Propagation of multi-cosine-Laguerre-Gaussian correlated Schell-model beams in free space and atmospheric turbulence

Jie Zhu; Xiaoli Li; Huiqin Tang; Kaicheng Zhu

doi:10.1364/OE.25.020071

1. Introduction

As is well known, for a partially coherent beam, the properties of the spectral degree of coherence (SDOC) or spatial correlation function significantly affect its propagation properties, such as the intensity distribution profiles and the propagation factors [1]. Due to some constraints in the choice of the mathematical form of the SDOCs for partially coherent optical fields, earlier work mainly concentrated on the partially coherent beams with the conventional SDOCs being Gaussian-correlated Schell-model functions, but studies of partially coherent beams with non-conventional SDOCs were quite few in number and confined to only the Bessel-correlated Schell-model beams [2–4]. In order to fulfil the physical limits of the SDOCs, Gori and Santarsiero generally established the sufficient condition for devising a genuine SDOC of a scalar or electromagnetic non-conventional partially coherent beam based on the theory of reproducing kernel Hilbert spaces [5]. Since then plenty of partially coherent beams with non-conventional SDOCs have been explored, including circular or rectangular Laguerre-Gaussian correlated Schell-model (LGCSM) beam [4, 6–16], cos-Gaussian correlated Schell-model (cGCSM) beam [17–24], multi-cos-Gaussian correlated Schell-model (McGCSM) beam [25–31], and so on [32–40]. Theoretical analyses have demonstrated that these novel beams exhibit extraordinary propagation characteristics. For example, a LGCSM beam with circular symmetry displays a single ring-shaped beam profile in the far field or a controllable optical cage near the focal plane and a McGCSM beam displays robust optical lattice patterns in the far field, although both beams are Gaussian beam profiles in the source plane. Experiments have been performed that have verified these theoretical predications [7–10, 15, 19, 23, 32, 33, 35, 38–40]. Very recently, we introduced a novel kind of partially coherent beam whose complicated SDOC contains both non-conventional cosine- and Hermite-type components [41]. Such beams have been found to exhibit cascade self-splitting properties for two successive times during propagation in free space. Additionally, the propagation of partially coherent beams with conventional or non-conventional SDOCs in turbulent atmosphere has also been studied in detail. It is generally believed that modulating the initial coherence properties is perhaps one of the most efficient way for reducing the negative influence of the turbulence [4, 11, 14, 15, 18, 20–22, 24, 26, 27, 33, 36, 42–45].

On the other hand, ring-shaped beams (also known as dark hollow beams (DHBs)) and optical cages have attracted considerable attentions recently due to their potential applications in optical trapping [46–49], optical imaging [50, 51], and optical cloaking [52, 53]. Partially coherent DHBs with conventional or non-conventional SDOCs have also been introduced and some of them have been experimentally realized. As mentioned above, a ring-shaped beam profile or an optical cage can be arrived in the far field (on near the focal plane) by employing a circular LGCSM beam [8, 46–53]. However, note that most of the existing technologies are limited to generate only one single ring-shaped beam or optical cage, which is not sufficient for meeting the engineering requirement. For instance, it was found that for the optical imaging in a stimulated emission depletion (STED) microscopy, a regular single spot STED setup has only a quarter of scanning speed compared to the parallelized STED setup in which four ring-shaped spots are combined to reduce the recording time [54]. Finding an efficient way to generate robust multiple ring-shaped beams or optical cages is therefore highly desirable. Very recently, using a full polarization-controlled method the optical cage array with tunable number and positions was first obtained by focusing a so-called Dammann vector beam [55]. In this paper, we propose a new kind of non-conventional partially coherent beam named as multi-cosine-Laguerre-Gaussian correlated Schell-model (McLGCSM) beam whose non-conventional SDOC is a Gaussian function modulated by both Laguerre- and multi-cosine-related factors. By adjusting the initial coherence states such beams are able to naturally create a robust ring-shaped beam array with prescribed number and tunable positions in the far field during propagation.

The paper is organized as follows: in section 2, we introduce the theoretical model for a McLGCSM beam and determine its genuine SDOC function. In section 3, the analytical expression for the cross-spectral density (CSD) function of a McLGCSM beam propagating in atmospheric turbulence is derived based on the extended Huygens-Fresnel integral. In section 4, we numerically study the evolution properties of the spectral density of a McLGCSM beam and compare the results with a McGCSM beam and LGCSM beam. While the propagation factor of the McLGCSM beam is discussed in section 5, some extraordinary and useful results are summarized in section 6.

2. Model for multi-cosine-Laguerre-Gaussian-correlated Schell-model (McLGCSM) source

Let $W^{(0)} ({r^{'}}_{1}, {r^{'}}_{2}, ω)$ denote the CSD at two spatial positions ${r^{'}}_{1} = ({x^{'}}_{1}, {y^{'}}_{1})$ and ${r^{'}}_{2} = ({x^{'}}_{2}, {y^{'}}_{2})$ which are two arbitrary transverse position vectors. For brevity, the dependence of W⁽⁰⁾ on frequency ω is not explicitly shown in the following equations. The CSD must have a non-negative definite kernel, meaning that, for any choice of f (r′), the following inequality must be established [5]:

\iint d^{2} {r^{'}}_{1} d^{2} {r^{'}}_{2} W^{(0)} ({r^{'}}_{1}, {r^{'}}_{2}) f ({r^{'}}_{1}) f ({r^{'}}_{2}) \geq 0.

