Distribution of intensity and M<sup>2</sup> factor for a partially coherent flat-topped beam in bidirectional turbulent atmosphere and plasma connection

Xinzhuang Wang; Xinzhuang Wang; Xinzhuang Wang; Hui Zhang; Hui Zhang; Hui Zhang; Yaru Gao; Yaru Gao; Yaru Gao; Dongmei Wei; Dongmei Wei; Dongmei Wei; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangsheng Yuan; Yangsheng Yuan; Yangsheng Yuan; Yangsheng Yuan

doi:10.1364/OE.514420

1. Introduction

The unique structure of the flat-topped beam is widely applied in high-power laser processing [1], laser nuclear physics [2], biomedical engineering [3], chromatic aberrations correction [4], and free space optical (FSO) communications [5]. Some theoretical models are proposed to describe the beams with the flat-topped profile [6–8]. Analytical expressions for the superposition of multiple Gaussian beams with varying beam width have been introduced to conveniently analyze the propagation properties of these beams [8]. Experimental generation methods include an electro-optically Q-switched [9] or cavity-dumped burst-mode Nd: YAG lasers [10], diffractive optical elements [11], aspheric lens with a long-pitch reflective diffraction gratings [12], fiber-bundle prisms [13], and all solid antiresonant fibers [14].

The turbulent atmosphere has a disruptive effect on the signal carried by these beams, impacting the propagation properties [15]. Scholars have proposed partially coherent beams as a solution to mitigate the turbulent atmosphere evolution properties induced the. Subsequently, a model for partially coherent flat-topped beams was proposed, and its propagation properties were studied [16]. In recent years, various kinds of partially coherent flat-topped beam and other structurally partial beams have been explored [17–26], including flat-topped beams propagating through uniaxial crystals and paraxial optical systems [20,21]. Additionally, the polarization state of the partially coherent flat-topped beams in turbulent atmosphere has been analyzed [22]. Furthermore, the beam properties of the partially and truncated partially coherent flat-topped beams disturbed by the turbulent atmosphere have been examined [23–26]. The M² factor is an important parameter of the laser beams propagation in free space and/or random media [27–30], serving as a key metric for evaluating the transmission properties of the beams.

Plasma, the fourth state of matter generated by a strong electro-magnetic field and ultra-high temperature ionization, find diverse applications in the atomic physics, new particle accelerators and radiation sources [31–35]. The propagation properties of ultra-short laser pulses in rarefied plasma have been studied, revealing that the propagation of relativistic self-focused supercritical power laser pulses is almost the same as that in a vacuum [36]. The propagation characteristics of ultra-short laser pulses in parabolic plasma channels were analyzed to describe the influence of transverse mass dynamic nonlinearity on the propagation characteristics of laser pulses [37]. The effect of the hollow channels drilled by long-pulse lasers in inhomogeneous plasma for the propagation of the intense laser pulses have been studied [38]. The transverse distribution of laser pulses in plasma [39] and the self-focusing of intense optical beam propagation in plasma have also been explored [40]. Self-focusing of Hermite–Gaussian laser beams in plasma under plasma density ramp was investigated, and localized upward plasma density ramp was used to overcome the defocusing of the beam [41]. The evolution properties of a Gaussian beam in a turbulent plasma sheath were investigated, revealing that the amplitude and phase variations of the Gaussian beam are determined by the plasma sheath turbulence [42]. Additionally, the propagation characteristics for the speckle field of the laser beam were examined, when the speckle propagates in the plasma, many kinds of nonlinear effects can be restrained [43].

The plasma can also be generated when the aircraft travels at supersonic speeds. In this letter, we studied the intensity distribution and M ² factor of a partially coherent flat-topped beam propagating in a bidirectional turbulent atmosphere and plasma. The results show that the propagation properties of this beam can be controlled by the beam order, wave length, beam width, and transverse spatial coherence width of the beam. Our results can be applied in FSO communications.

2. Theory

2.1 Intensity distribution

The distribution of laser beam intensities undergoes changes as it propagates through turbulent atmospheres or other random media. The analysis of the evolving properties of intensity distribution can be analyzed using the cross spectral density (CSD) function [15]. The CSD function for partially coherent flat-topped beams at the source plane is described as follows [16]:

(1)$$\begin{aligned} W({{{\boldsymbol \rho}_1},{{\boldsymbol \rho}_2},0} )&= \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)} } \\ &\times \left\{ {\exp \left( { - \frac{{n{\boldsymbol \rho}_1^2}}{{w_0^2}} - \frac{{m{\boldsymbol \rho}_2^2}}{{w_0^2}}} \right)\exp \left[ { - \frac{{{{({{{\boldsymbol \rho}_1} - {{\boldsymbol \rho}_2}} )}^2}}}{{2\sigma_0^2}}} \right]} \right\} \end{aligned}$$

where, ${{\boldsymbol \rho}_1} = ({\rho _{1x}},{\rho _{1y}})$ and ${{\boldsymbol \rho}_2} = ({\rho _{2x}},{\rho _{2y}})$ are the arbitrary position vectors at the source plane, (N, n)^T= N!/n!(N-n)! and (N, m)^T= N!/m!(N-m)! are the binomial coefficients, N is the beam order of the partially coherent flat-topped beam, w₀ is beam width, and ${\sigma _0}$ represents the transverse spatial coherence width.

