Abstract

In this paper, we have derived the analytical formulae for the cross-spectral densities of partially coherent Gaussian vortex beams propagating in a gradient-index (GRIN) fiber. In numerical analysis, the variations of the intensity and the phase distributions are demonstrated to illustrate the change in singularities within a GRIN fiber. It turns out that the beam intensity and phase distribution change periodically in the propagation process. The partially coherent Gaussian vortex beams do not typically possess the center intensity zero in the focal plane, which usually called ‘hidden’ singularities in intensities detection. We demonstrated the phase singularities more clearly by the phase distribution, one finds that the phase vortex of a partially coherent beam will crack near the focus, and opposite topological charge will be generated, we attribute to the wave-front decomposition and reconstruction of the vortex beams by the GRIN fiber. Our results show that the change in phase singularities not only affected by the GRIN fiber, but also by the initial coherence of the beam source, and high initial coherence will be more conducive to maintaining the phase singularities in the propagation. Our results may find applications in singular optics, wave-front reconstruction and optical fiber communications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

As a prominent branch of modern optics, the research objects of singular optics including phase singularities, polarization singularities and coherent singularities [13]. As is well known, the phase plays a key role in the spatiotemporal evolution of an optical field [4]. Since Nye and Berry demonstrated that optical fields might contain wave-front dislocations (phase singularities) [5], where the phase is indeterminate and the wave amplitude is zero. There has been a growing interest in studying the fundamental properties of phase singularities (dislocations, defects, topological charges, orbital-angular-momentum, optical vortices, etc.) [612]. It is useful and meaningful to generate or design the phase singularities in optical fields. Over the past thirty years, phase singularities were generated by various methods [13], such as mode conversions [14], computer-generated holograms [15], Pancharatnam-Berry phase optical elements [16], inhomogeneous anisotropic media [17], microfabricated wedge with spatial filtering technique [18], spiral phase plate [19], nano-antennas [20], a binary array of pinholes [21] and so on.

Recently, light beams possessing vortex phase (known as vortex beams) have become the focus on many investigations because of their attractive for technical applications of optical manipulation, atom trapping, quantum information and entanglement, biological tissues, optical communications and so on [2231]. A prominent example of vortex beams is present in coherent laser beams, like the familiar Laguerre-Gauss (LG) modes, they are eigenmodes of the paraxial wave equation in polar coordinates system [9,32]. The LG modes have a separable phase with a simple helicoidal structure around the beam center, where the intensity is zero. On the other hand, for the partially coherent case. By using the coherent mode decomposition [33], a class of partially coherent beams possessing optical vortices can be represented as an incoherent superposition of coherent LG modes [34,35]. An alternative way of obtaining partially coherent vortex beams based on the LG modes is to assume that the cross-spectral density (CSD) function has the Gaussian Schell-model (GSM) form [36]. The partially coherent vortex beams do not typically possess regions of zero intensity and hence do not possess any obvious phase singularities [3,3436], the ‘hidden’ singularities are closely related to the beam coherence [37]. After the ‘hidden’ singularity was revealed, research on partially coherent vortex beams became a hotspot and have shown advantages over a fully coherent beam or a nonvortex beam in many applications [3841]. For example, the focus intensity distribution of the partially coherent GSM vortex beam can be modulated by its initial spatial coherence width from dark hollow to flat-topped, which is useful for trapping particles with different refractive index [38,39]. Compared with partially coherent nonvortex beams, partially coherent vortex beams can effectively overcome the scintillations and beam spreading caused by atmospheric turbulence [40]. The partially coherent vortex beam has strong self-reconstruction, which gives it the potential applications for information recovery and encryption [41].

To solve the limit of data traffic capacity caused by the nonlinear effects of optical fiber, a class of spatial orbital angular momentum modes multiplex in fiber communication is proposed and can enhance the data traffic capacity to terabit-scale [31]. Multimode optical fibers can transmit multiple modes at a given wavelength, according to its refractive index distribution it is divided into gradient-index (GRIN) type and step-index type [42]. The GRIN fiber with a continuous distribution, quadratic index dependence of the refraction index, and hold lower pulse dispersion than step-index fiber. With the self-focusing properties, the GRIN fiber has broad prospects for the application of the focusing and image formation, optical communication, optical sensing technology, and optical fiber manufacturing [4346]. For the propagation of the beams in GRIN medium, it is essentially a problem of the interaction between light fields and the particles of the medium. Recently, great interest has arisen in the study of beam propagation in a GRIN medium, different kinds of beams including the Gaussian beams, the partially coherent flat-topped beams, Airy-Gaussian vortex beams, the chirped Airy beams and Bessel beams have been investigated [4752]. To the best of our knowledge, there is little work on the propagation of a partially coherent Gaussian vortex beams in a GRIN fiber, which may show some interesting features especially in phase singularities.

Based on the generalized Huygens-Fresnel integral formula, this paper derived analytical expressions of the partially coherent Gaussian vortex beams propagating in a GRIN fiber, such fiber can be characterized by a paraxial ABCD system. The changes in phase singularities of a partially coherent Gaussian vortex beam within a GRIN fiber have been investigated in detail.

2. Theoretical model

For configurations with cylindrical symmetry the LG mode is the natural choice [32]. Let us begin by recalling that the electric field of a standard LG beam in the source plane, $z = 0$, is expressed as follows [9,3236]

$$E({{\textbf s},\;0} )= {\left( {\frac{{\sqrt 2 \;s}}{{{w_0}}}} \right)^m}L_n^m\left( {\frac{{2{s^2}}}{{w_0^2}}} \right)\exp \left( { - \frac{{{s^2}}}{{w_0^2}}} \right)\exp ({im\theta } )$$
where ${\textbf s}(s,\theta )$ is a position vector, s and $\theta $ are the radial and azimuthal coordinates. It is clearly seen, the separable phase is specified by term exp(imθ), by this term we mean the field whose wave front is endowed with vortex structure. m is the azimuthal mode index (also called the topological charge), and n is the order of the Laguerre polynomial $L_n^m$. For $m \ne 0$ and $n = 0$, Eq. (1) degenerates to the electric field of a Gaussian vortex beam [3841]; For $m = 0$ and $n = 0$, Eq. (1) degenerates to the electric field of a fundamental Gaussian beam [53], ${w_0}$ is the spot size at the waist of the fundamental Gaussian mode.

