Partially coherent vortex cosh-Gaussian beam and its paraxial propagation

Abstract

In this work, the propagation properties of a partially coherent vortex cosh-Gaussian beam (PCvChGB) have been investigated. Within the framework of the extended Huygens–Fresnel diffraction integral, an analytical formula for a PCvChGB propagating in a paraxial ABCD optical system is derived. Based on the obtained formula, the influences of the spatial coherence, decentered parameter and vortex charge on the propagation properties of a PCvChGB in free space are illustrated numerically and analyzed. The obtained results could be beneficial for applications of PCvChG beams in optical communications, remote sensing and atom optics.

Appendix

The derivation process of Eq. (9) is presented in detail in following.

The cross-spectral density of a PCvChGB propagating in free space is given by Mandel and Wolf (1995), Collins (1970)

$$W\left( {\vec{r}_{1} ,\vec{r}_{2} ,z} \right) = \left( {\frac{k}{2\pi B}} \right)^{2} \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {W\left( {\vec{r}_{01} ,\vec{r}_{02} ,z = 0} \right)\exp \left[ {\frac{ik}{{2B}}\left\{ {A\left( {\vec{r}_{02}^{2} - \vec{r}_{01}^{2} } \right) - 2\left( {\vec{r}_{02} \vec{r}_{2} - \vec{r}_{01} \vec{r}_{1} } \right) + D\left( {\vec{r}_{2}^{2} - \vec{r}_{1}^{2} } \right)} \right\}} \right]dr_{01} dr_{02} } } .$$

(A1)

Recalling the following expansions (Gradshteyn and Ryzhik 1994)

$$\left( {x + iy} \right)^{M} = \sum\limits_{l = 0}^{M} {C_{l}^{M} } x^{l} \left( {iy} \right)^{M - l} ,$$

(A2)

where

$$C_{l}^{M} = {\kern 1pt} \frac{M!}{{l!\left( {M - l} \right){\kern 1pt} !}},$$

(A3)

the cross-spectral density can be rearranged as

$$W\left( {\vec{r}_{1} ,\vec{r}_{2} ,z} \right) = \left( {\frac{k}{2\pi B}} \right)^{2} \exp \left\{ {\frac{ikD}{{2B}}\left( {\vec{r}_{2}^{2} - \vec{r}_{1}^{2} } \right)} \right\}\sum\limits_{l = 0}^{M} {C_{l}^{M} \left( i \right)^{M - l} \sum\limits_{n = 0}^{M} {C_{n}^{M} \left( { - i} \right)^{M - n} } {\kern 1pt} } U_{l,n} \left( {x_{1} ,x_{2} } \right)U_{M - l,M - n} \left( {y_{1} ,y_{2} } \right),$$

(A4)

where $U_{l,n} \left( {s_{1} ,s_{2} } \right)$ is the typical integral expression given by

$$\begin{aligned} U_{l,n} \left( {s_{1} ,s_{2} } \right) = & \int\limits_{ - \infty }^{ + \infty } {s_{01}^{l} \cosh \left( {\frac{{b\,s_{01} }}{{\omega_{0} }}} \right)\exp \left\{ { - \alpha^{ * } s_{01}^{2} + \frac{{ik\,s_{1} }}{B}s_{01} } \right\}} \\ & \;\; \times \left[ {\int\limits_{ - \infty }^{ + \infty } {s_{02}^{n} \cosh \,\left( {\frac{{b\,s_{02} }}{{\omega_{0} }}} \right)\exp \left\{ { - \alpha \,s_{02}^{2} + \left( {\frac{{s_{01} }}{{\sigma_{0}^{2} }} - \frac{{ik\,s_{2} }}{B}} \right)\,s_{02} } \right\}ds_{02} ,} } \right]\,ds_{01} , \\ \end{aligned}$$

(A5)

where s represents either x or y (hereafter), and

$$\alpha = \frac{1}{{\omega_{0}^{2} }} + \frac{1}{{2\sigma_{0}^{2} }} - \frac{ikA}{{2B}}.$$

(A6)

Using the definition of the cosh (.) function

$$\cosh \left( x \right) = \frac{{e^{x} + e^{ - x} }}{2},$$

and recalling the following integral equation (Belafhal et al. 2020)

$$\int\limits_{ - \infty }^{ + \infty } {x^{n} \exp \left( { - px^{2} + 2qx} \right)} dx = \sqrt {\frac{\pi }{p}} \,\exp \left( {\frac{{q^{2} }}{p}} \right)\left( {\frac{1}{2i\sqrt p }} \right)^{n} H_{n} \left( {\frac{iq}{{\sqrt p }}} \right),$$

