but only modes with different topological charges mappear [39]. Then we write the complex amplitude of the perturbed beam in the form

where Mmis a normalized factor and σz=1−iz/z0.

It is important to notice the intensity moments Jp.qare degenerate with respect to the sign of the vortex topological charge min the case of axial symmetry of the modes in Eq. (12). To remove degeneracy of the intensity moments, it is important to change symmetry of singular beams, but without destroying of structure beams so that the vortex modes with opposite topological charges in sign have distinct geometric contours. Such a requirement provides by the astigmatic transformation with a cylindrical lens of the Laguerre-Gauss beam, which was considered detail in [34] (the general theory of paraxial astigmatic transformations can be found in [41]). We suppose that the cylindrical lens with a focal length fis located at the plane z=0. A paraxial beam with complex amplitude (12) is projected at the lens input. The plane of the beam waist is matched with the plane of the lens. Besides, we restrict our study to zero initial phases of mode beams, i.e., amplitudes Cmare real values. Then, following [34], we write the complex amplitude of a combined beam at the wave diffraction zone as

where Am=−iz0z1qq0−w0q0m1−q02q2,Φ=ik2zx2+y2−z0zw0q2x2−z0zw0q02y2,F=z0w0ziq0qx−qq0yq2−q02,q02=1−iz0z,q2=1+iz0z1,z1=zfz−f..

Let us consider new normalized coordinates so that the beam axes are directed by an angle π/4relative to the axes of the cylindrical lens: u=x+y/w0,v=x−y/w0, we will consider field of paraxial beam in double focus plane z=2f, and also require that z0/2f=1. Then distribution of the beam intensity is written as

where Hnxis Hermite polynomials, Am,n=−1n+meim−n4π, Nn,m2=π4wo2n+m2n+m+1n!m!. As can be seen from Eq. (14), the terms corresponding to optical vortices with positive and negative topological charges are only partially divided. The last two terms characterize the cross terms C−nСmand CnС−m. Besides, the first two terms also contain cross amplitudes C−nС−mand CnСmn≠m. On the other hand, our task is to measure only the squared amplitudes Cm2. It is important to note, that the factors Am,n=−An,m=iwith the difference of the indices m−n=2in Eq. (14) are imaginary, therefore the cross terms of the beam intensity distribution with such indices difference disappear.

It seems that for solving this problem it is reasonable to choose intensity moment functions in the form of Hermite polynomials Eq. (14) as Mp,0u=Hpu/2and M0,qv=Hqv/2. Next, one can use the orthogonality condition of the HG beams and write a system of linear equations, the number of which is equal to the number of variables, as was done above for a non-degenerate vortex array. However, as shown by the assessed computations, as the indices pand qincrease along with the 2Nvariables Cm2, the 2N−6cross terms C±nС±mand C±nС∓mare also added. The system of Eq. (14) cannot be closed. However, one can act otherwise. First, we make two Fourier transforms of the intensity distribution ℑxyin the form

In particular, the terms ℑ−m.−nin the intensity distribution ℑare

Thus, after the Fourier transform the distribution of intensity ℑξηcontains all the squared amplitudes, and, also, the cross terms with the difference m−n=±4. The cross terms ℑ−т,mand ℑm,−nrefer to vortices with positive and negative charges. Besides, the terms ℑ−т,−n+2mξincludes amplitudes С−n2with the similar coefficients, and the member ℑт,n+2mξincludes amplitudes Сn2with only different coefficients. The expressions are received for the terms, that ℑ−т,−n+2mηand ℑт,n+2mη, if we swap the signs of the indices.

The equations for squared amplitudes С±n2will be found, if we will make use the full basis of the Laguerre-Gaussian modes for expansion of the intensity moments:

where J00is a total beam intensity and Lpqxis Laguerre polynomials. Then, using [39], we find a system of linear equations

The system of Eq. (18) enables us to find the values of the squared amplitudes via the measured values of the intensity moments Jp,q±. Each equation contains 2Nterms for amplitudes Cm2and N−3cross terms CmCn. Increasing the value of the indices pand qdoes not change the number of variables, so the system of Eq. (18) can be closed. Note that each of the Eq. (18) can be solved independently of each other and find the amplitudes Cm2and C−m2. The total number of linear equations is relative to signs ±, so that the systems can be easy solved for finding the squared amplitudes in real time using modern computer software.