To satisfy the constraint given by inequality (1) a sufficiency condition is that the CSD has to be expressed as a superposition integral of the form [5]

W^{(0)} ({r^{'}}_{1}, {r^{'}}_{2}) = \iint d^{2} v p (v) H_{0}^{*} ({r^{'}}_{1}, v) H_{0} ({r^{'}}_{2}, v),

where H₀ is an arbitrary kernel and p(v) is a non-negative function.

If the kernel H₀ is chosen as a Fourier-like structure, i.e.,

H_{0} (r^{'}, v) = τ (r^{'}) \exp (- i r^{'} \cdot v),

where

τ (r^{'})

is a generally complex profile function, then Eq. (2) becomes

W^{(0)} ({r^{'}}_{1}, {r^{'}}_{2}) = τ^{*} ({r^{'}}_{1}) τ ({r^{'}}_{2}) \iint d^{2} v p (v) \exp [i v \cdot ({r^{'}}_{1} - {r^{'}}_{2})] = τ^{*} ({r^{'}}_{1}) τ ({r^{'}}_{2}) μ ({r^{'}}_{1} - {r^{'}}_{2})

where

μ ({r^{'}}_{1} - {r^{'}}_{2})

represents the two-dimensional Fourier transform of p(v) and characterizes the SDOC.

Depending on the choice of p(v), a family of sources with different SDOCs are determined. In this paper we use the following expression for the non-negative function p(v)

\begin{matrix} p (v_{x}, v_{x}) = \frac{N}{4} \sum_{l_{x} = - M_{x}}^{M_{x}} \sum_{l_{y} = - M_{y}}^{M_{y}} [p_{L} (v_{x} + \frac{l_{x} β_{x}}{δ}, v_{y} + \frac{l_{y} β_{y}}{δ}) + p_{L} (v_{x} + \frac{l_{x} β_{x}}{δ}, v_{y} - \frac{l_{y} β_{y}}{δ}) \\ + p_{L} (v_{x} - \frac{l_{x} β_{x}}{δ}, v_{y} + \frac{l_{y} β_{y}}{δ}) + p_{L} (v_{x} - \frac{l_{x} β_{x}}{δ}, v_{y} - \frac{l_{y} β_{y}}{δ})], \end{matrix}

with

p_{L} (v_{x}, v_{y}) = {(\frac{v_{x}^{2} + v_{y}^{2}}{δ^{2}})}^{n} \exp [- \frac{δ^{2} (v_{x}^{2} + v_{y}^{2})}{2}],

where β_x, β_y and δ are real constants, n is a positive integer, M_x = (P_x−1)/2 and M_y = (P_y−1)/2 with P_x and P_y being positive integers, and

N = 1 / \max [μ ({r^{'}}_{1} - {r^{'}}_{2})]

to make the maximum values of

μ ({r^{'}}_{1} - {r^{'}}_{2})

being 1. In fact, p_L (v_x, v_y) is just the function associated with a LGCSM beam defined in [6–8].

Let us assume $τ (r^{'}) = \exp (- \frac{{r^{'}}^{2}}{4 w_{0}^{2}})$ to be a Gaussian with w₀ being the beam width, after substituting Eqs. (5) and (6) into Eq. (4), and making use of the shift theorem of the Fourier transformation [56]

\int_{- \infty}^{\infty} f (z - z_{0}) \exp (i x z) d z = \exp (i z_{0} x) \int_{- \infty}^{\infty} f (y) \exp (i x y) d y,

we obtain the following CSD

W^{(0)} ({r^{'}}_{1}, {r^{'}}_{2}) = \exp (- \frac{{r^{'}}_{s}^{2}}{2 w_{0}^{2}} - \frac{{r^{'}}_{d}^{2}}{8 w_{0}^{2}}) μ ({r^{'}}_{d}),

where

μ ({r^{'}}_{d}) = N L_{n} (\frac{{r^{'}}_{d}^{2}}{δ^{2}}) \exp (- \frac{{r^{'}}_{d}^{2}}{2 δ^{2}}) \sum_{l_{x} = - M_{x}}^{M_{x}} \sum_{l_{y} = - M_{y}}^{M_{y}} \cos (\frac{l_{x} β_{x} {x^{'}}_{d}}{δ}) \cos (\frac{l_{y} β_{y} {y^{'}}_{d}}{δ}),

with

{r^{'}}_{s} = ({r^{'}}_{1} + {r^{'}}_{2}) / 2

,

{r^{'}}_{d} = {r^{'}}_{1} - {r^{'}}_{2}

being the “sum” and “difference” coordinates in the source plane, respectively, and L_n() denoting the Laguerre polynomial of mode order n. Such a random source with CSD function determined by Eqs. (8) and (9) is termed as a McLGCSM source and represents a more general source model. It can reduce to a circular LGCSM source (for β_x = β_y = 0), a rectangular McGCSM source (for n = 0) and a well-known Gaussian-Schell model (GSM) source (for β_x = β_y = n = 0). As was proved, a circular LGCSM beam displays a ring-shaped beam profile [6–8] and a rectangular McGCSM beam exhibits a self-splitting beam spot array [25–27] in the far field due to their non-conventional correlation functions.

Figure 1 comparatively shows the density plot of the square of the modulus of the SDOC for a McGCSM source, LGCSM source, and McLGCSM source. It is interesting to note that the SDOC of the McLGCSM source significantly differs from that of the counterpart LGCSM source or McGCSM source. Although all of these beams have the same Gaussian intensity distributions in the source plane, the variations in the source structured coherence state lead to extraordinary propagation properties of the McLGCSM beam as shown later.

Fig. 1 Density plots of the square of the modulus of the SDOC for (a) a McGCSM source with n = 0 and β_x = β_y = 8, (b) a LGCSM source with n = 5 and β_x = β_y = 0, and (c) a McLGCSM source with n = 5 and β_x = β_y = 8. The other parameters are P_x = P_y = 3.