The evolution properties of the CSD function for the partially coherent flat-topped beam propagated through the random media can be obtained by the Collins formula, as follows [44,45]:

(2)$$\begin{aligned} W({{\boldsymbol r}_1},{{\boldsymbol r}_2},z) &= \frac{1}{{{\lambda ^2}{B^2}}}\int\!\!\!\int\!\!\!\int {\int\limits_{ - \infty }^\infty {W({{\boldsymbol \rho}_1},{{\boldsymbol \rho}_2},0)} } \exp \left\{ { - \frac{{ik}}{{2B}}[{A(\rho_{1x}^2 - \rho_{2x}^2 + \rho_{1y}^2 - \rho_{2y}^2)} } \right.\;\\ & + D(x_1^2 - x_2^2 + y_1^2 - y_2^2) { { - 2({x_1}{\rho_{1x}} - {x_2}{\rho_{2x}} + {y_1}{\rho_{1y}} - {y_2}{\rho_{2y}})} ]} \}\\ &\times \exp [{H({{\boldsymbol r}_1},{{\boldsymbol r}_2},{{\boldsymbol \rho}_1},{{\boldsymbol \rho}_2},z)} ]d{\rho _{1x}}d{\rho _{1y}}d{\rho _{2x}}d{\rho _{2y}} \end{aligned}$$

where (r₁= x₁, y₁) and (r₂= x₂, y₂) are the arbitrary position vectors at the receiver plane, z is the propagation distance, the asterisk denotes the corresponding complex conjugate, k = 2π/λ is the wave number with the wavelength λ, and A, B, and D are corresponding matrix elements of the optical transmission system. The term exp[H(r, ρ, z)] denotes the contribution of the turbulent atmosphere.

To analyze the evolution properties of a partially coherent flat-topped beam propagating in a random medium, we introduce the sum and difference vector notations, as follows:

(3)$${\rho _d} = {\rho _1} - {\rho _2},\textrm{ }{\rho _s} = \frac{{{\rho _1} + {\rho _2}}}{2},\textrm{ }{r_d} = {r_1} - {r_2},\textrm{ }{r_s} = \frac{{{r_1} + {r_2}}}{2},$$

which can be combined with Eq. (3) to rewrite Eqs. (1) and (2) as follows:

(4)$$\begin{aligned} W({{\boldsymbol \rho}_s},{\rho _d},0) &= \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)\exp \left( { - \frac{{{\rho_d}^2}}{{2\sigma_0^2}}} \right)} } \\ &\times \exp \left[ { - \frac{{n{{({{\rho_s} + {\rho_d}/2} )}^2}}}{{w_0^2}} - \frac{{m{{({{\rho_s} - {\rho_d}/2} )}^2}}}{{w_0^2}}} \right] \end{aligned}$$

and

(5)$$\begin{aligned} &W({r_s},{r_d},z) = {(\frac{k}{{2\pi B}})^2}\int\!\!\!\int {W({\rho _s},{\rho _d},0)} \;\exp \left\{ {\frac{{ik}}{B}[{A({\rho_s} \cdot {\rho_d})} } \right.\\ &{ + D({r_s} \cdot {r_d}) - ({\rho_d}{r_s} + {\rho_s}{r_d})} ]\exp \{{ - H({r_d},{\rho_d},z)} \}\;{d^2}{\rho _s}{d^2}{\rho _d} \end{aligned}$$

The last term in Eq. (5) can be expressed as follows

(6)$$H({r_d},{\rho _d},z) = 4{\pi ^2}{k^2}z\int\limits_0^1 {} \int\limits_0^\infty {} [{1 - {J_0}({\mathrm{\kappa }|{{\rho_d}\xi + {r_d}(1 - \xi )} |} )} ]{\Phi _n}(\mathrm{\kappa })\mathrm{\kappa }d\mathrm{\kappa }d\xi$$

where, $\mathrm{\kappa }$ is the spatial frequency, ${J_0}$ is the zero-order Bessel function, and ${\Phi _n}$ is the one-dimensional power spectrum of the refractive index fluctuation for the turbulent atmosphere.

