Asymmetric cosine-Gaussian Schell-model sources

Yawei Jiang; Zhangrong Mei

doi:10.1364/OE.522151

1. Introduction

In recent years, generalized stationary random sources have received extensive attention from researchers. It has been shown that the coherence state of a light source causes its radiated light field to develop various novel properties during the propagation process, which is called structural coherence in statistical optics [1]. Therefore, it is valuable to study the coherence states of all kinds of light sources, and the research has been widely used in free space optical communication, optical experiments, material processing, and other fields [2,3].

The proposal of sufficient conditions for designing a class of genuine CSD functions has aroused great interest of researchers in this field [4]. In the following period, using this method, a large number of interesting and novel partially coherent sources were proposed and experimentally generated. The most common sources include the Multi-Gaussian Shell-model beams [5–7] and the Sinc-Gaussian Shell-model beams [8], whose spectral density in the far field can be arbitrarily converted between the flat-top distribution and the Gaussian distribution; the Gaussian Schell-model arrays beams [9,10], which can precisely control the number and distribution of beams during propagation; the non-uniformly varying Gaussian pseudo-Schell-model beams [11,12] are able to form self-focusing properties in the far field; the Cosine-Gaussian Schell-model beams with self-splitting characteristics [13–15]; and so on. However, in the study of these beams, the complex degree of coherence (CDC) of most of the beams is selected as a real-value function, so that the distribution of the far-field spectral density of the beams exhibits Cartesian symmetry, limiting the possibility of a wider class of beams.

It is widely recognized that the appearance of the twisted phases and the vortex phases is a special case of the CDC with complex-value function [16–18], which can break the Cartesian symmetry. Recently, a novel approach in order to construct the CDC, derived from the self-convolution of the Fourier transform sliding function with complex valued phases, was proposed [19]. Based on this theory, the research of asymmetric coherent light has made great progress [20–22]. This kind of beam has important application value in information storage, optical communication, and other fields. And the propagation characteristics of the asymmetric multi-Gaussian Schell-model beams [23] and the asymmetric array beams [24] have also been studied. However, as far as is known, there is little research on asymmetric coherent light with self-splitting properties. In this paper, a cosine function will be added to the CDC with a complex value phase, in order to realize the splitting property of the beam during propagation. The far-field spectral density of such beams is derived. In addition, using free-space propagation as an illustration, the behavior of such a light source at different propagation distances is analyzed. And we also introduce the center of gravity shift function to investigate the change of center of gravity position of the beam during its propagation. The results demonstrate that when such beams are propagated, they will not only produce the self-splitting property, but also produce the asymmetric property of center of gravity shift.

2. Asymmetric cosine-Gaussian Schell-model (ACGSM) sources

The CSD is the most important way to characterize the statistical properties of partially coherent beams at any pair of points in a given optical field domain. According to the derivation of the Helmholtz equation, if each CSD function satisfies the same equation, then the total CSD function obtained by superposition of them will still satisfy a pair of Helmholtz equations, which is also a condition that the genuine CSD must satisfy [22]. Therefore, in the one-dimensional source plane, the CSD of any two points ${x_1}$ and ${x_2}$ can be defined as

(1)$${W^{(0)}}({{x_1},{x_2};\omega } )= \frac{1}{C}\sum\limits_{n = 1}^N {{c_n}} \,W_n^{(0)}({{x_1},{x_2};\omega } ),$$

where $\omega = {{2\pi c} / \lambda }$ is the angular frequency associated with the wavelength $\lambda$, c is the speed of light in a vacuum, ${c_n}$ is the positive weighting factor, and $C = \sum\nolimits_{n = 1}^N {{c_n}}$ is the normalization function. We will leave out the correlated quantities’ dependency on the angular frequency $\omega$ for the sake of brevity in the following. It is known that for a Schell-model source (the CDC is only related to the difference between two arbitrary points in space), the following form can be used to express its CSD [5]

(2)$${W^{(0)}}({{x_1},{x_2}} )= \sqrt {S({x_1})} \sqrt {S({x_2})} \mu ({x_d}),$$

where $S(x)$ represents the spectral density at position x in the source plane, $\mu ({x_d})$ is the CDC with ${x_d} = {x_2} - {x_1}$, which plays a leading role in light field regulation. Then, Eq. (1) and Eq. (2) can be combined to get the following equation:

