Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Influences of edge dislocation on optical vortex transmission

Penghui Gao, Meihong Lu, and Jingying Li

Open Access Open Access

Abstract

Through theoretical calculation, the analytical expression for the cross-spectral density function of vortex beam with and without edge dislocation during transmission in turbulent atmosphere and free space is obtained. The calculation result is used for researching the influences of edge dislocation on optical vortex transmission. The research shows that due to the edge dislocation, when the optical vortex's topological charge is greater than +1, the optical vortex will no longer carry out steady transmission in the free space transmission. Instead, it will divide into two optical vortices, and the distance between them will gradually increase as the transmission distance increases. Optical vortex will split in turbulent atmosphere propagation. Due to the edge dislocation, when the topological charge of optical vortex is greater than +2, it is found that the distance between one optical vortex and other optical vortices is much larger than that between other optical vortices. Besides, when there's an edge dislocation, the greater the light wavelength and the structure constant are, the smaller the distance between the optical vortex and the edge dislocation on the source plane is, and the evolution of the optical vortex will be accelerated.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Singular optics, as a new branch in contemporary optics, mainly involves the phase singularity in the wave field and the topological structure of the wavefront. Singularity beam mainly has two types of wavefront dislocation structure, i.e., edge dislocation and screw dislocation (optical vortex) [1]. Optical vortex is related to the orbital angular momentum [2]. At the center of optical vortex, the phase is uncertain, and the light intensity is zero. Edge dislocation means that the phase has π changes at a wavefront line of the singularity beam [3]. The theories of singular optics are so important and widely applied to various fields, such as optical communication [4–7], optical microcontrol [8–10], optical data storage and quantum cryptography [11,12]. As a result, they have attracted the extensive research interest. Ilyenkov, et al. have verified in view from theory and experiment that laser can generate optical vortex through passing photorefractive crystals [13]. Petrov has reported the generation of edge dislocation and specified the generation conditions of edge dislocation [14]. Leach, et al. have observed the phase structure distribution of the vortex beam with fractional topological charges after a certain transmission distance and discovered that there are an integer number of optical vortices with the same topological charge symbols [15]. Wang pointed out that using optical vortex for optical communication provides another possible approach to improving information transmission capacity and spectrum effectiveness [16]. Zhang, et al. reported experimentally the generation of a perfect vortex beam and discussed the influences of the beam's wavelength and topological charges on light intensity evolution during transmission [17]. Porfirev, et al. have carried out a research on the influences of topological charges on the transmission of vortex beam in aerosol in view from theory and experiment [18]. Fang, et al. studied the transmission characteristics of symmetric Airy vortex beam in free space and pointed out that the light field rotates during the transmission when the optical vortex is at the center of the beam, and the larger the topological charges are, the more apparent the rotation will be [19]. Srinivas, et al. pointed out in their report of the transmission characteristics of vector vortex beam that if a part of the beam on the source plane is covered, the beam will be able to restore to its initial light field distribution characteristics after a certain transmission distance [20]. Hamedi, et al. probed into the formation of a combined vortex utilizing the overlapping of two vortex beams [21]. Dorrah, et al. provided the light intensity distribution and phase distribution of Bessel vortex beam when it's transmitted to a specific distance based on both theoretical and experimental results [22]. Li, et al. pointed out that the vortex phase plate can produce the ideal terahertz vortex beam with mode purity beyond 90% [23]. Chen, et al. reported the focusing characteristics of partial coherent radially polarized vortex beam [24].

However, according to our knowledge, there is no document research on the influences of edge dislocation on optical vortex transmission. The objective of this paper is to propose a detailed research on the influences of edge dislocation on optical vortex transmission. The second section will take vortex beam with and without edge dislocation as an example and respectively list out their analytical expression for the cross-spectral density functions in the turbulent atmosphere. The influences of edge dislocation on optical vortex transmission in free space will be analyzed in the third section. In the fourth section, there is a research on the influences of edge dislocation in turbulent atmosphere on optical vortex transmission. Last but not least, the fifth section provides a summary of the paper's major research results.

2. Transmission model

The field expression of vortex beam on the source plane is [25]

$${E_{01}}({{\boldsymbol s}\textrm{, }z = 0} )= \sum\limits_{m = 0}^l {{2^{ - l}}{\textrm{i}^m}} C_l^m{H_{l - m}}\left( {\frac{{\sqrt 2 {s_x}}}{{{w_0}}}} \right){H_m}\left( {\frac{{\sqrt 2 {s_y}}}{{{w_0}}}} \right)\exp \left( { - \frac{{s_x^\textrm{2} + s_y^\textrm{2}}}{{w_0^2}}} \right), $$
the field expression of vortex beam with edge dislocation on the source plane is [26]
$${E_{02}}({{\boldsymbol s}\textrm{, }z = 0} )= \sum\limits_{m = 0}^l {\frac{{l!s_x^m{{({\textrm{i}{s_y}} )}^{l - m}}({{s_x} - a} )}}{{m!({l - m} )!w_0^{l + 1}}}} \exp \left( { - \frac{{s_x^\textrm{2} + s_y^\textrm{2}}}{{w_0^2}}} \right). $$

In formulas (1) and (2), s = (sx, sy) is the position coordinates of the source plane, w0 is the waist width of the beam, l is the topological charge of the optical vortex, a is the distance between the optical vortex and the edge dislocation and Hm() is the Hermite polynomial. According to formulas (1) and (2), Fig. 1 shows the phase distributions and average normalized light intensity distributions of the beam with optical vortex and the beam with optical vortex and edge dislocation on the source plane. Calculation parameters: w0 = 5 mm, l = 1, a = 5 mm. It can be seen from Fig. 1(a) that the optical vortex has an increase of 2π in the phase by rotating anticlockwise around a phase singularity. It can be learned from Fig. 1(d) that there is an edge dislocation close to the optical vortex. It can be seen from Fig. 1(b, c, e, f) that the light intensity at the center of the optical vortex and the edge dislocation is zero.

 figure: Fig. 1.

Fig. 1. Phase distributions (a, d) and average normalized light Intensity distributions (b, c, e, f) of the beam with optical vortex (a-c) and the beam with optical vortex and edge dislocation (d-f) on the source plane. Blue indicates that the phase value is -π or the light intensity is zero. Yellow indicates that the phase value is π or the average normalized light intensity is 1.