Substituting Eqs. (4) and (6) into Eq. (5), the cross-spectral density function of the partially coherent flat-topped beam propagating through ABCD optical system in turbulent atmosphere can be obtained as follows:

(7)$$\begin{aligned} &W({r_1},{r_2},z) = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{1}{{{\lambda ^2}{B^2}}}\frac{{{\pi ^2}}}{{{a_0}{b_0}}}\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)\exp \left[ { - \frac{{ikD}}{{2B}}(r_1^2 - r_2^2)} \right]} } \\ &\times \exp \left[ { - \frac{{{\pi^2}{k^2}zT}}{3}(r_1^2 + r_2^2) + \frac{{2{\pi^2}{k^2}zT}}{3}{r_1}{r_2}} \right]\exp \left\{ {\frac{1}{{4{b_0}}}S_1^2r_1^2 + \frac{1}{{4{b_0}}}S_2^2r_2^2 + \frac{1}{{2{b_0}}}{S_1}{S_2}{r_1}{r_2}} \right\}\\ &\times \exp \left[ {\frac{1}{{4{a_0}}}{{(\frac{{ik}}{B} - \frac{{{\pi^2}{k^2}zT}}{3})}^2}r_1^2 + \frac{1}{{4{a_0}}}{{(\frac{{{\pi^2}{k^2}zT}}{3})}^2}r_2^2 + \frac{1}{{2{a_0}}}\frac{{{\pi^2}{k^2}zT}}{3}(\frac{{ik}}{B} - \frac{{{\pi^2}{k^2}zT}}{3}){r_1}{r_2}} \right] \end{aligned}$$

where

(8)$$T = \int\limits_0^\infty {} {\Phi _n}({\kappa }){{\kappa }^3}d{\kappa }\;$$

and

(9a)$${a_0} = \frac{n}{{w_0^2}} + \frac{1}{{2\sigma _0^2}} + \frac{{{\pi ^2}{k^2}zT}}{3} + \frac{{ikA}}{{2B}}$$

(9b)$${b_0} = \frac{m}{{w_0^2}} + \frac{1}{{2\sigma _0^2}} + \frac{{{\pi ^2}{k^2}zT}}{3} - \frac{{ikA}}{{2B}} - \frac{1}{{4{a_0}}}{(\frac{1}{{\sigma _0^2}} + \frac{{2{\pi ^2}{k^2}zT}}{3})^2}$$

(9c)$${S_1} = \frac{1}{{2{a_0}}}(\frac{1}{{\sigma _0^2}} + \frac{{2{\pi ^2}{k^2}zT}}{3})(\frac{{ik}}{B} - \frac{{{\pi ^2}{k^2}zT}}{3}) + \frac{{{\pi ^2}{k^2}zT}}{3}\;$$

(9d)$${S_2} = \frac{1}{{2{a_0}}}(\frac{1}{{\sigma _0^2}} + \frac{{2{\pi ^2}{k^2}zT}}{3})\frac{{{\pi ^2}{k^2}zT}}{3} - (\frac{{ik}}{B} + \frac{{{\pi ^2}{k^2}zT}}{3})$$

We use the following integral formula to obtain the above derivation:

(10)$$\int\limits_{ - \infty }^\infty {\exp ( - {q^2}{x^2} \pm sx)} dx = \frac{{\sqrt \pi }}{q}\exp (\frac{{{s^2}}}{{4{q^2}}})$$

Equation (7) describes the intensity distribution of the partially coherent flat-topped beam propagating through the random medium when the parameter r₁, r₂ are fulfilled by the condition r₁= r₂= r. We use Eq. (7) to analyze the evolution properties of the intensity distribution of the partially coherent flat-topped impacted by the turbulent atmosphere and plasma.

2.2 Propagation factor (M² -factor)

To obtain the M ² factor of the partially coherent flat-top propagation in the turbulent atmosphere and plasma, we adopt the similar methods used in previous studies [28,29], resulting in Eqs. (4) and (5) rewritten in the following form:

(11)$$\begin{aligned} &W({{\rho ^{\prime}}_s},\frac{1}{A}({r_d} + \frac{B}{k}{\kappa _d}),0) = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)} } \\ &\quad \times \exp \left\{ { - n{{\left[ {{{\rho^{\prime}}_s} + \frac{1}{{2A}}({r_d} + \frac{B}{k}{\kappa_d})} \right]}^2}/w_0^2} \right.\\ &\quad\left. { - m{{\left[ {{{\rho^{\prime}}_s} - \frac{1}{{2A}}({r_d} + \frac{B}{k}{\kappa_d})} \right]}^2}/w_0^2 - {{\left( {\frac{1}{A}({r_d} + \frac{B}{k}{\kappa_d})} \right)}^2}/2\sigma_0^2} \right\} \end{aligned}$$