(3)$${W^{(0)}}({{x_1},{x_2}} )= \frac{1}{C}\sum\limits_{n = 1}^N {{c_n}} \,W_n^{(0)}({{x_1},{x_2}} )= \frac{1}{C}\sum\limits_{n = 1}^N {{c_n}} \sqrt {S({x_1})} \sqrt {S({x_2})} {\mu _n}({x_d}),$$

where ${\mu _n}({x_d})$ denotes the CDC of the nth CSD $W_n^{(0)}({x_1},{x_2})$. Then the relationship between the total CDC and each CDC can be expressed as

(4)$$\mu ({{x_d}} )= \frac{1}{C}\sum\limits_{n = 1}^N {{c_n}} \,{\mu _n}({{x_d}} ).$$

A class of the CDC with complex-valued phases has been modeled in the literature [22]. Here, on the basis of literature [22], let us consider the case where each CDC is modulated by a cosine function, i.e.,

(5)$${\mu _n}({{x_d}} )= \,\textrm{exp} \left( { - \frac{{{x_d}^2}}{{4{\delta^2}}} + i{\kern 1pt} {a_n}{x_d}} \right)\,\cos \left( {\frac{{b{\kern 1pt} {x_d}}}{{2\delta }}} \right),$$

where $\delta$ is a correlation width, ${a_n}$ is an arbitrary real-valued function closely related to n, $\cos (x)$ is a typical cosine function, and the parameter b can take any value greater than zero. According to Eq. (5), Eq. (4) can be easily expressed as

(6)$$\mu ({{x_d}} )= \frac{1}{C}\sum\limits_{n = 1}^N {{c_n}} \textrm{exp} \left( { - \frac{{{x_d}^2}}{{4{\delta^2}}} + i{\kern 1pt} {a_n}{x_d}} \right)\,\cos \left( {\frac{{b{\kern 1pt} {x_d}}}{{2\delta }}} \right).$$

In general, the CSD must satisfy a sufficient condition to ensure that it is physically genuine, and this condition is expressed by the following integral form [4]

(7)$${W^{(0)}}({{x_1},{x_2}} )= \int {\,p(v )} \,H_0^\ast ({{x_1},v} )\,{H_0}({{x_2},v} )\textrm{d}v,$$

where $p(v)$ is an arbitrary non-negative function, which can represent the spectral density distribution in the far-field to a certain extent, ${H_0}(x,v)$ is an arbitrary kernel function, and the asterisk refers to the complex conjugate of function H. Without losing generality, choose the kernel function to be in the Fourier-like form, i.e.,

(8)$${H_0}({x,v} )= \tau (x )\,\textrm{exp} ({ - 2\pi ivx} ),$$

where $\tau (x)$ denotes a regular complex profile function. By bringing Eq. (8) into Eq. (7), we obtain

(9)$${W^{(0)}}({{x_1},{x_2}} )= \,{\tau ^\ast }({{x_1}} )\tau ({{x_2}} ){\cal F}[{p(v )} ]= {\tau ^\ast }({{x_1}} )\tau ({{x_2}} )\mu ({x_d}),$$

where the symbol ${\cal F}$ represents the Fourier transform of the function contained in parentheses. It is obvious that the relationship between the far-field spectral density $p(v)$ and the source-field CDC $\mu ({x_d})$ is a Fourier transform pair. The inverse Fourier transform of Eq. (6) can therefore be used to represent $p(v)$ in the following way

(10)$$p(v )= {{\cal F}^{ - 1}}[\mu ({x_d})] = \frac{{2\delta \sqrt \pi }}{C}\sum\limits_{n = 1}^N {{c_n}} \,\textrm{exp} \left[ { - {\delta^2}{{({2\pi v + {a_n}} )}^2} - \frac{{{b^2}}}{4}} \right]\,\cosh [{b\delta ({2\pi v + {a_n}} )} ].$$

Now, let us discuss the non-negativity of the $p(v)$ function. By observing the Eq. (10), it is obvious that as long as ${c_n}$ takes a positive value, then there is never a negative value for the $p(v)$ function. If the value of ${c_n}$ is negative, then the non-negative case of the $p(v)$ function needs to be discussed separately.

