Abstract

We theoretically show that optical vortices conserve the integer topological charge (TC) when passing through an arbitrary aperture or shifted from the optical axis of an arbitrary axisymmetric carrier beam. If the beam contains a finite number of off-axis optical vortices with same-sign different TC, the resulting TC of the beam is shown to equal the sum of all constituent TCs. If the beam is composed of an on-axis superposition of Laguerre-Gauss modes (n, 0), the resulting TC equals that of the mode with the highest TC. If the highest positive and negative TCs of the constituent modes are equal in magnitude, the “winning” TC is the one with the larger absolute value of the weight coefficient. If the constituent modes have the same weight coefficients, the resulting TC equals zero. If the beam is composed of two on-axis different-amplitude Gaussian vortices with different TC, the resulting TC equals that of the constituent vortex with the larger absolute value of the weight coefficient amplitude, irrespective of the correlation between the individual TCs. In the case of equal weight coefficients of both optical vortices, TC of the entire beam equals the greatest TC by absolute value. We have given this effect the name “topological competition of optical vortices”.

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

1. Introduction

Laser optical vortices (OV) are a particular type of laser beams that carry an orbital angular momentum (OAM) [1]. The OAM associated with paraxial, nonparaxial, and vector beams has been amply studied, as can be seen from works published in 2019 alone [28]. Well-known examples of laser OV are presented by Laguerre-Gauss modes [9], Bessel [10] and Bessel-Gauss [11] beams, Hyper-geometric [12] and Circular [13] beams. The listed radially symmetric beams carry the same OAM normalized to the beam power, which is equal to the beams’ integer TC, n. Non-axially symmetric OVs have also been described and known to carry different OAM, for which a variety of formulae have been deduced [14,15]. As well as carrying OAM, optical vortices are also characterized by the topological charge (TC), which was defined in [16]. The literature dealing with the calculation of TC of composite OVs is very scarce. For example, in [17], TC was shown to conserve in medium-turbulence atmosphere over a distance of several kilometers, while in [18], TC variations were numerically studied of vortex soliton in a nonlinear medium. Sometimes authors identify TC and OAM, moreover, sometimes they assert that TC can be changed via diffraction by simple dielectric obstacles. On the other hand, in the articles [19,20] it was shown that the sectorial aperture can significantly change the OAM while the TC remains constant and equal to the initial value. Thus, it became necessary to provide some clarification on this issue.

In our article, we focused on some noteworthy examples of the TC behavior in optical vortex arrays that show a cautious approach to calculating TC. In particular, we demonstrate that TC of an optical vortex is conserved in spite of an amplitude distortion and shift of the OV center across the carrier beam. It is also shown that in a linear superposition of simple OVs whose amplitude is given by A(r)exp(inφ) (where (r, φ) are the polar coordinates in the beam cross-section) the constituent beams enter a “competition": TC of the resulting beam is defined by both magnitude and sign of the constituent vortex, +n, –n, as well as being dependent on the amplitude of weight coefficients of the linear combination.

2. TC of an OV after passing an amplitude mask

Below, we analyze changes in TC resulting from “cut-off” of a sector-shaped portion from an optical vortex. OVs with a “cut-off” sector have been discussed in detail by A. Volyar and colleagues [19]. This work has given an impetus to studying a topic of TC conservation following different types of distortions and transformation of an OV. The definition of TC of an OV (and an arbitrary paraxial light field) was given by M. Berry [16] and J. Nye [21]. For an arbitrary light field with complex amplitude E(r, φ), where (r, φ) are the polar coordinates, can be written in the form [16]:

$$TC = \frac{{\lim }}{{r \to \infty }}\frac{1}{{2\pi }}\int\limits_0^{2\pi } {d\varphi \frac{\partial }{{\partial \varphi }}} \arg E({r,\varphi } )= \frac{1}{{2\pi }}\frac{{\lim }}{{r \to \infty }}{\mathop{\rm Im}\nolimits} \int\limits_0^{2\pi } {d\varphi \frac{{\partial E({r,\varphi } )/\partial \varphi }}{{E({r,\varphi } )}}} . $$
This means that the TC monochromatic beam is specified the total number of optical vortices in the transverse cross section of a light beam taking into account their signs.