and

(12)$$\begin{aligned} W({r_s},{r_d},z) &= {(\frac{1}{{2\pi A}})^2}\int\!\!\!\int {W({{\rho ^{\prime}}_s},\frac{1}{A}({r_d} + \frac{B}{k}{\kappa _d}),0)} \exp \left[ {i\frac{k}{B}(D - \frac{1}{A}){r_s}{r_d}} \right.\\ &\quad \times \left. { - i\frac{1}{A}{r_s}{\kappa_d} + i{\kappa_d} - H({r_d},\frac{1}{A}({r_d} + \frac{B}{k}{\kappa_d}),z){{\rho^{\prime}}_s}} \right]{d^2}{\kappa _d}{d^2}{{\rho ^{\prime}}_s}\; \end{aligned}$$

where ${\kappa _d} = ({\kappa _{dx}},{\kappa _{dy}})$ represents the position vector.

Substituting Eqs. (6) and (11) into Eq. (12), the CSD function of the partially coherent flat-topped beams through an ABCD optical system with a turbulent atmosphere and plasma can be obtained as follows:

(13)$$\begin{aligned} W({r_s},{r_d},z) &= {(\frac{1}{{2\pi A}})^2}\int\!\!\!\int {\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)} } } \\ &\quad \times \exp \left( { - \left\{ {\frac{{(n + m)\rho^{\prime 2}_s}}{{w_0^2}} + \frac{{(n - m)}}{{Aw_0^2}}{{\rho^{\prime}}_s}({r_d} + \frac{B}{k}{\kappa_d})} \right.} \right.\\ &\quad\left. {\left. { + \frac{1}{{{A^2}}}\left[ {\frac{{(n + m)}}{{4w_0^2}} + \frac{1}{{2\sigma_0^2}}} \right]{{\left|{{r_d} + \frac{B}{k}{\kappa_d}} \right|}^2}} \right\}} \right)\\ &\quad \times \exp \left[ {i\frac{k}{B}(D - \frac{1}{A}){r_s}{r_d} - i\frac{1}{A}{r_s}{\kappa_d} + i{\kappa_d}{{\rho^{\prime}}_s}} \right]\\ &\quad \times \exp \left\{ { - H\left[ {{r_d},\frac{1}{A}({r_d} + \frac{B}{k}{\kappa_d}),z} \right]} \right\}\;{d^2}{\kappa _d}{d^2}{{\rho ^{\prime}}_s}\; \end{aligned}$$

In general, Wigner distribution is used to calculate M ² factor of the laser beam propagation in random media. The expression of the Wigner distribution can be obtained with CSD function, as follows

(14)$$h({{\boldsymbol r}_s},{\boldsymbol \theta },z) = \frac{{{k^2}}}{{4{\pi ^2}}}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {W({{\boldsymbol r}_s},{{\boldsymbol r}_d},z)\exp ({ - ik{\boldsymbol \theta } \cdot {{\boldsymbol r}_d}} )} } {d^2}{{\boldsymbol r}_d}$$

where θ=(θ_x, θ_y) is the angle between the z direction and the associated vector. kθ_x and kθ_y are the wave vector components in x and y directions, respectively.

Substituting Eq. (13) into Eq. (14), we obtain the following expression:

(15)$$\begin{aligned} h({{\boldsymbol r}_s},{\boldsymbol \theta },z) &= \frac{{{k^2}\pi w_0^2}}{{16{\pi ^4}{A^2}}}\;\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)\frac{1}{{(n + m)\;}}} } \\ &\quad \times \int\!\!\!\int {\exp \{{ - ar_d^2 - b\kappa_d^2 + c{r_d}{\kappa_d}} } + ie{r_s}{r_d} - i\frac{1}{A}{r_s}{\kappa _d}\\ &\quad\left. {\textrm{ } - ik\theta {r_d} - H({r_d},\frac{1}{A}({r_d} + \frac{B}{k}{\kappa_d}),z)} \right\}\;\;{d^2}{\kappa _d}{d^2}{r_d} \end{aligned}$$

where

(16a)$$a = \frac{1}{{{A^2}w_0^2}}\left[ {\frac{{nm}}{{(n + m)\;}} + \frac{1}{{2{\alpha^2}}}} \right]$$

(16b)$$b = \left( {\frac{{w_0^2}}{{4(n + m)\;}} + \frac{1}{{{A^2}}}\frac{{{B^2}}}{{{k^2}w_0^2}}\left[ {\frac{{nm}}{{(n + m)}} + \frac{1}{{2{\alpha^2}}}} \right] + \frac{{iB}}{{2kA\;}}\frac{{(n - m)}}{{(n + m)}}} \right)$$

(16c)$$c = \left( { - \frac{1}{{{A^2}}}\frac{{2B}}{{kw_0^2}}\left[ {\frac{{nm}}{{(n + m)}} + \frac{1}{{2{\alpha^2}}}} \right] - \frac{1}{A}\frac{{i(n - m)}}{{2(n + m)\;}}} \right)$$