Download Full Size | PDF

Based on the generalized Huygens-Fresnel principle [27,28], the cross-spectral density function expression of the beam transmission through turbulent atmosphere is

$$\begin{array}{c} W({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )= {\left( {\frac{k}{{2\mathrm{\pi }z}}} \right)^2}\int {\int {\int {\int {{W_0}} } } } ({{{\boldsymbol s}_1},{{\boldsymbol s}_2},z = 0} )\exp \left\{ { - \frac{{\textrm{i}k}}{{2z}}[{{{({{{\boldsymbol s}_1} - {{\boldsymbol \rho }_1}} )}^2} - {{({{{\boldsymbol s}_2} - {{\boldsymbol \rho }_2}} )}^2}} ]} \right\}\\ \times \textrm{exp}\left[ { - \frac{{{{({{{\boldsymbol s}_1} - {{\boldsymbol s}_2}} )}^2} + {{({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )}^2} + ({{{\boldsymbol s}_1} - {{\boldsymbol s}_2}} )({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )}}{{{{({0.545C_n^2{k^2}z} )}^{ - {6 / 5}}}}}} \right]d{{\boldsymbol s}_1}d{{\boldsymbol s}_2} \end{array}, $$
wherein, ρ1 and ρ2 are the 2D plane vectors on the transmission plane z, k is the wave number, $C_n^2$ is the structure constant and W0(s1, s2, z = 0) is the cross-spectral density of the beam on the source plane. The cross-spectral density function expression of vortex beam on the source plane is
$$\begin{aligned} {W_{01}}({{{\boldsymbol s}_1},{{\boldsymbol s}_2},z = 0} )&= \left\langle {{E_{01}}^\mathrm{\ast }({{\boldsymbol s}_1},0){E_{01}}({{\boldsymbol s}_2},0)} \right\rangle \\& = \sum\limits_{{m_1} = 0}^l {\sum\limits_{{m_2} = 0}^l {{2^{ - 2l}}} {{({ - \textrm{i}} )}^{{m_1}}}{\textrm{i}^{{m_2}}}} C_l^{{m_1}}C_l^{{m_2}}\exp \left( { - \frac{{s_{\textrm{2}x}^\textrm{2} + s_{\textrm{2}y}^\textrm{2} + s_{\textrm{1}x}^\textrm{2} + s_{\textrm{1}y}^\textrm{2}}}{{w_0^2}}} \right)\\& \times {H_{l - {m_1}}}\left( {\frac{{\sqrt 2 {s_{\textrm{1}x}}}}{{{w_0}}}} \right){H_{{m_1}}}\left( {\frac{{\sqrt 2 {s_{\textrm{1}y}}}}{{{w_0}}}} \right)\left( {\frac{{\sqrt 2 {s_{2x}}}}{{{w_0}}}} \right){H_{{m_2}}}\left( {\frac{{\sqrt 2 {s_{2y}}}}{{{w_0}}}} \right) \end{aligned}, $$
the cross-spectral density function expression of vortex beam with edge dislocation on the source plane is
$$\begin{aligned} {W_{02}}({{{\boldsymbol s}_1},{{\boldsymbol s}_2},z = 0} )&= \left\langle {{E_{02}}^\mathrm{\ast }({{\boldsymbol s}_1},0){E_{02}}({{\boldsymbol s}_2},0)} \right\rangle \\& = \sum\limits_{{m_1} = 0}^l {\sum\limits_{{m_2} = 0}^l {\frac{{{{({l!} )}^2}s_{1x}^{{m_1}}s_{2x}^{{m_2}}{{({ - \textrm{i}{s_{\textrm{1}y}}} )}^{l - {m_1}}}{{({\textrm{i}{s_{\textrm{2}y}}} )}^{l - {m_2}}}({{s_{\textrm{1}x}} - a} )({{s_{\textrm{2}x}} - a} )}}{{{m_1}!({l - {m_1}} )!{m_2}!({l - {m_2}} )!w_0^{2l + 2}}}} } \\& \times \exp \left( { - \frac{{s_{\textrm{2}x}^\textrm{2} + s_{\textrm{2}y}^\textrm{2} + s_{\textrm{1}x}^\textrm{2} + s_{\textrm{1}y}^\textrm{2}}}{{w_0^2}}} \right) \end{aligned}.$$

Use the calculation formula below [29]