On the other hand, when rectangular geometry is more appropriate, the proper description is in terms of Hermite-Gaussian (HG) modes [53]. By using the relations between a Laguerre polynomial and pairs of Hermite polynomials [36,54],

$$\exp ({im\theta } ){s^m}L_n^m({{s^2}} )= \frac{{{{({ - 1} )}^n}}}{{{2^{2n + m}}n\;!}}\sum\limits_{t = 0}^n {\sum\limits_{r = 0}^m {{i^r}} } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ r \end{array}} \right){H_{2t + m - r}}({s_x}){H_{2n - 2t + r}}({s_y})$$
we can expand a LG mode when it has to be integrated over Cartesian coordinates. ${H_.}(.)$ being the Hermite polynomial, $\left( {\begin{array}{{c}} .\\ . \end{array}} \right)$ being binomial coefficients. Equation (1) can be expressed in following alternative form in Cartesian coordinates ${\textbf s}({s_x},{s_y})$
$$E({{\textbf s},\;0} )= \frac{{{{({ - 1} )}^n}}}{{{2^{2n + m}}n\;!}}\sum\limits_{t = 0}^n {\sum\limits_{r = 0}^m {{i^r}} } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ r \end{array}} \right){H_{2t + m - r}}\left( {\frac{{\sqrt 2 {s_x}}}{{{w_0}}}} \right){H_{2n - 2t + r}}\left( {\frac{{\sqrt 2 {s_y}}}{{{w_0}}}} \right)\exp \left( { - \frac{{{{\textbf s}^2}}}{{w_0^2}}} \right)$$
Although we cannot directly observe the separable phase exp(imθ) in HG modes, the wave front of the field still holding a vortex structure as like the LG mode, and two or three HG modes are sufficient for the expansion of a LG mode of low or moderate order [55].

We now extend the standard LG beam to a partially coherent case. On the basis of second-order coherence theory in the space-frequency domain, the second-order statistical properties of a partially coherent beam are generally characterized by the CSD function ${W_0}({{\textbf s}_1},{{\textbf s}_2},0)\textrm{ = < }{E^{\ast }}({{{\textbf s}_1},\;0} )E({{{\textbf s}_2},\;0} )\textrm{ > }$, where the asterisk denotes complex conjugation and the angular brackets indicate ensemble average [4]. For a partially coherent standard LG beam generated by a fully coherent standard LG beam (at $z = 0$), the CSD function can be expressed in the following form [30,36]

$$\begin{array}{l} {W_0}({{{\textbf s}_1},{{\textbf s}_2},0} )= \frac{1}{{{2^{4n + 2m}}{{({n\;!} )}^2}}}\sum\limits_{{t_1} = 0}^n {\sum\limits_{{r_1} = 0}^m {\sum\limits_{{t_2} = 0}^n {\sum\limits_{{r_2} = 0}^m {{i^{{r_2}\textrm{ - }{r_1}}}\left( {\begin{array}{{c}} n\\ {{t_1}} \end{array}} \right)\left( {\begin{array}{{c}} m\\ {{r_1}} \end{array}} \right)\left( {\begin{array}{{c}} n\\ {{t_2}} \end{array}} \right)\left( {\begin{array}{{c}} m\\ {{r_2}} \end{array}} \right)} } } } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times {H_{2{t_1} + m - {r_1}}}\left( {\frac{{\sqrt 2 {s_{1x}}}}{{{w_0}}}} \right){H_{2n - 2{t_1} + {r_1}}}\left( {\frac{{\sqrt 2 {s_{1y}}}}{{{w_0}}}} \right){H_{2{t_2} + m - {r_2}}}\left( {\frac{{\sqrt 2 {s_{2x}}}}{{{w_0}}}} \right){H_{2n - 2{t_2} + {r_2}}}\left( {\frac{{\sqrt 2 {s_{2y}}}}{{{w_0}}}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \textrm{exp}\left( { - \frac{{{\textbf s}_1^2 + {\textbf s}_2^2}}{{w_0^2}}} \right)\textrm{exp}\left[ { - \frac{{{{({{{\textbf s}_1} - {{\textbf s}_2}} )}^2}}}{{2\sigma_0^2}}} \right] \end{array}$$
where ${\sigma _0}$ is the transverse coherence width. In Eq. (4), the intensity part is based on the standard LG mode while the degree of coherence (DOC) has Gaussian profile [4], the phase structure of the partially coherent fields is similar to the fully coherent LG beam. Under the condition of ${\sigma _0} \to \infty $, Eq. (4) reduces to the expression of a fully coherent standard LG beam. Under the condition of $m = 0$ and $n = 0$, Eq. (4) reduces to the classical partially coherent Gaussian Schell-model beam. Under the condition of $m \ne 0$ and $n = 0$, Eq. (4) reduces to a partially coherent Gaussian vortex beam.

In the following, let us concentrate our attention on the paraxial propagation of such beam through a stigmatic ABCD optical system, the CSD function can be evaluated by the generalized Collins formula [4851]