(16d)$${\alpha ^2} = \frac{{\sigma _0^2}}{{w_0^2}}\;\;\;\;\;\;\;e = \frac{k}{B}(D - \frac{1}{A})$$

The arbitrary order moments of the laser beams in random media can be obtained by the Wigner distribution function as follows:

(17)$$\left\langle {{x^{{n_1}}},{y^{{n_2}}},\theta_x^{{m_1}},\theta_y^{{m_2}}} \right\rangle = \frac{1}{P}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{x^{{n_1}}}{y^{{n_2}}}\theta _x^{{m_1}}\theta _y^{{m_2}}} } h({{\boldsymbol r}_s},{\boldsymbol \theta },z){d^2}{{\boldsymbol r}_s}{d^2}{\boldsymbol \theta }$$

and

(18)$$P = \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {h({{\boldsymbol r}_s},{\boldsymbol \theta },z){d^2}{{\boldsymbol r}_s}{d^2}{\boldsymbol \theta }} }$$

where P is the total power of the beam.

Substituting Eq. (15) into Eqs. (17) and (18), we can obtain the second order moments of the partially coherent flat-topped beams through an ABCD optical system with a turbulent atmosphere and plasma as follows:

(19a)$$\left\langle {{r_s}^2} \right\rangle = 2Aw_0^2\frac{{{e_1}}}{{{e_3}}} + \frac{1}{{{A^2}}}\frac{{2{B^2}}}{{{k^2}w_0^2}}\left[ {4A\;\frac{{{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right] + \frac{1}{{{A^2}}}\frac{{4{\pi ^2}z{B^2}T}}{3}$$

(19b)$$\begin{aligned} \left\langle {{\theta^2}} \right\rangle &= \left\{ {{A^2}{e^2}\left[ {\frac{{w_0^2}}{{{k^2}A}}\frac{{{e_1}}}{{{e_3}}} + \frac{{{B^2}}}{{{A^2}{k^4}w_0^2}}(\frac{4}{A}\frac{{{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}) + \frac{{{B^2}}}{{{A^2}{k^2}}}\frac{{2{\pi^2}zT}}{3}} \right]} \right.\\ &\quad + Ae\left[ {\frac{1}{{{A^2}}}\frac{{2B}}{{{k^3}w_0^2}}\left[ {\frac{4}{A}\frac{{{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right] + \frac{{2{\pi^2}zT}}{3}(\frac{1}{A} + \frac{2}{{{A^2}}})\frac{B}{k}} \right]\\ &\quad\left. {\textrm{ } + \left[ {\frac{1}{{{A^2}{k^2}w_0^2}}\left[ {\frac{4}{A}\frac{{{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right] + \frac{{2{\pi^2}zT}}{3}(1 + \frac{1}{A} + \frac{1}{{{A^2}}})} \right]} \right\} \end{aligned}$$

(19c)$$\begin{aligned} \left\langle {{{\boldsymbol r}_s} \cdot {\boldsymbol \theta }} \right\rangle &= \frac{{2Aew_0^2}}{k}\;\frac{{{e_1}}}{{{e_3}}} + 2e\frac{1}{A}\frac{{{B^2}}}{{{k^3}w_0^2}}\left[ {\;4\frac{{{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right]\\ &\quad + 4e\frac{{{\pi ^2}zT{B^2}}}{{3k}}\frac{1}{A} + 2\frac{1}{{{A^2}}}\frac{B}{{{k^2}w_0^2}}\left[ {\;4\frac{{{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right] + \frac{{2{\pi ^2}zTB}}{3}(\frac{1}{A} + \frac{2}{{{A^2}}}) \end{aligned}$$

The above expression of the second order moments is derived from the following integral formula:

(20a)$$\delta (s )= \frac{1}{{2\pi }}\int_{ - \infty }^{ + \infty } {\exp ({ - isx} )dx}$$

(20b)$${\delta ^{(n)}}(s )= \frac{1}{{2\pi }}\int_{ - \infty }^{ + \infty } {{{( - ix)}^n}\exp ({ - isx} )} dx\;\;\;\;\;(n = 1,2)$$

(20c)$$\int_{ - \infty }^{ + \infty } {f(x )} {\delta ^{(n )}}(x )dx = {({ - 1} )^n}{f^{(n )}}(0 )\;\;\;\;\;\;\;(n = 1,2)$$

The M²-factor of the partially coherent flat-topped beams propagating through an ABCD optical system with a turbulent atmosphere and plasma is obtained as follows:

