Propagation of coherence-OAM matrix of an optical beam in vacuum and turbulence

Fei Wang; Weihao Li; Dan Wu; Lin Liu; Olga Korotkova; Olga Korotkova; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.489324

1. Introduction

Vortex beam is a highly directional optical field with spiral phase distribution, exp(ilφ). It is known to carry the well-defined orbital angular momentum (OAM), lh / 2π per photon, where l is the topological charge, φ is the azimuthal variable in the polar coordinate system and h is the Plank constant [1]. Therefore, vortex beams are also referred to as OAM carrying beams. The spatial intensity profile of a vortex beam exhibits unique doughnut shape because its phase singular (undefined) at the on-axis point. In addition, vortex phases with different topological charges are mutually orthogonal in the Hilbert space, which offers additional degrees of freedom in light-field manipulations. Due to such peculiar characteristics, vortex beams have found diverse applications in optical trapping, quantum information, object detection/identification, optical communications, etc. [2–11].

Any deterministic beam-like field can be represented as a linear superposition of its OAM modes, i.e., by a vector in the spiral phase Fourier basis [12]. In partially coherent fields, different OAM modes may have non-trivial spatial correlations. However, the widely used two-point spatial correlation function (mutual intensity in space-time domain or cross-spectral density (CSD) in space-frequency domain) [13] cannot characterize such correlations directly. In order to resolve this issue, the Coherence-OAM (COAM) matrix was introduced [14], which makes it possible to characterize the correlations for pairs of the OAM modes, resolved at radial positions r₁ and r₂. The COAM matrix can be regarded as the generalized version of the previously introduced pair-mode OAM correlations [8,15,16] that ignored spatial radial resolution or assumed a specific basis in the radial variable. The mathematical properties of the COAM matrix and the method for its experimental measurement were addressed in detail in [14,17]. In addition, the relations between certain structured coherence states (including spatially uniform or non-uniform) and the structure of the COAM matrix were revealed and the method for designing new optical sources with genuine COAM matrices, based on Bochner’s theorem, was introduced [18–20]. The transformations of the COAM matrices passing through thin devices and spatially extended systems were also formulated [21,22].

Importantly, the phase of a light beam employed in the OAM-based free-space optical communication and remote sensing systems, operating in the presence of atmospheric turbulence, is randomly redistributed in the transverse cross-sections along the propagation path. Such phase degradations were shown to result in the OAM mode dispersion and crosstalk [23,24]. Over the past decades, the OAM mode diffusion of various kinds of laser (deterministic) beams in different types of atmospheric turbulence have been extensively studied both theoretically and experimentally [25–33]. However, these studies solely targeted evolution of spiral/OAM spectrum in turbulence, i.e., the energy contained in the individual OAM modes, neglecting the possible correlations among pairs of the OAM modes. Although in [22] the general propagation formula of the COAM matrix for deterministic or random light beams in linear random media was derived, its application to turbulence has not yet been revealed.

This paper’s aim is to reveal the new correlation phenomena occurring on propagation of the COAM matrix of random light beams (and deterministic ones, as a limiting case) in atmospheric turbulence, using the extended Huygens-Fresnel principle. The selection law which governs dispersion of the COAM matrix elements in isotropic turbulence is derived. Further, we develop a numerical simulation approach that involves multi-phase screen method, modal representation method and coordinate transformation to evaluate the COAM matrix’s content in free space and in atmospheric turbulence. The paper is organized as follows: (1) in section 2, the definition of the COAM matrix is briefly reviewed. (2) In section 3, the propagation formula for the COAM matrix based on the extended Huygens-Fresnel principle in atmospheric turbulence is established and the selection law is revealed. (3) In section 4, the numerical simulation method based on wave optical simulation is introduced and thoroughly described. (4) In section 5, the COAM matrix of the circular and elliptical Gaussian Schell-model (GSM) beam in turbulence are comparatively studied as numerical examples and section 6 concludes the work.

2. Definition of coherence-orbital angular momentum matrix in partially coherent beams

In this section, we briefly review the definition of the COAM matrix in partially coherent beams and its characteristics. Let U(r, z; ω) be a realization of a statistically stationary partially coherent beam, propagating along the z-axis, with r denoting the transverse position vector at propagation distance z and ω being the angular frequency of light. The second-order correlations of this field’s realizations are characterized by the CSD function $W({{\bf r}_1},{{\bf r}_2},z;\omega ) = \left\langle {{U^\ast }({{\bf r}_1},z;\omega )U({{\bf r}_2},z;\omega )} \right\rangle$, where the asterisk and angle brackets denote complex conjugate and ensemble average over the field fluctuations. The CSD characterizes the correlation state of the total field between two spatial points across the plane z, but it cannot directly express individual OAM mode-to-mode correlations. To characterize mode-to-mode correlations the realization U is first decomposed in terms of its angular Fourier harmonics with coefficients U_l(r, z), i.e.,

(1)$$U({\bf r},z;\omega ) = \sum\limits_{l ={-} \infty }^\infty {{U_l}(r,z;\omega ){e^{il\varphi }}} ,$$

where r and φ are radial and azimuthal variables in the polar coordinate system. Here l is an integer, denoting the topological charge. Hereafter, the angular frequency is suppressed for brevity. The COAM matrix describes the ensemble average of any two coefficients U_l(r₁, z) and U_m(r₂, z) at points r₁ and r₂, with its elements defined as: $W_{lm}^{(\textrm{OAM})}({r_1},{r_2},z) = \left\langle {U_l^\ast ({r_1},z){U_m}({r_2},z)} \right\rangle$ [14]. In general, the COAM matrix can be used to characterize the correlation of two coefficients between two points in 3D space, i.e., (r₁, z₁) and (r₂, z₂), but here we only focus on its properties in the transverse planes at fixed z. By applying Eq. (1) and with the help of the definition of the CSD and COAM matrix, one may establish the relations between them via the following formulae

(2)$$W({{\bf r}_1},{{\bf r}_2},z) = \sum\limits_{l ={-} \infty }^\infty {\sum\limits_{m ={-} \infty }^\infty {W_{lm}^{\textrm{(OAM)}}({r_1},{r_2},z){e^{ - il{\varphi _1} + im{\varphi _2}}}} }, $$