Here, let us take ${a_n} = \alpha n$ and ${c_n} = \textrm{exp} ( - \beta n)$ as an example where $\alpha$ is any real number and $\beta$ is any non-negative real number. And in the following experimental analysis, unless otherwise specified, two parameters are assumed to have the following values: $\delta = 1\textrm{mm}$ and $N = 15$.

Figure 1 shows the cosine-modulated CDC distribution (including its magnitude, argument, real part and imaginary part along with the change of ${x_d}$), the evolution of the corresponding coherent curve and the corresponding far-field spectral density distribution for various values of the parameter b, as defined by Eq. (6) and Eq. (10), respectively. As the value of parameter b increases, the oscillation degree of the CDC gradually increases, which is most obviously reflected in that the argument changes from the smooth curve at the beginning to multiple broken lines, and the number of broken lines gradually increases to become more compact, as can be observed in the left column in Fig. 1. The coherence curve presented in the middle column in Fig. 1 becomes increasingly distorted as a result. The corresponding far-field spectral density is reflected in the appearance of dark regions, and the intensity of the CDC oscillation directly determines the area size of dark regions, as shown in the right column in Fig. 1.

Fig. 1. The CDC (left column), the corresponding coherence curves (middle column) and the corresponding far-field spectral density (right column), which was calculated using Eq. (6) and Eq. (10) with $\alpha = {\pi / 4}$ and $\beta = 0.3$ for various values of the parameter b. (a)–(c) $b = 0$; (d)–(f) $b = 4\,\pi$; (g)–(i) $b = 6\,\pi$; (j)–(l) $b = 8\,\pi$.

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Similarly, according to Eq. (6) and Eq. (10), the evolution of the CDC, the corresponding coherence curves, along with the corresponding far-field spectral density are shown in Fig. 2 and Fig. 3 along with changes in the values of the parameters $\alpha$ and $\beta$ when $b = 6\pi$, respectively. Figures 2(a) to 2(c) indicate that the beam has the same characteristics as the Cosine-Gaussian Schell-model beams [13] when $\alpha = 0$. As the values of $\alpha$ gradually increases, the CDC's phase jumps are getting more and more severe, the coherence curve is distorted from a straight line and it becomes more obvious that the far-field spectral density distribution is asymmetric in Figs. 2(d) to 2(f). When the value of $\alpha$ is large enough, the far-field spectral density distribution begins to appear sidelobes (Airy-like effect in Fig. 2(f)), and the number of sidelobes can be adjusted based on the value of N. Figure 3(c) indicates that its far-field spectral density distribution is presented by two-beam displaced flat-top Gaussian profiles when $\beta = 0$ and $\alpha$ is small enough. With the gradual increase of the values of $\beta$, the magnitude distribution of the CDC gradually tends to that of the Cosine-Gaussian Schell-model source [13], the coherence curve is gradually extended in all directions and the far-field spectral density distribution gradually tends to the split two-beam perfect Gaussian profiles in Figs. 3(d) to 3(f).

Fig. 2. Same as Fig. 1, but with $b = 6\,\pi$ and $\beta = 0.3$ for different values of the parameter $\alpha$. (a)–(c) $\alpha = 0$; (d)–(f) $\alpha = {{3\pi } / 4}$.

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Fig. 3. Same as Fig. 1, but with $b = 6\,\pi$ and $\alpha = {\pi / 4}$ for different values of the parameter $\beta$. (a)–(c) $\beta = 0$; (d)–(f) $\beta = 3$.

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Figure 4 shows the impact of this parameter $\delta $ on the CDC, the corresponding coherence curves and the corresponding far-field spectral density. It can be seen that for smaller value of the parameter $\delta$, the lateral distribution of the CDC appears more dense and the coherence curve is concentrated in the first and third quadrants, while for larger value of the parameter $\delta$, the coherence state is more dispersed and the coherence curve is also separate in four quadrants. The corresponding beam width of far-field spectral density and the distance between the split beams decrease with the increase of $\delta$. Therefore, by adjusting the value of parameter $\delta$, it is possible to modulate the coherence state of the source-field, thereby simultaneously controlling the beam width and the distance of the split beam in the far-field.