Let us write the complex amplitude En(r, φ) with a cut-off sector as

$$E({r,\varphi } )= A(r )\exp ({in\varphi } )f(\varphi ), $$
where the sector function reads as
$$f(\varphi )= \left\{ \begin{array}{l} 1,\; - \alpha < \varphi < \alpha ,\\ \delta \ll 1\textrm{,}\;\textrm{otherwise}\textrm{.}\; \end{array} \right.$$
Substituting (2) into (1) yields:
$$\begin{array}{c} TC = \frac{{\lim }}{{r \to \infty }}\frac{{{\mathop{\rm Im}\nolimits} }}{{2\pi }}\int\limits_0^{2\pi } {d\varphi \frac{{inE({r,\varphi } )+ A(r){e^{in\varphi }}\frac{{\partial f(\varphi )}}{{\partial \varphi }}}}{{E({r,\varphi } )}}} \\ = \frac{{{\mathop{\rm Im}\nolimits} }}{{2\pi }}\frac{{\lim }}{{r \to \infty }}\int\limits_0^{2\pi } {d\varphi \left( {in + \frac{{\partial f(\varphi )}}{{\partial \varphi }}\frac{1}{{f(\varphi )}}} \right)} = n. \end{array}$$
The final equality in (4) reflects the fact that the second term within the brackets is real. We may infer that if the aperture is only φ-angle dependent, TC of an OV remains unchanged. Although, if strictly δ = 0 in (3), then instead of (4) it should be written that TC = αn/π. We have two different physical situations here. If δ > 0 in Eq. (3), the light field passes through the diaphragm at any angle φ. Therefore, the integral in Eq. (4) can be evaluated from 0 to 2π. If, however, δ = 0, integration in Eq. (4) can be done only where the light passes, i.e. from –α till α. In the latter case, we obtain the fractional TC: TC = αn/π. But fractional TC can only be in the initial plane. After propagation in free space, the light field should have an integer topological charge in order to be continuous. Therefore, TC is not always conserved on propagation (see, for instance, [22]), in difference with the orbital angular momentum, which is always conserved.

The proof of Eq. (4) can easily be repeated for an arbitrarily-shaped amplitude filter (3), which is defined by both angle φ and radius r:

$$f({r,\varphi } )= \left\{ \begin{array}{l} 1,\;({r,\varphi } )\in \Omega ,\\ \delta \ll 1,\;\;({r,\varphi } )\notin \Omega , \end{array} \right.$$
where Ω is the diaphragm cut-off area. Then, instead of Eq. (4), we obtain a similar relation:
$$TC = \frac{{{\mathop{\rm Im}\nolimits} }}{{2\pi }}\frac{{\lim }}{{r \to \infty }}\int\limits_0^{2\pi } {d\varphi \left( {in + \frac{{\partial f({r,\varphi } )}}{{\partial \varphi }}\frac{1}{{f({r,\varphi } )}}} \right)} = n. $$
A weak transmission (δ << 1) was introduced in (3) and (5) in the region where the diaphragm should not transmit light in order to avoid the 0/0 uncertainty in the division of E(r, φ) in (4) and (6) on E(r, φ). We note that the derivative ∂()/∂φ is equal to n only if the angle φ can be arbitrary (0 < φ < 2π). This means that instead of conditions in Eqs. (3) and (5) it is sufficient that a closed curve existed around the singular point (OV center) with nonzero field amplitude on this curve. Simulation (Fig. 1) confirms this requirement. An indirect confirmation of the conservation of TC of an optical vortex with a cut out sector (4), (6) is that the OAM of such a beam is equal to the topological charge. Indeed, the OAM [6,7] normalized to the beam power
$$\begin{aligned}{J_z} &= {\mathop{\rm Im}\nolimits} \int\limits_0^\infty {\int\limits_0^{2\pi } {\bar{E}({r,\varphi ,z} )\left( {\frac{{\partial E({r,\varphi ,z} )}}{{d\varphi }}} \right)rdrd\varphi } } \\ &= {\mathop{\rm Im}\nolimits} \int\limits_0^\infty {\int\limits_{ - \alpha }^\alpha {A(r ){e^{in\varphi }}({in\bar{A}(r ){e^{ - in\varphi }}} )rdrd\varphi } } = 2\alpha n\int\limits_0^\infty {{{|{A(r)} |}^2}rdr,} \\ \frac{{{J_z}}}{W} &= n\,,\quad W = 2\alpha \int\limits_0^\infty {{{|{A(r)} |}^2}rdr.} \end{aligned}$$
In Eq. (7) $\bar{E}$ and $\bar{A}$ are complex conjugates of E and A. The orbital angular momentum from Eq. (7) is conserved at any z. It is seen from Eq. (6) that multiplying the complex amplitude (2) of the OV by any real function does not change TC of the original OV, as a real function does not change the complex amplitude argument in Eq. (1). In other words, since the TC of an optical vortex is determined solely by the phase and since the phase is not affected by amplitude distortions in the diaphragm plane, these amplitude distortions do not affect the TC, although, strictly speaking, the sign of the real function, describing distortions, can be alternating (for instance, the alternating function $f(\varphi ) = \cos (m\varphi )$ [23] in Eq. (3) does not change the TC from Eq. (2)). Alternating real-valued function in Eq. (3) distorts also the phase of the field (2) since it has jumps by π. However, these phase jumps do not change the TC of the field (2).