(21)$$\begin{aligned} {M^2}(z) &= k\sqrt {\left\langle {{r_s}^2} \right\rangle \left\langle {{\theta^2}} \right\rangle - {{\left\langle {{r_s},\theta } \right\rangle }^2}} \\ &\quad = k\left\{ {\left\{ {\left[ {\frac{{2Aw_0^2{e_1}}}{{{e_3}}} + \frac{{2{B^2}}}{{{A^2}{k^2}w_0^2}}\left( {\;\frac{{4A{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right) + \frac{{4{\pi^2}z{B^2}T}}{{3{A^2}}}} \right]} \right\}} \right.\\ &\quad \times 2\left\{ {{A^2}{e^2}\left[ {\frac{{w_0^2{e_1}}}{{{k^2}A{e_3}}} + \frac{{{B^2}}}{{{A^2}{k^4}w_0^2}}(\frac{{4{e_2}}}{{A{e_3}}} + \frac{1}{{{\alpha^2}}}) + \frac{{2{\pi^2}zT{B^2}}}{{3{A^2}{k^2}}}} \right]} \right.\\ &\quad + Ae\left[ {\frac{{2B}}{{{A^2}{k^3}w_0^2}}\left( {\frac{{4{e_2}}}{{A{e_3}}} + \frac{1}{{{\alpha^2}}}} \right) + \frac{{2{\pi^2}zBT}}{{3k}}\left( {\frac{1}{A} + \frac{2}{{{A^2}}}} \right)} \right]\\ &\quad\left. { + \left[ {\frac{1}{{{A^2}{k^2}w_0^2}}\left( {\frac{{4{e_2}}}{{A{e_3}}} + \frac{1}{{{\alpha^2}}}} \right) + \frac{{2{\pi^2}zT}}{3}\left( {1 + \frac{1}{A} + \frac{1}{{{A^2}}}} \right)} \right]} \right\}\\ &\quad - \left[ {\frac{{2Ae{e_1}w_0^2}}{{k{e_3}}}\; + \frac{{2e{B^2}}}{{A{k^3}w_0^2}}\left( {\;\frac{{4{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right) + \frac{{4e{\pi^2}zT{B^2}}}{{3Ak}}} \right.\\ &\quad{\left. {{{\left. {\textrm{ } + \frac{{2B}}{{{k^2}w_0^2{A^2}}}\left( {\;\frac{{4{e_2}}}{{{e_3}}} + \frac{1}{{{\alpha^2}}}} \right) + \frac{{2{\pi^2}zTB}}{3}\left( {\frac{1}{A} + \frac{2}{{{A^2}}}} \right)} \right]}^2}} \right\}^{\frac{1}{2}}} \end{aligned}$$

where

(22a)$${e_1} = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)\frac{1}{{{{(n + m)}^2}\;}}} }$$

(22b)$${e_2} = \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)\frac{{nm}}{{{{(n + m)}^2}\;}}} }$$

(22c)$${e_3} = 2\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{( - 1)}^{n + m}}}}{{{N^2}}}\left( \begin{array}{l} N\\ n \end{array} \right)\left( \begin{array}{l} N\\ m \end{array} \right)\frac{1}{{(n + m)\;}}} }$$

Equations (7) and (21) are the primary analytical results of this study, which are primarily determined by the value of the beam, the strength of a turbulent atmosphere, and plasma parameters. The derived expression with the appropriate parameter can also be used to analyze the partially coherent or fully coherent flat-topped beam propagation in free space.

2.3 Transmission matrix

The plasma is generated around the machine when the vehicle is moving at a hypersonic speed. Consequently, the optical channel of the communication link between the ground-based station and the vehicle becomes filled with a turbulent atmosphere and plasma. The thickness of the plasma is relatively thin. In Fig. 1, a schematic shows the concept of the optical channel with a turbulent atmosphere and plasma. The channel operates in two modes: forward transmission, where laser beam propagates through the turbulent atmosphere and plasma, reverse transmission, which involves transmission in the opposite direction. To analyze this optical, we derive the elements of the transmission matrix. The optical transmission matrix is $\left[ {\begin{array}{*{20}{c}} 1&z\\ 0&1 \end{array}} \right]$ for the propagation distance z in free space. The matrix of density gradient plasma is $\left[ {\begin{array}{*{20}{c}} 1&{{L_z}}\\ 0&{{1 / {{n_z}}}} \end{array}} \right]$, where ${n_z}$ is the refractive index of the gradient plasma, which is a linear function of the propagation distance z, and ${L_z} = \int {[1/n(z)]} dz$ denotes a function of the optical path [43]. Therefore, the transmission matrix of the forward and reverse transmission of bidirectional propagation can be obtained as follows:

(23)$$\left[ {\begin{array}{*{20}{c}} A&B\\ C&D \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&{{L_z}}\\ 0&{1/{n_z}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&{{z_1}}\\ 0&1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&{{z_1} + {L_z}}\\ 0&{1/{n_z}} \end{array}} \right]$$

and

(24)$$\left[ {\begin{array}{*{20}{c}} {A^{\prime}}&{B^{\prime}}\\ {C^{\prime}}&{D^{\prime}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&{{z_2}}\\ 0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&{{L_z}}\\ 0&{1/{n_z}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&{{z_2}/{n_z} + {L_z}}\\ 0&{1/{n_z}} \end{array}} \right]$$

The intensity distribution and M ² of a partially coherent flat-top beam propagated in a bidirectional turbulent atmosphere and plasma can be analyzed when the above two matrices are substituted into the Eqs. (7) and (21), respectively.