Fig. 4. Same as Fig. 1, but with $b = 6\,\pi$, $\alpha = {\pi / 4}$ and $\beta = 3$ for different values of the parameter $\delta$. (a)–(c) $\delta = 0.4\textrm{mm}$; (d)–(f) $\delta = 2.4\textrm{mm}$.

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3. ACGSM beams and their evolution characteristics

By bring the Eq. (6) into the Eq. (9), it is easy to obtain a one-dimensional expression for the CSD between any two points ${x_1}$ and ${x_2}$ of the ACGSM sources in the source plane:

(11)$${W^{(0)}}({{x_1},{x_2}} )= {\tau ^\ast }({x_1})\,\tau ({x_2})\,\frac{1}{C}\sum\limits_{n = 1}^N {{c_n}} \,\textrm{exp} \left( { - \frac{{{x_d}^2}}{{4{\delta^2}}} + i{\kern 1pt} {a_n}{x_d}} \right)\,\cos \left( {\frac{{b{\kern 1pt} {x_d}}}{{2\delta }}} \right).$$

Let us make the assumption that the complex profile function $\tau (x)$ corresponds to a typical gaussian profile. That is, $\tau (x) = {{{x^2}} / {4{\sigma ^2}}}$, and $\sigma$ denotes the initial width of the beam. Then we found the source-field CSD has the following form:

(12)$${W^{(0)}}({{x_1},{x_2}} )= \frac{1}{C}\textrm{exp} \left( { - \frac{{x_1^2 + x_2^2}}{{4{\sigma^2}}}} \right)\,\sum\limits_{n = 1}^N {{c_n}} \,\textrm{exp} \left( { - \frac{{{x_d}^2}}{{4{\delta^2}}} + i{\kern 1pt} {a_n}{x_d}} \right)\,\cos \left( {\frac{{b{\kern 1pt} {x_d}}}{{2\delta }}} \right).$$

It is simple to generalize the one-dimensional field to the two-dimensional field. According to the literature [25], the expression for the CSD function in a two-dimensional field between any two vector points, ${\boldsymbol{\mathrm{\rho}}_1} = ({x_1},{y_1})$ and ${\boldsymbol{\mathrm{\rho}}_2} = ({x_2},{y_2})$, can be represented as

(13)$$\begin{aligned} {W^{(0)}}({{\boldsymbol{\mathrm{\rho}}_\textrm{1}},{\boldsymbol{\mathrm{\rho}}_\textrm{2}}} )&= \prod\limits_{\xi = x,y} {{W^{(0)}}({{\xi_1},{\xi_2}} )} \\ &= \textrm{exp} \left( { - \frac{{\boldsymbol{\mathrm{\rho}}_1^2 + \boldsymbol{\mathrm{\rho}}_2^2}}{{4{\sigma^2}}}} \right)\,\prod\limits_{\xi = x,y} {\frac{1}{{{C_\xi }}}} \sum\limits_{{n_\xi } = 1}^{{N_\xi }} {{c_{{n_\xi }}}} \textrm{exp} \left[ { - \frac{{{{({{\xi_2} - {\xi_1}} )}^2}}}{{4{\delta_\xi }^2}} + i{\kern 1pt} {a_{{n_\xi }}}({{\xi_2} - {\xi_1}} )} \right]\,\\ &\times \cos \left[ {\frac{{{b_\xi }({{\xi_2} - {\xi_1}} )}}{{2{\delta_\xi }}}} \right]. \end{aligned}$$

Using the expanded Huygens-Fresnel theory as foundation, when the ACGSM beam propagates in free space, the CSD function for a pair of points ${{\textbf r}_1} = ({\boldsymbol{\mathrm{\rho}}_1},z)$ and ${{\textbf r}_2} = ({\boldsymbol{\mathrm{\rho}}_2},z)$ in the receiving plane z has the following expression:

