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Photonics
, Volume 9 (3) – Mar 6, 2022

/lp/multidisciplinary-digital-publishing-institute/depolarization-of-vector-light-beams-on-propagation-in-free-space-2clkHvUOCQ

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hv photonics Article Depolarization of Vector Light Beams on Propagation in Free Space Nikolai Petrov Scientiﬁc and Technological Centre of Unique Instrumentation of the Russian Academy of Sciences, 15 Butlerova Str., 117342 Moscow, Russia; petrovni@mail.ru Abstract: Nonparaxial propagation of the vector vortex light beams in free space was investigated theoretically. Propagation-induced polarization changes in vector light beams with different spa- tial intensity distributions were analyzed. It is shown that the hybrid vector Bessel modes with polarization-OAM (orbital angular momentum) entanglement are the exact solutions of the vector Helmholtz equation. Decomposition of arbitrary vector beams in the initial plane z = 0 into these polarization-invariant beams with phase and polarization singularities was used to analyze the evo- lution of the polarization of light within the framework of the 2 2 coherency matrix formalism. It is shown that the 2D degree of polarization decreases with distance if the incident vector beam is not the modal solution. The close relationship of the degree of polarization with the quantum-mechanical purity parameter is emphasized. Keywords: polarization; degree of polarization; depolarization; purity parameter; vortex light; vector Bessel modes; phase and polarization singularities; polarization invariance; entropy 1. Introduction The polarization of light reﬂects the vector nature of electromagnetic ﬁelds and plays a very important role in optics and physics of the light–matter interaction [1–3]. The polariza- tion properties of ﬁelds should be taken into account in many problems of light propagation Citation: Petrov, N. Depolarization of in various media. It is known that the polarization of a plane wave in a homogeneous Vector Light Beams on Propagation in and isotropic non-dispersion medium does not change during propagation. Although Free Space. Photonics 2022, 9, 162. a polarized plane wave propagates in free space without changing its polarization state, https://doi.org/10.3390/ structured light beams undergo depolarization and a change in the polarization state during photonics9030162 propagation due to diffraction and spin–orbit interaction. Signiﬁcant changes in the state Received: 7 February 2022 of polarization and the degree of polarization occur during the propagation of radiation in Accepted: 4 March 2022 inhomogeneous media. Physically, depolarization occurs due to a decrease in the degree Published: 6 March 2022 of correlation between the different ﬁeld components. In [4,5] depolarization of light in randomly inhomogeneous media was investigated. Two mechanisms of depolarization Publisher’s Note: MDPI stays neutral were analyzed: diffractional and geometrical, which are caused by diffraction and the with regard to jurisdictional claims in published maps and institutional afﬁl- Rytov rotation [6,7] of the polarization vector, respectively. In single-mode optical ﬁbers, iations. the change in the input polarization is usually due to nonlinearity and birefringence of the medium [8–10]. However, depolarization takes place also in optical waveguides without birefringence. In particular, it was shown in [11] that the polarization degree of the linearly polarized light in a graded-index isotropic optical ﬁber decreases with increasing distance. Copyright: © 2022 by the author. In [12], it was shown theoretically that the polarization degree of the linearly polarized light Licensee MDPI, Basel, Switzerland. in an isotropic optical ﬁber with parabolic distribution of refractive index decreases with This article is an open access article increasing distance due to the Rytov rotation of the polarization vector, but the degree of distributed under the terms and polarization of circularly polarized light is retained with increasing distance. However, in conditions of the Creative Commons the experiments [13], preservation of the polarization degree for circularly polarized light Attribution (CC BY) license (https:// in an isotropic optical ﬁber has been not observed. In [14–16], the quantum-mechanical for- creativecommons.org/licenses/by/ malism of coherent states was applied to study the evolution of polarization in a multimode 4.0/). Photonics 2022, 9, 162. https://doi.org/10.3390/photonics9030162 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 162 2 of 16 isotropic graded-index medium. It was shown that the effects of diffraction and spin–orbit interaction are responsible for depolarization of light in an isotropic graded-index medium. The rotation of the plane of linear polarization during propagation in an optical ﬁber is considered. It is shown that the axial displacement and the angle of tilt of the incident beam to the ﬁber axis inﬂuence on the rotation angle of the polarization plane. Intraﬁbre rotation of the plane of polarization was also demonstrated in [17]. In past decades, there has been considerable interest in studying changes in the degree of polarization of electromagnetic beams on propagation in free space [18–26]. It has been shown that, in general, the degree of polarization varies with propagation even in free space [18,19]. In [25], the possibility of controlling the degree of polarization of light during propagation in free space by changing the coherence of the light source was demonstrated experimentally. The results were in good agreement with the theory [24]. In some cases, the spectral degree of polarization remains propagation-invariant along the axis of a Gaussian Shell-model beam [27,28]. Recently, the study of the propagation and focusing of nonuni- formly totally polarized beams has been of great interest [29–34]. Of particular interest is the study of the invariance on propagation of the characteristics of vector beams, such