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 Ωis a scale parameter, δpstands for the perturbation of the holographic grating responsible for shaping the beam, Mis a topological charge of the non-perturbed forked grating. Factors cosmπ/2or sinmπ/2thin out the vortex spectrum remaining the modes with either even or odd m−indices. This makes the dips in the OAM spectrum more distinct. The complex amplitude (12) corresponds to the OAM in the form

Figure 1(b) and (c) displays the case of even vortices beams in the curves of ℓzp. The spectrum of OAM in Figure 1(b) characterizes the general view of resonant bursts and dips. The Ωparameter of scale admits displace along the axis of the resonant regions p=M+δp. Figure 1(c) (curves 1 and 2) describes their characteristic features. In the even vortex modes of integer topological charge are located resonant dips. Swaps of the cosine with the sine in Eq. (19) displaces along the axis pof the resonant regions by one. Increase of the topological charge Mis accompanied by resonances compression, until their depth gets become approximately equal to Mat Ω=18and p<6. For large parameters Ω>>20, the OAM is doubled ℓz=2Min range of the initial topological charges M. As was shown in [47, 48], a perturbation δpcauses a local distortion of the forked grating defect for the large values parameter pshown in Figure 3(a).

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 δp∼10−2of the grating cause an avalanche of optical vortices. The satellites appear in the OAM spectrum, the maxima of which has topological charges less than the initial M. The energy of the basic mode Mflows partially into the spectral satellites, that are generated both in the region of positive m>0and negative m>0topological charges.

For large perturbations δp∼0.1, a vortex avalanche picks almost all the energy out of the basic mode. The vortex avalanche at small deviations δpof the topological charge from integer values Mimmediately causes a resonance dip in the OAM spectrum (Figure 5(a)). With increasing Mthe depth of the dip decreases while its shape is distorted (see Figure 5(b)).

Absolutely other situation arises when the holographic grating with M>10and a forked defect is subjected to a small perturbation. As shown in Figure 3(c), even a weak perturbation δp=−0.001changes drastically the entire relief of the holographic grating although the intensity distribution. In the result, even small differences in the grating relief affects inevitably the shape of the vortex spectrum Cmm2and Ω>10induce the vortex avalanche (see Figure 4(b)). However, the avalanche type differs significantly from that in the previous case. Indeed, as well as in the above case, the perturbation provokes appearing resonance satellites with both positive and negative topological charges (TC). However, their values are always greater than those of the initial TC. In this case, energy is also pumped from the basic mode to the satellites but with higher topological charges under the condition of the resonance m=p. In the OAM spectrum, it looks like the OAM burst at the resonance as shown in Figure 5(c) and (d) but the maxima ℓzare at the region of fractional values p. As the scale parameter increases, the height of the flash decreases, and the resonance circuit broadens (Figure 5(d)).

Further increasing the parameter p(Ω=const) is accompanied by smoothing the curve ℓzpso that ℓzmax→2pexcept for the OAM dips at the integer values m=p.

4.1 Sector aperture

Let us consider propagation a scalar beam of Laguerre-Gaussian LG0mwith a zero radial index p=0and an azimuthal index (a topological charge) mthrough the rough regular sector aperture obstacle, that an angle shown in Figure 1. The edge of sector touches axis of the beam. The field of beam at the initial plane z=0can be represented as [49].

where r=ρ/w. wis a beam waist radius at the plane z=0, ρand φrepresent polar coordinates. We rewrite the beam field (1) formed by the rigid-edges aperture with the half angle αas a sum of non-normalized Laguerre-Gauss beams LGpmin the form

where the beam amplitudes are

where Γxis a Gamma function. The terms in the series (23) with radial indices p≠0disappear due to orthogonality of the LG modes.

The complex amplitude (22) can describe the spatial evolution of the perturbed beam if we replace r→r/σand multiply the sum by σwhere σ=1−iz/z0. Figure 6(b) and (c) shows a computer simulation (Figure 6(b)) and experiment (Figure 6(c)) of the intensity distribution ℑrφz=ΨΨ∗along the vortex beam length with m=15modulated by the sector aperture with α=π/4. It is important to note, that the cut out dark sector from the beam intensity distribution does not lead to its self-healing. When the beam propagates, the dark sector rotates synchronously. We showed, both theoretically and experimentally, that beam self-healing does not occur at any beam length zeven at a very small sector angle α.