$$\begin{array}{l} W({{{{\boldsymbol {\rho}} }_1},{{{\boldsymbol {\rho}} }_2},z} )= {\left( {\frac{k}{{2\pi B}}} \right)^2}\int {\int {\int {\int {\textrm{d}{s_{1x}}\textrm{d}{s_{1y}}\textrm{d}{s_{2x}}\textrm{d}{s_{2y}}{W_0}({{{\textbf s}_1},{{\textbf s}_2},0} )} } } } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left\{ { - \frac{{ik}}{{2B}}[{A({{\textbf s}_1^2 - {\textbf s}_2^2} )- 2({{{\textbf s}_1} \cdot {{{\boldsymbol {\rho}} }_1} - {{\textbf s}_2} \cdot {{{\boldsymbol {\rho}} }_2}} )+ D({{{\boldsymbol {\rho}} }_1^2 - {{\boldsymbol {\rho}} }_2^2} )} ]} \right\} \end{array}$$
where ${{{\boldsymbol {\rho}} }_1}\textrm{ = (}{x_1}\textrm{,}{y_1}\textrm{),}\;{{{\boldsymbol {\rho}} }_2}\textrm{ = (}{x_2}\textrm{,}{y_2}\textrm{)}$ are the position vectors in the receiver plane, and k denotes the wave number with $\lambda $ being the wavelength. A, B, C, and D are elements of the transfer matrix of the optical system. On substituting from Eq. (4) into Eq. (5), we obtain the CSD function in the receiving plane
$$\begin{array}{l} W({{{{\boldsymbol {\rho}} }_1},{{{\boldsymbol {\rho}} }_2},z} )= {\left( {\frac{k}{{2\pi B}}} \right)^2}\frac{1}{{{2^{4n + 2m}}{{({n\;!} )}^2}}}\frac{{{\pi ^2}}}{{{M_1}{M_2}}}{\left( {\frac{{1 - {G^2}}}{2}} \right)^{\frac{{2n + m}}{2}}}\exp \left[ { - \frac{{ikD}}{{2B}}({{{\boldsymbol {\rho}} }_1^2 - {{\boldsymbol {\rho}} }_2^2} )} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left[ { - {{\left( {\frac{{k{{{\boldsymbol {\rho}} }_1}}}{{2B\sqrt {{M_2}} }} - \frac{{k{{{\boldsymbol {\rho}} }_2}}}{{4{M_1}\sqrt {{M_2}} B\sigma_0^2}}} \right)}^2} - \frac{{{k^2}{{\boldsymbol {\rho}} }_2^2}}{{4{M_1}{B^2}}}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \sum\limits_{{t_1} = 0}^n {\sum\limits_{{r_1} = 0}^m {\sum\limits_{{t_2} = 0}^n {\sum\limits_{{r_2} = 0}^m {\left( {\begin{array}{{@{}c@{}}} n\\ {{t_1}} \end{array}} \right)\left( {\begin{array}{{@{}c@{}}} m\\ {{r_1}} \end{array}} \right)\left( {\begin{array}{{@{}c@{}}} n\\ {{t_2}} \end{array}} \right)\left( {\begin{array}{{@{}c@{}}} m\\ {{r_2}} \end{array}} \right)} } } } \sum\limits_{{c_1} = 0}^{\left[ {\frac{{2{t_1} + m - {r_1}}}{2}} \right]} {\sum\limits_{{d_1} = 0}^{2{t_2} + m - {r_2}} {\sum\limits_{{e_1} = 0}^{\left[ {\frac{{{d_1}}}{2}} \right]} {\sum\limits_{{c_2} = 0}^{\left[ {\frac{{2n - 2{t_1} + {r_1}}}{2}} \right]} {\sum\limits_{{d_2} = 0}^{2n - 2{t_2} + {r_2}} {\sum\limits_{{e_2} = 0}^{\left[ {\frac{{{d_2}}}{2}} \right]} {{i^{{r_2}\textrm{ - }{r_1}}}} } } } } } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times {( - 1)^{{c_1} + {c_2} + {e_1} + {e_2}}}\left( {\begin{array}{{@{}c@{}}} {2{t_2} + m - {r_2}}\\ {{d_1}} \end{array}} \right)\left( {\begin{array}{{@{}c@{}}} {2n - 2{t_2} + {r_2}}\\ {{d_2}} \end{array}} \right)\frac{{{d_1}\;!{d_2}\;!}}{{{e_1}\;!{e_2}\;!({{d_1} - 2{e_1}} )\;!({{d_2} - 2{e_2}} )\;!}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \frac{{({2{t_1} + m - {r_1}} )\;!}}{{{c_1}\;!({2{t_1} + m - {r_1} - 2{c_1}} )\;!}}\frac{{({2n - 2{t_1} + {r_1}} )\;!}}{{{c_2}\;!({2n - 2{t_1} + {r_1} - 2{c_2}} )\;!}}{\left( {\frac{{2\sqrt 2 }}{{{w_0}}}} \right)^{2n + m - 2{c_1} - 2{c_2}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times {\left[ {\frac{{\sqrt 2 G}}{{\sigma_0^2\sqrt {{M_1}({1 - {G^2}} )} }}} \right]^{{d_1} + {d_2} - 2{e_1} - 2{e_2}}}{\left( {\frac{1}{{2i\sqrt {{M_2}} }}} \right)^{2n + m - 2{c_1} - 2{c_2} + {d_1} + {d_2} - 2{e_1} - 2{e_2}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \times {H_{2{t_2} + m - {r_2} - {d_1}}}\left[ { - \frac{{ik\sqrt 2 G{x_2}}}{{2B\sqrt {{M_1}(1 - {G^2})} }}} \right]{H_{2{t_1} + m - {r_1} - 2{c_1} + {d_1} - 2{e_1}}}\left( { - \frac{{k{x_1}}}{{2\sqrt {{M_2}} B}} + \frac{{k{x_2}}}{{4{M_1}\sqrt {{M_2}} B\sigma_0^2}}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \times {H_{2n - 2{t_2} + {r_2} - {d_2}}}\left[ { - \frac{{ik\sqrt 2 G{y_2}}}{{2B\sqrt {{M_1}(1 - {G^2})} }}} \right]{H_{2n - 2{t_1} + {r_1} - 2{c_2} + {d_2} - 2{e_2}}}\left( { - \frac{{k{y_1}}}{{2\sqrt {{M_2}} B}} + \frac{{k{y_2}}}{{4{M_1}\sqrt {{M_2}} B\sigma_0^2}}} \right) \end{array}$$
with ${M_1} = \frac{1}{{w_0^2}} + \frac{1}{{2\sigma _0^2}} - \frac{{ikA}}{{2B}},{M_2} = \frac{1}{{w_0^2}} + \frac{1}{{2\sigma _0^2}} + \frac{{ikA}}{{2B}} - \frac{1}{{4{M_1}\sigma _0^4}},G = \frac{{\sqrt 2 }}{{{w_0}\sqrt {{M_1}} }}.$ In above derivations, we have used the following integral and expansion formulae [56,57]
$$\int {\exp [{ - {{({x - y} )}^2}} ]} \;{H_n}({ax} )\textrm{d}x = \sqrt \pi {({1 - {a^2}} )^{n/2}}{H_n}\left[ {\frac{{ay}}{{{{({1 - {a^2}} )}^{1/2}}}}} \right]$$
$$\int {{x^n}\exp [{ - {{(x - \beta )}^2}} ]} \;\textrm{d}x = {(2i)^{ - n}}\sqrt \pi {H_n}({i\beta } )\;$$
$${H_n}({x + y} )= \frac{1}{{{2^{n/2}}}}\sum\limits_{k = 0}^n {\left( {\begin{array}{{c}} n\\ k \end{array}} \right)} {H_k}\left( {\sqrt 2 x} \right){H_{n - k}}\left( {\sqrt 2 y} \right)\;$$
$${H_n}(x )= \sum\limits_{m = 0}^{[{n/2} ]} {{{( - 1)}^m}\frac{{n\;!}}{{m\;!({n - 2m} )\;!}}{{(2x)}^{n - 2m}}} \;$$
Equation (6) is the main result in this paper, the phase information is characterized by the superposition of Hermite polynomial which cannot be directly observed. Equation (6) can be used for evaluating the propagation characteristics of any order beams through any ABCD optical systems. In Sec. 3, we will carefully study the changes in phase singularities of a partially coherent Gaussian vortex beam propagating in a GRIN fiber. The spectral intensity at point ${{\boldsymbol {\rho}} }$ and the degree of coherence (DOC) in the output plane are defined as follows [4,8]
$$I({{{\boldsymbol {\rho}} },z} )= W({{{\boldsymbol {\rho}} },{{\boldsymbol {\rho}} },z} )$$
$$\mu ({{{{\boldsymbol {\rho}} }_1},{{{\boldsymbol {\rho}} }_2},z} )= \frac{{W({{{{\boldsymbol {\rho}} }_1},{{{\boldsymbol {\rho}} }_2},z} )}}{{{{[{I({{{{\boldsymbol {\rho}} }_1},z} )I({{{{\boldsymbol {\rho}} }_2},z} )} ]}^{1/2}}}}\;\;$$
and the position of phase singularities is determined by
$${\mathop{\rm Re}\nolimits} [{\mu ({{{{\boldsymbol {\rho}} }_1},{{{\boldsymbol {\rho}} }_2},z} )} ]= 0\;$$
$${\mathop{\rm Im}\nolimits} [{\mu ({{{{\boldsymbol {\rho}} }_1},{{{\boldsymbol {\rho}} }_2},z} )} ]= 0\;$$
where Re and Im stand for the real and imaginary parts of $\mu ({{{{\boldsymbol {\rho}} }_1},{{{\boldsymbol {\rho}} }_2},z} )$, respectively. The topological charge and its sign of vortices are determined by the sign principle [8], the phase vortex anticlockwise increase corresponds to the sign of ‘+’, otherwise it is ‘−’.