as the state and degree of polarization [35–38]. In [27,36,37], the conditions under which the polarization invariance is observed in the propagation of partially coherent electro- magnetic beams are analyzed. It is known that the spirally polarized beams [29,30,39,40] remain invariant in polarization during propagation. Note that the radial and azimuthal polarizations are the particular cases of the spirally polarized beams. In recent years, the study of the vector vortex beams with the phase and polarization singularities has been of particular interest [41–46]. Various types of cylindrical vector beams (CVBs) with helical wavefronts and spatially nonuniform state of polarization, such as Laguerre–Gauss (LG) beams which combine spin and orbital angular momentum (OAM), were proposed. It should be noted that cylindrically polarized LG beams can be considered as modal solutions in free space only in the paraxial approximation. Paraxial and nonparax- ial propagation properties of cylindrically polarized beams have been considered in [47,48]. In [49–51], generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere was investigated. In [52], an overview of the latest results on the generation and observa- tion of polarization singularities in metaphotonics is presented. A review on polarization optics and polarimetry for recent biomedical and clinical applications is presented in [53]. In [54], the phase and polarization singularity sheets using metasurfaces experimentally are realized. Recent developments of wave ﬁeld multidimensional manipulations based on artiﬁcial microstructures are presented in [55]. In [56] the spin-decoupled metasurface for simultaneous detection of SAM (spin angular momentum) and OAM via momentum transformation was proposed. Various approaches to generate vector vortex beams have been proposed [31], such as interferometry, subwavelength gratings, conical Brewster prism, twisted nematic liquid crystals, etc. In [51], to generate an arbitrary vector vortex beam on the hybrid-order Poincaré sphere, a simple approach using the combination of an inhomogeneous birefrin- gent q-plate and a spiral phase plate is proposed. A passive device consisting of a pair of polarization-selective cylindrical lenses for generating cylindrical vector beams with arbitrary azimuthal and radial topological charges is demonstrated in [44]. The usual description of polarization is based on the Stokes parameters or on 2 2 polarization matrices. In [57,58] the uniﬁed theory of coherence and polarization is formu- lated which can be used to study the changes of polarization at propagation of random electromagnetic beams. Usually, the calculations of the coherence matrix elements are carried out in a paraxial approximation. However, for tightly focused beams, nonparaxial effects become signiﬁcant. In this paper, a theoretical analysis of the propagation of polarized vortex light beams in free space is carried out using the vector mode decomposition method. It is found that the hybrid vector Bessel modes with radial and azimuthal indices are the solutions of the Maxwell equations. Polarization-invariant vector Bessel beams with phase and polarization Photonics 2022, 9, 162 3 of 16 singularities which combine spin and OAM are proposed. Change in the polarization state and the degree of polarization on propagation is shown for an arbitrary incident beam that is not a modal solution of the Maxwell equations. 2. Problem Formulation The equation describing the propagation of monochromatic light in free space for a vector electric ﬁeld E can be obtained from Maxwell’s equations as ! ! rr E k E = 0, (1) ! ! ! ! ! ! ! ! where E(r, j, z) = e + e z ex p(ibz), e = e e + e e , e = r e , k = 2p/l is z r r j j z ? ? ? ? the wavenumber, and b is the propagation constant. The radial, azimuthal and longitudinal components of the electric ﬁeld, deﬁned in cylindrical polar coordinates r, j, z, are described by the equations: 2 2 ¶e 1 ¶ ¶e 1 2 1 ¶ e ¶ e 2 r r r r e + + + k e = 0, 2 r 2 2 2 2 r r ¶r ¶r r r ¶j r ¶j ¶z 2 2 ¶e ¶ e ¶ e 1 ¶ j 1 2 ¶e 1 j j r 2 r e + + + + k e = 0, (2) 2 j 2 2 2 2 j r ¶r ¶r ¶j r r r ¶j ¶z 2 2 1 ¶ ¶e 1 ¶ e ¶ e z z z 2 r + + + k e = 0. 2 2 2 z r ¶r ¶r r ¶j ¶z Below we consider the beams with spatial transverse dimensions signiﬁcantly exceed- ing the wavelength, so the longitudinal ﬁeld component e e can be neglected because the electromagnetic ﬁeld in this case is transverse. For such ﬁelds, the polarization is well described by Stokes parameters determined from the 2 2 coherency matrix [1]. Cylindrical hybrid vector Bessel beams with phase (scalar) and polarization (vector) singularities are the exact solutions of the vector two-dimensional Helmholtz Equation (2): ! e 1 1 1 r ilj+ij ij ilj+ij ij ib z 0 0 pl e r, j, z = = y r e e + e e e , (3) ( ) ( ) pl e 2 i i pl where y (r) = A J m r/R , A = 1/ p R J m , b = k 1 , m are pl l pl 0 0 l+1 pl pl pl kR the positive zeros of the Bessel function J (z); p = 1, 2, . . . ; l = 0, 1, 2, . . . ; j is the l 0 initial angle. Solutions (3) are the combination of Bessel functions of different order, and they consist of different polarization states. Vector modes are hybrid, i.e., they include both components of the ﬁeld. Note that the solution (3) satisﬁes the Maxwell’s divergence equation r e (r, j, z) = 0. It follows from the orthogonality condition for Bessel functions [59] 0 2 J m r/R J m r/R rdr = J m d , (4) ( ) [ ( )] m i 0 m j 0 m+1 i i j that the normalized functions y (r) satisfy the equation pl Z Z 2p R y (r, j)y (r, j)rdrdj = 1. pl pl 0 0 These solutions form a complete set of mutually orthogonal functions in the given interval [0, R ]. Hence, any ﬁeld in the initial plane z = 0 can be decomposed into these modal solutions. Solutions (3) are propagation-invariant in free space, i.e., they are non-diffractive Bessel vector beams with radial and azimuthal indices possessing polarization and phase singularities. Note that the non-diffractive scalar Bessel beams in free space were proposed in [60]. In [61], three-dimensional