(3)$$W_{lm}^{\textrm{(OAM)}}({r_1},{r_2},z) = \frac{1}{{{{(2\pi )}^2}}}\int_0^{2\pi } {\int_0^{2\pi } {{e^{il{\varphi _1}}}W({{\bf r}_1},{{\bf r}_2},z){e^{ - im{\varphi _2}}}d{\varphi _1}d{\varphi _2}} } .$$

Owing to the fact that the in a Hilbert space topological charges can generally have infinitely many values, the COAM matrix is, therefore, a square, infinite-dimensional matrix. In practical situations, one must consider the truncated version of the COAM matrix with its dimension (l_max – l_min + 1) × (m_max – m_min + 1), where l(m) with subscript max and min represent the maximum and minimum number considered. Because of Hermiticity of the CSD, any two elements of the COAM matrix with indices l and m must satisfy relation $W_{lm}^{\mathrm{(OAM)\ast }}({r_1},{r_2},z) = W_{ml}^{\textrm{(OAM)}}({r_2},{r_1},z)$. It can also be shown that the COAM matrix is non-negative definite. In addition, one also can obtain the information of the total energy of each OAM mode from the COAM matrix by means of the integrals ${c_l} = 2\pi \int {rW_{ll}^{(\textrm{OAM})}(r,r,z)dr}$. The sum of all c_l coefficients must equal to the total energy of the beam in the cross-section of interest.

3. Propagation of the COAM matrix in free space and in turbulent atmosphere

3.1 Huygens-Fresnel integral method

The most straightforward approach for acquiring the COAM matrix on propagation in free space is to perform integration, as given in Eq. (3), provided that the CSD of a partially coherent beam in the plane z is already known. Nevertheless, if only the information of the CSD in a certain transverse plane, e.g., in the source plane (z = 0), is known, the CSDs in the observation plane and the source plane can be related by the Huygens-Fresnel integral within the paraxial approximation as [14]

(4)$$\begin{aligned} W({{\bf r}_1},{{\bf r}_2},z) &= \frac{{{k^2}}}{{{{(2\pi )}^2}{z^2}}}\int {\int {{d^2}{{\bf r}_{10}}{d^2}{{\bf r}_{20}}{W_0}({{\bf r}_{10}},{{\bf r}_{20}})} } \\ \textrm{ } &\times \textrm{exp} \left[ { - \frac{{ik}}{{2z}}{{({{{\bf r}_{10}} - {{\bf r}_1}} )}^2}} \right]\textrm{exp} \left[ {\frac{{ik}}{{2z}}{{({{{\bf r}_{20}} - {{\bf r}_2}} )}^2}} \right], \end{aligned}$$

where W₀ and r_i₀ (i = 1,2) is the CSD and transverse position vector in the source plane. Here k = 2π / λ is the wavenumber with λ being the wavelength. By inserting Eq. (4) into Eq. (3), we obtain the expression for the COAM matrix elements after integrating over φ₁₀, φ₂₀, φ₁, φ₂:

(5)$$\begin{aligned} W_{lm}^{(\textrm{OAM})}({r_1},{r_2},z) &= \frac{{{k^2}}}{{{z^2}}}\int_0^\infty {\int_0^\infty {\textrm{exp} \left[ { - \frac{{ik}}{{2z}}({r_{10}^2 - r_{20}^2 + r_1^2 - r_2^2} )} \right]} } \\ \textrm{ } &\times W_{0lm}^{(\textrm{OAM})}({r_{10}},{r_{20}}){J_l}\left( {\frac{{k{r_{10}}{r_1}}}{z}} \right){J_m}\left( {\frac{{k{r_{20}}{r_2}}}{z}} \right){r_{10}}{r_{20}}d{r_{10}}d{r_{20}}, \end{aligned}$$

where J_α is the first kind Bessel function of order α. In the derivation of Eq. (5), the following formula is applied

(6)$$\textrm{exp} (ix\sin \varphi ) = \sum\limits_{m ={-} \infty }^\infty {{J_m}(x)\textrm{exp} (im\varphi )} .$$

Equation (5) implies that in free-space propagation each element of the COAM matrix obeys the propagation law independently. Calculation using Eq. (5) involves double integral operation in the source plane or additional double integral operation is required if the analytical expression of the COAM matrix in the source is unknown.

Let us now discuss propagation of the COAM matrix in atmospheric turbulence. The propagation of a partially coherent beam in homogeneous atmosphere can be treated by the extended Huygens-Fresnel integral [34]

(7)$$\begin{aligned} W({{\bf r}_1},{{\bf r}_2},z) &= \frac{{{k^2}}}{{{{(2\pi )}^2}{z^2}}}\int {\int {{d^2}{{\bf r}_{10}}{d^2}{{\bf r}_{20}}{W_0}({{\bf r}_{10}},{{\bf r}_{20}})\textrm{exp} \left[ { - \frac{{ik}}{{2z}}{{({{{\bf r}_{10}} - {{\bf r}_1}} )}^2}} \right]} } \\ \textrm{ } &\times \textrm{exp} \left[ {\frac{{ik}}{{2z}}{{({{{\bf r}_{20}} - {{\bf r}_2}} )}^2}} \right]{\left\langle {\textrm{exp} [{{\psi^\ast }({{\bf r}_{10}},{{\bf r}_1},z) + \psi ({{\bf r}_{20}},{{\bf r}_2},z)} ]} \right\rangle _t}, \end{aligned}$$

where the angle brackets with subscript t denotes the ensemble average over the fluctuations of atmospheric turbulence. Here $\psi ({{\bf r}_{10}},{{\bf r}_1},z)$ represents the complex phase perturbations of a spherical wave propagating from (r₁₀, 0) to (r₁, z). Assuming that the turbulence is also isotropic and is governed by the Kolmogorov power spectrum, the second-order statistics of the complex phase perturbation can be expressed, under the quadratic approximation, in the form [35]