3. Change in singularities of a partially coherent Gaussian vortex beam propagating in a GRIN fiber

In this section, our primary purpose is to characterize the change in singularities of a partially coherent vortex beam in a GRIN fiber. Let us consider a GRIN fiber with rotational symmetry around the z-axis, whose spatial dependence of the square of the index of refraction has a parabolic profile [4351]

$${n^2} = \left\{ {\begin{array}{{c}} {n_0^2({1 - {\beta^2}{\rho^2}} ),\;\;\;\;\;\;{\rho^2} \le R_0^2}\\ {n_0^2({1 - {\beta^2}R_0^2} ),\;\;\;\;\;\;{\rho^2} \ge R_0^2} \end{array}} \right.\;$$
Here ${R_\textrm{0}}$ is the core radius, ${n_\textrm{0}}$ is the refractive index at the center of the fiber, $\beta $ is the radial gradient of the refractive index. If ${n_\textrm{1}}$ is the refractive index at the boundary of the fiber, the radial gradient of the refractive index is given by the expression
$$\beta = \frac{1}{{{R_0}}}{\left[ {1 - \frac{{n_1^2}}{{n_0^2}}} \right]^{1/2}}\;$$
The ABCD matrix for paraxial ray propagation through a GRIN fiber is given by [4851]
$${\left[ {\begin{array}{{cc}} A&B\\ C&D \end{array}} \right]_z} = \left[ {\begin{array}{{cc}} {\cos ({\beta z} )}&{\frac{{\sin ({\beta z} )}}{{{n_0}\beta }}}\\ { - {n_0}\beta \sin ({\beta z} )}&{\cos ({\beta z} )} \end{array}} \right]\;$$
It is easy to find Eq. (13) has periodicity ${\left[ {\begin{array}{{cc}} A&B\\ C&D \end{array}} \right]_z} = {\left[ {\begin{array}{{cc}} A&B\\ C&D \end{array}} \right]_{z + aL}}$ with $a\textrm{ = }1,2,3\ldots $ is a positive integer and $L\textrm{ = 2}\pi \textrm{/}\beta$. As a typical GRIN fiber whose cladding made of pure SiO2 and core center made of silica glass doped with 7.9% GeO2, the refractive index at the boundary/center of the fiber is given by the Sellmeier formula [52]
$${n^2}(\omega )= 1 + \sum\limits_{j = 1}^3 {\frac{{{B_j}\omega _j^2}}{{\omega _j^2 - {\omega ^2}}}} \;$$
where ωj=2πc/λj. For the boundary refractive index (pure SiO2): B1=0.6961663, B2=0.4079426, B3=0.8974794, λ1=0.684043µm, λ2=0.1162414µm, λ3=9.896161µm. For the center refractive index (doped with 7.9% GeO2), the parameters are set as: B1=0.7136824, B2=0.4254807, B3=0.896422, λ1=0.0617167µm, λ2=0.1270814µm, λ3=9.896161µm. As an example, we provide an intuitive numerical analysis of a partially coherent Gaussian vortex beam propagating through such GRIN fiber. The parameters used in following calculations are set as: R0=25µm, λ=632.8 nm, n = 0, w0=100/k, σ0=100/k, (unless stated otherwise). By using Eqs. (16) and (18), the parameters of the GRIN fiber are given as n0=1.46977, n1=1.45702, and β=5.25798mm-1.

We calculate in Fig. 1 the normalized intensity distribution I/I0max of the partially coherent Gaussian vortex beams for different topological charge (m = 1,2,3,5,8) along the y-z plane, within a GRIN fiber. Figure 1 clearly shows the self-focusing properties of the partially coherent vortex beams propagating through the GRIN fiber, as expected [4352]. The intensity distribution focus and diverge periodically in the propagation process, and the periodical distance is L = 0.597mm. In Figs. 1(a)–1(d), the partially coherent Gaussian vortex beams do not typically possess dark hollow in the beam center, and the phase singularities seem to be ‘hidden’ [37]. For the case of m = 1, the intensity distribution is flat-topped in the focal plane (z = 0.5aL, a = 1,2,3…). As m increases, the intensity of the beam center in the focal plane gradually decreases. Figure 1(e) indicates that when topological charge m is large enough, the intensity on the axis is always zero during the propagation. One finds that the ‘hidden’ singularity is closely related to the topological charge, it cannot be directly observed in the intensity distribution unless topological charge is large enough. Moreover, the beam transverse width increases with m increases, which means that a larger diameter fiber core is required to transmit a high order vortex beam.

 figure: Fig. 1.

Fig. 1. Normalized intensity distribution I/I0max for different topological charge m along the y-z plane, within a GRIN fiber with a parabolic refractive-index.

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The effect of initial coherence on intensity distribution is also be considered. Figure 2 illustrates the normalized intensity distribution I/I0max of a partially coherent Gaussian vortex beams with m = 2 for different initial coherence width (${\sigma _0} = 50/k,\;100/k,\;200/k,\;500/k$), within a GRIN fiber. One finds that the partially coherent Gaussian vortex beams with different initial coherence width have the same beam transverse width at propagation distance z = aL. Compare with Fig. 1, increasing initial coherence width has the similar effect as m increase. In Fig. 2(a), with the low coherence, the intensity of the beam center is flat-topped in the focal plane. In Figs. 2(b)–2(d), as initial coherence width increases the intensity of the beam center gradually evolved into zero in the focal plane. It means that the ‘hidden’ singularity can be directly observed with control both topological charge and initial coherence width within a GRIN fiber.

 figure: Fig. 2.

Fig. 2. Normalized intensity distribution I/I0max for different coherence width σ0 along the y-z plane, within a GRIN fiber with a parabolic refractive-index.