self-imaging electromagnetic ﬁelds that are exact solu- Photonics 2022, 9, 162 4 of 16 tions of Maxwell’s equations were analyzed. In [62], the amplitude components of the vector non-diffractive beams were obtained as solutions to the vector Helmholtz wave equation. It was shown in [63] that the normalized Bessel beams with radial indices are the solutions of the scalar Helmholtz equation. Using Dirac notations, the vector hybrid modes (3) can be represented as: ij ij ib z pl jY (r, j, z)i = j p, li jl, Rie +j l, Lie e (5) pl 1 1 ilj+ij ilj+ij 0 0 where j p, li = y (r), jl, Ri = e and j l, Li = e . pl i i When l = 0, Equation (5) reduces to the electric ﬁeld for the linearly polarized Bessel beams with non-zero intensity in the center. When l = 1, Equation (5) describes the electric ﬁeld of a radially or azimuthally polarized beam with annular intensity distribution. Light beams with radial and azimuthal polarizations are the most well-known cylindrical vector beams with phase (scalar) and polarization (vector) singularities. The phase singularity is associated with a phase (scalar) vortex of a beam or with the helical wavefront. The polar- ization singularity occurs in beams with spatially varying polarization distribution [45,46]. Cartesian components of the ﬁeld are related to polar by the relationship: e cosj sinj e 1 x r ib z pl Y r, j, z = = = j p, li jl, Ri+j l, Li e . (6) ( ) ( ) pl e sinj cosj e 2 y j Thus, the vector mode represents the superposition of beams in orthogonal polar- ization (spin) and OAM states. A spin part is associated with polarization and an orbital part is associated with spatial distribution. There is an entanglement or non-separability between spatial, polarization and orbital angular momentum degrees of freedom. Arbitrary incident beam at z = 0 with spatial transverse dimensions signiﬁcantly exceeding the wavelength can be decomposed in the series of modal solutions: ! a pl E(r, j, 0) = e (r, j, 0), (7) pl pl where Z Z ¥ 2p a = he (r, j, 0)jE (r, j, 0)i = E (r, 0)e (r, j, 0)rdrdj, r r r pl 0 0 Z Z ¥ 2p a = he (r, j, 0)jE (r, j, 0)i = E (r, 0)e (r, j, 0)rdrdj, j j j pl 0 0 and Z Z ¥ 2p E (r, 0)E(r, 0)rdrdj = 1. 0 0 The evolution of the incident ﬁeld is determined by the expression pl ib z pl E(r, j, z) = Y (r, j, 0)e , (8) å pl pl pl where b are the propagation constants of the modes. pl Degree of Polarization The degree of polarization is determined from the calculations of the coherency ma- trix [1]: hE E i hE E i r j r r F = (9) hE E i hE E i r j j j Photonics 2022, 9, 162 5 of 16 The angular brackets deﬁne the ensemble average taken over the statistical ensemble representing the randomly varying electromagnetic ﬁeld. The degree of polarization is determined by the expression [1]: 1/2 4detF P = 1 , (10) S p F where F = he jF(r, z)je i, i, j = r, j, i,j i j h i rr r r F (r, z) = A Y (r, j)Y (r, j)cos b b 0 0 z , rr 0 0 0 0 å 0 pl p l pl p l pl p l p p h i rj j F (r, z) = A Y (r, j)Y (r, j)cos b b 0 0 z , rj 0 0 0 0 å 0 pl p l pl p p pl p l p l h i jr j F (r, z) = A Y (r, j)Y (r, j)cos b b 0 0 z , jr 0 0 å 0 0 0 pl p l p l p p pl p l pl h i jj j j F (r, z) = A Y (r, j)Y (r, j)cos b b 0 0 z , jj å 0 0 0 0 pl p l pl p l pl p l p p i j j i rr rr r r A = a a . . . . A = a = a a . 0 0 0 0 p p0 pl p0l0 pl p l pl p0l0 pl p l In the experiments, the degree of polarization P can be determined from the measure- ments of the Stokes parameters [1,2]: 1/2 2 2 2 P = S + S + S , 2 3 where S = I + I is the total intensity, x y S = I I , x y S = I + I , 45 45 (11) S = I I , 3 RCP LCP S = S + iS = A ex p if , ( ) 12 1 2 12 12 f = tan (S /S ) is the Stokes phase. 12 2 1 Note that, at the polarization singularities, the Stokes phase f is the undeﬁned quantity [45]. There is an analogy between the coherency matrix and the quantum-mechanical density matrix [64]: r r rr rj r = , (12) r r jr jj where r is the density matrix corresponding to the normalized coherence function L = F/S p(F). This analogy allows such thermodynamic parameters as entropy, temperature, etc. for the optical radiation to be introduced [64]. The entropy of the optical beam can be determined as S = w lnw , (13) n n hnjr ˆjni where w = is the probability of excitation of a given mode, i.e., the fraction of S pr ˆ radiation energy carried by this mode. It is known that the entropy is the measure of the lack of information about the system. The entropy reaches its minimum (S = 0) in the case of totally polarized pure state and has a maximum (S = ¥) for unpolarized and incoherent radiation. The density matrix operator has a trace S pr = 1 and for the mixed (partially polarized) state S pr 1. Photonics 2022, 9, 162 6 of 16 The value S pr is a measure of quantum-mechanical purity. The purity parameter or impurity can be introduced to describe the partially polarized beams [65]: z = 1 S pr , (14) where S pr = r r , (15) å pl, p0l0 p0l0, pl pl, p0l0 r 0 0 = hYjr ˆjYi, hYj = Y (r, j), jYi = Y (r, j), S pr = r = 1. 0 0 å p p pl, p l pl p p l There is a close relationship between the degree of polarization and the quantum purity parameter of the light beam, so the degree of polarization can be represented by [65]: P = 1 2(1 S p(r ˆ ) . (16) It follows from (15) that the trace of the matrix r is given by the expression 2 2 S pr = 1 + P , (17) and, hence, the equality S pr = 1 is the necessary and sufﬁcient condition for ﬁnding a beam in a pure state. The degree of polarization P = 1 corresponds to a pure state and P = 0 corresponds to an unpolarized beam. Note that we use here the term “purity” with respect to coherent ﬁelds. We show that the purity of a beam is preserved during propagation if only this beam is a polarization-invariant modal solution of Maxwell’s equations. 