$$\int {{x^n}} \exp ({ - p{x^2} + 2qx} )dx = n!\exp \left( {\frac{{{q^2}}}{p}} \right)\sqrt {\frac{\mathrm{\pi }}{p}} {\left( {\frac{q}{p}} \right)^n}\sum\limits_{k = 0}^{\left[ {\frac{n}{2}} \right]} {\frac{1}{{({n - 2k} )!k!}}} {\left( {\frac{p}{{4{q^2}}}} \right)^k}, $$
$$\int {\exp [{ - {{({x - y} )}^2}} ]} {H_n}({ax} )\textrm{d}x = \sqrt {\pi } {({1 - {a^2}} )^{\frac{n}{2}}}{H_n}\left[ {ay{{({1 - {a^2}} )}^{ - \frac{1}{2}}}} \right], $$
$$\int {{x^n}\exp [{ - {{({x - \beta } )}^2}} ]} \textrm{d}x = {({2\textrm{i}} )^{ - n}}\sqrt {\pi } {H_n}({\textrm{i}\beta } ), $$
$${H_n}({x + y} )= {2^{ - \frac{n}{2}}}\sum\limits_{k = 0}^n {C_n^k} {H_k}\left( {\sqrt 2 x} \right){H_{n - k}}\left( {\sqrt 2 y} \right), $$
$${H_n}(x )= \sum\limits_{m = 0}^{[\frac{n}{2}]} {{{({ - 1} )}^m}} \frac{{n!}}{{m!({n - 2m} )!}}{({2x} )^{n - 2m}}, $$
through integral computation, we get that the cross-spectral density function of vortex beam without and with edge dislocation during transmission in turbulent atmosphere is respectively W1(ρ1, ρ2, z) and W2(ρ1, ρ2, z). The analytical expressions of W1(ρ1, ρ2, z) and W2(ρ1, ρ2, z) are as follows
$$\begin{aligned} {W_1}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )&= {\left( {\frac{k}{{{2^{l + 1}}\mathrm{\pi }z}}} \right)^2}\sum\limits_{{m_1} = 0}^l {\sum\limits_{{m_2} = 0}^l {{{({ - \textrm{i}} )}^{{m_1}}}{\textrm{i}^{{m_2}}}C_l^{{m_1}}C_l^{{m_2}}\exp \{{ - B[{{{({{\rho_{1x}} - {\rho_{2x}}} )}^2} + {{({{\rho_{1y}} - {\rho_{2y}}} )}^2}} ]} \}} } \\& \times \sum\limits_{{n_1} = 0}^{[\frac{{l - {m_1}}}{2}]} {\sum\limits_{{n_2} = 0}^{l - {m_2}} {\sum\limits_{{n_3} = 0}^{[\frac{{{n_2}}}{2}]} {\frac{{C_{l - {m_2}}^{{n_2}}}}{{{{({2\textrm{i}} )}^{l - {m_1} - 2{n_1} + {n_2} - 2{n_3}}}}}} } } {\left( {\frac{1}{2} - \frac{1}{{w_0^2 + A}}} \right)^{\frac{{l - {m_2}}}{2}}}{\left( {\frac{1}{{\sqrt D }}} \right)^{l - {m_1} - 2{n_1} + {n_2} - 2{n_3} + 1}}\\& \times \frac{{\mathrm{\pi }({l - {m_1}} )!{n_2}!}}{{{n_1}!({l - {m_1} - 2{n_1}} )!{n_3}!({{n_2} - 2{n_3}} )\sqrt A }}{\left( {\frac{{4B}}{{\sqrt {w_0^2{A^2} - 2A} }}} \right)^{{n_2} - 2{n_3}}}{\left( {\frac{{2\sqrt 2 }}{{{w_0}}}} \right)^{l - {m_1} - 2{n_1}}}\\& \times {H_{l - {m_2} - {n_2}}}\left[ {\frac{{Bz({{\rho_{1x}} - {\rho_{2x}}} )- \textrm{i}k{\rho_{2x}}}}{{z\sqrt {w_0^2{A^2} - 2A} }}} \right]\exp \left\{ {\frac{1}{{4A}}{{\left[ {B({{\rho_{1x}} - {\rho_{2x}}} )- \frac{{ik{\rho_{2x}}}}{z}} \right]}^2} + \frac{{{C^2}}}{{4D}}} \right\}\\& \times {H_{l - {m_1} - 2{n_1} + {n_2} - 2{n_3}}}\left( {\frac{{\textrm{i}C}}{{2\sqrt D }}} \right)\sum\limits_{{n_4} = 0}^{[\frac{{{m_1}}}{2}]} {\sum\limits_{{n_5} = 0}^{{m_2}} {\sum\limits_{{n_6} = 0}^{[\frac{{{n_5}}}{2}]} {\frac{{C_{{m_2}}^{{n_5}}{{({ - 1} )}^{{n_4} + {n_6}}}}}{{{{({2\textrm{i}} )}^{{m_1} - 2{n_4} + {n_5} - 2{n_6}}}}}} } } {\left( {\frac{1}{2} - \frac{1}{{w_0^2 + A}}} \right)^{\frac{{{m_2}}}{2}}}\\& \times \frac{{\mathrm{\pi }({{m_1}} )!{n_5}!}}{{{n_4}!({{m_1} - 2{n_4}} )!{n_6}!({{n_5} - 2{n_6}} )\sqrt A }}{\left( {\frac{{4B}}{{\sqrt {w_0^2{A^2} - 2A} }}} \right)^{{n_5} - 2{n_6}}}{\left( {\frac{1}{{\sqrt D }}} \right)^{{m_1} - 2{n_4} + {n_5} - 2{n_6} + 1}}\\& \times {H_{{m_2} - {n_5}}}\left[ {\frac{{Bz({{\rho_{1y}} - {\rho_{2y}}} )- \textrm{i}k{\rho_{2y}}}}{{z\sqrt {w_0^2{A^2} - 2A} }}} \right]\exp \left[ { - \frac{{\textrm{i}k}}{{2z}}({{\rho_{1x}}^2 + {\rho_{1y}}^2 - {\rho_{2x}}^2 - {\rho_{2y}}^2} )} \right]\\& \times {H_{{m_1} - 2{n_4} + {n_5} - 2{n_6}}}\left( {\frac{{\textrm{i}E}}{{2\sqrt D }}} \right)\exp \left\{ {\frac{1}{{4A}}{{\left[ {B({{\rho_{1y}} - {\rho_{2y}}} )- \frac{{\textrm{i}k{\rho_{2y}}}}{z}} \right]}^2} + \frac{{{E^2}}}{{4D}}} \right\}{\left( {\frac{{2\sqrt 2 }}{{{w_0}}}} \right)^{{m_1} - 2{n_4}}} \end{aligned}$$
$$\begin{aligned} {W_2}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )&= {\left( {\frac{k}{{2{\pi }zw_0^{l + 1}}}} \right)^2}\sum\limits_{{m_1} = 0}^l {\sum\limits_{{m_2} = 0}^l {\frac{{{{({l!} )}^2}{{({ - \textrm{i}} )}^{l - {m_1}}}{\textrm{i}^{l - {m_2}}}}}{{{m_1}!({l - {m_1}} )!{m_2}!}}} } \sum\limits_{{n_1} = 0}^{[\frac{{l - {m_2}}}{2}]} {\sum\limits_{{n_2} = 0}^{l - {m_2} - 2{n_1}} {\exp \left[ {\frac{{ - \textrm{i}k({{\rho_{1x}}^2 + {\rho_{1y}}^2} )}}{{2z}}} \right]} } \\& \times \frac{{{\pi }{A^{{m_2} - l + {n_1} - \frac{1}{2}}}({l - {m_1} + {n_2}} )!{B^{{n_2}}}}}{{{4^{{n_1}}}{n_1}!{n_2}!({l - {m_2} - 2{n_1} - {n_2}} )!\sqrt F }}{\left[ { - \frac{{\textrm{i}k{\rho_{2y}}}}{{2z}} + \frac{{B({{\rho_{1y}} - {\rho_{2y}}} )}}{2}} \right]^{l - {m_2} - 2{n_1} - {n_2}}}\\& \times \exp \left\{ {\frac{1}{{4A}}{{\left[ { - \frac{{\textrm{i}k{\rho_{2y}}}}{z} + B({{\rho_{1y}} - {\rho_{2y}}} )} \right]}^2} + \frac{{{E^2}}}{{4F}}} \right\}{\left( {\frac{E}{{2F}}} \right)^{l - {m_1} + {n_2}}}\sum\limits_{{n_3} = 0}^{[\frac{{l - {m_1} + {n_2}}}{2}]} {{{\left( {\frac{F}{{{E^2}}}} \right)}^{{n_3}}}} \\& \times \frac{{\exp \{{ - B[{{{({{\rho_{1x}} - {\rho_{2x}}} )}^2} + {{({{\rho_{1y}} - {\rho_{2y}}} )}^2}} ]} \}}}{{({l - {m_1} + {n_2} - 2{n_3}} )!{n_3}!}}\left\{ {\left[ {\sum\limits_{{n_4} = 0}^{[\frac{{{m_2} + 1}}{2}]} {\frac{{\sqrt {\pi } {A^{{n_4} - {m_2} - 1 - \frac{1}{2}}}({{m_2} + 1} )!}}{{{4^{{n_4}}}({{m_2} + 1 - 2{n_4}} )!{n_4}!}}} } \right.} \right.\\& \times \sum\limits_{{n_5} = 0}^{{m_2} + 1 - 2{n_4}} {\frac{{({{m_2} + 1 - 2{n_4}} )!{B^{{n_5}}}({{I_{{b_1}}} - a{I_{{b_2}}}} )}}{{{n_5}!({{m_1} + 1 - 2{n_4} - {n_5}} )!}}{{\left( {\frac{G}{2}} \right)}^{{m_1} + 1 - 2{n_4} - {n_5}}}\left. {\exp \left( {\frac{{{G^2}}}{{4A}}} \right)} \right]} \\& - a\left[ {\sum\limits_{{n_8} = 0}^{[\frac{{{m_2}}}{2}]} {\frac{{\sqrt {\pi } {A^{{n_4} - {m_2} - \frac{1}{2}}}{m_2}!}}{{{4^{{n_4}}}({{m_2} - 2{n_8}} )!{n_8}!}}} } \right.\sum\limits_{{n_9} = 0}^{{m_2} - 2{n_8}} {\frac{{({{m_2} - 2{n_8}} )!{B^{{n_9}}}({{I_{{c_1}}} - a{I_{{c_2}}}} )}}{{{n_5}!({{m_1} + 1 - 2{n_8} - {n_9}} )!}}} {\left( {\frac{G}{2}} \right)^{{m_1} - 2{n_8} - {n_9}}}\\& \times \left. {\left. {\exp \left( {\frac{{{G^2}}}{{4A}}} \right)} \right]} \right\}\exp \left[ {\frac{{ - \textrm{i}k({{\rho_{2x}}^2 + {\rho_{2y}}^2} )}}{{2z}}} \right] \end{aligned}.$$