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In fact, the McLGCSM source with the SDOC given by Eqs. (8) and (9) can be considered as a superposition of P_xP_y LGCSM sources [7, 8] respectively with tilting factors exp[ ± i(l_xβ_xx ± l_yβ_yy)/δ]. Hence, it is straight that the required condition for such a source representing a beam is $\frac{1}{δ^{2}} + \frac{1}{4 w_{0}^{2}} < < \frac{2 π^{2}}{λ^{2}}$ , that is, just the same as that of a LGCSM source [6–8] or a cGCSM source [17, 20, 21].

3. Propagation of a McLGCSM beam in atmospheric turbulence

Based on the extended Huygens–Fresnel principle, paraxial propagation of the CSD of a McLGCSM beam propagating in turbulent atmosphere can be written as [11, 14, 20–22, 26, 33, 36, 42–45]

\begin{matrix} W (r_{1}, r_{2}, z) = \frac{1}{λ^{2} z^{2}} \exp [\frac{i k}{2 z} (r_{2}^{2} - r_{1}^{2})] \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} {r^{'}}_{1} d^{2} {r^{'}}_{2} W^{(0)} ({r^{'}}_{1}, {r^{'}}_{2}) \\ \times \exp [- \frac{i k}{2 z} ({r^{'}}_{1}^{2} - {r^{'}}_{2}^{2}) + \frac{i k}{z} ({r^{'}}_{1} \cdot r_{1} - {r^{'}}_{2} \cdot r_{2})] \cdot 〈 \exp [Φ ({r^{'}}_{1}, r_{1}) + Φ^{*} ({r^{'}}_{2}, r_{2})] 〉, \end{matrix}

where r₁ = (x₁, y₁) and r₂ = (x₂, y₂) are two arbitrary transverse position vectors in the output plane, z is the propagation distance, and k = 2π/λ is the wave number with λ being the wavelength. The asterisk (*) denotes the complex conjugation and

Φ ({r^{'}}_{i}, r_{i}) (i = 1, 2)

represents the random part of the complex phase of a spherical wave due to the atmospheric turbulence. The last term in the integrand of Eq. (10) can be expressed as [11, 26, 33, 42–45]

〈 \exp [Φ ({r^{'}}_{1}, r_{1}) + Φ^{*} ({r^{'}}_{2}, r_{2})] 〉 = \exp [- T_{α} ({r^{'}}_{d}^{2} + {r^{'}}_{d} \cdot r_{d} + r_{d}^{2})],

where r_s = (r₁ + r₂)/2 and r_d = r₁−r₂ are the “sum” and “difference” coordinates in the receiving plane, respectively, and the symbol T_α reflects the intensity of the turbulence. In this paper, the Kolmogorov spectrum and a quadratic approximation of the 5/3 power law for the Rytov’s phase structure function are employed, which are accepted to be valid not only for weak fluctuations but also for strong ones. Then

T_{α} = ρ_{0}^{- 2}

where

ρ_{0} = {(0.545 C_{n}^{2} k^{2} z)}^{- 3 / 5}

denotes the spatial coherence radius of a spherical wave propagating in a turbulent atmosphere with

C_{n}^{2}

being the refraction index structure constant and describing the strength of the atmospheric turbulence [11, 26, 33, 42–45]. Therefore, Eq. (10) can be rewritten in the following form

\begin{matrix} W (r_{1}, r_{2}, z) = \frac{N}{λ^{2} z^{2}} \exp (\frac{i k}{z} r \cdot r_{d} - T_{α} r_{d}^{2}) \sum_{l_{x} = 1}^{M_{x}} \sum_{l_{y} = 1}^{M_{y}} \int \int_{- \infty}^{\infty} d^{2} {r^{'}}_{s} d^{2} {r^{'}}_{d} L_{n} (\frac{{r^{'}}_{d}^{2}}{2 δ^{2}}) \\ \times \cos (\frac{Q_{l_{x}} {x^{'}}_{d}}{δ}) \cos (\frac{Q_{l_{y}} {y^{'}}_{d}}{δ}) \exp [- \frac{{r^{'}}_{s}^{2}}{2 w_{0}^{2}} - (\frac{1}{δ^{2}} + \frac{1}{4 w_{0}^{2}} + 2 T_{α}) \frac{{r^{'}}_{d}^{2}}{2}] \\ \times \exp [- \frac{i k}{z} {r^{'}}_{s} \cdot {r^{'}}_{d} + \frac{i k}{z} (r_{d} \cdot {r^{'}}_{s} + r \cdot {r^{'}}_{d}) - T_{α} r_{d} \cdot {r^{'}}_{d}] . \end{matrix}

When Τ_α→0, Eq. (12) represents the propagation of the beam within free space.