3. Simulation Results Consider the incident vector beam at z = 0 with Gaussian and Bessel–Gauss (BG) spatial distributions of the intensity: 2 1 cosj 2 2 E(r, j, 0) = ex p r /a , (18) sinj p a cosj 2 2 E(r, j, 0) = A ex p r /w J (gr) , (19) 0 0 sinj h i 1/2 2 1 1 2 2 1 2 2 where a is the radius of a Gaussian beam, A = ex p g w I g w , 0 0 0 p w 8 0 4 0 w /m = g is the effective width of the BG beam. B 1 These beams are linearly polarized, so they can be represented by the decomposition into the modal solutions with zero OAM (l = 0). The modal amplitude coefﬁcients for these incident beams can be calculated analytically. The expressions for modal coefﬁcients have the form: ( ) 2 2 a m 2a j 0 0 a = a = ex p , (20) pl pl 2 R J m 4R 0 1 p 2 h i A pw ag j 0 r 0 2 2 2 2 a = a = ex p a + g w /4 I w , (21) pl 0 0 pl R J m 2 0 1 p where m are the positive zeros of the Bessel function J (z), and I (z) is the modiﬁed Bessel 0 0 function of the ﬁrst kind. Note that polarization-invariant linear and circular polarized vector beams have no vortices (l = 0), i.e., such beams have non-zero intensity on the axis and no OAM. On the contrary, polarization-invariant radially and azimuthally polarized vector beams have non-zero angular orbital momentum and annular intensity distributions. Photonics 2022, 9, 162 7 of 16 Photonics 2022, 9, 162 3.1. Purity 7 of 15 Photonics 2022, 9, 162 7 of 15 The quantum-mechanical purity of a beam can be calculated using (14). Initial density ! ! matrix operator for a pure state is given by the incident vector beams: r = j Eih Ej. Purity Parameter Purity Parameter Purity Parameter In Figure 1, the dependences of the quantum-mechanical purity 𝜌 and the purity In Figure 1, the dependences of the quantum-mechanical purity 𝜌 and the purity In Figure 1, the dependences of the quantum-mechanical purity S pr and the purity parameter (impurity) 𝜁= 1 − 𝑝𝑆𝜌 on the propagation distance are presented for differ- parameter (impurity) 𝜁= 1 − 𝑝𝑆𝜌 on the propagation distance are presented for differ- parameter (impurity) z = 1 S pr on the propagation distance are presented for differ- ent widths of linearly polarized Gaussian beam with zero OAM. Figure 1b shows the de- ent widths of linearly polarized Gaussian beam with zero OAM. Figure 1b shows the de- ent widths of linearly polarized Gaussian beam with zero OAM. Figure 1b shows the pendence of the quantum purity parameter of the beam on the distance. It can be seen that pendence of the quantum purity parameter of the beam on the distance. It can be seen that dependence of the quantum purity parameter of the beam on the distance. It can be seen if the beam is in a pure state in the initial plane, then, during propagation, the beam trans- if the beam is in a pure state in the initial plane, then, during propagation, the beam trans- that if the beam is in a pure state in the initial plane, then, during propagation, the beam forms into a mixed state. forms into a mixed state. transforms into a mixed state. 0.6 0.6 1.0 1.0 1 1 1 1 1 1 0.4 0.4 0.8 0.8 0.2 0.6 0.2 0.6 2 3 0.4 0.0 0.4 0 5 10 15 20 25 30 35 0.0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 z, mm z, mm z, mm z, mm (a) (b) (a) (b) Figure 1. The purity (a) and impurity (b) as function of distance. 1 − 𝑎 = 30 μm; 2 − 𝑎 = 50 μm; 3 Figure 1. The purity (a) and impurity (b) as function of distance. 1 − 𝑎 = 30 μm; 2 − 𝑎 = 50 μm; 3 Figure 1. The purity (a) and impurity (b) as function of distance. 1 a = 30 m; 2 a = 50 m; 0 0 − 𝑎 = 100 μm. λ = 0.63 μm. − 𝑎 = 100 μm. λ = 0.63 μm. 3 a = 100 m. l = 0.63 m. In Figure 2a, the dependence of the impurity on distance is shown for a beam radius In F In Figur igure 2a e 2a, , the dependence of the dependence of the i them impurity purity on di on distance stance isis sh shown own fo for r a bea a beam m radi radius us 𝑎 = 15 μm. In Figure 2b, the dependence of an entropy of a Gaussian beam as function 𝑎 = 15 μm. In Figure 2b, the dependence of an entropy of a Gaussian beam as function a = 15 m. In Figure 2b, the dependence of an entropy of a Gaussian beam as function of of propagation distance is shown. It can be seen that there is a significant change in en- of p propagation ropagation dist distance ance is is shown. shown. It It can cabe n b seen e seen t thathther at th eere is is a signiﬁcant a significant change chanin ge in entr en- opy tropy at a length of the order of the diffraction length. tropy a at a length t a length of of the or the order of the dif der of the diffraction fracti length. on length. 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 5 10 15 0 5 10 15 z, mm z, mm (a) (b) (a) (b) Figure 2. (a) Impurity as function of distance. (b) Entropy as function of distance. 𝑎 = 15 μm. Figure 2. (a) Impurity as function of distance. (b) Entropy as function of distance. a = 15 m. Figure 2. (a) Impurity as function of distance. (b) Entropy as function of distance. 𝑎 = 15 μm. 3.2. Degree of Polarization 3.2. Degree of Polarization 3.2. Degree of Polarization 3.2.1. Linear Polarization 3.2.1. Linear Polarization 3.2.1. Linear Polarization The degree of polarization can be calculated using the expression (15). Figure 3a shows The degree of polarization can be calculated using the expression (15). Figure 3a The degree of polarization can be calculated using the expression (15). Figure 3a the change in the degree of polarization of linearly polarized Gaussian light beams with a shows the change in the degree of polarization of linearly polarized Gaussian light beams shows the change in the degree of polarization of linearly polarized Gaussian light beams distance for different beam radii. It can be seen from the ﬁgure that a strong change in the with a distance for different beam radii. It can be seen from the figure that a strong change with a distance for different beam radii. It can be seen from the figure that a strong change degree of polarization occurs at distances of the order of the diffraction length l = ka /2. in the degree of polarization occurs at distances of the order of the diffraction length 𝑙 = in the degree of polarization occurs at distances of the order of the diffraction length 𝑙 = This means that the focused beams are depolarized at a shorter distance. Figure 3b shows 𝑘𝑎 ⁄2. This means that the focused beams are depolarized at a shorter distance. Figure 𝑘𝑎 ⁄2. This means that the focused beams are depolarized at a shorter distance. Figure 3b shows the change in the degree of polarization of a linearly polarized Gaussian light 3b shows the change in the degree of polarization of a linearly polarized Gaussian light beam with a distance for the beam radius 𝑎 = 15 mm. beam with a distance for the beam radius 𝑎 = 15 mm. Purity Purity Entropy Entropy Impurity Impurity 𝑆𝑝 𝑆𝑝 Photonics 2022, 9, 162 8 of 16 Photonics 2022, 9, 162 8 of 15 Photonics 2022, 9, 162 8 of 15 the change in the degree of polarization of a linearly polarized Gaussian light beam with a distance for the beam radius a = 15 mm. 1.0 1.0 1.0 0.8 1.0 0.8 0.8 0.6 0.8 0.6 0.6 0.4 0.6 0.4 0.4 0.2 0.4 0.2 0.2 0.0 0.2 0.0 0 5 10 15 20 25 30 35 0 5 10 15 0.0 z, mm 0.0 z, mm 0 5 10 15 20 25 30 35 0 5 10 15 z, mm (a) (b z,) mm (a) (b) Figure 3. (a) Degree of polarization as function of distance. 