(14)$$W({{{\textbf r}_\textrm{1}},{{\textbf r}_\textrm{2}},z} )= \,{\left( {\frac{k}{{2\pi z}}} \right)^2}\int\!\!\!\int {{W^{(0)}}({{\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2}} )} \,\textrm{exp} \left\{ { - \frac{{ik}}{{2z}}[{{{({{{\textbf r}_1} - {\boldsymbol{\mathrm{\rho}}_1}} )}^2} - {{({{{\textbf r}_2} - {\boldsymbol{\mathrm{\rho}}_2}} )}^2}} ]} \right\}{\textrm{d}^2}{\boldsymbol{\mathrm{\rho}}_1}{\textrm{d}^2}{\boldsymbol{\mathrm{\rho}}_2},$$

where $k = {{2\pi } / \lambda }$ is the number of waves and $z \gg {z_R}$, where ${z_R} = {{k{\sigma ^2}} / 2}$ is the Reileigh length. Bring the Eq. (13) into the Eq. (14), after long calculations, the following formula is obtained:

(15)$$\begin{aligned} W({{{\textbf r}_\textrm{1}},{{\textbf r}_\textrm{2}},z} )&= \,\frac{{{k^2}{\sigma ^2}}}{{4{z^2}}}\textrm{exp} \left[ {\frac{{ik}}{{2z}}({{\textbf r}_2^2 - {\textbf r}_1^2} )} \right]\,\textrm{exp} \left[ { - \frac{{{k^2}{\sigma^2}}}{{2{z^2}}}{{({{{\textbf r}_2} - {{\textbf r}_1}} )}^2}} \right]\,\prod\limits_{\xi = x,y} {\frac{1}{{{A_\xi }{C_\xi }}}} \sum\limits_{{n_{\xi = 1}}}^{{N_\xi }} {{c_{{n_\xi }}}} \\ &\times \,\left[ {\textrm{exp} \left( { - \frac{{\gamma_{{\xi_ + }}^2}}{{2{A_\xi }^2}}} \right) + \textrm{exp} \left( { - \frac{{\gamma_{{\xi_ - }}^2}}{{2{A_\xi }^2}}} \right)} \right], \end{aligned}$$

where

(16)$${A_\xi }^2 = \frac{1}{{2{\delta _\xi }^2}} + \frac{1}{{4{\sigma ^2}}} + \frac{{{k^2}{\sigma ^2}}}{{{z^2}}},$$

(17)$${\gamma _{{\xi _ \pm }}} = \frac{{i{k^2}{\sigma ^2}}}{{{z^2}}}({{\xi_2} - {\xi_1}} )+ \frac{k}{{2z}}({{\xi_2} + {\xi_1}} )+ \left( {{a_{{n_\xi }}} \pm \frac{{{b_\xi }}}{{2{\delta_\xi }}}} \right).$$

According to the analytical expressions of the source-field CSD function and the far-field CSD function expressed by Eq. (13) and Eq. (15) respectively, it is easy to study the source-field coherence state $\mu (\boldsymbol{\mathrm{\rho}}, - \boldsymbol{\mathrm{\rho}})$ changes and the corresponding far-field spectral density $S({\textbf r},z)$ changes when propagating in free space for the ACGSM sources. The expressions for both are as follows:

(18)$$\mu ({\boldsymbol{\mathrm{\rho}}, - \boldsymbol{\mathrm{\rho}}} )= \frac{{{W^{(0)}}(\boldsymbol{\mathrm{\rho}}, - \boldsymbol{\mathrm{\rho}})}}{{\sqrt {{W^{(0)}}(\boldsymbol{\mathrm{\rho}},\boldsymbol{\mathrm{\rho}})\;{W^{(0)}}( - \boldsymbol{\mathrm{\rho}}, - \boldsymbol{\mathrm{\rho}})} }} = \prod\limits_{\xi = x,y} {\frac{1}{{{C_\xi }}}} \sum\limits_{{n_\xi } = 1}^{{N_\xi }} {{c_{{n_\xi }}}} \textrm{exp} \left( { - \frac{{{\xi^2}}}{{{\delta_\xi }^2}} - 2i{\kern 1pt} {a_{{n_\xi }}}\xi } \right)\,\cos \left( {\frac{{{b_\xi }\xi }}{{{\delta_\xi }}}} \right),$$