(A7)

where $H_{n} \left( {.{\kern 1pt} {\kern 1pt} } \right)$ is the Hermite polynomial of nth-order, Eq. (A5) can be expressed as

$$\begin{aligned} U_{l,n} \left( {s_{1} ,s_{2} } \right) = & \frac{1}{2}\sqrt {\frac{\pi }{\alpha }} \left( {\frac{1}{2i\sqrt \alpha }} \right)^{n} \int\limits_{ - \infty }^{ + \infty } {s_{01}^{l} \cosh \left( {\frac{{b\,s_{01} }}{{\omega_{0} }}} \right)\exp \left\{ { - \alpha^{ * } s_{01}^{2} + \frac{{ik\,s_{1} }}{B}s_{01} } \right\}} \\ & \;\; \times \left[ {\exp \left( {\frac{{q_{ + }^{2} \left( {s_{01} ,s_{2} } \right)}}{\alpha }} \right)H_{n} \left( {\frac{{iq_{ + } \left( {s_{01} ,s_{2} } \right)}}{\sqrt \alpha }} \right) + \exp \left( {\frac{{q_{ - }^{2} \left( {s_{01} ,s_{2} } \right)}}{\alpha }} \right)H_{n} \left( {\frac{{iq_{ - } \left( {s_{01} ,s_{2} } \right)}}{\sqrt \alpha }} \right)} \right]\,ds_{01} , \\ \end{aligned}$$

(A8)

where

$$q_{ \pm } \left( {s_{01} ,s_{2} } \right) = \frac{{s_{01} }}{{2\sigma_{0}^{2} }} - \frac{{iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }},$$

(A9)

then Eq. (A8) can be written as

$$U_{l,n} \left( {s_{1} ,s_{2} } \right) = \frac{1}{4}\sqrt {\frac{\pi }{\alpha }{\kern 1pt} {\kern 1pt} } \left( {\frac{1}{2i\sqrt \alpha }} \right)^{n} \left[ {F^{ + } \left( {s_{1} ,s_{2} } \right) + F^{ - } \left( {s_{1} ,s_{2} } \right) + G^{ + } \left( {s_{1} ,s_{2} } \right) + G^{ - } \left( {s_{1} ,s_{2} } \right)} \right]{\kern 1pt} ,$$

(A10)

where

$$F^{ \pm } \left( {s_{1} ,s_{2} } \right) = \int\limits_{ - \infty }^{ + \infty } {s_{01}^{l} \exp \left\{ { - \alpha^{ * } s_{01}^{2} + \left( {\frac{{iks_{1} }}{B} + \frac{b}{{\omega_{0} }}} \right)s_{01} } \right\}} \exp \left\{ {\frac{{q_{ \pm }^{2} \left( {s_{01} ,s_{2} } \right)}}{\alpha }} \right\}H_{n} \left( {\frac{{iq_{ \pm } \left( {s_{01} ,s_{2} } \right)}}{\sqrt \alpha }} \right)ds_{01} ,$$

(A11)

and

$$G^{ \pm } \left( {s_{1} ,s_{2} } \right) = \int\limits_{ - \infty }^{ + \infty } {s_{01}^{l} \exp \left\{ { - \alpha^{ * } s_{01}^{2} + \left( {\frac{{iks_{1} }}{B} - \frac{b}{{\omega_{0} }}} \right)s_{01} } \right\}} \exp \left\{ {\frac{{q_{ \pm }^{2} \left( {s_{01} ,s_{2} } \right)}}{\alpha }} \right\}H_{n} \left( {\frac{{iq_{ \pm } \left( {s_{01} ,s_{2} } \right)}}{\sqrt \alpha }} \right)ds_{01} {\kern 1pt} .$$

(A12)