1 − 𝑎 = 30 μm; 2 − 𝑎 = 50 μm; 3 − 𝑎 = 100 μm. (b) Degree of polarization as function of distance. 𝑎 =15 μm. Figure 3. (a) Degree of polarization as function of distance. 1 − 𝑎 = 30 μm; 2 − 𝑎 = 50 μm; 3 − 𝑎 Figure 3. (a) Degree of polarization as function of distance. 1 a = 30 m; 2 a = 50 m; 0 0 = 100 μm. (b) Degree of polarization as function of distance. 𝑎 =15 μm. 3 a = 100 m. (b) Degree of polarization as function of distance. a = 15 m. 0 0 Note that a coherence-induced decrease in the degree of polarization with the prop- agation distance for partially coherent beams in free space was shown in [19,25,26]. It was Note that a coherence-induced decrease in the degree of polarization with the prop- Note that a coherence-induced decrease in the degree of polarization with the prop- found that for a completely spatially coherent beam, its degree of polarization does not agation distance for partially coherent beams in free space was shown in [19,25,26]. It was agation distance for partially coherent beams in free space was shown in [19,25,26]. It change during propagation [19,25,26]. The spectral degree of polarization remains propa- found that for a completely spatially coherent beam, its degree of polarization does not was found that for a completely spatially coherent beam, its degree of polarization does gation-invariant along the axis of a Gaussian Shell-model beam [27,28]. Here, we show change d not change urin during g proppr agopagation ation [19,25 [,19 26,] 25 . T ,26 he sp ]. The ectra spectral l degree of p degree olaof rizat polarization ion remains p remains ropa- that the decrease in the degree of polarization occurs also for spatially coherent beams. gat propagation-invariant ion-invariant along t along he axis o the faxis a Gof auss a Gaussian ian Shell-m Shell-model odel beam beam [27,28]. [27 Here, ,28]. we show Here, we Unlike the coherence-induced mechanism of polarization changes, the decrease in the de- show that the decrease in the degree of polarization occurs also for spatially coherent that the decrease in the degree of polarization occurs also for spatially coherent beams. gree of polarization with distance arises also due to the interference between the modes Unlike beams. the co Unlike herence-induced mech the coherence-induced anism of po mechanism larization ch of polarization anges, the decre changes,athe se in the d decrease e- having different propagation constants. A decrease in the degree of correlation between gree of in the degr pola ee rizat of polarization ion with distwith ance distance arises also arises due to the interfe also due to theren interfer ce between the modes ence between the different modes takes place due to a difference in propagation velocities of Bessel vector modes having different propagation constants. A decrease in the degree of correlation having different propagation constants. A decrease in the degree of correlation between modes with different radial indices. between different modes takes place due to a difference in propagation velocities of Bessel different modes takes place due to a difference in propagation velocities of Bessel vector vector Note tha modes with t for very sm differentaradial ll radii indices. of the incident beam (𝑎 << λ), the two-dimensional modes with different radial indices. descript Note ion o that f pol for a very rizatsmall ion is incomp radii oflet the e. It incident was sho beam wn in [ (a 6<< 6,67] ), th the at knowled two-dimensional ge of the Note that for very small radii of the incident beam (𝑎 0 << λ), the two-dimensional description vector polari of zat polarization ion of radiat is ion incomplete. alone is not It s was uffic shown ient to f in ul [l66 y det ,67]ethat rmine t knowledge he polariz of atthe ion description of polarization is incomplete. It was shown in [66,67] that knowledge of the vector state of polarization a focused lof ight radiation beam. Add alone itiis onnot al p suf ara ﬁcient meters ( to tfully ensor pola determine rizatithe on components) polarization vector polarization of radiation alone is not sufficient to fully determine the polarization state of a focused light beam. Additional parameters (tensor polarization components) should be taken into account. It is shown in [66] that the vector and tensor degrees of state of a focused light beam. Additional parameters (tensor polarization components) should be taken into account. It is shown in [66] that the vector and tensor degrees of polarization are sufficient for a complete description of the three-component field of a should be taken into account. It is shown in [66] that the vector and tensor degrees of polarization are sufﬁcient for a complete description of the three-component ﬁeld of a light beam. In Figure 4, the dependences of the degree of polarization on the propagation polarization are sufficient for a complete description of the three-component field of a light beam. In Figure 4, the dependences of the degree of polarization on the propagation distance are shown for Gaussian and BG beams with approximately similar effective light beam. In Figure 4, the dependences of the degree of polarization on the propagation distance are shown for Gaussian and BG beams with approximately similar effective widths. widths. distance are shown for Gaussian and BG beams with approximately similar effective widths. 