Completing integration over r_s, we obtain

\begin{matrix} W (r_{1}, r_{2}, z) = \frac{2 π w_{0}^{2} N}{λ^{2} z^{2}} \exp [\frac{2 i z_{R}}{z w_{0}^{2}} r \cdot r_{d} - (\frac{2 z_{R}^{2}}{z^{2} w_{0}^{2}} + T_{α}) r_{d}^{2}] \\ \times \sum_{l_{x} = - M_{x}}^{M_{x}} \sum_{l_{y} = - M_{y}}^{M_{y}} [W_{l, + +} (r_{1}, r_{2}) + W_{l, + -} (r_{1}, r_{2}) + W_{l, - +} (r_{1}, r_{2}) + W_{l, - -} (r_{1}, r_{2})], \end{matrix}

where

W_{l, u v} (r_{1}, r_{2}) = \frac{1}{4} \int_{- \infty}^{\infty} d^{2} {r^{'}}_{d} L_{n} (\frac{{r^{'}}_{d}^{2}}{2 δ^{2}}) \exp [- \frac{Ω {r^{'}}_{d}^{2}}{8 w_{0}^{2}} + \frac{i k}{z} (b_{l_{x}, x u} {x^{'}}_{d} + b_{l_{y}, y v} {y^{'}}_{d})],

Ω = 1 + \frac{4}{δ_{w}^{2}} + 8 w_{0}^{2} T_{α} + \frac{16 z_{R}^{2}}{z^{2}},

b_{q \pm} = q + i (\frac{T_{α} z}{k} - \frac{2 z_{R}}{z}) q_{d} \pm \frac{z w_{0}}{2 z_{R} δ_{w}} l_{q} β_{q}, (q = x, y u, v = \pm),

with

δ_{w} = δ / w_{0}

and

z_{R} = π w_{0}^{2} / λ

being the Rayleigh distance.

Using the following formulae [56]

L_{n} (x) = \sum_{p = 0}^{n} (\begin{matrix} n \\ p \end{matrix}) \frac{{(- 1)}^{p}}{p!} x^{p}

{(x^{2} + y^{2})}^{p} = \sum_{m = 0}^{p} (\begin{matrix} p \\ m \end{matrix}) x^{2 (p - m)} y^{2 m},

\int_{- \infty}^{\infty} z^{n} \exp [- {(z - β)}^{2}] d z = {(2 i)}^{- n} \sqrt{π} H_{n} (i β),

and performing tedious mathematical operations, Eq. (14) turns to be

\begin{matrix} W_{l, u v} (r_{1}, r_{2}) = \frac{2 π w_{0}^{2}}{Ω} \exp [- \frac{2 k^{2} w_{0}^{2} (b_{l_{x}, x u}^{2} + b_{l_{y}, y v}^{2})}{Ω z^{2}}] \sum_{p = 0}^{n} (\begin{matrix} n \\ p \end{matrix}) \frac{1}{p!} {(\frac{1}{Ω δ_{w}^{2}})}^{p} \\ \times \sum_{m = 0}^{p} (\begin{matrix} p \\ m \end{matrix}) H_{2 (p - m)} (\sqrt{\frac{2}{Ω}} \frac{k w_{0}}{z} b_{l_{x}, x u}) H_{2 m} (\sqrt{\frac{2}{Ω}} \frac{k w_{0}}{z} b_{l_{y}, y v}), \end{matrix}

where H_n denotes the Hermite polynomial of mode order n. Applying the following formulae [56–58]

\sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) H_{2 m} (x) H_{2 n - 2 m} (y) = {(- 4)}^{n} n! L_{n} (x^{2} + y^{2}),

L_{n} (b x) = {(1 - b)}^{n} \sum_{m = 0}^{n} {(\frac{b}{1 - b})}^{m} (\begin{matrix} n \\ m \end{matrix}) L_{m} (x),

Equation (20) can be further simplified to

\begin{matrix} W_{l, u v} (r_{1}, r_{2}) = \frac{2 π w_{0}^{2}}{Ω} \exp [- \frac{2 k^{2} w_{0}^{2} (b_{l_{x}, x u}^{2} + b_{l_{y}, y v}^{2})}{Ω z^{2}}] \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) {(- \frac{4}{Ω δ_{w}^{2}})}^{m} L_{m} [\frac{2 k^{2} w_{0}^{2} (b_{l_{x}, x u}^{2} + b_{l_{y}, y v}^{2})}{Ω z^{2}}] \\ = \frac{2 π w_{0}^{2}}{Ω} {(\frac{Ω δ_{w}^{2}}{Ω δ_{w}^{2} - 4})}^{n} \exp [- \frac{2 k^{2} w_{0}^{2} (b_{l_{x}, x u}^{2} + b_{l_{y}, y v}^{2})}{Ω z^{2}}] L_{m} [\frac{8 k^{2} w_{0}^{2} (b_{l_{x}, x u}^{2} + b_{l_{y}, y v}^{2})}{Ω z^{2} (4 - Ω δ_{w}^{2})}] . \end{matrix}

Equations (13) and (23) represent the CSD of a McLGCSM beam in the output plane. Letting ρ₁ = ρ₂ = ρ, we obtain the following explicit expressions for the average intensity of a McLGCSM beam propagating in turbulent atmosphere

I (r, z) = W (r, r, z) = \frac{π N w_{0}^{2}}{z^{2} z_{R}^{2}} \sum_{l_{x} = 1}^{M_{x}} \sum_{l_{y} = 1}^{M_{y}} [I_{l, + +} (r, z) + I_{l, + -} (r, z) + I_{l, - +} (r, z) + I_{l, - -} (r, z)],

where

I_{l, \pm \pm} (r, z) = {(\frac{Ω δ_{w}^{2}}{Ω δ_{w}^{2} - 4})}^{n} \exp (- \frac{8 z_{R}^{2}}{Ω z^{2}} R_{l_{x}, l_{y}}) L_{n} [\frac{32 z_{R}^{2} δ_{w}^{2}}{z^{2} (4 - Ω δ_{w}^{2})} R_{l_{x}, l_{y}}]

with

R_{l_{x}, l_{y}} = {(x_{w} \pm \frac{z l_{x} β_{x}}{2 z_{R} δ_{w}})}^{2} + {(y_{w} \pm \frac{z l_{y} β_{y}}{2 z_{R} δ_{w}})}^{2}

and (x_w, y_w) = (x, y)/w₀ being the scaled transversal coordinates.

