Propagation of the autofocusing Lommel–Gaussian vortex beam with I-Bessel beam in turbulent atmosphere

Abstract

By the Fourier transformation upon an autofocusing Lommel–Gaussian vortex beam (LGVB) with J-Bessel beam, a novel autofocusing LGVB with I-Bessel beam is generated, and then the generated beam propagation in turbulence-free channel and turbulent atmosphere is investigated. Results demonstrate that under similar beam intensity profile parameters, a LGVB with J-Bessel beam has stronger anti-diffraction effect than a LGVB with I-Bessel beam in short-distance transmission, while a LGVB with I-Bessel beam has better autofocusing properties in long-distance transmission. Also, with the increase in topological charge, the intensity profile of the LGVB with I-Bessel beam remains almost unchanged within a certain distance, while it enlarges and the maximum intensity reduces beyond the certain distance. Additionally, when the ring radius approaches to a Gaussian beam waist, a LGVB with I-Bessel beam would degenerate into a LGVB with J-Bessel beam. Besides, impact of refractive index structure parameter and wavelength on the received probability of the LGVB with I-Bessel beam are also studied, and it is showed that the received probability decreases with the increase in the refractive index structure parameter or wavelength. This work could extend potential applications of LGVB with I-Bessel beam in free-space optical communication.

Appendix

On Substitution from Eqs. (6.2), (11) into Eq. (10) and utilizing Eq. (12), the integral formula for the average probability distribution at the receiving plane becomes:

$$\begin{aligned} \left\langle {\left| {R_{l,m} \left( {\rho ,z} \right)} \right|^{2} } \right\rangle & = \frac{1}{2\pi }\int_{0}^{2\pi } {\int_{0}^{2\pi } {E_{I} \left( {\rho ,\theta ,z} \right)} E_{I}^{ * } \left( {\rho ^{\prime},\theta ^{\prime},z} \right) \times \left\langle {\exp \left[ {\psi \left( {\rho ,\theta ,z} \right) + \psi^{ * } \left( {\rho ^{\prime},\theta ^{\prime},z} \right)} \right]} \right\rangle {\text{d}}\theta {\text{d}}\theta ^{\prime}} \\ & =\frac{1}{2\pi }\sum\limits_{q = 0}^{\infty } {\left( {ic} \right)^{2q} } \sum\limits_{q^{\prime} = 0}^{\infty } {\left( {ic} \right)^{2q^{\prime}} } \int_{0}^{2\pi } {\int_{0}^{2\pi } {\frac{{k^{2} }}{{4n_{1} n_{1}^{*} z^{2} }}\exp \left( {\frac{{\alpha_{1}^{2} - \beta_{1}^{2} }}{{4n_{1} }}} \right)J_{l} \left( {\frac{{\alpha_{1} \beta_{1} }}{{2n_{1} }}} \right)} } \\ & \quad \times \frac{k}{{2n_{1} z}}\exp \left[ {i2\left( {q - q^{\prime}} \right)\theta } \right]\exp \left( {\frac{{\alpha_{1}^{2} - \beta_{1}^{2} }}{{4n_{1}^{*} }}} \right)J_{l} \left( {\frac{{\alpha_{1} \beta_{1} }}{{2n_{1}^{*} }}} \right) \\ & \quad \times \exp \left[ { - i\left( {m - m_{0} - 2q^{\prime}} \right)\left( {\theta - \theta ^{\prime}} \right)} \right]\exp \left( { - \frac{{2\rho^{2} - 2\rho^{2} \cos \left( {\theta - \theta ^{\prime}} \right)}}{{\sigma_{0}^{2} }}} \right){\text{d}}\theta {\text{d}}\theta ^{\prime}, \\ \end{aligned}$$

(15)

where l = m₀ + 2q is topological charge. Utilizing the following formula:

$$\begin{aligned} & \int_{0}^{2\pi } {\exp \left[ { - ix\theta + y\cos \left( {\theta - \theta ^{\prime}} \right)} \right]{\text{d}}\theta = 2\pi \exp \left( { - ix\theta ^{\prime}} \right)I_{x} \left( y \right)} , \\ & \int_{0}^{2\pi } {\exp \left( {is\phi } \right){\text{d}}\phi = \left\{ {\begin{array}{*{20}c} {2\pi } & {{\text{if}}\;m = 0,} \\ { 0} & {{\text{if}}\;m \ne 0,} \\ \end{array} } \right.} \\ \end{aligned}$$

(16)

a more convenient analytical expression can be obtained

$$\left\langle {\left| {R_{l,m} \left( {\rho ,z} \right)} \right|^{2} } \right\rangle = \sum\limits_{q = 0}^{\infty } {c^{4q} \frac{{k^{2} \pi }}{{2n_{1} n_{1}^{ * } z^{2} }}} \exp \left( {\frac{{\alpha_{1}^{2} - \beta_{1}^{2} }}{{4n_{1} }} + \frac{{\alpha_{1}^{2} - \beta_{1}^{2} }}{{4n_{1}^{ * } }} - \frac{{2\rho^{2} }}{{\sigma_{0}^{2} }}} \right)J_{l} \left( {\frac{{\alpha_{1} \beta_{1} }}{{2n_{1} }}} \right)J_{l} \left( {\frac{{\alpha_{1} \beta_{1} }}{{2n_{1}^{ * } }}} \right)I_{m - l} \left( {\frac{{2r^{2} }}{{\sigma_{0}^{2} }}} \right).$$

(17)