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0 5 10 15 0.0 z, mm 0 5 10 15 z, mm Figure Figure 4. 4. Degr Degree ee of polar of polarization ization as function as function of of distance. distance. G Gaussian aussian beam beam (b (black lack line) line), , 𝑎 a = = 30 30 μm; m; BG beam (red line), 𝑤 = 200 μm, 𝑤 = 40 μm. BG beam (red line), w = 200 m, w = 40 m. 0 B Figure 4. Degree of polarization as function of distance. Gaussian beam (black line), 𝑎 = 30 μm; BG beam (red line), 𝑤 = 200 μm, 𝑤 = 40 μm. Degree of polarization Degree of polarization Degree of polarization Degree of polarization Degree of polarization Degree of polarization Photonics 2022, 9, 162 9 of 16 Photonics 2022, 9, 162 9 of 15 It can be seen that a faster decrease in the degree of polarization occurs for the Gaussian It can be seen that a faster decrease in the degree of polarization occurs for the Gauss- beam (17). The degree of polarization of the BG beam (18) decreases more slowly than for a ian beam (17). The degree of polarization of the BG beam (18) decreases more slowly than Gaussian beam if w w . Indeed, in this case, the BG beam becomes more identical to for a Gaussian beam 0 if 𝑤 ≫ B 𝑤 . Indeed, in this case, the BG beam becomes more identi- the Bessel beam. The intensity proﬁles of the incident Gaussian (17) and Bessel–Gauss (18) cal to the Bessel beam. The intensity profiles of the incident Gaussian (17) and Bessel– beams are shown in Figure 5. Gauss (18) beams are shown in Figure 5. 1.0 0.8 0.6 0.4 0.2 0.0 050 100 150 r, μm Figure 5. Intensity proﬁles of Gaussian (black line) and BG (red line) beams. a = 30 m, w = 200 m, Figure 5. Intensity profiles of Gaussian (black line) and BG (red line) beams. 𝑎 = 30 μm, 𝑤 = 200 0 0 μm, 𝑤 = 40 μm. w = 40 m. 3.2.2. Radial Polarization 3.2.2. Radial Polarization Consider the radially polarized incident vector beams at z = 0 with annular Gaussian Consider the radially polarized incident vector beams at z = 0 with annular Gaussian and Bessel–Gauss spatial distributions of the intensity and OAM l = 1: and Bessel–Gauss spatial distributions of the intensity and OAM l = 1: ⃗ 2 r 1 𝐸 (𝑟, 𝜑, 0 ) = 𝑒𝑥𝑝 (−𝑟 2 /𝑎 2 ) , (22) E(r, j, 0) = p ex p r /a , (22) p a 𝐸 (𝑟, 𝜑, 0 ) = 𝐴 𝑒𝑥𝑝 (−𝑟 /𝑤 ) 𝐽 (𝛾𝑟 ) , (23) 2 2 E(r, j, 0) = A ex p r /w J (gr) , (23) 0 1 where 𝑎 is the radius of a Gaussian beam, 𝐴 = 𝑒𝑥𝑝 𝛾 𝑤 𝐼 𝛾 𝑤 , h i 1/2 and 𝑤 /𝜇 =𝛾 is the effective width of the BG beam. 2 1 1 1 2 2 2 2 where a is the radius of a Gaussian beam, A = ex p g w I g w , 0 0 1 0 0 p w 8 4 For the modal coefficients we have and w /m = g is the effective width of the BG beam. B 1 For the modal coefﬁcients we have 𝑎 =𝑎 = 𝑒𝑥𝑝 − , (24) ( ) ( ) 2 2 a m a m p p j √ 0 r 0 a = a = ( ex p ) , (24) 𝑎 =𝑎 = 𝑒𝑥𝑝 − 𝛼 +𝛾 𝑤 /4 𝐼 𝑤 , (25) pl pl 2 2 R J m 4R 2 p 0 0 where 𝜇 are the positive zeros of the Bessel function 𝐽 (𝑧) , and 𝐼 (𝑧 ) is the modified 2 h i A pw ag j 0 r 0 2 2 2 2 Bessel function of the first kind. a = a = ex p a + g w /4 I w , (25) pl 0 0 pl R J m 0 2 p In Figure 6a, the dependences of the degree of polarization on the propagation dis- tance are presented for the beams with annular Gaussian intensity distribution (21) and where m are the positive zeros of the Bessel function J (z), and I (z) is the modiﬁed Bessel p 1 1 OAM l = 1. function of the ﬁrst kind. It can be seen that the smaller the radius of the beam, the smaller the distance at In Figure 6a, the dependences of the degree of polarization on the propagation dis- which the noticeable depolarization of the beam occurs. In Figure 6b, the degrees of po- tance are presented for the beams with annular Gaussian intensity distribution (21) and larization as function of distance are presented for Gaussian (21) and BG (22) beams with OAM l = 1. OAM l = 1 and with approximately similar effective widths (Figure 7). It can be seen that the smaller the radius of the beam, the smaller the distance at which It follows that the decrease in the degree of polarization of the BG beam (22) occurs the noticeable depolarization of the beam occurs. In Figure 6b, the degrees of polarization much slower than for a Gaussian beam (21) if the value 𝑤 exceeds the effective radius as function of distance are presented for Gaussian (21) and BG (22) beams with OAM l = 1 of the BG beam 𝑤 . In this case, the behavior of the BG beam is similar to the vector Bessel and with approximately similar effective widths (Figure 7). beam, which is a modal solution. The closer the beam is to the modal solution, the weaker It follows that the decrease in the degree of polarization of the BG beam (22) occurs the beam depolarization with distance. much slower than for a Gaussian beam (21) if the value w exceeds the effective radius of the BG beam w . In this case, the behavior of the BG beam is similar to the vector Bessel Photonics 2022, 9, 162 10 of 16 Photonics 2022, 9, 162 10 of 15 Photonics 2022, 9, 162 10 of 15 beam, which is a modal solution. The closer the beam is to the modal solution, the weaker the beam depolarization with distance. 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 5 10 15 20 25 30 35 0 5 10 15 0 5 10 15 20 25 30 35 0 5 10 15 z, mm z, mm z, mm z, mm (a) (b) (a) (b) Figure 6. (a) Degree of polarization as function of distance. 1 − 𝑎 = 50 μm; 2 − 𝑎 = 70 μm; 3 − 𝑎 Figure 6. (a) Degree of polarization as function of distance. 1 a = 50 m; 2 a = 70 m; Figure 6. (a) Degree of polarization as function of distance. 1 − 𝑎 = 50 0 μm; 2 − 𝑎 = 70 0μm; 3 − 𝑎 = 100 μm. λ = 0.63 μm. (b) Degree of polarization as function of distance. Gaussian beam (black line), 3 a = 100 m. = 0.63 m. (b) Degree of polarization as function of distance. Gaussian beam = 100 μm. λ = 0.63 μm. (b) Degree of polarization as function of distance. Gaussian beam (black line), 𝑎 = 45 μm; BG beam (red line), 𝑤 = 70 μm; 𝑤 = 200 μm. λ = 0.63 μm. (black 𝑎 = 45 line), μm;a BG = b 45 eam m; (red l BG beam ine), 𝑤 (red = line), 70 μm; w 𝑤 = = 70 200 m;μ w m.