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 Сm,n2for the initial topological charges n=m=5with α=45°. The short part of the beam intensity redistributed symmetric at the neighboring vortices orders. But, if we increase the angle α, we see a violation of the symmetric distribution of intensity among the vortex modes. There formed a second maximum in the spectrum range of negative topological charges n<0. It is important to note, that the authors of Ref. [50], taking into consideration the optical uncertainty principle, also plotted the vortex spectrum for a topologically neutral beam m=0and a beam with a small topological charge m=2at the angle of aperture α=45°. The authors did not detect a maximum of spectra in the negative region of topological charges. As we presented above, the emergence of the second intensity maximum is possible only at sufficiently large angles of overlap of beam αand also large topological charges mof the initial singular beam.

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 Ψmof the Laguerre-Gauss beams so that the squared amplitudes Сm,n2and С¯m,n2are obeyed a simple relation С¯m,n2=2−n−2n!Сm,n2. The changes of OAM ℓzαmafter increasing the aperture angle α(decreasing the adjacent angle β=π−α) is presented by Figure 7(a) for topological charges m=5,m=10and m=15of the initial vortex beam.

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 β≈2°.

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 LGnmmodes with various radial indices n, but a constant topological charge m[54]. Such modes are sorted using the holographic grating techniques and 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 LG0mbeam at the z=0plane with a ring aperture so that its complex amplitude is written in the form (see also Figure 10a)

If the axis of the annular aperture coincides with the beam axis then the perturbation excites only LG beams with the same TC (m=const) but different radial indices n(this follows from the condition of orthogonally of LG beams). The perturbed beam field can be represented as an expansion over LG beams

The LG vortex modes amplitudes restricted by the ring of thickness h, we find for the field difference the beams passing simultaneously through the circular aperture of radius R+hand the opaque disk of radius R. As a result we obtain

where 1F1is a confluent hypergeometric function and we used the integrals from [56]. The expression (28) together with Eq. (27) allows covering three cases: (1) a circular aperture, R=0,h=R0; (2) an opaque disk R=R0,h=0; and (3) an annular aperture R,h≠0. An important property of mode amplitudes (28) is that the perturbation of a singular beam with a defined TC via these types of axial apertures does not excite vortex modes with other TC. This means that the OAM ℓzdoes not change due to such a perturbation process. Does this mean that the perturbed vortex beam completely restores its initial properties during propagation, i.e., possesses the self-healing effect? We will peer into this process carefully.

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 Mp,qrφshould choose the forms Mp,q=sinrLpk2r2or Mp,q=cosrLpk2r2while in the intensity distribution ℑrφuse the complex amplitude (26). Variation of the indices p and k enables us to obtain a closed system of linear equations for the squared amplitudes Cm,n2.

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 Cm,n2nis long spectral tails, which are omitted in the figures, but which make a significant contribution to the calculation of information entropy. Truncation of a topologically neutral beam (m=0) with a circular aperture in Figure 9(Ia) leads to overlapping many side rings, resulting in a wave-like form of the spectral tail. In the perturbed vortex beams shown in Figure 9(Ibc), a broadening of the vortex spectrum and a decrease in the tail amplitudes are observed. Such characteristic features of the vortex spectra insert significant uncertainty into information carried by the vortex beam, which is represented as the dependence of information entropy HIon the aperture radius R0in Figure 9(Id). It is noteworthy that even small variations in the aperture radius lead to changes in the entropy HIthat grow with increasing TC. Similar changes in the entropy HIhshown in Figure 9(IId) occur when a vortex beam is perturbed by a annular aperture, which are the result of transformations in the vortex spectra in Figure 9(a–c). The presented results show that any external interference in the beam immediately affects the uncertainty of the vortex beam state, the magnitude of which can be estimated by measuring informational entropy.

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 (mand −m). If the amplitudes of the beams are the same, then the complex amplitude of such a combined beam is described by Eq. (28) when replacing the phase factor expimφ→2cosmφ. Figure 10(b–d) illustrates the intensity distribution of such perturbed beams at the measurement plane. Each of these beams receives the same perturbations, regardless of the sign of their TC. Even if the beam amplitudes are different, they receive the same amount of the vortex state uncertainty. However, this apparent indistinguishability of the modes can be easily detected experimentally due to opposite phase circulation of the fields, and the modes can be sorted out in different memory cells.