(8)$${\left\langle {\textrm{exp} [{{\psi^\ast }({{\bf r}_{10}},{{\bf r}_1},z) + \psi ({{\bf r}_{20}},{{\bf r}_2},z)} ]} \right\rangle _t} = \textrm{exp} \left( { - \frac{{{\bf r}_d^2 + {{\bf r}_d}\cdot {{\bf r}_{d0}} + {\bf r}_{d0}^2}}{{\rho_0^2}}} \right),$$

where r_d = r₂ - r₁ and r_d₀= r₂₀ - r₁₀; ${\rho _0} = {({0.545C_n^2{k^2}z} )^{ - 3/5}}$ is the coherence width of a spherical wave propagating in atmosphere, where C_n² is the structure constant, a measure of the local strength of turbulence. On substitution from Eqs. (7,8) into Eq. (3) and after integration, the elements of the COAM matrix become (see Appendix A)

(9)$$W_{lm}^{(\textrm{OAM})}({r_1},{r_2},z) = \frac{{{k^2}}}{{{{(2\pi )}^4}{z^2}}}\sum\limits_{p ={-} \infty }^\infty {\sum\limits_{q ={-} \infty }^\infty {\int_0^\infty {\int_0^\infty {W_{0pq}^{(\textrm{OAM})}({r_{10}},{r_{20}}){H_{pqlm}}({r_1},{r_2},{r_{10}},{r_{20}})} } d{r_{10}}d{r_{20}}} } ,$$

where H_pqlm can be viewed as the Pair-Impulse Response (PAIR) matrix [22] of the atmospheric channel [see Appendix A, Eq. (A5)]. In the process of the derivation of Eq. (9), the OAM indices in the source and output planes must satisfy the condition [also see Eq. (A6)]

(10) $$m - l = q - p.$$

Equation (10) reveals the dispersion mechanism of the PAIR matrix of a homogeneous and isotropic turbulent medium. Unlike free space, producing no coupling among the COAM matrix elements, a homogeneous and isotropic turbulent medium disperses any element at the input COAM matrix to those at the output plane that have the same index difference. We may term Eq. (10) the selection rule governing transformation of the COAM matrix in turbulence. Figure 1 illustrates the selection rule using a generic example of the input COAM matrix with non-trivial major and two minor diagonals. The differences of two indices of the elements marked with red, yellow and purple elements are 0, 3 and −3, respectively. According to selection rule, only coupling among the elements with the same color occurs in isotropic turbulence. The COAM matrix retains its three-diagonal structure upon propagation, as was set in the source. In other words, if the COAM matrix of the source has zero values for the elements W_0pq(r₁₀, r₂₀) with q – p = t, the output elements W_lm(r₁, r₂) with m – l = t still remain zero. We also note that Fig. 1 does not contain any information regarding power redistribution among the elements on three diagonals. As we will numerically show, as propagation distance and/or local strength of turbulence (C_n²) grows, the power is gradually driven from modes with more power to adjacent ones (on the same diagonal) with less power, gradually spilling into more and more modes.

Fig. 1. Dispersion of the COAM matrix propagating in isotropic turbulence under the selection rule. Red, yellow and purple squares represent the elements with indices differences 0, 3, and −3, respectively.

Download Full Size | PDF

Although Eqs. (9) and (10) establish the propagation law of the COAM matrix of the partially coherent beam in homogeneous and isotropic turbulence, the calculation of each element with each radial variable pair (r₁, r₂) should involve 8-fold summation operation and double integral operation, which is time consuming especially for acquiring the 2D distribution of each element. Therefore, in what follows, we succumb to wave-optics simulation for speedy while comprehensive illustrations of the discussed effects of turbulence on the COAM matrix.

3.2 Wave-optics numerical simulation

One of the widely used wave-optics simulation (WOS) methods for generating random realizations of optical beams propagating in atmospheric turbulence is the multi-phase screen (MPS) method [36] in which the split-step fast Fourier transform algorithm is applied. When one applies the MPS method to a partially coherent beam, the CSD in the source is first decomposed into incoherent superposition of modes,

(11)$${W_0}({{\bf r}_{10}},{{\bf r}_{20}}) = \sum\limits_{n = 1}^{{N_1}} {{\alpha _n}E_n^\ast ({{\bf r}_{10}}){E_n}({{\bf r}_{20}})} ,$$

where α_n are non-negative eigenvalues and N₁ is the number of modes in the truncated series (the theoretical sum generally has infinite upper limit). Note that the optical modes for representation of the CSD can be coherent modes, pseudo modes, random modes or others. The similarities and differences of the three aforementioned modal representations for Gaussian Schell-model (GSM) beams were studied in [37].

For each optical mode, the propagation scenario in the MPS method is illustrated in Fig. 2(a). The turbulence is modeled as a series of random phase screens with desired statistics, and the transmission path is equally separated into N_T sub-paths each with distance Δz = z / N_T. At the end of each sub-path, a random phase screen is placed to simulate the effect of the atmospheric turbulence in this sub-path. The optical mode propagates from the source to the end of the first sub-path without turbulence, multiplies the random phase to mimic turbulence in the first sub-path, then propagates to the next sub-path. This process is repeated, until the optical mode arrives at the observation plane. Hence, the optical mode with its data represented as a square matrix in the Cartesian coordinate system is obtained.

Fig. 2. (a) Computational propagation model for random beams in turbulent atmosphere. RPS_n denotes random phase screens. (b) Schematic diagram for mapping of the modes from Cartesian to polar coordinates and acquiring element U_nl. Repeating this mapping and changings topological charge l, a series of elements with different topological charges are obtained.

Download Full Size | PDF

In order to evaluate the COAM matrix, we should map the obtained data from Cartesian to polar coordinates. The process is shown in Fig. 2(b). At the first step, the obtained mode in Cartesian system is multiplied by exp(-ilφ) with φ = arctan(y / x), and then is mapped to the polar coordinate. Second, the element U_l(r, z) with column matrix (vector) is obtained after numerically integrating over φ. Repeating the above two steps and changing the topological charge l, a series of elements with different indices l are obtained. Finally, the elements of the COAM matrix in one optical mode and one realization of turbulence are acquired by W_nlm^(OAM)(r₁, r₂, z)= [U_nl*(r₁, z)]^TU_nm(r₂, z), where the superscript T denotes the transpose of the matrix.