In formulas (11) and (12)

$$A = \frac{1}{{w_0^2}} - \frac{{\textrm{i}k}}{{2z}} + B,B = {({0.545C_n^2{k^2}z} )^{{6 / 5}}}$$
$$C = \frac{{\textrm{i}k{\rho _{1x}}}}{z} - B\left( {{\rho _{1x}} - {\rho _{2x}}} \right) + \frac{B}{A}\left[ {B\left( {{\rho _{1x}} - {\rho _{2x}}} \right) - \frac{{\textrm{i}k{\rho _{2x}}}}{z}} \right],D = A - \frac{{{B^2}}}{A}$$
$$E = \frac{{\textrm{i}k{\rho _{1y}}}}{z} - B({{\rho_{1y}} - {\rho_{2y}}} )+ \frac{B}{A}\left[ {B({{\rho_{1y}} - {\rho_{2y}}} )- \frac{{\textrm{i}k{\rho_{2y}}}}{z}} \right], $$
$$F = \frac{1}{{w_0^2}} + \frac{{\textrm{i}k}}{{2z}} + B - \frac{{{B^2}}}{A},G ={-} \frac{{\textrm{i}k{\rho _{2x}}}}{z} + B({{\rho_{1x}} - {\rho_{2x}}} ),$$
$${I_{{b_i}}} = {b_i}!\exp \left( {\frac{{{C^2}}}{{4F}}} \right)\sqrt {\frac{\mathrm{\pi }}{F}} {\left( {\frac{C}{{2F}}} \right)^{{b_i}}}\sum\limits_{{n_{i + 5}} = 0}^{[\frac{{{b_i}}}{2}]} {\frac{{{{({F/{C^2}} )}^{{n_{i + 5}}}}}}{{({{b_i} - 2{n_{i + 5}}} )!{n_{i + 5}}!}}} ({i = 1,2,{b_1} = {m_1} + {n_5} + 1;{b_2} = {b_1} - 1} ),$$
$${I_{{c_i}}} = {c_i}!\exp \left( {\frac{{{C^2}}}{{4F}}} \right)\sqrt {\frac{\mathrm{\pi }}{F}} {\left( {\frac{C}{{2F}}} \right)^{{c_i}}}\sum\limits_{{n_{i + 9}} = 0}^{[\frac{{{c_i}}}{2}]} {\frac{{{{({F/{C^2}} )}^{{n_{i + 9}}}}}}{{({{c_i} - 2{n_{i + 9}}} )!{n_{i + 9}}!}}} ({i = 1,2,{c_1} = {m_1} + {n_9} + 1;{c_2} = {c_1} - 1} ).$$

When $C_n^2$=0, formulas (11) and (12) degrade into the analytical expression for the cross-spectral density function of the beam during transmission in free space.

According to the definition of spectral coherence degree [30]

$$\mu ({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )= \frac{{W({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )}}{{{{[{I({{{\boldsymbol \rho }_1},z} )I({{{\boldsymbol \rho }_2},z} )} ]}^{{1 / 2}}}}}, $$
wherein, I(ρi, z)=W(ρi, ρi, z) (i = 1, 2) represents the light intensity at point (ρi, z). In the process of formula derivation, the correlation length has been taken to infinity. When the correlation length is taken to infinity, the coherence vortex evolves into an optical vortex [31,32]. Thus, the position of the center of optical vortex is determined through the equation set below [33]
$$\left\{ {\begin{array}{{c}} {\textrm{R}\textrm{e}[{\mu ({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )} ]= 0}\\ {\textrm{Im}[{\mu ({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},z} )} ]= 0} \end{array}} \right., $$
wherein, Re and Im respectively indicate the computation of real part and imaginary part.

The definition of topological charge is [1]

$$l = \frac{1}{{2\mathrm{\pi }}}\oint_Q {\nabla \varphi (r )} \textrm{d}r, $$
wherein, φ(r) represents the phase distribution and Q represents the anticlockwise closed curve around optical vortex. After one circle of clockwise rotating around the center of optical vortex, the phase increases by 2|l|π, and the topological charge value l < 0. After one circle of anticlockwise rotating around the center of optical vortex, the topological charge value l > 0 [34].