Eqs. (A11) and (A12) can also be developed as

$$\begin{aligned} F^{ \pm } \left( {s_{1} ,s_{2} } \right) = & \exp \left\{ {\frac{1}{\alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)^{2} } \right\}\int\limits_{ - \infty }^{ + \infty } {s_{01}^{l} \exp \left\{ { - \alpha^{ * } s_{01}^{2} + \left( {\frac{{iks_{1} }}{B} + \frac{b}{{\omega_{0} }}} \right)s_{01} } \right\}} \\ & \;\;\exp \left\{ {\frac{1}{\alpha }\left( {\frac{{s_{01}^{2} }}{{4\sigma_{0}^{2} }} + \frac{1}{{\sigma_{0}^{2} }}\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)s_{01} } \right)} \right\}H_{n} \left( {\frac{i}{{2\sigma_{0}^{2} \sqrt \alpha }}s_{01} + \frac{i}{\sqrt \alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)} \right)ds_{01} , \\ \end{aligned}$$

(A13)

and

$$\begin{aligned} G^{ \pm } \left( {s_{1} ,s_{2} } \right) = & \exp \left\{ {\frac{1}{\alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)^{2} } \right\}\int\limits_{ - \infty }^{ + \infty } {s_{01}^{l} \exp \left\{ { - \alpha^{ * } s_{01}^{2} + \left( {\frac{{iks_{1} }}{B} - \frac{b}{{\omega_{0} }}} \right)s_{01} } \right\}} \\ & \;\;\;\exp \left\{ {\frac{1}{\alpha }\left( {\frac{{s_{01}^{2} }}{{4\sigma_{0}^{2} }} + \frac{1}{{\sigma_{0}^{2} }}\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)s_{01} } \right)} \right\}H_{n} \left( {\frac{i}{{2\sigma_{0}^{2} \sqrt \alpha }}s_{01} + \frac{i}{\sqrt \alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)} \right)ds_{01} . \\ \end{aligned}$$

(A14)

Now, by using the following addition formula of the Hermite polynomials (Magnus et al. 1966)

$$H_{n} \left( {x + y} \right) = \sum\limits_{j = 0}^{n} {C_{j}^{n} \left( {2x} \right)^{j} } H_{n - j} \left( y \right),$$

(A15)

we find the expressions of Eqs. (A13) and (A14)

$$\begin{aligned} F^{ \pm } \left( {s_{1} ,s_{2} } \right) = & \exp \left\{ {\frac{1}{\alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)^{2} } \right\}\sum\limits_{h = 0}^{n} {C_{h}^{n} } \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)^{h} H_{n - h} \left( {\frac{i}{\sqrt \alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)} \right) \\ & \;\;\;\int\limits_{ - \infty }^{ + \infty } {s_{01}^{l + h} \exp \left\{ { - \eta s_{01}^{2} + 2q_{1s}^{ \pm } s_{01} } \right\}} ds_{01} , \\ \end{aligned}$$

(A16)

and

$$\begin{aligned} G^{ \pm } \left( {s_{1} ,s_{2} } \right) = & \exp \left\{ {\frac{1}{\alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)^{2} } \right\}\sum\limits_{h = 0}^{n} {C_{h}^{n} } \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)^{h} H_{n - h} \left( {\frac{i}{\sqrt \alpha }\left( {\frac{{ - iks_{2} }}{2B} \pm \frac{b}{{2\omega_{0} }}} \right)} \right) \\ & \;\;\int\limits_{ - \infty }^{ + \infty } {s_{01}^{l + h} \exp \left\{ { - \eta s_{01}^{2} + 2q_{2s}^{ \pm } s_{01} } \right\}} ds_{01} , \\ \end{aligned}$$

(A17)