In atmospheric optical propagation problems and applications, it is commonly assumed that the integration time of the detector (detector’s response) rate is much lower than the fluctuation rate of the partially coherent radiation but is much higher than that of turbulence. As a result, the detector can only detect fluctuations induced by the turbulence. Hence, the simulation procedure for evaluating the COAM matrix propagating in atmospheric turbulence is as follows: (1) N₁ number of optical modes for representing the CSD of a partially coherent beam are prepared and synthesized. (2) N_T number of random phase screens for simulating turbulence are synthesized and placed with equal distance in the propagation path. (3) Each mode synthesized in (1) propagates in turbulence with “frozen” phase screens, resulting in a propagated mode in the observation plane. The mode multiplied by vortex phase exp(-ilφ) is then mapped into the polar coordinate system and numerically integrated over φ, populating column matrix U_nl(r, z) with various indices l. The elements of the COAM matrix in one realization of turbulence are calculated as ${[W_{lm}^{(\textrm{OAM})}({r_1},{r_2},z)]_{{n_2}}} = \sum\limits_{n = 1}^{{N_1}} {{{[U_{nl}^\ast ({r_1},z)]}^T}{U_{nm}}({r_2},z)}$ with subscript n₂ (n₂ = 1, 2, 3…) denoting the elements corresponding to n₂ realization of the turbulence. (4) Loop over (2)-(3) for N₂ times is set and finally the COAM matrix is acquired by averaging over the fluctuations of turbulence: $W_{lm}^{(\textrm{OAM})}({r_1},{r_2},z) = \sum\limits_{{n_2} = 1}^{{N_2}} {{{[W_{lm}^{(\textrm{OAM})}({r_1},{r_2},z)]}_{{n_2}}}}$. It is worth mentioning here that Eqs. (7) and Eq. (8) are obtained under the condition of ensemble averaging over the fluctuations of turbulence. Hence, in order to compare the results between the extended Huygens-Fresnel integral and the WOS method, number N₂ must be large enough in step (4).

4. Numerical results

In this section, we examine propagation characteristics of the COAM matrix of circular and elliptical GSM beams both in free space and in turbulent atmosphere, and demonstrate the feasibility of the WOS method.

With the help of Eq. (7), the CSD of the (more general) elliptical GSM beams propagating at distance z in turbulence can be found, under the quadratic approximation, from analytical expression [38]

(12)$$W({{\bf r}_1},{{\bf r}_2},z) = {W_x}({x_1},{x_2},z){W_y}({y_1},{y_2},z),$$

with

(13)$${W_x}({x_1},{x_2},z) = \frac{1}{{\sqrt {{\Delta _x}} }}\textrm{exp} \left( { - \frac{{x_1^2 + x_2^2}}{{4\sigma_x^2}}} \right)\textrm{exp} \left[ { - \frac{{{{({{x_1} - {x_2}} )}^2}}}{{2\delta_x^2}}} \right]\textrm{exp} \left[ { - \frac{{ik(x_1^2 - x_2^2)}}{{2{R_x}}}} \right],$$

where ${\Delta _x} = 1 + \left[ {\frac{1}{{4{k^2}\sigma_0^4}} + \frac{1}{{{k^2}\sigma_0^2}}\left( {\frac{1}{{\delta_{x0}^2}} + \frac{2}{{\rho_0^2}}} \right)} \right]{z^2},$, ${\delta _x} = {\left( {\frac{1}{{\delta_{x0}^2{\Delta _x}}} + \frac{2}{{\rho_0^2}}\left( {1 + \frac{2}{{{\Delta _x}}}} \right) - \frac{{{z^2}}}{{{k^2}\sigma_0^2\rho_0^4{\Delta _x}}}} \right)^{ - 1/2}},\;{\sigma _x} = {\sigma _0}\sqrt {{\Delta _x}}$, and ${R_x} = z + \frac{{\sigma _0^2z - {z^3}/{k^2}\rho _0^2}}{{({{\Delta _x} - 1} )\sigma _0^2 + {z^2}/{k^2}\rho _0^2}}.$ Here σ₀ and δ_x₀ are the beam width and the coherence width along the x direction in the source plane, respectively. Component W_y has the same form as W_x (with replacement of x with y). We assume that in Eq. (12) the beam profile in the source plane is of circular Gaussian shape, but the coherence widths in x and y directions are different. In particular, for the circular GSM beam case, i.e., when δ_x₀ = δ_y₀= δ₀, the COAM matrix is given, by means of Eq. (2), by analytical expression:

(14)$$W_{ll}^{(\textrm{OAM})}({r_1},{r_2},z) = \frac{1}{{{\Delta _x}}}\textrm{exp} \left[ { - \left( {\frac{1}{{4\sigma_x^2}} + \frac{1}{{2\delta_x^2}}} \right)(r_1^2 + r_2^2)} \right]{I_l}\left( {\frac{{{r_1}{r_2}}}{{\delta_x^2}}} \right)\textrm{exp} \left[ {\frac{{ik(r_2^2 - r_1^2)}}{{2{R_x}}}} \right],$$

for l = m, while W_lm^(OAM)(r₁, r₂, z) = 0 otherwise. In this case, the COAM matrix remains diagonal on propagation in free space or in homogeneous and isotropic turbulent atmosphere. Evidently, it obeys the selection rule. In the elliptical GSM beam case, the analytical expression for the COAM matrix is not known, but one can study its propagation numerically via Eq. (2) or by the WOS method introduced in section 3.

In the WOS method, the CSD of the elliptical GSM beam is represented as the incoherent superposition of the Hermite-Gaussian (HG) modes (coherent mode representation) [39]. The accuracy of this representation is discussed in [37]. In free space (in the absence of turbulence), the numerical grid of 512 × 512 elements is chosen and only one screen located exactly in the middle of the path is set with constant unit value for all points.