3. Transmission of optical vortex in free space

Figures 2–5 show the phase distributions of spectral coherence degree of vortex beam and vortex beam with edge dislocation on the source plane z = 0 and when transmitted to z = 10 m, 20 m and 50 m in free space under the condition of the topological charge l = + 1, + 2 and +3. The calculation parameters are as follows: λ = 632.8 nm, w0 = 5 mm, ρ1 = (1 cm, 1 cm), a = 5 mm, $C_n^2$= 0. It can be seen from Fig. 2(a) that the vortex beam has an optical vortex with a topological charge of +1 on the source plane. We can see from Fig. 2(b-d) that optical vortex always exist when the vortex beam is transmitted to z = 10 m, 20 m and 50 m. It can be concluded from Fig. 2(e) that there is an edge dislocation near the optical vortex on the source plane. Seeing from Fig. 2(f-h), when there's an edge dislocation near the optical vortex, the evolution of optical vortex is similar to that shown in Fig. 2(b-d). The optical vortex exists throughout the transmission.

 figure: Fig. 2.

Fig. 2. Phase distributions of spectral coherence degree of vortex beam (a-d) and vortex beam with edge dislocation (e-h) on the source plane and during transmission in the free space. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 1.

Download Full Size | PDF

 figure: Fig. 3.

Fig. 3. Phase distributions of spectral coherence degree of vortex beam (a-d) and vortex beam with edge dislocation (e-h) on the source plane and during transmission in the free space. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 2.

Download Full Size | PDF

 figure: Fig. 4.

Fig. 4. Phase distributions of spectral coherence degree of vortex beam (a-d) and vortex beam with edge dislocation (e-h) on the source plane and during transmission in the free space. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 3.

Download Full Size | PDF

 figure: Fig. 5.

Fig. 5. Influences of light wavelength (a) and distance between edge dislocation and optical vortex (b) on optical vortex split distance d in the free space.

Download Full Size | PDF

We can learn from Fig. 3(a) that vortex beam has an optical vortex with a topological charge of +2 on the source plane z = 0. It can be seen from Fig. 3(b-d) that the optical vortex always exists when transmitted to z = 10 m, 20 m and 50 m. We can learn from Fig. 3(e) that there's an edge dislocation near the optical vortex with a topological charge of +2. It can be seen from Fig. 3(f-h) that due to the edge dislocation, the optical vortex splits into two optical vortices with a topological charge of +1 during the transmission in free space. As the transmission distance increases, the distance between the two optical vortices gradually increases. We can conclude from Fig. 4(a) that vortex beam has an optical vortex with a topological charge of +3 when z = 0. According to Fig. 4(b-d), when z = 10 m, 20 m and 50 m, the optical vortex still exists in a steady way in the free space. In Fig. 4(e), we can see that there's an edge dislocation near the optical vortex. It can be seen from Fig. 4(f-h) that due to the edge dislocation, the optical vortex with a topological charge of +3 splits into one optical vortex with a topological charge of +1 and one optical vortex with a topological charge of +2 during the transmission in free space. As the transmission distance increases, the transmission of the two optical vortices remains stable, and the distance between the two optical vortices gradually increases.

Figure 5 specifies the influences of light wavelength and the distance between the edge dislocation and the optical vortex on the evolution of optical vortex when there's an edge dislocation in the free space. Calculation parameters: a = 5 mm (Fig. 5(a)), λ = 632.8 nm (Fig. 5(b)). The rest parameters are the same as those in Fig. 4 or are shown in the Fig. 5. According to Fig. 5(a), at the same transmission distance z, the distance d between the two split optical vortices increases as the wavelength increases. From Fig. 5(b), we know that at the same transmission distance z, the larger the distance between the edge dislocation and the optical vortex on the source plane is, the smaller the distance d between the two optical vortexes will be.

4. Transmission of optical vortex in turbulent atmosphere

Figures 6–8 show the phase distributions of spectral coherence degree of vortex beam and vortex beam with edge dislocation in turbulent atmosphere respectively when transmitted to z = 10 m, 20 m and 50 m under the condition of the topological charge l = + 1, + 2 and +3. Calculation parameter: $C_n^2$= 10−15 m−2/3. Other calculation parameters are as shown in Fig. 2. It can be concluded from Fig. 6(a-c) and 6(d-f) that when the optical vortex has a topological charge of +1, as the transmission distance increases, the evolution of optical vortex with edge dislocation is similar to the evolution of optical vortex without edge dislocation, and optical vortex exists stably. It can be seen from Fig. 7(a-c) and Fig. 7(d-f) that during the transmission of the optical vortex with a topological charge of +2, the optical vortex may evolve into two optical vortices with a topological charge of +1 no matter with and without edge dislocation. The difference is when there's edge dislocation, the distance between the two evolved optical vortices is greater at the same transmission distance. It is indicated in Fig. 8(a-c) and Fig. 8(d-f) that the optical vortex with a topological charge of +3 may evolve into 3 optical vortices with a topological charge of +1 during transmission, with or without edge dislocation. The difference is when there's edge dislocation, the distance of one of three split optical vortices from the other two may be way greater than that between the other two.

 figure: Fig. 6.

Fig. 6. Phase distributions of spectral coherence degree of vortex beam (a-c) and vortex beam with edge dislocation (d-f) during transmission in turbulent atmosphere. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 1.

Download Full Size | PDF

 figure: Fig. 7.

Fig. 7. Phase distributions of spectral coherence degree of vortex beam (a-c) and vortex beam with edge dislocation (d-f) during transmission in turbulent atmosphere. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 2.

Download Full Size | PDF

 figure: Fig. 8.

Fig. 8. Phase distributions of spectral coherence degree of vortex beam (a-c) and vortex beam with edge dislocation (d-f) during transmission in turbulent atmosphere. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 3.

Download Full Size | PDF

From Figs. 6–8, we have learned that optical vortex will split during transmission in turbulent atmosphere. With edge dislocation, there is an optical vortex whose distance from the other optical vortices is much greater than the distance between the other optical vortices. Figure 9 takes the changes of the largest distance between optical vortices as an example and specifies the influences of the distance between the optical vortex and the edge dislocation, the light wavelength and the structure constant on the evolution of optical vortex when there's edge dislocation in turbulent atmosphere. Calculation parameters: $C_n^2$=10−15 m−2/3 (Fig. 9(a, b)), λ = 632.8 nm and a = 5 mm (Fig. 9(c)). The other parameters are the same as those in Fig. 5. When there's edge dislocation in the transmission of optical vortices in the turbulent atmosphere, we can conclude from Fig. 9(a, c) that as the light wavelength and the structure constant increase, the largest distance between optical vortices at the same transmission distance will become greater. Based on Fig. 9(b), we can summarize that when the transmission distance remains the same, the larger the distance between the optical vortex and the edge dislocation is, the smaller the largest distance between optical vortices will be.

 figure: Fig. 9.