and then Eq. (A10) can be written as

$$U_{l,n} \left( {s_{1} ,s_{2} } \right) = \frac{1}{4}\sqrt {\frac{\pi }{\alpha }{\kern 1pt} {\kern 1pt} } \left( {\frac{1}{2i\sqrt \alpha }} \right)^{n} \left[ \begin{gathered} \left\{ \begin{gathered} \exp \left\{ {\frac{1}{\alpha }\left( {\frac{{ - iks_{2} }}{2B} + \frac{b}{{2\omega_{0} }}} \right)^{2} } \right\}\sum\limits_{h = 0}^{n} {C_{h}^{n} } \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)^{h} H_{n - h} \left( {\frac{i}{\sqrt \alpha }\left( {\frac{{ - iks_{2} }}{2B} + \frac{b}{{2\omega_{0} }}} \right)} \right) \hfill \\ \times \left( {\int\limits_{ - \infty }^{ + \infty } {s_{01}^{l + h} \exp \left\{ { - \eta s_{01}^{2} + 2q_{1s}^{ + } s_{01} } \right\}} ds_{01} + \int\limits_{ - \infty }^{ + \infty } {s_{01}^{l + h} \exp \left\{ { - \eta s_{01}^{2} + 2q_{2s}^{ + } s_{01} } \right\}} ds_{01} } \right) \hfill \\ \end{gathered} \right\} \hfill \\ + \left\{ \begin{gathered} \exp \left\{ {\frac{1}{\alpha }\left( {\frac{{iks_{2} }}{2B} + \frac{b}{{2\omega_{0} }}} \right)^{2} } \right\}\sum\limits_{t = 0}^{n} {C_{t}^{n} } \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)^{t} H_{n - t} \left( {\frac{ - i}{{\sqrt \alpha }}\left( {\frac{{iks_{2} }}{2B} + \frac{b}{{2\omega_{0} }}} \right)} \right) \hfill \\ \times \left( {\int\limits_{ - \infty }^{ + \infty } {s_{01}^{l + t} \exp \left\{ { - \eta s_{01}^{2} + 2q_{1s}^{ - } s_{01} } \right\}} ds_{01} + \int\limits_{ - \infty }^{ + \infty } {s_{01}^{l + t} \exp \left\{ { - \eta s_{01}^{2} + 2q_{2s}^{ - } s_{01} } \right\}} ds_{01} } \right) \hfill \\ \end{gathered} \right\} \hfill \\ \end{gathered} \right]{\kern 1pt} ,$$

(A18)

By applying again the integral Eq. (A7), we get

$$\begin{aligned} U_{l,n} \left( {s_{1} ,s_{2} } \right) = & \frac{1}{4}\sqrt {\frac{\pi }{\alpha }{\kern 1pt} {\kern 1pt} } \sqrt {\frac{\pi }{\eta }{\kern 1pt} {\kern 1pt} } \left( {\frac{1}{2i\sqrt \alpha }} \right)^{n} \exp \left\{ {\frac{ikD}{{2B}}\left( {s_{2}^{2} - s_{1}^{2} } \right)} \right\} \\ & \;\;\;\;\left\{ \begin{gathered} \exp \left\{ {\frac{1}{4\alpha }\left( {\frac{b}{{\omega_{0} }} - \frac{{iks_{2} }}{B}} \right)^{2} } \right\}\sum\limits_{h = 0}^{n} {C_{h}^{n} \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)}^{h} \left( {\frac{1}{2i\sqrt \eta }} \right)^{l + h} H_{n - h} \left( {\frac{i}{2\sqrt \alpha }\left( {\frac{b}{{\omega_{0} }} - \frac{{iks_{2} }}{B}} \right)} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \times \left[ {\exp \left\{ {\frac{{\left( {q_{1s}^{ + } } \right)^{2} }}{\eta }} \right\}H_{l + h} \left( {\frac{{iq_{1s}^{ + } }}{\sqrt \eta }} \right) + \exp \left\{ {\frac{{\left( {q_{2s}^{ + } } \right)^{2} }}{\eta }} \right\}H_{l + h} \left( {\frac{{iq_{2s}^{ + } }}{\sqrt \eta }} \right)} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \exp \left\{ {\frac{1}{4\alpha }\left( {\frac{b}{{\omega_{0} }} + \frac{{iks_{2} }}{B}} \right)^{2} } \right\}\sum\limits_{t = 0}^{n} {C_{t}^{n} \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)}^{t} \left( {\frac{1}{2i\sqrt \eta }} \right)^{l + t} H_{n - t} \left( {\frac{ - i}{{2\sqrt \alpha }}\left( {\frac{b}{{\omega_{0} }} + \frac{{iks_{2} }}{B}} \right)} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \times \left[ {\exp \left\{ {\frac{{\left( {q_{1s}^{ - } } \right)^{2} }}{\eta }} \right\}H_{l + t} \left( {\frac{{iq_{1s}^{ - } }}{\sqrt \eta }} \right) + \exp \left\{ {\frac{{\left( {q_{2s}^{ - } } \right)^{2} }}{\eta }} \right\}H_{l + t} \left( {\frac{{iq_{2s}^{ - } }}{\sqrt \eta }} \right)} \right] \hfill \\ \end{gathered} \right\}, \\ \end{aligned}$$