First, let us examine the elements of the COAM matrix of the circular and elliptical GSM beam in the source plane (z = 0). In this case, we directly map the represented HG modes from the Cartesian coordinates to polar coordinates and process them as shown in Fig. 2(b). In order to gain insight into possible contributions of various HG modes to the COAM matrix, we show in Fig. 3 the color density plots of three elements W₀₀^(OAM), W₁₁^(OAM) and W₂₂^(OAM) calculated from four selected HG mode components, i.e., HG₀₀, HG₀₁, HG₀₂ and HG₁₁, of the circular GSM source with δ_x₀ = δ_y₀= 10 mm. The intensity patterns of these HG modes are also illustrated in Fig. 3(a)-(d). In the calculations the beam parameters are σ₀ = 10 mm and λ = 632.8 nm and are fixed in the following analysis unless other values are specified. As shown in Fig. 3(e)-(p), different HG modes play the distinct roles in contributing to the various COAM matrix elements. For instance, the HG₀₀ mode, also being the eigenstate of the OAM mode with l = 0, dominates the element W₀₀^(OAM), whereas no contribution is present to elements W₁₁^(OAM) and W₂₂^(OAM). On the other hand, element W₁₁^(OAM) mainly comes from modes HG₀₁ and HG₁₀, while W₂₂^(OAM) depends on modes HG₀₂, HG₂₀ and HG₁₁. Other high-order HG modes also have certain contributions to the COAM matrix, however, their impact gradually disappears since the mode weight decreases with the increase of mode order.

Fig. 3. (a)-(d) Intensity profiles of four HG modes in the circular GSM source with δ_x₀ = δ_y₀= 10 mm. (e)-(p) Density plots of three COAM matrix elements W₀₀^(OAM), W₁₁^(OAM) and W₂₂^(OAM) of the four HG modes presented in the corresponding columns.

Download Full Size | PDF

Figure 4 shows color density plots of three elements of the COAM matrix of the circular GSM source [(a)-(c)] and elliptical GSM sources [(d)-(i)]. The coherence widths in Figs. 4(a)-(c), 3(d)-(f) and 4(g)-(i) are δ_x₀ = δ_y₀= 10 mm, δ_x₀ = 2δ_y₀= 10 mm, and 2δ_x₀ = δ_y₀= 10 mm, respectively. Under this circumstance, the number of HG modes are chosen to be N₁ = 100, as a good approximation for the representation of the CSD. In circular GSM source case, the COAM matrix is diagonal, hence, three diagonal elements are presented in Figs. 3(a)-(c). It is shown that the patterns are symmetric with respect to (r₁, r₂). The maximum value of W₀₀^(OAM) is at r₁= r₂= 0, whereas the maximum value of other diagonal elements at this point is zero. This is only because the shape of radial mode with l = 0 has a solid core, others are doughnut-like owing to the phase singularity at on-axis point. The maximum value in W_ll^(OAM) decreases as the topological charge l increases, implying that the correlations between high-order OAM modes reduce as l increases. In elliptical GSM source case, the non-zero off-diagonal elements W₂₀^(OAM) appear [see Fig. 4(f) and 4(i)], while the element W₁₀^(OAM) still remains zero. It was already revealed in [19] that if the degree of coherence is real and has rectangular symmetry, the elements W_lm^(OAM) with l – m being odd number are trivial.

Fig. 4. Density plots of the elements of the COAM matrix of (a)-(c) circular GSM source with δ_x₀ = δ_y₀= 10 mm, (d)-(f) elliptical GSM source with δ_x₀ = 10 mm and δ_y₀= 5 mm, (g)-(i) elliptical GSM source with δ_x₀ = 5 mm and δ_y₀= 10 mm.

Download Full Size | PDF

To demonstrate the accuracy of the WOS method, the cross lines of W₀₀^(OAM) of the circular GSM source at r₁ = 10 mm and W₂₀^(OAM) of two elliptical GSM sources at r₁= 10 mm are shown in Fig. 5. The green solid curves and red dotted curves correspond to the numerical calculations from Eqs. (3) and the WOS method, respectively. The results from the WOS method agree well with those from direct integrations, except for the region where r is close to zero. This is because the number of discrete points near r = 0 in Cartesian coordinate system is much fewer than that near r = 0 in polar coordinate system, which leads to too large error in interpolation operation during the coordinate transformation.

Fig. 5. Variation of the elements of the COAM matrix of circular and elliptical GSM source with radial position r₂ with fixed r₁ = 10 mm. The solid green curves are calculated from integral in Eq. (3) directly. The red dotted curves are obtained from the WOS method (in the source, the mapping method is directly applied).

Download Full Size | PDF

We will now turn to evolution of the COAM matrix of the elliptical GSM beam with δ_x₀ = 10 mm and δ_y₀= 5 mm in vacuum. Since the spherical phase factor induced by propagation must be incorporated into the COAM matrix, the elements become complex-valued. Hence, we only investigate their absolute values on propagation. Figures 6(a)-(i) illustrate the 2D plots of three COAM matrix elements at different propagation distances. It is to be shown that the spatial profile of the diagonal elements |W₀₀^(OAM)| and |W₁₁^(OAM)| remain unchanged during propagation, while the size is enlarged due to the diffraction effect. The correlation strength (maximum value) is also weakened as the propagation distance increases. The off-diagonal element |W₂₀^(OAM)| changes its spatial shape upon propagation. Figures 6(j)-(l) show that the results obtained from the WOS method agree well with those from direct integrations.

Fig. 6. (a)-(i) Density plots of magnitudes of three COAM matrix elements of the elliptical GSM beam at different propagations distances. (j) –(i) The cross lines of the absolute of W₀₀^(OAM), W₁₁^(OAM) and W₂₀^(OAM) at r = r₁ = r₂. The solid curves are obtained from the WOS method. The dotted curves are calculated from direct integration.

Download Full Size | PDF

In the presence of turbulence, 10 phase screens are inserted in the propagation path and the power spectrum of refractive index fluctuations are adoped via the Kolmogorov spectrum ${\Phi _n}(\kappa ) = 0.033C_n^2{\kappa ^{ - 11/3}}$. The number of the realizations of turbulence is chosen to be N₂ = 600. Figure 7 presents the simulation results of the |W₀₀^(OAM)|, |W₁₁^(OAM)| and |W₁₀^(OAM)| of the elliptical GSM beam (δ_x₀ = 10 mm, δ_y₀ = 5 mm) in different strengths of turbulence at distance z = 1 km. One can be seen from Figs. 7(a)-(f) that with the increase of the structure constant (strength of the turbulence), not only the correlation area but also the correlation strengths are substantially reduced under the effect of turbulence. This phenomenon is quite different from that was observed in free space. The main reason for this is that turbulence deteriorates the coherence state of the propagated beam, resulting in the decrease of the coherence width/area of the beam. Therefore, the correlation area also decreases. In addition, the fluctuations of turbulence and of optical field are uncorrelated, hence turbulence acting as noise for the beam. As a consequence, the correlation strength drops with increase of the turbulence strength. In Figs. 7(g)-(i), it shows that the correlation strength between two OAM modes with topological charges 1 and 0 always maintains a low level (less than 0.004), irrespective of the structure constant. According to the selection law, this element is strictly zero in isotropic turbulence. Therefore, our numerical results support the theoretical analysis. The residual correlation may come from the accumulated errors of numerical simulations.