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The phase singularities can not only be observed by the points of intensity zero, it can be more clearly detected by the phase distribution [3741]. To learn more about the phase singularities, we calculate in Fig. 3 the phase distribution and the positions of phase singularities (Re(μ) = 0 and Im(μ) = 0) at different propagation distance within a GRIN fiber, where the black spots mean the topological charge of ‘+1’ while white spots mean ‘−1’, respectively. The parameters are set in accordance with Figs. 1(a)–1(c). It is easy to find that the phase distribution also focusing and diverging periodically in the propagation process within a GRIN fiber. For the case m = 1, from Figs. 3(a1)–3(a10) we see that a phase vortex ‘+1’ always staying around the optical axis, and a new relative topological charge ‘-1’ will be generated near the focus. Compare with Fig. 1(a), one finds that because of the mutual influence of the opposite topological charge, the beam center intensity in the focal plan will not be zero, which we called ‘hidden’ singularity in the intensity detection. For the case of $m \ne 1$, from Figs. 3(b) and 3(c) we see that the topological charge m is cracked into m independent phase vortex ‘+1’ around the beam center near by the focus, and relative m topological charge ‘-1’ will be generated. This is quite interesting compared to the lens focusing properties [38,39], and we attribute to the wave-front decomposition and reconstruction of the light field modes by the GRIN fiber [53,58]. The effect of coherence on phase distribution is also be considered in Fig. 4 for the low, high and fully coherence with m = 2. Figures 4(a1)–4(a10) display the low coherence case, we can obviously see the phase vortex splitting into two ‘+1’, and a pair of topologically charge ‘−1’ rotate around the beam center like satellites during transmission. From Figs. 4(b)–4(d) we see that as the coherence increases, the wave-front splitting becomes less obvious. For the case of fully coherence, the GRIN fiber does not perform phase vortex splitting of the wave-front, the phase singularities can reconstruction in the focal plane (z = 0.5L), and the phase distribution is exactly the same as the source field. This can be explained by means of the Wolf’s modal decomposition theory of coherence [3335], which has indicated that a partially coherent beam possessing optical vortices can be represented as an incoherent superposition of coherent modes. For the case of a fully coherence beam, Eq. (6) degenerates to HG modes, and it is well known that the modes of a parabolic-index medium (GRIN fiber) are HG functions [53], which allows a fully coherence beam to propagating well within a GRIN fiber. On the other hand, for the case of partially coherent beam within a GRIN fiber, the change in phase singularities not only affected by the low coherence of the beam source, but also by the GRIN fiber, the phase singularities propagation process is more complicated.

 figure: Fig. 3.

Fig. 3. The phase distribution and the positions of phase singularities (Re(μ) = 0and Im(μ) = 0) for different m within a GRIN fiber, where black spots denote the topological charge ‘+1’ and white spots denote ‘−1’ of the optical vortices.

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 figure: Fig. 4.

Fig. 4. The phase distribution and the positions of phase singularities (Re(μ) = 0and Im(μ) = 0) for different ${\sigma _0}$ within a GRIN fiber, where black spots denote the topological charge ‘+1’ and white spots denote ‘−1’ of the optical vortices.

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Figure 5 gives curves of Re(μ) = 0 (red solid curves) and Im(μ) = 0 (black dashed curves) and the normalized intensity distribution in the source plane and focal plane. The zeros of the real and imaginary parts of the degree of coherence were computed numerically for fixed ρ1=(0, 0) and a variety of values of ρ2=(x, y). The parameters are set in accordance with Figs. 1(a)–1(d). In Figs. 5(a1)–5(d1), for all topological charge m, the curves of Re(μ) = 0 and Im(μ) = 0 intersect at the beam center, i.e., the position of the singularities at the source plane are (0,0). From Figs. 5(a2)–5(d2) we see that the partially coherent Gaussian vortex beams do not typically possess the center intensity zero in the focal plane, which are usually called as “hidden” singularities along the intensity detection [37]. However, we can demonstrate the singularities by the curves of Re(μ) = 0 and Im(μ) = 0, the number of curves intersections is 2m which determines the value of the topological charge is m, and the position of the curves intersections can be used to determine the position of the singularities.

 figure: Fig. 5.

Fig. 5. The normalized intensity of cross section and the positions of phase singularities for different m within a GRIN fiber, where red solid curves denote Re(μ) = 0 and black dashed curves denote Im(μ) = 0.

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4. Summary

In this paper, based on the extended Huygens-Fresnel principle, we have derived the analytical formulae for the cross-spectral densities of partially coherent Gaussian vortex beams propagating through ABCD optical systems, and used to study the change in singularities of such vortex beams within a GRIN fiber. For numerical analysis, the variations of the intensity propagation trajectories and the phase distributions are demonstrated to illustrate the change in singularities. It turns out that the beam intensity and phase distribution focus and diverge periodically in the propagation process, and the periodical distance is L. The partially coherent Gaussian vortex beams do not typically possess the center intensity zero in the focal plane, and the phase singularities seem to be hidden for the case of small topological charge or low coherence. However, the ‘hidden’ singularities can be directly observed by controlling both topological charge and initial coherence width. We demonstrated the phase singularities more clearly by the phase distribution, one finds that the phase vortex will crack near the focus, and opposite topological charge will be generated, we attribute to the wave-front decomposition and reconstruction of the vortex beams by the GRIN fiber. The effect of coherence on phase distribution shows that the change in phase singularities not only affected by the GRIN fiber, but also by the initial coherence of the beam source. As the initial coherence increases, the phase singularities remain more stable within a GRIN fiber. The results obtained in this work will deepen the understanding of the change in phase singularities within a GRIN fiber, and can be employed in singular optics, wave-front reconstruction and optical fiber communications.

Funding

National Natural Science Foundation of China (11904253); The Outstanding Young Scholars of Shanxi Province (201801D211006); The Fund for Shanxi “1331 Project” Key Innovative Research Team (1331KIRT); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2019L0646); Taiyuan University of Science and Technology Scientific Research Initial Funding (20192004).

Disclosures

The authors declare no conflicts of interest.

References

1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001). [CrossRef]  

2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009). [CrossRef]  

3. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009). [CrossRef]  

4. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, 2013).