Obviously, Eqs. (24) and (25) will go back to the average intensity of a LGCSM beam for β_x = β_y = 0 [6–8] or of a McGCSM beam for n = 0 [25–27]. Moreover, we point out that if β_x ≠ 0 or β_y ≠ 0, the four terms in the summation of Eq. (24) represent the average intensity patterns of LGCSM beams respectively centered at $\frac{z}{2 z_{R} δ_{w}} (\pm l_{x} β_{x}, \pm l_{y} β_{y})$ , implying that the McLGCSM beam can behave self-splitting behavior when the parameters β_x and β_y are sufficiently large. Therefore, the results clearly reveal that for the proposed McLGCSM beam on propagation, modulating the multi-cosine factor of the total correlation function can induce the self-splitting behavior, and adjusting the Laguerre factor of the total correlation function can give rise to the formation of the ring-shaped beam profile.

The SDOC of a McLGCSM beam in the output plane is obtained as

μ (r_{1}, r_{2}, z) = \frac{W (r_{1}, r_{2}, z)}{\sqrt{W (r_{1}, r_{1}, z) W (r_{2}, r_{2}, z)}} .

Applying Eqs. (13), (23)-(26), the propagation properties of the CSD and SDOC of a McLGCSM beam in free space or Kolmogorov turbulence can be numerically demonstrated in a convenient way.

4. Numerical examples for the spectral density of a McLGCSM beam in free space and atmospheric turbulence

In this section, we will numerically study the evolution of the spectral density of a McLGCSM beam on propagation in free space and turbulent atmosphere. Because we only emphasize on the evolution of the beam pattern configurations but do not care about the size of the beam patterns, in the following graphical discussions arbitrary units for transversal coordinates in different z-planes are used, although they have the same transversal dimensions in the same z-plane. In addition, the beam parameters are set as λ = 632.8nm, and β_x = β_y = β.

Figure 2 shows the normalized intensity distributions of a McLGCSM, McGCSM, and LGCSM beams in free space at several propagation distances with w₀ = 5mm, P_x = P_y = 3, and δ_w = 1. For one typical case of β = 9, n = 3 as shown in Fig. 2(a), the McLGCSM beam exhibits a ring-shaped beam array during propagation in free space, i.e., the initial single Gaussian beam spot is first self-split apart into a beam spot array at short propagation distance z₁ (z₁ = 0.1km); then as the travelling distance increases the beam spot array further turns into a ring-shaped beam array at farther propagation distance z₂ (z₂ = 0.2km). Although the size of the patterns increases with increasing propagation distance, the configuration of the ring-shaped beam array is very robust and remains unchanged till z = 40km. Figures 2(b) and 2(c) show that a McGCSM beam is split into a beam spot array and a LGCSM beam is shaped into a single ring-shaped beam profile in the far field, which agrees well with the results that have been reported in [6–8, 25–27]. Note that the propagation behaviors of the proposed McLGCSM beam significantly differs from that of the McGCSM beam or LGCSM beam; or more specifically, it can be regarded as a superposition of the self-splitting of the McGCSM beam and self-shaping of the LGCSM beam. As a result, the number of the ring-shaped beams obtained by a McLGCSM beam is equal to P_xP_y, which is just same as the number of the self-splitting beam spots produced by a McGCSM beam [25–27]. Moreover, since previous studies have reported that by focusing a LGCSM beam the ring-shaped beam pattern created in free space can evolve into an optical cage near the focal plane [8], it is reasonable to infer that a tunable optical cage array can also be generated by focusing the proposed McLGCSM beam. To the best of our knowledge, such a phenomenon of a single beam spot self-splitting and self-shaping into a ring-shaped beam array in the far field has not been reported elsewhere.

Fig. 2 Intensity distribution patterns of a McLGCSM beam (a), McGCSM beam (b), and LGCSM beam (c) at several propagation distances in free space with w₀ = 5mm, P_x = P_y = 3, and δ_w = 1. The x and y axes taken as the transverse coordinates are in arbitrary units.

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Figure 3 shows the influence of the beam parameters β and δ_w on the formation of the ring-shaped beam array of a McLGCSM beam on propagation in free space with w₀ = 5mm, P_x = P_y = 3, and n = 4. One finds from Fig. 3(b) that only for sufficiently large values of β and relatively small values of δ_w, the McLGCSM beam exhibits a ring-shaped beam array in the far field. Comparing Fig. 3(b) with Fig. 3(c) reveals that, when δ_w is sufficiently large, only self-splitting behavior can be observed during propagation, indicating that the self-shaping of a McLGCSM beam originates from the Laguerre-related factor of the entire SDOC. Furthermore, contrasting Fig. 3(a) with 3(b) shows that, when β is appropriately small, only self-shaping phenomenon can be observed during propagation, demonstrating that the self-splitting of a McLGCSM beam is induced by the multi-cosine-related factor of the entire SDOC. Obviously, these results are in good agreement with the analytical conclusions aforesaid in the previous section.

Fig. 3 Intensity distribution patterns of a McLGCSM beam at several propagation distances in free space for different values of β and δ_w with w₀ = 5mm, P_x = P_y = 3, and n = 4.