= λ200 = 0.63 m. μm. = 0.63 m. 0 0 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 050 100 150 050 100 150 r, μm r, μm Figure 7. Intensity profiles of Gaussian (black line) and BG (red line) beams. 𝑎 = 45 μm; 𝑤 = 200 Figure 7. Intensity profiles of Gaussian (black line) and BG (red line) beams. 𝑎 = 45 μm; 𝑤 = 200 Figure 7. Intensity proﬁles of Gaussian (black line) and BG (red line) beams. a = 45 m; w = 200 m, 0 0 μm, 𝑤 = 70 μm. μm, 𝑤 = 70 μm. w = 70 m. It should be noted that the simulation results presented above correspond to the de- It should be noted that the simulation results presented above correspond to the degree It should be noted that the simulation results presented above correspond to the de- gree of polarization, which is averaged over the cross section of the beam. The values of of polarization, which is averaged over the cross section of the beam. The values of the gree of polarization, which is averaged over the cross section of the beam. The values of the local degree of polarization determined at a given point of the transverse plane are local degree of polarization determined at a given point of the transverse plane are usually the local degree of polarization determined at a given point of the transverse plane are usually considered. It is known that the local degree of polarization varies with displace- considered. It is known that the local degree of polarization varies with displacement from usually considered. It is known that the local degree of polarization varies with displace- ment from the axis of the beam, i.e., the polarization is inhomogeneous in cross section the axis of the beam, i.e., the polarization is inhomogeneous in cross section [19]. However, ment from the axis of the beam, i.e., the polarization is inhomogeneous in cross section [19]. However, the average degree of polarization is usually determined during measure- the average degree of polarization is usually determined during measurements. [19]. However, the average degree of polarization is usually determined during measure- ments. ments. 4. Discussion 4. Discussion Thus, polarization-invariant hybrid vector Bessel beams with phase and polariza- 4. Discussion tion singularities are proposed, which combine spin and OAM and are the solutions of Thus, polarization-invariant hybrid vector Bessel beams with phase and polarization Thus, polarization-invariant hybrid vector Bessel beams with phase and polarization sing Maxwell’s ularities equations are propose in d fr,ee which combine spin space. These modal and solutions OAM an with d are the solution discrete azimuthal s of Mand ax- singularities are proposed, which combine spin and OAM and are the solutions of Max- radial indices form a complete set of mutually orthogonal functions. Consequently, an arbi- well’s equations in free space. These modal solutions with discrete azimuthal and radial well’s equations in free space. These modal solutions with discrete azimuthal and radial trary vector beam in the initial plane z = 0 can be decomposed into these modal solutions, indices form a complete set of mutually orthogonal functions. Consequently, an arbitrary indices form a complete set of mutually orthogonal functions. Consequently, an arbitrary the evolution of which is determined by the propagation constant b . The modal approach vector beam in the initial plane 𝑧= 0 can be decomposed into these moda pl l solutions, the vector beam in the initial plane 𝑧= 0 can be decomposed into these modal solutions, the provides clear physical insight into the depolarization mechanism of vector beams in free evolution of which is determined by the propagation constant 𝛽 . The modal approach evolution of which is determined by the propagation constant 𝛽 . The modal approach space and computational simpliﬁcation in the analysis. provides clear physical insight into the depolarization mechanism of vector beams in free provides clear physical insight into the depolarization mechanism of vector beams in free space and computational simplification in the analysis. space and computational simplification in the analysis. One of the important properties of hybrid vector beams, which are the modal solu- One of the important properties of hybrid vector beams, which are the modal solu- tions, is the entanglement of phase and spin degrees of freedom describing the orbital tions, is the entanglement of phase and spin degrees of freedom describing the orbital angular momentum and the polarization vector in orthogonal basis vectors of circular angular momentum and the polarization vector in orthogonal basis vectors of circular Degree of polarization Intensity Intensity Degree of polarization Degree of polarization Degree of polarization Photonics 2022, 9, 162 11 of 16 One of the important properties of hybrid vector beams, which are the modal solu- tions, is the entanglement of phase and spin degrees of freedom describing the orbital angular momentum and the polarization vector in orthogonal basis vectors of circular polarization. These complex modes cannot simply be expressed through the TE and TM modes. The entanglement property is inherent in quantum systems and may have im- plications for future quantum networks [68–71]. In [70], it was demonstrated that the entanglement, or non-separability, between the spatial and polarization degrees of freedom also experiences self-healing. Hybrid vector modes have non-separable spatial, phase and polarization degrees of freedom. Hence, there is an analogy between the non-separability of such modes and entanglement of quantum states. Here, we found that quantum-mechanical purity is a measure of the average degree of polarization. In turn, the purity is related to the entropy. Note that the vector beam with separable spin and OAM degrees of freedom in the initial plane z = 0 transforms into the polarization-OAM entanglement beam during propagation [72,73]. Degree of polarization of pure states, which are the modal solutions of the Maxwell equations, remains