(A19)

with

$$\eta = \alpha^{ * } - \frac{1}{{4\alpha \sigma_{0}^{4} }},$$

(A20)

$$q_{1s}^{ \pm } = \left( {\frac{{iks_{1} }}{2B} + \frac{b}{{2\omega_{0} }}} \right) + \frac{1}{{4\alpha \sigma_{0}^{2} }}\left( {\frac{{ - iks_{2} }}{B} \pm \frac{b}{{\omega_{0} }}} \right),$$

(A21)

and

$$q_{2s}^{ \pm } = \left( {\frac{{iks_{1} }}{2B} - \frac{b}{{2\omega_{0} }}} \right) + \frac{1}{{4\alpha \sigma_{0}^{2} }}\left( {\frac{{ - iks_{2} }}{B} \pm \frac{b}{{\omega_{0} }}} \right).$$

(A22)

After the substitution of Eq. (A18) into Eq. (A4), the cross-spectral density of a PCvChGB propagating through a paraxial optical system is obtained as

$$\begin{aligned} W\left( {\vec{r}_{1} ,\vec{r}_{2} ,z} \right) = & \frac{1}{16}\frac{1}{\alpha \eta }\left( \frac{k}{2z} \right)^{2} \left( {\frac{1}{2i\sqrt \alpha }} \right)^{M} \exp \left\{ {\frac{ikD}{{2B}}\left( {x_{2}^{2} - x_{1}^{2} + y_{2}^{2} - y_{1}^{2} } \right)} \right\} \\ & \;\; \times \sum\limits_{l = 0}^{M} {\sum\limits_{n = 0}^{M} {C_{l}^{M} C_{n}^{M} \left( i \right)^{n - l} {\rm P}_{l,n} \left( {x_{1} ,x_{2} } \right)\,{\rm P}_{M - l,M - n} \left( {y_{1} ,y_{2} } \right)} } , \\ \end{aligned}$$

(A23)

where

$$\begin{aligned} {\rm P}_{l,n} \left( {s_{1} ,s_{2} } \right) = & \exp \left\{ {\frac{1}{4\alpha }\left( {\frac{b}{{\omega_{0} }} - \frac{{iks_{2} }}{B}} \right)^{2} } \right\}\sum\limits_{h = 0}^{n} {C_{h}^{n} \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)}^{h} \left( {\frac{1}{2i\sqrt \eta }} \right)^{l + h} H_{n - h} \left( {\frac{i}{2\sqrt \alpha }\left( {\frac{b}{{\omega_{0} }} - \frac{{iks_{2} }}{B}} \right)} \right) \\ & \;\; \times \left[ {\exp \left\{ {\frac{{\left( {q_{1s}^{ + } } \right)^{2} }}{\eta }} \right\}H_{l + h} \left( {\frac{{iq_{1s}^{ + } }}{\sqrt \eta }} \right) + \exp \left\{ {\frac{{\left( {q_{2s}^{ + } } \right)^{2} }}{\eta }} \right\}H_{l + h} \left( {\frac{{iq_{2s}^{ + } }}{\sqrt \eta }} \right)} \right] \\ & \;\; + \exp \left\{ {\frac{1}{4\alpha }\left( {\frac{b}{{\omega_{0} }} + \frac{{iks_{2} }}{B}} \right)^{2} } \right\}\sum\limits_{t = 0}^{n} {C_{t}^{n} \left( {\frac{i}{{\sigma_{0}^{2} \sqrt \alpha }}} \right)}^{t} \left( {\frac{1}{2i\sqrt \eta }} \right)^{l + t} H_{n - t} \left( {\frac{ - i}{{2\sqrt \alpha }}\left( {\frac{b}{{\omega_{0} }} + \frac{{iks_{2} }}{B}} \right)} \right) \\ & \;\; \times \left[ {\exp \left\{ {\frac{{\left( {q_{1s}^{ - } } \right)^{2} }}{\eta }} \right\}H_{l + t} \left( {\frac{{iq_{1s}^{ - } }}{\sqrt \eta }} \right) + \exp \left\{ {\frac{{\left( {q_{2s}^{ - } } \right)^{2} }}{\eta }} \right\}H_{l + t} \left( {\frac{{iq_{2s}^{ - } }}{\sqrt \eta }} \right)} \right]{\kern 1pt} . \\ \end{aligned}$$

(A24)