Fig. 9. Influences of light wavelength (a), distance between edge dislocation and optical vortex (b) and structure constant (c) on optical vortex split distance d in turbulent atmosphere.

Download Full Size | PDF

5. Conclusion

Based on the generalized Huygens-Fresnel principle, this paper provides the cross-spectral density function expression of vortex beam with and without edge dislocation during transmission in turbulent atmosphere and free space and further studies the influences of edge dislocation on the evolution of optical vortex based on the definition of spectral coherence degree. The research results show that optical vortex exist stably during the transmission in the free space. When there's edge dislocation, the optical vortex with a topological charge l of above +1 will split into two optical vortices as the transmission distance increases. As the beam transmission distance continues to increase, the distance between the two optical vortices gradually increases.

During the transmission of optical vortex in the turbulent atmosphere, the optical vortex will split as the transmission distance increases. Throughout the transmission, the topological charges are conserved. When there's edge dislocation, there is an optical vortex whose distance from the other optical vortices is much greater than the distance between the other optical vortices. Besides, when there's an edge dislocation, the greater the light wavelength and the structure constant are, the smaller the distance between the optical vortex and the edge dislocation is, and the evolution of the optical vortex will be accelerated.

Because the phase singularity is related to the orbital angular momentum [2], the change of the wavefront phase reflects the energy redistribution of the orbital angular momentum among different modes. These conclusions are of great significance in offering theoretical guidance to the transmission control of optical vortex and wireless optical communication.

Funding

Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (2022L504); Fundamental Research Program of Shanxi Province (202203021212169); National Natural Science Foundation of China (12204375).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author upon reasonable request.

References

1. J. F. Nye and M. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. A Math. Phys. Eng. Sci. 336(2), 165–190 (1974). [CrossRef]  

2. L. Allen, M. Beijersbergen, R. Spreeuw, et al., “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]  

3. M. Soskin and M. Vasnetsov, “Singular optics,” Prog. opt. 42(9), 219–276 (2001). [CrossRef]  

4. P. Gao and L. Bai, “Spiral spectrum of the phase singularity beam in the source plane and atmospheric turbulence,” OSA Continuum 3(10), 2797 (2020). [CrossRef]  

5. P. Gao, L. Bai, and J. Li, “Comparison of propagation properties of circular edge dislocation beams and circularlinear edge dislocation beams,” OSA Continuum 3(11), 2997 (2020). [CrossRef]  

6. S. Chang, Y. Song, Y. Dong, et al., “Spreading properties of a multi-Gaussian Schell-model vortex beam in slanted atmospheric turbulence,” Opt. Appl. L 50(1), 83–94 (2020). [CrossRef]  

7. G. Guo, X. Li, Q. Wang, et al., “Beam alignments based on the spectrum decomposition of orbital angular momentums for acoustic-vortex communications,” Chinese Phys. B 31(12), 124302 (2022). [CrossRef]  

8. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]  

9. R. M. Pettit, W. Ge, P. Kumar, et al., “An optical tweezer phonon laser,” Nat. photonics 13(6), 402–405 (2019). [CrossRef]  

10. X. Liu, D. Zhang, L. Zhao, et al., “Abruptly autofocusing property of circular Pearcey vortex beams with different initial launch angles in harmonic potentials,” Opt. Appl. LII 52(3), 417–428 (2022). [CrossRef]  

11. R. J. Voogd, M. Singh, and J. J. Braat, “The use of orbital angular momentum of light beams for optical data storage,” Proc. SPIE 5380, 387–392 (2004). [CrossRef]  

12. M. D. Levenson, T. Ebihara, G. Dai, et al., “Optical vortex masks for via levels,” J. Micro/Nanolith. Mems. Moems. 3(2), 293–304 (2004). [CrossRef]  

13. A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, et al., “Birth and evolution of wave-front dislocations in a laser beam passed through photorefractive LiNbO3 crystal,” Appl. Phys. B 62(5), 465–471 (1996). [CrossRef]  

14. D. V. Petrov, “Second harmonic generation by optical beams with edge phase dislocations,” Opt. Commun. 192(1-2), 101–106 (2001). [CrossRef]  

15. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004). [CrossRef]  

16. J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016). [CrossRef]  

17. Y. Zhang, W. Liu, J. Gao, et al., “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018). [CrossRef]  

18. A. P. Porfirev, M. S. Kirilenko, S. N. Khonina, et al., “Study of propagation of vortex beams in aerosol optical medium,” Appl. Opt. 56(11), E8–E15 (2017). [CrossRef]  

19. Z. X. Fang, Y. Chen, Y. X. Ren, et al., “Interplay between topological phase and self-acceleration in a vortex symmetric Airy beam,” Opt. Express 26(6), 7324–7335 (2018). [CrossRef]  

20. P. Srinivas, C. Perumangatt, N. Lal, et al., “Investigation of propagation dynamics of truncated vector vortex beams,” Opt. Lett. 43(11), 2579–2582 (2018). [CrossRef]  

21. H. R. Hamedi, J. Ruseckas, E. Paspalakis, et al., “Transfer of optical vortices in coherently prepared media,” Phys. Rev. A 99(3), 033812 (2019). [CrossRef]  

22. A. H. Dorrah, C. Rosales-Guzmán, A. Forbes, et al., “Evolution of orbital angular momentum in three-dimensional structured light,” Phys. Rev. A 98(4), 043846 (2018). [CrossRef]  

23. W. Y. Li, R. Sun, J. Y. Liu, et al., “Broadband and high efficiency terahertz metasurfaces for anomalous refraction and vortex beam generation,” Chinese Phys. B 31(10), 108701 (2022). [CrossRef]  

24. K. Chen, Z. Y. Ma, and Y. Y. Hu, “Tightly focused properties of partially coherent radially polarized power-exponent-phase vortex beam,” Chinese Phys. B 32(2), 024208 (2023). [CrossRef]  

25. J. Qu, Y. Zhong, Z. Cui, et al., “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010). [CrossRef]  

26. D. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quant. Electron. 34(8), 759–773 (2002). [CrossRef]  

27. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media (SPIE Press, 2005).

28. S. Wang and M. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]  

29. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Acadenic Press, 2007).

30. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

31. K. Cheng and B. Lü, “Composite coherence vortices in coherent and incoherent superpositions of two off-axis partially coherent vortex beams,” J. Mod. Opt. 55(17), 2751–2764 (2008). [CrossRef]  

32. K. Cheng and B. Lü, “Composite coherence vortices and their propagation in free space,” Optik 121(7), 589–594 (2010). [CrossRef]  

33. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003). [CrossRef]  

34. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)
Fig. 1.
Fig. 1. Phase distributions (a, d) and average normalized light Intensity distributions (b, c, e, f) of the beam with optical vortex (a-c) and the beam with optical vortex and edge dislocation (d-f) on the source plane. Blue indicates that the phase value is -π or the light intensity is zero. Yellow indicates that the phase value is π or the average normalized light intensity is 1.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Phase distributions of spectral coherence degree of vortex beam (a-d) and vortex beam with edge dislocation (e-h) on the source plane and during transmission in the free space. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 1.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Phase distributions of spectral coherence degree of vortex beam (a-d) and vortex beam with edge dislocation (e-h) on the source plane and during transmission in the free space. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 2.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Phase distributions of spectral coherence degree of vortex beam (a-d) and vortex beam with edge dislocation (e-h) on the source plane and during transmission in the free space. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 3.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Influences of light wavelength (a) and distance between edge dislocation and optical vortex (b) on optical vortex split distance d in the free space.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Phase distributions of spectral coherence degree of vortex beam (a-c) and vortex beam with edge dislocation (d-f) during transmission in turbulent atmosphere. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 1.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. Phase distributions of spectral coherence degree of vortex beam (a-c) and vortex beam with edge dislocation (d-f) during transmission in turbulent atmosphere. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 2.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. Phase distributions of spectral coherence degree of vortex beam (a-c) and vortex beam with edge dislocation (d-f) during transmission in turbulent atmosphere. Blue indicates that the phase value is –π, and yellow indicates that the phase value is π. l = + 3.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. Influences of light wavelength (a), distance between edge dislocation and optical vortex (b) and structure constant (c) on optical vortex split distance d in turbulent atmosphere.