5. J. Nye and M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336(1605), 165–190 (1974). [CrossRef]  

6. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992). [CrossRef]  

7. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993). [CrossRef]  

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

9. L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]  

10. K. M. Iftekharuddin and M. A. Karim, “Heterodyne detection by using diffraction-free beam: tilt and offset effects,” Appl. Opt. 31(23), 4853–4856 (1992). [CrossRef]  

11. S. Klewitz, F. Brinkmann, S. Herminghaus, and P. Leiderer, “Bessel-beam-pumped tunable distribution-feedback laser,” Appl. Opt. 34(33), 7670–7673 (1995). [CrossRef]  

12. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]  

13. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]  

14. M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]  

15. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef]  

16. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27(21), 1875–1877 (2002). [CrossRef]  

17. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef]  

18. X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007). [CrossRef]  

19. V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008). [CrossRef]  

20. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]  

21. Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019). [CrossRef]  

22. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]  

23. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]  

24. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef]  

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

26. M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014). [CrossRef]  

27. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef]  

28. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011). [CrossRef]  

29. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015). [CrossRef]  

30. J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017). [CrossRef]  

31. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef]  

32. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

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

34. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998). [CrossRef]  

35. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001). [CrossRef]  

36. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef]  

37. G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004). [CrossRef]  

38. F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011). [CrossRef]  

39. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef]  

40. J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11(4), 045710 (2009). [CrossRef]  

41. X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017). [CrossRef]  

42. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000). [CrossRef]  

43. S. A. Ponomarenko, “Self-imaging of partially coherent light in graded-index media,” Opt. Lett. 40(4), 566–568 (2015). [CrossRef]  

44. S. H. Song, S. Park, C. H. Oh, P. S. Kim, M. H. Cho, and Y. S. Kim, “Gradient-index planar optics for optical interconnections,” Opt. Lett. 23(13), 1025–1027 (1998). [CrossRef]  

45. J. R. Leger and W. M. Kunkel, “Gradient-index design for mode conversion of diffracting beams,” Opt. Express 24(12), 13480–13488 (2016). [CrossRef]  

46. C. Gomez-Reino, M.V. Perez, and C. Bao, Gradient-index optics: fundamentals and applications (Springer Science & Business Media, 2002).

47. J. Alda and G. D. Boreman, “On-axis and off-axis propagation of Gaussian beams in gradient index media,” Appl. Opt. 29(19), 2944–2950 (1990). [CrossRef]  

48. V. Arrizon, F. Soto-Eguibar, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. Soc. Am. A 32(6), 1140 (2015). [CrossRef]  

49. R. Zhao, F. Deng, W. Yu, J. Huang, and D. Deng, “Propagation properties of Airy–Gaussian vortex beams through the gradient-index medium,” J. Opt. Soc. Am. A 33(6), 1025–1031 (2016). [CrossRef]  

50. L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017). [CrossRef]  

51. Z. Cao, C. Zhai, S. Xu, Y. Chen, and Y, “Propagation of on-axis and off-axis Bessel beams in a gradient-index medium,” J. Opt. Soc. Am. A 35(2), 230–235 (2018). [CrossRef]  

52. H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23(4), 940–948 (2006). [CrossRef]  

53. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).

54. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]  

55. J. Vickers, M. Burch, R. Vyas, and S. Singh, “Phase and interference properties of optical vortex beams,” J. Opt. Soc. Am. A 25(3), 823–827 (2008). [CrossRef]  

56. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

57. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

58. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012). [CrossRef]  

References

  • View by:

  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
    [Crossref]
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
    [Crossref]
  3. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
    [Crossref]
  4. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, 2013).
  5. J. Nye and M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336(1605), 165–190 (1974).
    [Crossref]
  6. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
    [Crossref]
  7. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
    [Crossref]
  8. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
    [Crossref]
  9. L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  10. K. M. Iftekharuddin and M. A. Karim, “Heterodyne detection by using diffraction-free beam: tilt and offset effects,” Appl. Opt. 31(23), 4853–4856 (1992).
    [Crossref]
  11. S. Klewitz, F. Brinkmann, S. Herminghaus, and P. Leiderer, “Bessel-beam-pumped tunable distribution-feedback laser,” Appl. Opt. 34(33), 7670–7673 (1995).
    [Crossref]
  12. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
    [Crossref]
  13. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
    [Crossref]
  14. M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
    [Crossref]
  15. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [Crossref]
  16. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27(21), 1875–1877 (2002).
    [Crossref]
  17. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
    [Crossref]
  18. X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
    [Crossref]
  19. V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008).
    [Crossref]
  20. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
    [Crossref]
  21. Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
    [Crossref]
  22. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [Crossref]
  23. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
    [Crossref]
  24. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
    [Crossref]
  25. J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016).
    [Crossref]
  26. M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
    [Crossref]
  27. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref]
  28. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
    [Crossref]
  29. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015).
    [Crossref]
  30. J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017).
    [Crossref]
  31. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
    [Crossref]
  32. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.
  33. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  34. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
    [Crossref]
  35. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
    [Crossref]
  36. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
    [Crossref]
  37. G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004).
    [Crossref]
  38. F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
    [Crossref]
  39. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
    [Crossref]
  40. J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11(4), 045710 (2009).
    [Crossref]
  41. X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
    [Crossref]
  42. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
    [Crossref]
  43. S. A. Ponomarenko, “Self-imaging of partially coherent light in graded-index media,” Opt. Lett. 40(4), 566–568 (2015).
    [Crossref]
  44. S. H. Song, S. Park, C. H. Oh, P. S. Kim, M. H. Cho, and Y. S. Kim, “Gradient-index planar optics for optical interconnections,” Opt. Lett. 23(13), 1025–1027 (1998).
    [Crossref]
  45. J. R. Leger and W. M. Kunkel, “Gradient-index design for mode conversion of diffracting beams,” Opt. Express 24(12), 13480–13488 (2016).
    [Crossref]
  46. C. Gomez-Reino, M.V. Perez, and C. Bao, Gradient-index optics: fundamentals and applications (Springer Science & Business Media, 2002).
  47. J. Alda and G. D. Boreman, “On-axis and off-axis propagation of Gaussian beams in gradient index media,” Appl. Opt. 29(19), 2944–2950 (1990).
    [Crossref]
  48. V. Arrizon, F. Soto-Eguibar, A. Zuñiga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. Soc. Am. A 32(6), 1140 (2015).
    [Crossref]
  49. R. Zhao, F. Deng, W. Yu, J. Huang, and D. Deng, “Propagation properties of Airy–Gaussian vortex beams through the gradient-index medium,” J. Opt. Soc. Am. A 33(6), 1025–1031 (2016).
    [Crossref]
  50. L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
    [Crossref]
  51. Z. Cao, C. Zhai, S. Xu, Y. Chen, and Y, “Propagation of on-axis and off-axis Bessel beams in a gradient-index medium,” J. Opt. Soc. Am. A 35(2), 230–235 (2018).
    [Crossref]
  52. H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23(4), 940–948 (2006).
    [Crossref]
  53. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).
  54. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
    [Crossref]
  55. J. Vickers, M. Burch, R. Vyas, and S. Singh, “Phase and interference properties of optical vortex beams,” J. Opt. Soc. Am. A 25(3), 823–827 (2008).
    [Crossref]
  56. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  57. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
  58. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
    [Crossref]