invariant during propagation, i.e., it does not change on propagation. This indicates that the purity parameter and entropy for the modal solutions remain unchanged. A decrease in the degree of polarization with distance arises due to the interference between the modes having different propagation constants. Due to differences in propagation velocities of Bessel vector modes with different radial indices, a decrease in the degree of correlation between the different modes takes place. The more propagating modes are excited, the more the degree of polarization decreases with distance. Note, it was shown in [24–28], that the change in the degree of polarization of the beam is inﬂuenced by the coherence properties of the ﬁeld in the source plane. Here, we have demonstrated that, unlike the depolarization of partially coherent beams, the depolarization of coherent beams occurs due to interference between propagating modes. Here we considered 2 2 matrices to describe the polarization of vector beams. Indeed, in conventional paraxial optics, the ﬁelds are transverse, so the polarization is well described by Stokes parameters determined from the 2 2 coherency matrix [1–3]. Although the electromagnetic ﬁeld in the far zone is transverse, many problems of near-ﬁeld optics, data storage, optical microscopy, and focusing and scattering need to be analyzed, taking into account the longitudinal component of the ﬁeld E . For strongly focused beams, the Stokes 2 2 formalism should be extended to 3 3 coherency matrices [74–79]. In [67,72,73], the evolution of three-dimensional electromagnetic ﬁelds in a graded-index medium was studied. The polarization state of the most general two-dimensional ﬁeld can be completely described by three parameters. This is related to as vector polarization. For a three-dimensional ﬁeld except for 3 components of the vector polarization, there are 5 components of tensor polarization. The complexity of 3D electromagnetic ﬁeld arises from the tensor polarization [66,67]. In contrast to the 2D case, where any beam can be represented as the superposition of totally polarized and unpolarized light, in the 3D case, an arbitrary beam cannot be represented as a mixture of an unpolarized beam (one parameter) and a polarized beam (5 parameters) because nine independent parameters are needed to characterize the coherence matrix. We have considered the polarization properties of coherent vector beams. Analysis of the propagation of partially polarized and partially coherent beams is important for elucidating the physical mechanisms of depolarization. Future research will be related to polarization effects of partially coherent vector vortex beams [80–84]. Of particular interest is the study of the effects of non-paraxiality in the propagation of focused vector beams [85–98], of evanescent waves [99,100] and the effects of coherence and polarization of pulsed beams [101]. Other important topics are the polarization effects in plasmonic structures, including subwavelength gratings [102], multilayer devices and microlens arrays for imaging systems [103–108], diffractive optical elements [109–111], etc. New Photonics 2022, 9, 162 12 of 16 directions in technology are opening up thanks to the possibilities of controlling the spin angular momentum and Berry phase [112,113]. Recently, new developments based on the transverse spin phenomenon, which occurs in focused ﬁelds, have been of great interest [114–118]. New optical phenomena associated with the angular momentum have been demonstrated in [119]. It was shown in [115] that transverse spin manifests itself even in completely unpolarized ﬁelds. There is a connection between 3D polarization in the transverse spin [117] and non-paraxial ﬁelds [118] owing to the spin–orbit interaction of light. These phenomena can be used in telecommunications, nanoelectronics, optical imaging, etc. [119,120]. 5. Conclusions In summary, new hybrid vector Bessel beams with polarization-OAM entanglement, which are the modal solutions of the Maxwell equations, are proposed to study the evo- lution of vector beams in free space. Non-separability of degrees of freedom associated with the spatial coordinates, spin (polarization) and OAM (phase) is the main feature of the vector vortex beams which are the modal solutions. The vector mode decomposition method was developed for the analysis of the evo- lution of the polarization of arbitrary transverse incident beams with different spatial intensity distributions. Using the decomposition of the ﬁeld of an arbitrary incident vector beam into the hybrid modal solutions, the change in the 2D degree of polarization, the quantum purity parameter and the entropy of the beam with distance was investigated. The change in the degree of polarization of vector vortex beams on propagation in free space due to interference between different propagating modes has been demonstrated. A signiﬁcant decrease in the degree of polarization occurs at the diffraction length l for incident beams that are not vector modal solutions. Close connection of the degree of polarization with the quantum-mechanical purity parameter is emphasized. The results obtained may be useful in free-space optical communications, singular optics, photonics, optical coherence tomography, imaging, and may have implications for future quantum networks. Funding: This research was funded by the Ministry of Science and Higher Education of the Russian Federation under the State contract FFNS-2022-0009 and by the Russian Foundation for Basic Research, project number 19-29-11026. Conﬂicts of Interest: The author declares no conﬂict of interest. References 1. Wolf, E. 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Photonics – Multidisciplinary Digital Publishing Institute

**Published: ** Mar 6, 2022

**Keywords: **polarization; degree of polarization; depolarization; purity parameter; vortex light; vector Bessel modes; phase and polarization singularities; polarization invariance; entropy

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