View in Article | Download Full Size | PDF

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

E 01 ( s ,  z = 0 ) = ∑ m = 0 l 2 − l i m C l m H l − m ( 2 s x w 0 ) H m ( 2 s y w 0 ) exp ⁡ ( − s x 2 + s y 2 w 0 2 ) ,
E 02 ( s ,  z = 0 ) = ∑ m = 0 l l ! s x m ( i s y ) l − m ( s x − a ) m ! ( l − m ) ! w 0 l + 1 exp ⁡ ( − s x 2 + s y 2 w 0 2 ) .
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 ∫ ∫ ∫ ∫ W 0 ( s 1 , s 2 , z = 0 ) exp ⁡ { − i k 2 z [ ( s 1 − ρ 1 ) 2 − ( s 2 − ρ 2 ) 2 ] } × exp [ − ( s 1 − s 2 ) 2 + ( ρ 1 − ρ 2 ) 2 + ( s 1 − s 2 ) ( ρ 1 − ρ 2 ) ( 0.545 C n 2 k 2 z ) − 6 / 5 ] d s 1 d s 2 ,
W 01 ( s 1 , s 2 , z = 0 ) = ⟨ E 01 ∗ ( s 1 , 0 ) E 01 ( s 2 , 0 ) ⟩ = ∑ m 1 = 0 l ∑ m 2 = 0 l 2 − 2 l ( − i ) m 1 i m 2 C l m 1 C l m 2 exp ⁡ ( − s 2 x 2 + s 2 y 2 + s 1 x 2 + s 1 y 2 w 0 2 ) × H l − m 1 ( 2 s 1 x w 0 ) H m 1 ( 2 s 1 y w 0 ) ( 2 s 2 x w 0 ) H m 2 ( 2 s 2 y w 0 ) ,
W 02 ( s 1 , s 2 , z = 0 ) = ⟨ E 02 ∗ ( s 1 , 0 ) E 02 ( s 2 , 0 ) ⟩ = ∑ m 1 = 0 l ∑ m 2 = 0 l ( l ! ) 2 s 1 x m 1 s 2 x m 2 ( − i s 1 y ) l − m 1 ( i s 2 y ) l − m 2 ( s 1 x − a ) ( s 2 x − a ) m 1 ! ( l − m 1 ) ! m 2 ! ( l − m 2 ) ! w 0 2 l + 2 × exp ⁡ ( − s 2 x 2 + s 2 y 2 + s 1 x 2 + s 1 y 2 w 0 2 ) .
∫ x n exp ⁡ ( − p x 2 + 2 q x ) d x = n ! exp ⁡ ( q 2 p ) π p ( q p ) n ∑ k = 0 [ n 2 ] 1 ( n − 2 k ) ! k ! ( p 4 q 2 ) k ,
∫ exp ⁡ [ − ( x − y ) 2 ] H n ( a x ) d x = π ( 1 − a 2 ) n 2 H n [ a y ( 1 − a 2 ) − 1 2 ] ,
∫ x n exp ⁡ [ − ( x − β ) 2 ] d x = ( 2 i ) − n π H n ( i β ) ,
H n ( x + y ) = 2 − n 2 ∑ k = 0 n C n k H k ( 2 x ) H n − k ( 2 y ) ,
H n ( x ) = ∑ m = 0 [ n 2 ] ( − 1 ) m n ! m ! ( n − 2 m ) ! ( 2 x ) n − 2 m ,
W 1 ( ρ 1 , ρ 2 , z ) = ( k 2 l + 1 π z ) 2 ∑ m 1 = 0 l ∑ m 2 = 0 l ( − i ) m 1 i m 2 C l m 1 C l m 2 exp ⁡ { − B [ ( ρ 1 x − ρ 2 x ) 2 + ( ρ 1 y − ρ 2 y ) 2 ] } × ∑ n 1 = 0 [ l − m 1 2 ] ∑ n 2 = 0 l − m 2 ∑ n 3 = 0 [ n 2 2 ] C l − m 2 n 2 ( 2 i ) l − m 1 − 2 n 1 + n 2 − 2 n 3 ( 1 2 − 1 w 0 2 + A ) l − m 2 2 ( 1 D ) l − m 1 − 2 n 1 + n 2 − 2 n 3 + 1 × π ( l − m 1 ) ! n 2 ! n 1 ! ( l − m 1 − 2 n 1 ) ! n 3 ! ( n 2 − 2 n 3 ) A ( 4 B w 0 2 A 2 − 2 A ) n 2 − 2 n 3 ( 2 2 w 0 ) l − m 1 − 2 n 1 × H l − m 2 − n 2 [ B z ( ρ 1 x − ρ 2 x ) − i k ρ 2 x z w 0 2 A 2 − 2 A ] exp ⁡ { 1 4 A [ B ( ρ 1 x − ρ 2 x ) − i k ρ 2 x z ] 2 + C 2 4 D } × H l − m 1 − 2 n 1 + n 2 − 2 n 3 ( i C 2 D ) ∑ n 4 = 0 [ m 1 2 ] ∑ n 5 = 0 m 2 ∑ n 6 = 0 [ n 5 2 ] C m 2 n 5 ( − 1 ) n 4 + n 6 ( 2 i ) m 1 − 2 n 4 + n 5 − 2 n 6 ( 1 2 − 1 w 0 2 + A ) m 2 2 × π ( m 1 ) ! n 5 ! n 4 ! ( m 1 − 2 n 4 ) ! n 6 ! ( n 5 − 2 n 6 ) A ( 4 B w 0 2 A 2 − 2 A ) n 5 − 2 n 6 ( 1 D ) m 1 − 2 n 4 + n 5 − 2 n 6 + 1 × H m 2 − n 5 [ B z ( ρ 1 y − ρ 2 y ) − i k ρ 2 y z w 0 2 A 2 − 2 A ] exp ⁡ [ − i k 2 z ( ρ 1 x 2 + ρ 1 y 2 − ρ 2 x 2 − ρ 2 y 2 ) ] × H m 1 − 2 n 4 + n 5 − 2 n 6 ( i E 2 D ) exp ⁡ { 1 4 A [ B ( ρ 1 y − ρ 2 y ) − i k ρ 2 y z ] 2 + E 2 4 D } ( 2 2 w 0 ) m 1 − 2 n 4
W 2 ( ρ 1 , ρ 2 , z ) = ( k 2 π z w 0 l + 1 ) 2 ∑ m 1 = 0 l ∑ m 2 = 0 l ( l ! ) 2 ( − i ) l − m 1 i l − m 2 m 1 ! ( l − m 1 ) ! m 2 ! ∑ n 1 = 0 [ l − m 2 2 ] ∑ n 2 = 0 l − m 2 − 2 n 1 exp ⁡ [ − i k ( ρ 1 x 2 + ρ 1 y 2 ) 2 z ] × π A m 2 − l + n 1 − 1 2 ( l − m 1 + n 2 ) ! B n 2 4 n 1 n 1 ! n 2 ! ( l − m 2 − 2 n 1 − n 2 ) ! F [ − i k ρ 2 y 2 z + B ( ρ 1 y − ρ 2 y ) 2 ] l − m 2 − 2 n 1 − n 2 × exp ⁡ { 1 4 A [ − i k ρ 2 y z + B ( ρ 1 y − ρ 2 y ) ] 2 + E 2 4 F } ( E 2 F ) l − m 1 + n 2 ∑ n 3 = 0 [ l − m 1 + n 2 2 ] ( F E 2 ) n 3 × exp ⁡ { − B [ ( ρ 1 x − ρ 2 x ) 2 + ( ρ 1 y − ρ 2 y ) 2 ] } ( l − m 1 + n 2 − 2 n 3 ) ! n 3 ! { [ ∑ n 4 = 0 [ m 2 + 1 2 ] π A n 4 − m 2 − 1 − 1 2 ( m 2 + 1 ) ! 4 n 4 ( m 2 + 1 − 2 n 4 ) ! n 4 ! × ∑ n 5 = 0 m 2 + 1 − 2 n 4 ( m 2 + 1 − 2 n 4 ) ! B n 5 ( I b 1 − a I b 2 ) n 5 ! ( m 1 + 1 − 2 n 4 − n 5 ) ! ( G 2 ) m 1 + 1 − 2 n 4 − n 5 exp ⁡ ( G 2 4 A ) ] − a [ ∑ n 8 = 0 [ m 2 2 ] π A n 4 − m 2 − 1 2 m 2 ! 4 n 4 ( m 2 − 2 n 8 ) ! n 8 ! ∑ n 9 = 0 m 2 − 2 n 8 ( m 2 − 2 n 8 ) ! B n 9 ( I c 1 − a I c 2 ) n 5 ! ( m 1 + 1 − 2 n 8 − n 9 ) ! ( G 2 ) m 1 − 2 n 8 − n 9 × exp ⁡ ( G 2 4 A ) ] } exp ⁡ [ − i k ( ρ 2 x 2 + ρ 2 y 2 ) 2 z ] .
A = 1 w 0 2 − i k 2 z + B , B = ( 0.545 C n 2 k 2 z ) 6 / 5
C = i k ρ 1 x z − B ( ρ 1 x − ρ 2 x ) + B A [ B ( ρ 1 x − ρ 2 x ) − i k ρ 2 x z ] , D = A − B 2 A
E = i k ρ 1 y z − B ( ρ 1 y − ρ 2 y ) + B A [ B ( ρ 1 y − ρ 2 y ) − i k ρ 2 y z ] ,
F = 1 w 0 2 + i k 2 z + B − B 2 A , G = − i k ρ 2 x z + B ( ρ 1 x − ρ 2 x ) ,
I b i = b i ! exp ⁡ ( C 2 4 F ) π F ( C 2 F ) b i ∑ n i + 5 = 0 [ b i 2 ] ( F / C 2 ) n i + 5 ( b i − 2 n i + 5 ) ! n i + 5 ! ( i = 1 , 2 , b 1 = m 1 + n 5 + 1 ; b 2 = b 1 − 1 ) ,
I c i = c i ! exp ⁡ ( C 2 4 F ) π F ( C 2 F ) c i ∑ n i + 9 = 0 [ c i 2 ] ( F / C 2 ) n i + 9 ( c i − 2 n i + 9 ) ! n i + 9 ! ( i = 1 , 2 , c 1 = m 1 + n 9 + 1 ; c 2 = c 1 − 1 ) .
μ ( ρ 1 , ρ 2 , z ) = W ( ρ 1 , ρ 2 , z ) [ I ( ρ 1 , z ) I ( ρ 2 , z ) ] 1 / 2 ,
{ R e [ μ ( ρ 1 , ρ 2 , z ) ] = 0 Im [ μ ( ρ 1 , ρ 2 , z ) ] = 0 ,
l = 1 2 π ∮ Q ⁡ ∇ φ ( r ) d r ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use