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Generally, the control of the number, position and null field of the ring-shaped beam profiles can provide the ring-shaped beam array more possibilities for practical applications. As a result, numerical calculations are performed to study the relationship between the properties of the ring-shaped beam array and the beam parameters of the McLGCSM source. In particular, we find that the size of the null field region of each ring-shaped beam pattern can be gigantic and the velocity evolving into a ring-shaped beam array can be enhanced with increasing the order n of Laguerre-polynomial. Performing calculations also indicate that the separation distances among these ring-shaped beam patterns increase in proportion with the beam parameter β. As already mentioned, the number of the ring-shaped beam patterns in the array is determined by the product value of P_xP_y. However, note that increasing the coherence length δ can disrupt the evolution process and the self-shaping phenomenon fades away when δ is greater than the critical value of approximately 2 as shown in Fig. 3(c). In short, the results reveal that our method can give rise to a controllable ring-shaped beam array by modulating the source structured coherence state of a McLGCSM beam on propagation in free space.

Now we study the propagation properties of a McLGCSM beam in turbulent atmosphere. Figure 4 shows the average intensity distributions of a McLGCSM, McGCSM, and LGCSM beams in turbulence at several propagation distances with the identical beam parameters used in Fig. 2 and the structure constant $C_{n}^{2} = 5 \times 10^{- 14} m^{- 2 / 3}$ . It is interesting to find that at short propagation distance, the influence of turbulence are negligible and the propagation properties of these beams in turbulence are similar to those in free space (see Fig. 2), while at long propagation distance, the influence of turbulence is greatly enhanced and all of these beams exhibit combining properties. Particularly, one finds from Fig. 4(a) that for a McLGCSM beam in Kolmogorov turbulence, the initial single Gaussian beam spot still successively evolves into a beam spot array and a ring-shaped beam array at short propagation distance; however, due to the accumulative effect of turbulence with increasing the propagation distance, the ring-shaped beam array is being gradually destroyed and eventually merge to a single Gaussian-like beam spot at sufficiently large propagation distance. Of course, it should be pointed out that the size of the finally fused single Gaussian-like beam spot in the far field is much larger than the initial one in the source plane although the configurations of these beam spots are quite similar with each other.

Fig. 4 Average intensity distribution patterns of a McLGCSM beam (a), McGCSM beam (b), and LGCSM beam (c) at several propagation distances in atmospheric turbulence with w₀ = 5mm, P_x = P_y = 3, δ_w = 1, and $C_{n}^{2}$ = 5 × 10⁻¹⁴m^−2/3.

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To further illustrate the influence of the atmospheric turbulence on the average intensity characteristics of a McLGCSM beam, we calculate in Fig. 5 the average intensity distributions of a McLGCSM beam in turbulence at several propagation distances for different values of the structure constant $C_{n}^{2}$ and beam parameter δ_w with w₀ = 5mm, β = 10, and n = 5. When the structure constant $C_{n}^{2}$ increases as shown in Figs. 5(a) and 5(b), the ring-shaped beam array becomes more vulnerable and quickly evolves into a Gaussian-like configuration after travelling a relatively short distance. Evidently, this is caused by the fact that the propagation properties are determined by the free-space diffraction and the atmospheric disturbance together. For a weak or moderate turbulence ( $C_{n}^{2} = 8 \times 10^{- 15} m^{- 2 / 3}$ ), the free-space diffraction plays a dominant role and the ring-shaped beam patterns can be obviously observed at propagation distances z = 1km and z = 8km. However, for a stronger turbulence ( $C_{n}^{2} = 5 \times 10^{- 14} m^{- 2 / 3}$ ), the atmospheric disturbance is strengthened and the ring-shaped beam profiles degrade seriously at propagation distance z = 8km. Furthermore, as illustrated in Figs. 5(a) and 5(c), we demonstrate that the ring-shaped beam array can retain for a longer propagation distance as the coherence length δ is reduced. Therefore, it implies that producing a McLGCSM source with smaller coherence length δ will have a stronger capability of resisting the destructive effect of atmospheric turbulence on the ring-shaped beam array formation.

Fig. 5 Average intensity distribution patterns of a McLGCSM beam at several propagation distances in atmospheric turbulence for different values of $C_{n}^{2}$ and δ_w with w₀ = 5mm, β = 10, and n = 5.

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An analysis of the source structured coherence states suggests that the generation of the ring-shaped beam array derived from a McLGCSM beam can be attributed to its intrinsic coherence properties. It is shown in Fig. 1 that the SDOC distribution of a McLGCSM beam in the source plane presents a more complex structure compared with that of a LGCSM beam or a McGCSM beam. In fact, the model we proposed here is clearly more general: the non-conventional Laguerre- and multi-cosine-type components have been integrated together into one entire SDOC. By combining both non-conventional SDOC components to incorporate their individual excellent properties, the synthesis effect leads to the formation of the ring-shaped beam array for a McLGCSM beam in the far field. Hence, the principle for achieving a ring-shaped beam array here is based on modulating the spatial structure of non-conventional SDOC of the incident beam, which is totally different from that reported in [55] where multiple dark hollow patterns in the transverse plane are obtained with a full polarization-controlled method. In other words, careful design and proper tuning of the spatial coherence of a McLGCSM beam in the source plane can give rise to various beam patterns in the far field such as arriving a desired ring-shaped beam array. Since recently performed experiment indicates that the random Schell-model source can radiate any desired far-zone average intensity pattern with the help of the nematic, phase-only spatial light modulators [38], we believe that this technology can be directly applied here to generate this novel model source as well.

5. Propagation factor of a McLGCSM beam

The propagation factor (also named M₂ factor) is an important characteristic parameter of a beam and can be regarded as a beam quality factor in many practical applications. In this section, we are going to derive the analytical expressions for the propagation factors of such beams within free space or in a turbulent atmosphere from the second-order moments of the Wigner distribution function (WDF) for a McLGCSM beam.