(19)$$S({{\textbf r},z} )= W({{\textbf r},{\textbf r},z} )= \frac{{{k^2}{\sigma ^2}}}{{4{z^2}}}\,\prod\limits_{\xi = x,y} {\frac{1}{{{A_\xi }{C_\xi }}}} \sum\limits_{{n_{\xi = 1}}}^{{N_\xi }} {{c_{{n_\xi }}}} \,\left[ {\textrm{exp} \left( { - \frac{{{{\gamma^{\prime}}_{{\xi_ + }}}^2}}{{2{A_\xi }^2}}} \right) + \textrm{exp} \left( { - \frac{{{{\gamma^{\prime}}_{{\xi_ - }}}^2}}{{2{A_\xi }^2}}} \right)} \right],$$

where

(20)$${\gamma ^{\prime}_{{\xi _ \pm }}} = \frac{k}{z}\xi + \left( {{a_{{n_\xi }}} \pm \frac{{{b_\xi }}}{{2{\delta_\xi }}}} \right).$$

Due to the presence of parametric $\alpha$ in the CDC, the center of gravity of the beams will be offset as it propagates along the z-axis. In the literature [26], we learned that the calculation method of the offset of the beams center of gravity ${{\textbf r}_c} = ({x_c},{y_c})$ is as follows:

(21)$${x_c}(z) = \frac{{\int {x\,S({\textbf r},z){\textrm{d}^2}{\textbf r}} }}{{\int {S({\textbf r},z){\textrm{d}^2}{\textbf r}} }};\quad {y_c}(z) = \frac{{\int {y\,S({\textbf r},z){\textrm{d}^2}{\textbf r}} }}{{\int {S({\textbf r},z){\textrm{d}^2}{\textbf r}} }}.$$

Bring Eq. (19) into Eq. (21), the position of the beams center of gravity can be obtained:

(22)$${x_c}(z) ={-} \frac{z}{k}{{\sum\limits_{{n_x} = 1}^{{N_x}} {{c_{{n_x}}}{a_{{n_x}}}} } / {\sum\limits_{{n_x} = 1}^{{N_x}} {{c_{{n_x}}}} }};\quad {y_c}(z) ={-} \frac{z}{k}{{\sum\limits_{{n_y} = 1}^{{N_y}} {{c_{{n_y}}}{a_{{n_y}}}} } / {\sum\limits_{{n_y} = 1}^{{N_y}} {{c_{{n_y}}}} }}.$$

In this paper, let us set ${a_{{n_\xi }}} = {\alpha _\xi }\,{n_\xi }$ and ${c_{{n_\xi }}} = \textrm{exp} ( - {\beta _\xi }\,{n_\xi })$. In the study of the two-dimensional field, this article presents only the following case: ${\delta _x} = {\delta _y} = \delta$, ${b_x} = {b_y} = b$, ${\alpha _x} = {\alpha _y} = \alpha$, ${\beta _x} = {\beta _y} = \beta$ and ${N_x} = {N_y} = N$. For cases where the parameters are not equal, it is easy to infer from one of the above.

For different settings of the parameters b, $\alpha$ and $\beta$, Fig. 5 displays the change in the absolute value of the source-field CDC defined by Eq. (18). Obviously, by changing the value of b, the overall outline of the absolute value of the CDC does not change, but the lattice distribution is presented, and the number of lattices increases as the value of b increases (here, only the cases of $b = 0$ and $b = 6\,\pi$ are shown), as illustrated in Figs. 5(a) and 5(b). Figure 5(c) demonstrates that changing the value of $\alpha$ will change the overall outline of the absolute value of the CDC, that is, it presents a lattice distribution composed of multiple rectangular distributions. The absolute value of the CDC gradually forms a series of point-like distributions as the $\beta$ value decrease, as illustrated in Fig. 5(d). When the value of $\beta$ is large enough, the overall profile of the absolute value of the CDC for the source is close to that of the Cosine-Gaussian Schell-model source [13] (not shown in this paper).

Fig. 5. The absolute value of the CDC defined by Eq. (18) for different settings of the parameters b, $\alpha$ and $\beta$. (a) $b = 0$, $\alpha = {\pi / 4}$, $\beta = 0.3$; (b) $b = 6\,\pi$, $\alpha = {\pi / 4}$, $\beta = 0.3$; (c) $b = 6\,\pi$, $\alpha = 2\,\pi$, $\beta = 0.3$; (d) $b = 6\,\pi$, $\alpha = 2\,\pi$, $\beta = 0$.