Fig. 7. (a)-(i) Density plots of the |W₀₀^(OAM)|, |W₁₁^(OAM)| and |W₁₀^(OAM)| of the elliptical GSM beam (δ_x₀ = 10 mm, δ_y₀ = 5 mm) propagating in atmospheric turbulence with different structures at distance z = 1 km. The C_n² in the first, second and third rows are 1 × 10⁻¹⁴m^−2/3, 5 × 10⁻¹⁴m^−2/3 and 1 × 10⁻¹³m^−2/3, respectively. (j)-(l) the corresponding cross lines of three elements at r = r₁ = r₂.

Download Full Size | PDF

In the preceding analysis, we have investigated the average behavior of the COAM matrix in terms of 600 realizations of atmospheric turbulence. Note that in most practical applications the detectors are fast and hence it is crucial to appreciate the difference between the instantaneous and averaged statistics. We now question the instantaneous behavior of the COAM matrix in a single realization of turbulence. In other words, we will assume that a partially coherent beam passes through the “frozen” turbulence once. Does the selection law hold in this situation? Evidently, both the amplitude and the phase of the beam will be severely distorted passing through such a “frozen” turbulence. As a consequence, the beam spot may split into several speckles if turbulence is strong enough. One of the advantages using the WOS method is that it allows us to examine the statistical characteristics of partially coherent beams for the single realization of turbulence.

In the simulation, the elliptical GSM beam with the parameter being the same as those in Fig. 7 is used. The propagation distance is fixed at z = 1 km and the structure constant is set to C_n² = 5 × 10⁻¹⁴m^−2/3. Figure 8 presents two examples of the instantaneous intensity patterns in the output plane captured from two different realizations of the turbulence. The beam profiles are significantly affected by these “frozen” turbulence realizations. The centroid of the beam spot is generally displaced from optical axis and it indeed may split into several speckles [Fig. 8(b)], although the average intensity pattern displays the elliptical Gaussian shape with a smooth profile.

Fig. 8. Two examples of instantaneous intensities of the elliptical GSM beam (δ_x₀ = 10 mm, δ_y₀ = 5 mm) at z = 1 km after passing through two different realizations of the atmospheric turbulence.

Download Full Size | PDF

Figures 9 and 10 illustrate the absolute values (first column), real parts (second column) and imaginary parts (third column) of three elements W₀₀^(OAM), W₁₁^(OAM) and W₁₀^(OAM) in two different realizations of turbulence, corresponding to Fig. 8(a) and (b), respectively. It can be seen that these elements substantially differ as the realizations of turbulence vary. The off-diagonal element W₁₀^(OAM) [see Figs. 9(g)-(i) and Figs. 10(g)-(i)] exhibits the appreciable value in contrast to those in W₀₀^(OAM) or W₁₁^(OAM). Nevertheless, it is to be expected that the average value (average over the realizations of the turbulence) will tend to zero, and therefore, the correlation between two radial modes, l = 1 and m = 0, disappears. We should emphasize here that the selection law only describes the statistical (average) behavior of the dispersion of the COAM matrix in isotropic turbulence. For individual realizations of turbulence, this law is violated. We also notice that some features of the COAM matrix elements in frozen turbulence visually resemble those previously obtained for the non-uniformly correlated (NUC) sources [18]. Although the analytical model of the NUC sources is very different from that of turbulence, their ability to focus the radiated field off-axis is similar to a possible beam centroid displacement produced by some individual turbulence realizations. It is more apparent for example considered in Fig. 8(a) and Fig. 9 rather than that in Fig. 8(b) and Fig. 10.

Fig. 9. The absolute values (first column), real parts (second column) and imaginary parts (third column) of the elements W₀₀^(OAM), W₁₁^(OAM) and W₁₀^(OAM) calculated from a realization of turbulence corresponding to the case in Fig. 8(a).

Download Full Size | PDF

Fig. 10. The absolute values (first column), real parts (second column) and imaginary parts (third column) of the elements W₀₀^(OAM), W₁₁^(OAM) and W₁₀^(OAM) calculated from a realization of turbulence corresponding to the case in Fig. 8(b).

Download Full Size | PDF

5. Conclusion

In summary, we have investigated propagation of the COAM matrices of partially coherent beams in free space and in turbulent atmosphere. Exact analytic formula for describing propagation of the COAM matrix is established in atmospheric turbulence with the help of the extended Huygens-Fresnel principle. Although the formula is rather complicated, involving 8-fold summations and 2-fold integrations, it contains an important relation, named as a selection law, which governs dispersion of the elements of the COAM matrix propagating in homogeneous and isotropic turbulence. The selection law implies that only the elements with the same values of index difference l - m are coupled on propagation in isotropic turbulence. Obtaining exact analytical expressions for the COAM matrix elements is not straightforward, regardless of the availability of the exact propagation law. To circumvent such a tedious analytical work, we have modified the widely used WOS method that incorporates the multi-phase screen approach, the mode representation of a partially coherent beam and the Cartesian-polar coordinate transformation. This method, in principle, is applicable for any partially coherent beam including those with spatially uniform and non-uniform coherence states. In addition, as we illustrated, it can also be applied to two special cases: deterministic beams propagating in turbulence with averaging over many realizations and random beams propagating in frozen turbulence. Hence it is an excellent tool for conceptual analysis of beam-turbulence interactions.