Fig. 1. Schematic of a laser beam propagation in a bidirectional atmospheric turbulence and plasma.

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3. Numerical results

In this section, we use Tatarskii spectrum to study the evolution properties of the intensity distribution and M ² factor of a partially coherent flat-topped beam in a bidirectional turbulent atmosphere and plasma. Tatarskii spectrum is ${\Phi _n}({\kappa }) = 0.033C_n^2{{\kappa }^{ - 11/3}}\exp ( - {{\kappa }^2}/{\kappa }_m^2)\;$, where the parameter $C_n^2$ denotes the structure constant of the turbulent atmosphere refractive index fluctuation, and ${\kappa _m} = 5.92/{l_0}$ with the inner scale ${l_0}$. Subsequently, the parameter T in Eq.8 can be obtained as follows:

(25)$$T = 0.1661C_n^2{l_0}^{ - 1/3}$$

We selected the following parameters for our anlaysis: beam order N = 5, wavelength λ=632.8 nm, beam width w₀= 4 cm, the transverse spatial coherence width σ₀ = 4 cm, inner scale ${l_0}$=1 cm, and structure constant of the atmospheric turbulence $C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{ - 2/3}}$, with alternative values used for specific conditions. When the laser beam propagates through a plasma, the refractive index of the gradient plasma is assumed as n(z) = 1-0.8z/200 [43], the optical path of the laser beam propagation in the plasma can be derived to be L_z= 402.4μm.

Figures 2 and 3 show the intensity distribution of the partially coherent flat-topped beam impacted by the beam order and propagation distance for both the forward and reverse transmission, respectively. For the forward transmission, after propagating in a turbulent atmosphere, the partially coherent flat-topped beam is disturbed by the plasma. The beam is propagating in the plasma at first, then in a turbulent atmosphere for the reverse transmission. Compared to the turbulent atmosphere, the thickness of the plasma is significantly thinner. From Figs.2 and 3, the shape of the intensity distribution exhibits a flat-topped beam shape for the high-order beam at the source plane. However, as the beam propagates to the far field, it evolves into a Gaussian beam. The intensity distribution becomes the Gaussian distribution when the partially coherent flat-topped beam is propagating in a turbulent atmosphere and plasma over several kilometers for the forward transmission. In the case of reverse transmission, this transition occurs within a few hundred meters. The intensity distribution of the partially coherent flat-topped beam is influenced by the values B or B´, and D or D´ of the matrix elements, as shown in Eqs. (7), (23), and (24). As a result, the partially coherent flat-topped beam exhibits different propagation properties for the forward and reverse transmission.

Fig. 2. Intensity distribution of the partially coherent flat-topped beam for the forward transmission with different beam orders and propagation distances.

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Fig. 3. Intensity distribution of the partially coherent flat-topped beam for the reverse transmission with different beam orders and propagation distances.

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The beam shape maintains the approximate original shape of the flat-topped beam at the far field when the beam order is large for bidirectional propagation in a turbulent atmosphere and plasma (see Fig.2c3 and Fig.3c4).

Figures 4 and 5 show the intensity distributions of the partially coherent flat-topped beam being impacted by the transverse spatial coherence width and propagation distance for bidirectional propagation in a turbulent atmosphere and plasma. From Figs. 4 and 5, the intensity distributions of the partially coherent flat-topped beam become Gaussian distribution at the far field for bidirectional propagation. When the transverse spatial coherence width is small, that is, the beam with the low coherence, the beam shape evolves faster into Gaussian distribution for the forward and reverse transmission, respectively. The beams with the low coherent level have a large far-field divergence angle. Subsequently, the intensity distributions of the partially coherent flat-topped beam are influenced by the transverse spatial coherence width. From Figs.2-5, the plasma effects on the intensity distribution of the partially coherent flat-topped beam are larger than that in a turbulent atmosphere.

Fig. 4. Intensity distribution of the partially coherent flat-topped beam for the forward transmission with different transverse spatial coherence widths and propagation distances at the beam order N = 10.

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Fig. 5. Intensity distribution of the partially coherent flat-topped beam for the reverse transmission with different transverse spatial coherence widths and propagation distances at the beam order N = 10.