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Figures 6–9, respectively, show how the spectral density changes as propagation calculated by Eq. (19) corresponding to Figs. 5(a) to 5(d). Additionally, the center of gravity is calculated by Eq. (22) and marked on the images. It is easy to observe that as the propagation distance increases, the beam progressively shifts and deforms, resulting in the center of gravity gradually deviating from the maximum value of intensity. By comparing the images, it is easy to draw conclusions the following: the value of b mainly controls the distance between the split beams, the value of $\alpha$ mainly controls the beam shift and the appearance of the lattice, and the value of $\beta$ mainly controls whether the beam tends to a flat-top distribution or to a Gaussian distribution.

Fig. 6. With the parameters shown in Fig. 5(a), the corresponding spectral density at various propagation distances was calculated by Eq. (19).

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Fig. 7. With the parameters shown in Fig. 5(b), the corresponding spectral density at various propagation distances was calculated by Eq. (19).

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Fig. 8. With the parameters shown in Fig. 5(c), the corresponding spectral density at various propagation distances was calculated by Eq. (19).

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Fig. 9. With the parameters shown in Fig. 5(d), the corresponding spectral density at various propagation distances was calculated by Eq. (19).

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Obviously, the shift in the center of gravity of the beam is only related to the parameters z, $\alpha$, $\beta$ and N. Next, let us check how the beam's center of gravity shift varies with each parameter. Figure 10 shows the change of the above four parameters on the centroid shift, taking the x-axis as an example in Eq. (22). It is clear that the degree of deviation of the beam is linearly related to the propagation distance. Figure 10(a) shows that when ${\alpha _x} > 0$, it is inversely proportional. When ${\alpha _x} = 0$, there will be no offset. When ${\alpha _x} < 0$, it is proportional. With the increase of the absolute value of ${\alpha _x}$, the speed of the beam deviation is accelerated. In Fig. 10(b), the offset slope of the beam center of gravity gradually decreases to close to 0 as the ${\beta _x}$ value increases. As the value of ${N_x}$ grows, the offset slope of the beam center of gravity progressively gets larger, but the speed of increase slows down in Fig. 10(c).

Fig. 10. The center of gravity offset calculated by Eq. (22) for different parameters ${\alpha _x}$, ${\beta _x}$ and ${N_x}$. (a) ${\beta _x} = 0.3$, ${N_x} = 15$; (b) ${\alpha _x} = {\pi / 4}$, ${N_x} = 15$; (c) ${\alpha _x} = {\pi / 4}$, ${\beta _x} = 0.3$.

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4. Conclusions

In this paper, the coherence states and the corresponding spectral densities of the ACGSM sources in a one-dimensional field were studied. Subsequently, it was extended to a two-dimensional field and the propagation characteristics of the ACGSM sources in free space were analyzed. In addition, we also show how the center of gravity shift of the beam varies with the values of each parameter. It was found that the far-field spectral density will be split by changing the parameter b of the cosine function in the CDC. By changing the value of the parameter $\alpha$, the far-field spectral density may exhibit a grating (one-dimensional)/lattice (two-dimensional) distribution. By adjusting the value of the parameter $\beta$, the far-field spectral density can be adjusted between the flat-top distribution and the perfect Gaussian distribution. And the more the CDC distribution of the source field is closer to the Cosine-Gaussian Schell-model beams [13], the more the spectral density distribution of the far field is closer to the perfect Gaussian profile. When the parameter $\alpha \ne 0$ in the complex valued phase, an increase of the propagation distance will cause the beam center of gravity to shift, and the speed of the shift is affected by the parameters $\alpha$, $\beta$ and N.

The synthesis of such source may be readily implemented with the help of a nematic spatial light modulator described in Ref. [7]. From the above results, it can be seen that adjusting the parameters in the source coherence state can make the beam appear in the form of split, offset, Airy-like distribution, rectangular distribution, etc. This precise control of the beam will play a role in optical imaging, optical communication and other fields.

Funding

Natural Science Foundation of Zhejiang Province, China (LY23A040006); National Natural Science Foundation of China (11974107).

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.

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Asymmetric cosine-Gaussian Schell-model sources

Abstract

1. Introduction

2. Asymmetric cosine-Gaussian Schell-model (ACGSM) sources

3. ACGSM beams and their evolution characteristics

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (22)

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