As examples, the propagation characteristics of the COAM matrix of circular and elliptical GSM beams propagating in free space and in turbulence have been examined in detail, and the results are compared to those obtained from direct integrations. In free-space propagation, we show that the profiles of the diagonal COAM matrix’s elements remain invariant, only the size is enlarged due to diffraction. For the off-diagonal elements, the profiles vary upon propagation. In the case of isotropic turbulence, the distributions of the diagonal and off-diagonal elements change with the propagation distance. However, the selection rule is always obeyed. In addition, our numerical results show that the selection rule is only satisfied under the condition of ensemble averaging over turbulent fluctuations, but is violated in the case of the instantaneous turbulence. Our studies provide an alternative way to calculate the COAM matrix of partially coherent beams propagation in paraxial optical system, in linear random media and tight focusing system.

Appendix A

On Substitution from Eqs. (7) and (8) into Eq. (3) and making use of Eq. (2), the elements of the COAM matrix in the output plane becomes

(A1)$$\begin{aligned} W_{lm}^{(\textrm{OAM})}({r_1},{r_2},z) &= \frac{{{k^2}}}{{{{(2\pi )}^4}{z^2}}}\sum\limits_{p ={-} \infty }^\infty {\sum\limits_{q ={-} \infty }^\infty {} } \\ \textrm{ } &\times \int_0^{2\pi } {\int_0^{2\pi } {W_{0pq}^{(\textrm{OAM})}({r_{10}},{r_{20}}){H_{pqlm}}({r_1},{r_2},{r_{10}},{r_{20}})} } d{r_{10}}d{r_{20}}, \end{aligned}$$

with the Pair-Amplitude Impulse Response [PAIR] matrix

(A2)$$\begin{aligned} &{H_{pqlm}}({r_1},{r_2},{r_{10}},{r_{20}})= {r_{10}}{r_{20}}\textrm{exp} [{ - Q_1^\ast ({r_{20}^2 + r_2^2} )- {Q_1}({r_{10}^2 + r_1^2} )} ]\\ &\quad\times \int_0^{2\pi } {\int_0^{2\pi } {\int_0^{2\pi } {\int_0^{2\pi } {d{\varphi _{10}}d{\varphi _{20}}d{\varphi _1}d{\varphi _2}{e^{il{\varphi _1} - im{\varphi _2}}}{e^{ - ip{\varphi _{10}} + iq{\varphi _{20}}}}} } } } \textrm{exp} ({ - {Q^\ast }{{\bf r}_{10}}\cdot {{\bf r}_1}} )\\ &\quad\times \textrm{exp} ({ - Q{{\bf r}_2}\cdot {{\bf r}_{20}}} )\textrm{exp} \left( { - \frac{{{{\bf r}_2}\cdot {{\bf r}_1}}}{{\rho_0^2}}} \right)\textrm{exp} \left( { - \frac{{{{\bf r}_{20}}\cdot {{\bf r}_{10}}}}{{\rho_0^2}}} \right)\textrm{exp} \left( {\frac{{{{\bf r}_1}\cdot {{\bf r}_{20}}}}{{\rho_0^2}}} \right)\textrm{exp} \left( {\frac{{{{\bf r}_2}\cdot {{\bf r}_{10}}}}{{\rho_0^2}}} \right), \end{aligned}$$

where $Q = 1/\rho _0^2 + ik/z$, ${Q_1} = 1/\rho _0^2 + ik/2z$. Note that in the derivation of Eq. (A2), Eq. (8) is applied.

To simplify Eq. (A2), the exponential functions in Eq. (2) are expressed as the sum of the angular harmonics and Bessel functions of second kind I_s of order s,

(A3)$$\textrm{exp} (a{{\bf r}_{{j_1}}}\cdot {{\bf r}_{{j_2}}}) = \sum\limits_{s ={-} \infty }^\infty {{I_s}(a{r_{{j_1}}}{r_{{j_2}}})\textrm{exp} [is({\varphi _{{j_2}}} - {\varphi _{{j_1}}})]} ,({j_1},{j_2} = 1,2),$$

where a is a constant. On substitution Eq. (A3) into Eq. (A2) and integrating over φ₁₀, φ₂₀, φ₁, φ₂, the impulse response function yields

(A4)$$\begin{aligned} &{H_{pqlm}}({r_1},{r_2},{r_{10}},{r_{20}})= {(2\pi )^4}\textrm{exp} [{ - Q_1^\ast ({r_{20}^2 + r_2^2} )- {Q_1}({r_{10}^2 + r_1^2} )} ]\\ &\quad\sum\limits_{{s_1} ={-} \infty }^\infty {\sum\limits_{{s_2} ={-} \infty }^\infty {\sum\limits_{{s_3} ={-} \infty }^\infty {\sum\limits_{{s_4} ={-} \infty }^\infty {\sum\limits_{{s_5} ={-} \infty }^\infty {\sum\limits_{{s_6} ={-} \infty }^\infty {{I_{{s_1}}}} } } } } } ({ - {Q^\ast }{r_{10}}{r_1}} ){I_{{s_2}}}({ - Q{r_2}{r_{20}}} )\\ &\quad\times {I_{{s_3}}}\left( { - \frac{1}{{\rho_0^2}}{r_1}{r_2}} \right){I_{{s_4}}}\left( { - \frac{1}{{\rho_0^2}}{r_{10}}{r_{20}}} \right){I_{{s_5}}}\left( {\frac{1}{{\rho_0^2}}{r_1}{r_{20}}} \right){I_{{s_6}}}\left( {\frac{1}{{\rho_0^2}}{r_{10}}{r_2}} \right), \end{aligned}$$

and the following relations of the indices must satisfy

(A5)$$\begin{array}{l} {s_4} - {s_6} = p - {s_1};\textrm{ }{s_4} + {s_5} = q + {s_2};\\ {s_3} + {s_5} = {s_1} - l;\textrm{ } - {s_3} + {s_6} = m + {s_2}. \end{array}$$

According to Eq. (A5), we now obtain an important relation among the indices in the source plane and in the output plane:

(A6) $$m - l = q - p.$$

Funding

National Key Research and Development Program of China (2019YFA0705000, 2022YFA1404800); National Natural Science Foundation of China (11904247, 11974218, 12192254, 12274311, 92250304); Local Science and Technology Development Project of the Central Government (YDZX20203700001766).