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To learn about the normalized M ² factor of the partially coherent flat-topped beam in a bidirectional transmission disturbed by the turbulent atmosphere and plasma, we analyze various parameters related to the beam and the turbulent atmosphere.

Figure 6 shows the normalized M ² factor of the partially coherent flat-topped beam for the forward and reverse transmission with σ = 0.02 m and w₀= 0.02 m. From Fig. 6, it is evident that the normalized M ² factor of the partially coherent flat-topped through the turbulent atmosphere and plasma increases with the propagation distance. The normalized M ² factor of the partially coherent flat-topped beam with large beam order is smaller than the beam with low beam order. As the strength of the turbulent atmosphere increases, the normalized M ² factor increases. Under the same conditions, the normalized M ² factor of the partially coherent flat-topped beam for the forward transmission is smaller than that in reverse transmission. Meanwhile, the normalized M ² factor of the partially coherent flat-topped beam is smaller than that of the partially or fully coherent Gaussian beam through turbulent atmosphere and plasma.

Fig. 6. Normalized M ² factor of the partially coherent flat-topped beam for the forward (a) and reverse (b) transmission with different structural constants of the turbulent atmosphere and beam order.

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Figures 7–9 show the normalized M ² factor of the partially coherent flat-topped beam for the forward and reverse transmission influenced by the structure constant of turbulent atmosphere, transverse spatial coherence width, wavelength, and beam width. From Fig. 7–9, the normalized M ² factors increase as the propagation distance increases. In Fig. 7, with beam parameters N = 5 and w₀= 0.02 m, it is evident that the transverse spatial coherence width affects the normalized M ² factor of the partially coherent flat-topped beam, and the beam with the lower transverse spatial coherence width can mitigate the turbulent atmosphere and plasma disturbances, resulting in a smaller normalized M ² factor. In Fig. 8, with beam parameters N = 5, σ₀= 0.02 m, and w₀= 0.02 m, selecting a longer wavelength for the partially coherent flat-topped beam reduces the effects of the turbulent atmosphere and plasma, leading to a smaller normalized M ² factor. In Fig. 9, the beam parameters are N = 5, σ₀= 0.02 m, it can be found from Fig. 9 that the normalized M ² factor of the partially coherent flat-topped beam decreases when the beam width increases. This result indicates that the beam with the large beam width can relieve the turbulent atmosphere and plasma perturbed. Therefore, the partially coherent flat-topped beam has a smaller normalized M ² factor.

Fig. 7. Normalized M ² factor of the partially coherent flat-topped beam for the forward (a) and reverse (b) transmission with different structural constant of the turbulent atmosphere and transverse spatial coherence width.

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Fig. 8. Normalized M ² factor of the partially coherent flat-topped beam for the forward (a) and reverse (b) transmission with different structural constant of the turbulent atmosphere and wavelength.

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Fig. 9. Normalized M ² factor of the partially coherent flat-topped beam for the forward (a) and reverse (b) transmission with different structural constants of the turbulent atmosphere and beam width.

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From Fig. 6-9, the normalized M ² factor of the partially coherent flat-topped beam propagating in a bidirectional turbulent atmosphere and plasma is initially small at the near field. However, as the propagation distance increases, the normalized M ² factor rapidly increases. Additionally, as the strength of the turbulent atmosphere increases, the normalized M ² factor increases, and the beam quality becomes worse. At the same conditions, the normalized M ² factor of the partially coherent flat-topped beam in forward transmission is smaller than the beam in reverse transmission. The numerical results indicate that normalized M ² factor of the partially coherent flat-topped beam influenced by the plasma is larger than that in the turbulent atmosphere.

4. Conclusion

We theoretically investigated the intensity distribution and normalized M ² factor of the partially coherent flat-topped beam propagating in a bidirectional turbulent atmosphere and plasma link. The numerical results show that the intensity distribution and normalized M ² factor are influenced differently by the forward and reverse transmission. The values of the normalized M ² factor in forward transmission are smaller than that of the beam in the reverse transmission. It is evident that the partially coherent flat-topped beam with lower transverse spatial coherence width, longer wavelength, larger beam width and beam order is less affected by the turbulent atmosphere and plasma, and the partially coherent flat-topped beam is less affected by the atmospheric turbulence and plasma than fully and partially coherent Gaussian beams. Our results find useful applications in FSO communication systems in the special environments.

Funding

National Key Research and Development Program of China (2022YFA1404800, 2019YFA0705000); National Natural Science Foundation of China (12192254, 11974219, 12174227, 11904211, 92250304); Natural Science Foundation of Shandong Province (ZR2019MA028).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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Distribution of intensity and M² factor for a partially coherent flat-topped beam in bidirectional turbulent atmosphere and plasma connection

Abstract

1. Introduction