2019 (2)

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

2018 (1)

2017 (3)

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

2016 (3)

2015 (3)

2014 (1)

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

2013 (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

2012 (1)

2011 (4)

F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36(16), 3281–3283 (2011).
[Crossref]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[Crossref]

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

2009 (4)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
[Crossref]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[Crossref]

2008 (2)

2007 (2)

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

2006 (2)

2004 (2)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

2002 (1)

2001 (3)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
[Crossref]

2000 (1)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref]

1998 (2)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

S. H. Song, S. Park, C. H. Oh, P. S. Kim, M. H. Cho, and Y. S. Kim, “Gradient-index planar optics for optical interconnections,” Opt. Lett. 23(13), 1025–1027 (1998).
[Crossref]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

1995 (1)

1994 (1)

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

1993 (3)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
[Crossref]

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

1992 (4)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

K. M. Iftekharuddin and M. A. Karim, “Heterodyne detection by using diffraction-free beam: tilt and offset effects,” Appl. Opt. 31(23), 4853–4856 (1992).
[Crossref]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

1990 (1)

1974 (1)

J. Nye and M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336(1605), 165–190 (1974).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Agrawal, A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Agrawal, G. P.

Ahluwalia, B.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Alda, J.

Allen, L.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Anderson, I. M.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Arrizon, V.

Bao, C.

C. Gomez-Reino, M.V. Perez, and C. Bao, Gradient-index optics: fundamentals and applications (Springer Science & Business Media, 2002).

Barnett, S.

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

Bejersbergen, M. W.

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Berry, M.

J. Nye and M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336(1605), 165–190 (1974).
[Crossref]

Biener, G.

Boreman, G. D.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, 2013).

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

Brinkmann, F.

Bu, J.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Buchleitner, A.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015).
[Crossref]

Burch, M.

Burge, R.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Cai, Y.

Cao, Z.

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Chen, H.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Chen, Q.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Chen, Y.

Cheng, K.

Cho, M. H.

Courtial, J.

Deng, D.

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

R. Zhao, F. Deng, W. Yu, J. Huang, and D. Deng, “Propagation properties of Airy–Gaussian vortex beams through the gradient-index medium,” J. Opt. Soc. Am. A 33(6), 1025–1031 (2016).
[Crossref]

Deng, F.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Duan, M.

Duparré, M.

Elias, L. R.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Feng, L.

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

Flamm, D.

Forbes, A.

Franke-Arnold, S.

Freund, I.

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

Fu, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Gao, P.

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004).
[Crossref]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Gibson, G.

Gomez-Reino, C.

C. Gomez-Reino, M.V. Perez, and C. Bao, Gradient-index optics: fundamentals and applications (Springer Science & Business Media, 2002).

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Hasman, E.

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref]

Herminghaus, S.

Herzing, A. A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Hua, L.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

Huang, J.

Iftekharuddin, K. M.

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
[Crossref]

Karim, M. A.

Kats, M. A.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Kim, P. S.

Kim, Y. S.

Kimel, I.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Kleiner, V.

Klewitz, S.

Korotkova, O.

Kotlyar, V. V.

Kovalev, A. A.

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

Kunkel, W. M.

Leger, J. R.

Leiderer, P.

Leonhard, N. D.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015).
[Crossref]

Lezec, H. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Li, J.

J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017).
[Crossref]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

Lin, J.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Liu, L.

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Liu, Q.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Liu, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Liu, Y.

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

Lü, B.

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

Luo, M.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Mandel, L.

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

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

McClelland, J. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

McDuff, R.

McMorran, B. J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

Min, C.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Moya-Cessa, H. M.

Naidoo, D.

Niu, H.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Niv, A.

Nye, J.

J. Nye and M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336(1605), 165–190 (1974).
[Crossref]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Oh, C. H.

Padgett, M.

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Pang, Z.

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Park, S.

Pas’ko, V.

Peng, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Perez, M.V.

C. Gomez-Reino, M.V. Perez, and C. Bao, Gradient-index optics: fundamentals and applications (Springer Science & Business Media, 2002).

Ponomarenko, S. A.

Qiu, C. W.

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

Rosales-Guzmán, C.

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

Roychowdhury, H.

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

Schmidt, O. A.

Schulze, C.

Shatokhin, V. N.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015).
[Crossref]

Shen, Y.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Shvartsman, N.

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

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

Singh, S.

Smith, C. P.

Song, S. H.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Soto-Eguibar, F.

Spreeuw, R. J. C.

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Stuart, H. R.

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref]

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

Unguris, J.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

van Dijk, T.

Vanderveen, H.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

Vickers, J.

Visser, T. D.

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009).
[Crossref]

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004).
[Crossref]

Vyas, R.

Wang, F.

Wang, J.

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

Wang, L.

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

Wang, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Wolf, E.

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23(4), 940–948 (2006).
[Crossref]

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004).
[Crossref]

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

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, 2013).

Wu, G.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Xie, Z.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Xu, S.

Yang, X.

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

Yang, Y.

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

Yu, N.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Yu, W.

Yuan, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

Zhai, C.

Zhang, J.

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

Zhao, C.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[Crossref]

Zhao, D.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Zhao, Q.

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

Zhao, R.

Zhong, T.

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

Zhu, S.

Zuñiga-Segundo, A.

Appl. Opt. (3)

Appl. Phys. Lett. (2)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

J. Mod. Opt. (3)

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
[Crossref]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998).
[Crossref]

J. Opt. A: Pure Appl. Opt. (2)

G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6(5), S239–S242 (2004).
[Crossref]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A: Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

J. Opt. Soc. Am. A (7)

Light: Sci. Appl. (1)

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Nature (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Opt. Commun. (2)

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Feng, J. Zhang, Z. Pang, L. Wang, T. Zhong, X. Yang, and D. Deng, “Propagation properties of the chirped Airy beams through the gradient-index medium,” Opt. Commun. 402, 60–65 (2017).
[Crossref]

Opt. Express (5)

Opt. Lett. (7)

Photonics Res. (1)

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

Phys. Lett. A (1)

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Phys. Rev. A (4)

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

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

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Phys. Rev. Appl. (1)

Y. Yang, Q. Zhao, L. Liu, Y. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of Orbital-Angular-Momentum Spectrum Using Pinhole Plates,” Phys. Rev. Appl. 12(6), 064007 (2019).
[Crossref]

Phys. Rev. Lett. (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

J. Nye and M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336(1605), 165–190 (1974).
[Crossref]

Prog. Opt. (2)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

Science (4)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref]

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref]

Other (7)

C. Gomez-Reino, M.V. Perez, and C. Bao, Gradient-index optics: fundamentals and applications (Springer Science & Business Media, 2002).