Acknowledgments

OK acknowledges the UM support via the Cooper Fellowship.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]

2. N. Simpson, L. Allen, and M. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. 43(12), 2485–2491 (1996). [CrossRef]

3. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]

4. M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. 113(48), 13648–13653 (2016). [CrossRef]

5. S. Bernet, A. Jesacher, S. Furhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14(9), 3792–3805 (2006). [CrossRef]

6. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). [CrossRef]

7. L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3(3), e153 (2014). [CrossRef]

8. Z. Yang, O. Magana-Loaiza, M. Mirhosseini, Y. Zhou, B. Gao, L. Gao, S. M. H. Rafsanjani, G. Long, and R. Boyd, “Digital spiral object identification using random light,” Light: Sci. Appl. 6(7), e17013 (2017). [CrossRef]

9. X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14(2), 102–108 (2020). [CrossRef]

10. L. Li and F. Li, “Beating the Rayleigh limit: Orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E 88(3), 033205 (2013). [CrossRef]

11. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef]

12. H. Sroor, C. moodley, V. Rodriguez-Fajardo, Q. Zhan, and A. Forbes, “Modal description of paraxial structured light propagation: turorial,” J. Opt. Soc. Am. A 38(10), 1443–1449 (2021). [CrossRef]

13. L. Mandel and E. Wolf, Optical coherence and Quantum Optics, (Cambridge University Press1995).

14. O. Korotkova and G. Gbur, “Unified matrix representation for spin and orbital angular momentum in partially coherent beams,” Phys. Rev. A 103(2), 023529 (2021). [CrossRef]

15. A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Partial angular coherence and the angular Schmidt spectrum of entangled two-photon fields,” Phys. Rev. A 84(6), 063847 (2011). [CrossRef]

16. O. S. Magana-Loaiza, M. Mirhosseini, R. M. Cross, S. M. H. Rafsanjani, and R. W. Boyd, “Hanbury Brown and Twiss interferometry with twisted light,” Sci. Adv. 2(4), e1501143 (2016). [CrossRef]

17. Z. Yang, H. Wang, Y. Chen, F. Wang, G. Gbur, O. Korotkova, and Y. Cai, “Measurement of the coherence-orbital angular momentum matrix of a partially coherent beam,” Opt. Lett. 47(17), 4467–4470 (2022). [CrossRef]

18. D. Raveh and O. Korotkova, “Coherence-orbital angular momentum matrix of non-uniformly correlated sources,” Opt. Lett. 47(21), 5719–5722 (2022). [CrossRef]

19. F. Wang, Z. Yang, Y. Chen, O. Korotkova, and Y. Cai, “Coherence-orbital angular momentum matrix of Schell-model sources,” Opt. Lett. 47(11), 2826–2829 (2022). [CrossRef]

20. O. Korotkova, S. Pokharel, and G. Gbur, “Tailoring the coherence-OAM matrices,” Opt. Lett. 47(23), 6109–6112 (2022). [CrossRef]

21. O. Korotkova and G. Gbur, “Jones and Stokes-Mueller analogous calculi for OAM-transforming optics,” Opt. Lett. 46(11), 2585–2588 (2021). [CrossRef]

22. G. Gbur and O. Korotkova, “Orbital angular momentum transformations by non-local linear systems,” Opt. Lett. 47(2), 321–324 (2022). [CrossRef]

23. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communications,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef]

24. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009). [CrossRef]

25. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and B. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012). [CrossRef]

26. X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “Probability density of Orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25(13), 15286–15299 (2017). [CrossRef]

27. L. Yu, B. Hu, and Y. Zhang, “Intensity of vortex modes carried by Lommel beam in weak-to-strong non-Kolmogorov turbulence,” Opt. Express 25(16), 19538–19547 (2017). [CrossRef]

28. J. Zeng, X. Liu, C. Zhao, F. Wang, G. Gbur, and Y. Cai, “Spiral spectrum of a Laguerre-Gaussian beam propagating in anisotropic non-Kolmogorov turbulent atmosphere along horizontal path,” Opt. Express 27(18), 25342–25356 (2019). [CrossRef]

29. Y. Wang, B. Lu, J. Xie, D. Zhang, Q. Lv, and L. Guo, “Spiral spectrum of high-order elliptic Gaussian vortex beams in a non-Kolmogorov turbulent atmosphere,” Opt. Express 29(11), 16056–16072 (2021). [CrossRef]

30. Y. Li, Y. Xie, and B. Li, “Probability of orbital angular momentum for square Hermite-Gaussian vortex pulsed beam in oceanic turbulence channel,” Results Phys. 28, 104590 (2021). [CrossRef]

31. Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014). [CrossRef]

32. A. Klug, I. Nape, and A. Forbes, “The orbital angular momentum of a turbulent atmosphere and its impact on propagating structured light field,” New J. Phys. 23(9), 093012 (2021). [CrossRef]

33. D. Yang, Z. Hu, S. Wang, and Y. Zhu, “Influence of random media on orbital angular momentum quantum states of optical vortex beams,” Phys. Rev. A 105(5), 053513 (2022). [CrossRef]

34. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (SPIE, Bellingham, Washington, 1998).

35. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003). [CrossRef]

36. J. D. Schmidt, Numerical simulation of optical wave propagation, (SPIE, Bellingham, Washington, 2010).

37. F. Wang, H. Lv, Y. Chen, Y. Cai, and O. Korotkova, “Three modal decompositions of Gaussian Schell-model sources: comparative analysis,” Opt. Express 29(19), 29676–29689 (2021). [CrossRef]

38. F. Wang and O. Korotkova, “Random optical beam propagation in anisotropic turbulence along horizontal links,” Opt. Express 24(21), 24422–24434 (2016). [CrossRef]

39. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72(7), 923–928 (1982). [CrossRef]

Propagation of coherence-OAM matrix of an optical beam in vacuum and turbulence

Abstract

1. Introduction

2. Definition of coherence-orbital angular momentum matrix in partially coherent beams

3. Propagation of the COAM matrix in free space and in turbulent atmosphere

3.1 Huygens-Fresnel integral method

3.2 Wave-optics numerical simulation

4. Numerical results

5. Conclusion

Appendix A

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (20)

Optics Express