According to the method given in [42, 44, 45], the intensity moments of the order n₁ + n₂ + m₁ + m₂ of the WDF for partially coherent beams are defined as

\begin{matrix} {〈 x^{n_{1}} y^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} 〉}_{z} = \frac{1}{P} \int_{- \infty}^{\infty} d^{2} ρ d^{2} θ x^{n_{1}} y^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} h (ρ, θ, z) \\ = \frac{1}{P} \int_{- \infty}^{\infty} d^{2} ρ^{'} d^{2} θ^{'} G (ρ^{'}, θ^{'}, z) h (ρ^{'}, θ^{'}, 0), \end{matrix}

where

P = \int \int_{- \infty}^{\infty} h (ρ, θ, z) d^{2} ρ d^{2} θ

is the total power of the beam, and

h (ρ, θ, 0) = {(\frac{k}{2 π})}^{2} \int \int_{- \infty}^{\infty} W^{(0)} (ρ, ρ_{d}, 0) \exp (- i k ρ_{d} \cdot θ) d^{2} ρ_{d},

h (ρ, θ, z) = {(\frac{k}{2 π})}^{2} \int \int_{- \infty}^{\infty} W (ρ, ρ_{d}, z) \exp (- i k ρ_{d} \cdot θ) d^{2} ρ_{d},

are respectively the WDFs in the source plane z = 0 and the output z-plane, and

\begin{matrix} G (ρ^{'}, θ^{'}, z) = \frac{{(- i)}^{n_{1} + n_{2} + m_{1} + m_{2}} B^{n_{1} + n_{2}}}{k^{n_{1} + n_{2} + m_{1} + m_{2}}} \int_{- \infty}^{\infty} d^{2} ρ_{d} d^{2} {ρ^{'}}_{d} δ^{(n_{1})} ({x^{'}}_{d} - D x_{d}) δ^{(n_{2})} ({y^{'}}_{d} - D y_{d}) \\ \times δ^{(m_{1})} (x_{d}) δ^{(m_{2})} (y_{d}) \exp [\frac{i k}{B} ρ^{'} \cdot (A {ρ^{'}}_{d} - ρ_{d}) + i k θ^{'} \cdot {ρ^{'}}_{d} - H (ρ_{d}, {ρ^{'}}_{d}, z)], \end{matrix}

with δ and δ⁽ⁿ⁾ being the Dirac delta function and its nth derivative, respectively. For the propagation within free space or in a turbulent atmosphere it is clear that A = D = 1 and B = z.

Substituting Eqs. (13) and (20) into Eqs. (27)-(29) and after tedious integration, we obtain the following expressions for the second-order moments of the WDF of a McLGCSM beam in a turbulent atmosphere

{〈 x^{2} 〉}_{z} = \frac{2 z^{2}}{k^{2}} T_{α} + {〈 ρ_{x}^{2} 〉}_{0} + 2 z {〈 ρ_{x} θ_{x} 〉}_{0} + z^{2} {〈 θ_{x}^{2} 〉}_{0},

{〈 θ_{x}^{2} 〉}_{z} = \frac{6}{k^{2}} T_{α} + {〈 θ_{x}^{2} 〉}_{0},

{〈 x θ_{x} 〉}_{z} = \frac{3 z}{k^{2}} T_{α} + {〈 ρ_{x} θ_{x} 〉}_{0} + z {〈 θ_{x}^{2} 〉}_{0},

where

{〈 Q (r, θ) 〉}_{0} = \frac{1}{P} \int_{- \infty}^{\infty} d^{2} r d^{2} θ Q (r, θ) h (r, θ, 0) .

Completing some mathematical operations we finally obtain

{〈 x^{2} + y^{2} 〉}_{z} = (\frac{4}{k^{2}} T_{α} + \frac{M_{n, β}}{2 w_{0}^{2}}) z^{2} + 2 w_{0}^{2},

{〈 x θ_{x} + y θ_{y} 〉}_{z} = (\frac{6}{k^{2}} T_{α} + \frac{M_{n, β}}{2 w_{0}^{2}}) z,

{〈 θ_{x}^{2} + θ_{y}^{2} 〉}_{z} = \frac{12}{k^{2}} T_{α} + \frac{M_{n, β}}{2 w_{0}^{2}},

M_{n, β} = \frac{1}{k^{2}} [1 + \frac{4 w_{0}^{2}}{δ^{2}} (n + 1) + \frac{2 w_{0}^{2}}{δ^{2}} Q_{M, β}],

Q_{M, β} = \frac{β_{x}^{2}}{P_{x} - 1} \sum_{l_{x} = - M_{x}}^{M_{x}} l_{x}^{2} + \frac{β_{y}^{2}}{P_{y} - 1} \sum_{l_{y} = - M_{y}}^{M_{y}} l_{y}^{2} .

Obviously, for β_x = β_y = 0, the expressions obtained above can reduce to those for a LGCSM beam [11, 14], while for n = 0, the expressions can reduce to those for a McGCSM beam [26].

The propagation factor of a partially coherent beam in atmospheric turbulence is defined in terms of the second-order moments as follows [42, 45]

M^{2} (z) = k {({〈 ρ^{2} 〉}_{z} {〈 θ^{2} 〉}_{z} - {〈 ρ \cdot θ 〉}_{z}^{2})}^{1 / 2} .

Inserting Eqs. (35)-(39) into Eq. (40), we obtain the following expression for the propagation factor of a McLGCSM beam in atmospheric turbulent

M_{n, Q}^{2} (z) = {[M_{n, β} + \frac{2 T_{α}}{k^{2}} (12 w_{0}^{2} + \frac{6 T_{α}}{k^{2}} z^{2} + \frac{M_{n, β}}{w_{0}^{2}} z^{2})]}^{1 / 2} .