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

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

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, 2013).

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Figures (5)

Fig. 1.
Fig. 1. Normalized intensity distribution I/I0max for different topological charge m along the y-z plane, within a GRIN fiber with a parabolic refractive-index.
Fig. 2.
Fig. 2. Normalized intensity distribution I/I0max for different coherence width σ0 along the y-z plane, within a GRIN fiber with a parabolic refractive-index.
Fig. 3.
Fig. 3. The phase distribution and the positions of phase singularities (Re(μ) = 0and Im(μ) = 0) for different m within a GRIN fiber, where black spots denote the topological charge ‘+1’ and white spots denote ‘−1’ of the optical vortices.
Fig. 4.
Fig. 4. The phase distribution and the positions of phase singularities (Re(μ) = 0and Im(μ) = 0) for different ${\sigma _0}$ within a GRIN fiber, where black spots denote the topological charge ‘+1’ and white spots denote ‘−1’ of the optical vortices.
Fig. 5.
Fig. 5. The normalized intensity of cross section and the positions of phase singularities for different m within a GRIN fiber, where red solid curves denote Re(μ) = 0 and black dashed curves denote Im(μ) = 0.

Equations (18)

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E ( s , 0 ) = ( 2 s w 0 ) m L n m ( 2 s 2 w 0 2 ) exp ( s 2 w 0 2 ) exp ( i m θ )
exp ( i m θ ) s m L n m ( s 2 ) = ( 1 ) n 2 2 n + m n ! t = 0 n r = 0 m i r ( n t ) ( m r ) H 2 t + m r ( s x ) H 2 n 2 t + r ( s y )
E ( s , 0 ) = ( 1 ) n 2 2 n + m n ! t = 0 n r = 0 m i r ( n t ) ( m r ) H 2 t + m r ( 2 s x w 0 ) H 2 n 2 t + r ( 2 s y w 0 ) exp ( s 2 w 0 2 )
W 0 ( s 1 , s 2 , 0 ) = 1 2 4 n + 2 m ( n ! ) 2 t 1 = 0 n r 1 = 0 m t 2 = 0 n r 2 = 0 m i r 2  -  r 1 ( n t 1 ) ( m r 1 ) ( n t 2 ) ( m r 2 ) × H 2 t 1 + m r 1 ( 2 s 1 x w 0 ) H 2 n 2 t 1 + r 1 ( 2 s 1 y w 0 ) H 2 t 2 + m r 2 ( 2 s 2 x w 0 ) H 2 n 2 t 2 + r 2 ( 2 s 2 y w 0 ) × exp ( s 1 2 + s 2 2 w 0 2 ) exp [ ( s 1 s 2 ) 2 2 σ 0 2 ]
W ( ρ 1 , ρ 2 , z ) = ( k 2 π B ) 2 d s 1 x d s 1 y d s 2 x d s 2 y W 0 ( s 1 , s 2 , 0 ) × exp { i k 2 B [ A ( s 1 2 s 2 2 ) 2 ( s 1 ρ 1 s 2 ρ 2 ) + D ( ρ 1 2 ρ 2 2 ) ] }
W ( ρ 1 , ρ 2 , z ) = ( k 2 π B ) 2 1 2 4 n + 2 m ( n ! ) 2 π 2 M 1 M 2 ( 1 G 2 2 ) 2 n + m 2 exp [ i k D 2 B ( ρ 1 2 ρ 2 2 ) ] × exp [ ( k ρ 1 2 B M 2 k ρ 2 4 M 1 M 2 B σ 0 2 ) 2 k 2 ρ 2 2 4 M 1 B 2 ] × t 1 = 0 n r 1 = 0 m t 2 = 0 n r 2 = 0 m ( n t 1 ) ( m r 1 ) ( n t 2 ) ( m r 2 ) c 1 = 0 [ 2 t 1 + m r 1 2 ] d 1 = 0 2 t 2 + m r 2 e 1 = 0 [ d 1 2 ] c 2 = 0 [ 2 n 2 t 1 + r 1 2 ] d 2 = 0 2 n 2 t 2 + r 2 e 2 = 0 [ d 2 2 ] i r 2  -  r 1 × ( 1 ) c 1 + c 2 + e 1 + e 2 ( 2 t 2 + m r 2 d 1 ) ( 2 n 2 t 2 + r 2 d 2 ) d 1 ! d 2 ! e 1 ! e 2 ! ( d 1 2 e 1 ) ! ( d 2 2 e 2 ) ! × ( 2 t 1 + m r 1 ) ! c 1 ! ( 2 t 1 + m r 1 2 c 1 ) ! ( 2 n 2 t 1 + r 1 ) ! c 2 ! ( 2 n 2 t 1 + r 1 2 c 2 ) ! ( 2 2 w 0 ) 2 n + m 2 c 1 2 c 2 × [ 2 G σ 0 2 M 1 ( 1 G 2 ) ] d 1 + d 2 2 e 1 2 e 2 ( 1 2 i M 2 ) 2 n + m 2 c 1 2 c 2 + d 1 + d 2 2 e 1 2 e 2 × H 2 t 2 + m r 2 d 1 [ i k 2 G x 2 2 B M 1 ( 1 G 2 ) ] H 2 t 1 + m r 1 2 c 1 + d 1 2 e 1 ( k x 1 2 M 2 B + k x 2 4 M 1 M 2 B σ 0 2 ) × H 2 n 2 t 2 + r 2 d 2 [ i k 2 G y 2 2 B M 1 ( 1 G 2 ) ] H 2 n 2 t 1 + r 1 2 c 2 + d 2 2 e 2 ( k y 1 2 M 2 B + k y 2 4 M 1 M 2 B σ 0 2 )
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 ) = 1 2 n / 2 k = 0 n ( 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
I ( ρ , z ) = W ( ρ , ρ , z )
μ ( ρ 1 , ρ 2 , z ) = W ( ρ 1 , ρ 2 , z ) [ I ( ρ 1 , z ) I ( ρ 2 , z ) ] 1 / 2
Re [ μ ( ρ 1 , ρ 2 , z ) ] = 0
Im [ μ ( ρ 1 , ρ 2 , z ) ] = 0
n 2 = { n 0 2 ( 1 β 2 ρ 2 ) , ρ 2 R 0 2 n 0 2 ( 1 β 2 R 0 2 ) , ρ 2 R 0 2
β = 1 R 0 [ 1 n 1 2 n 0 2 ] 1 / 2
[ A B C D ] z = [ cos ( β z ) sin ( β z ) n 0 β n 0 β sin ( β z ) cos ( β z ) ]
n 2 ( ω ) = 1 + j = 1 3 B j ω j 2 ω j 2 ω 2

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