Under the condition of T_α → 0, Eq. (41) represents the propagation factor of a McLGCSM beam in free space and reduces to

M_{n, Q}^{2} (z) = M_{n, β}

. Note that the propagation factor of the beam in free space is independent of the propagation distance as expected. Based on Eq. (41), we obtain

M_{n, Q}^{2} (z) > M_{n, 0}^{2} (z) [M_{0, Q}^{2} (z)] > M_{0, 0}^{2} (z)

, indicating that the propagation factor of a McLGCSM beam is the largest compared with that of a LGCSM beam (with β = 0), McGCSM beam (with n = 0), and GSM beam (with β = n = 0).

The relative propagation factor is defined as

M_{r, n, Q}^{2} (z) = M_{n, Q}^{2} (z) / M_{n, β} .

Equation (42) describes the resistance of a beam to turbulence in comparison with the free-space propagation; i.e., the smaller the relative propagation factor is, the less the beam is affected by turbulence. Therefore, the relative propagation factor can be used to quantitatively compare the effect of turbulence for different partially coherent beams upon propagation. We comparatively analyze the relative propagation factors of a McLGCSM beam, LGCSM beam, McGCSM beam, and GSM beam, and generally confirm that

M_{r, 0, 0}^{2} (z) > M_{r, n, 0}^{2} (z) [M_{r, 0, Q}^{2} (z)] > M_{r, n, Q}^{2} (z) .

For example,

M_{r, n, Q}^{2} (z) - M_{r, n, 0}^{2} (z) = \frac{12 Ω (M_{n, 0}^{2} - M_{n, β}^{2}) (2 w_{0}^{2} + Ω z^{2} / k^{2})}{k^{2} M_{n, β} M_{n, 0} [M_{n, 0} M_{r, n, Q}^{2} (z) + M_{n, β} M_{r, n, 0}^{2} (z)]} < 0.

The theoretical predications are further demonstrated in Fig. 6 where the variations of the relative propagation factors against the propagation distance for a McLGCSM beam, LGCSM beam, McGCSM beam, and GSM beam in turbulent atmosphere are illustrated. It is shown that for all of these beams, the relative propagation factors grows rapidly from 1 with the increase of propagation distance, indicating the quality of these beams being degraded by the atmospheric turbulence. Moreover, we note that the growth of the relative propagation factor of a McLGCSM beam is the slowest compared with other beams, revealing that the McLGCSM beam is less affected by the atmospheric turbulence from the aspect of the propagation factor. Therefore, the results clearly show that the proposed McLGCSM beam has advantage in atmospheric turbulence, which will be useful in free-space optical communications.

Fig. 6 Relative propagation factors $M_{r, n, Q}^{2} (z)$ versus the propagation distance z in atmospheric turbulence for (G) a GSM beam with n = β_x = β_y = 0, (L) a LGCSM beam with n = 5 and β_x = β_y = 0, (Mc) a McGCSM beam with n = 0 and β_x = β_y = 8, and (McL) a McLGCSM beam with n = 5 and β_x = β_y = 8. The other parameter are w₀ = 10mm, δ_w = 1, P_x = P_y = 3, and $C_{n}^{2}$ = 5 × 10⁻¹⁵m^−2/3.

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6. Conclusions

We have introduced one kind of partially coherent beam, named as the McLGCSM beam whose SDOC in the source plane is a Gaussian function modulated by both non-conventional Laguerre- and multi-cosine-type factors, and studied its propagation properties in free space and atmospheric turbulence. Such a beam includes a LGCSM beam, McGCSM beam, and GSM beam as its special cases. The analytical expressions for the CSD and the propagation factor of a McLGCSM beam have been derived and its statistical properties, such as the spectral intensity and the propagation factor, have been illustrated numerically. We have found that a McLGCSM beam can form a ring-shaped beam array intensity profile in the far field on free-space propagation. Meanwhile, on propagation in atmospheric turbulence the McLGCSM beam still possesses a ring-shaped beam array profile at short distance but gradually merge to a Gaussian-like pattern at sufficiently large distance. Notably, differing from a LGCSM beam where only one single ring-shaped beam profile is realizable, with such a beam we can simultaneously obtain a ring-shaped beam array with adjustable number and positions by directly tuning the spatial structure of its SDOC. The physical mechanism behind the occurrence of the ring-shaped beam array derived from a McLGCSM beam is based on its peculiar source coherence state which combines the individual merits associated with each non-conventional SDOC component of the whole SDOC. Furthermore, we have generally proved that the propagation factor of a McLGCSM beam is minimally influenced by the atmospheric turbulence compared with that of a LGCSM, McGCSM and GSM beams. The work uncovers that the far-field ring-shaped beam array generated in free space can also be formed in atmospheric turbulence through synthesizing the non-conventional SDOC components to modulate the source spatial coherence properties, and the advantages of optical manipulation with controllable ring-shaped beam array will greatly benefit applications in micromanipulation of multiple particles, free-space optical communications and imaging in the atmosphere.

Funding

High level introduction of talent research start-up fund of Guizhou Institute of Technology

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Propagation of multi-cosine-Laguerre-Gaussian correlated Schell-model beams in free space and atmospheric turbulence

Abstract

1. Introduction

2. Model for multi-cosine-Laguerre-Gaussian-correlated Schell-model (McLGCSM) source

3. Propagation of a McLGCSM beam in atmospheric turbulence

4. Numerical examples for the spectral density of a McLGCSM beam in free space and atmospheric turbulence

5. Propagation factor of a McLGCSM beam

6. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (44)

Optics Express