 

Fig. 1. Distributions of intensity (a,c,e,g) and phase (b,d,f,h) of a Gaussian optical vortex bounded by a sector-shape diaphragm in the initial plane z = 0 (a-d) and after propagation in free space (e-h) for two different angles of the sector aperture α = π/6 (a,b,e,f) and α = π/4 (c,d,g,h). Red rings (f,h) show the circle over which the TC was calculated. Yellow text (e,g) shows the TC.

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When the spiral phase plate is bounded by a sector aperture (2), a Gaussian beam after passing it has the following complex amplitude in the Cartesian coordinates:

$$E({x,y,0} )= \exp \left[ { - \frac{{{x^2} + {y^2}}}{{{w^2}}} + in\arg ({x + iy} )} \right]{\mathop{\rm rect}\nolimits} \left\{ {\frac{{\arg [{({x - {x_0}} )+ i({y - {y_0}} )} ]}}{{2\alpha }}} \right\}, $$
where w is the Gaussian beam waist radius, n is the TC of the spiral phase plate, α from (3) is the half-angle of the sector aperture, (x0, y0) is the shift vector of the sector aperture (the singular point should be inside the sector), rect(x) = 1 at |x| ≤ 1/2 and rect(x) = 0 at |x| > 1/2, and arg(x) is meant as the principal value of the argument (i.e. –π < arg(x) ≤ π).

Figure 1 shows distributions of intensity (a,c,e,g) and phase (b,d,f,h) of a Gaussian optical vortex bounded by a sector-shape diaphragm in the initial plane z = 0 (a-d) and after propagation in free space (e-h) for two different angles of the sector aperture α = π/6 (a,b,e,f) and α = π/4 (c,d,g,h). Distributions in the initial plane are obtained by Eq. (1), while distributions at a distance are obtained by the Fresnel transform implemented numerically in Matlab as a convolution with using the fast Fourier transform. The following parameters are used in the calculations: wavelength λ = 532 nm, Gaussian beam waist radius w = 1 mm, TC of the spiral phase plate n = 5, vector of the sector diaphragm shift (x0, y0) = (–0.5, 0) mm, propagation distance z = z0/2 (z0 = kw2/2 is the Rayleigh range), calculation area –R ≤ x, y ≤ R (R = 12.5 mm, although Fig. 1 shows smaller areas), number of pixels N = 4096. The obtained values are 4.9668 for α = π/6 and 4.9693 for α = π/4.

3. TC of an off-axis optical vortex

In this section, we analyze how TC changes upon an off-axis shift of the OV center from the optical axis of a radially symmetric beam with amplitude A(r). May an OV be shifted by an arbitrary vector (r0, φ0). Then, instead of Eq. (2), the complex amplitude En(r, φ) takes the form:

$${E_n}({r,\varphi } )= {\left( {\frac{{r{e^{i\varphi }} - {r_0}{e^{i{\varphi_0}}}}}{w}} \right)^n}A(r ). $$
Substituting (9) into (1) yields:
$$TC = \frac{{\lim }}{{r \to \infty }}\frac{{{\mathop{\rm Im}\nolimits} }}{{2\pi }}\int\limits_0^{2\pi } {d\varphi \frac{{inr{e^{i\varphi }}}}{{r{e^{i\varphi }} - {r_0}{e^{i{\varphi _0}}}}}} = \frac{1}{{2\pi }}{\mathop{\rm Im}\nolimits} \frac{{\lim }}{{r \to \infty }}\int\limits_0^{2\pi } {d\varphi \frac{{inr{e^{i\varphi }}}}{{r{e^{i\varphi }} - {r_0}{e^{i{\varphi _0}}}}}} = n. $$
The final equality in (9) stems from the fact that for large radii (r >> r0), only the first term is retained in the denominator. It is seen from (9) that an off-axis shift of the OV center relative to a radially symmetric beam (e.g. a Gaussian beam) does not lead to a change in TC. In the meantime, for a beam with off-axis phase singularity center, the normalized OAM is lower than TC of the whole beam, with the former decreasing with increasing shift magnitude r0 [24,25].

In Fig. 2 shows the distribution of the intensity and phase of a Gaussian beam with an off-axis optical vortex in the initial plane and after propagation in space for different displacements of the vortex from the optical axis. The complex amplitude in the initial plane is ${E_n}({x,y} )= {[{{{({r{e^{i\varphi }} - {r_0}{e^{i{\varphi_0}}}} )} \mathord{\left/ {\vphantom {{({r{e^{i\varphi }} - {r_0}{e^{i{\varphi_0}}}} )} w}} \right.} w}} ]^n}\exp [{ - {{({{x^2} + {y^2}} )} \mathord{\left/ {\vphantom {{({{x^2} + {y^2}} )} {{w^2}}}} \right.} {{w^2}}}} ]$, where w is the waist radius of the Gaussian beam, n and (r0, φ0) are the topological charge of the optical vortex and the vector (in polar coordinates) of its displacement from the optical axis. The complex amplitude after propagation in space is calculated using a numerical Fresnel transform realized in the form of a convolution using the fast Fourier transform. The following calculation parameters were used: w = 1 mm, n = 7, φ0 = 0, r0 = w0/4 [Fig. 2(a,b)], r0 = w0/2 [Fig. 2(c,d)], r0 = 2w0 [Fig. 2(e,f)], space distance z = z0/2, computational domain –R ≤ x, y ≤ R (R = 5 mm). The TC in the initial plane, calculated numerically by formula (1) (along a ring of radius 0.8R), is 6.9997 for r0 = w0/4 and r0 = w0/2, 6.9995 for r0 = 2w0, i.e. in all cases about 7. Meanwhile, at a distance of z = z0/2, TC is 6.9989, 6.9989 and 6.9986, respectively.

 

Fig. 2. Distributions of intensity (a, c, e, g, i, k) and phase (b, d, f, h, j, l) of a Gaussian beam with an off-axis optical vortex in the initial plane (a, b, e, f, i, j) and after propagation in space (c, d, g, h, k, l) for different lateral displacements of the vortex from the optical axis. Calculation parameters: waist radius w = 1 mm, TC is n = 7, displacement r0 = w0/4 (a-d), r0 = w0/2 (e-h), r0 = 2w0 (i-l); φ0 = 0 in all figures, the propagation distance in space is z = z0/2 (z0 is the Rayleigh distance). The red (dashed) rings on the phase distributions denote the radius of the ring by which the TC was calculated by formula (1).

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An interesting case occurs when an optical vortex is bounded by a diaphragm in the initial plane and therefore it is impossible to use the limit r → ∞ like in Eq. (10). For example, if a spiral phase plate (SPP) is bounded by a circular diaphragm with a radius R and is shifted horizontally from the optical axis by a distance x0, a plane wave after passing such SPP acquires the following complex amplitude:

$$E(r,\varphi ) = {\mathop{\rm circ}\nolimits} \left( {\frac{r}{R}} \right)\exp \left[ {in\arctan \left( {\frac{{r\sin \varphi }}{{r\cos \varphi - {x_0}}}} \right)} \right],$$
where circ(r/R) = 1 for r ≤ R and circ(r/R) = 0 for r > R.

Topological charge [Eq. (1)] of the initial vortex field from Eq. (11) is given by

$$TC = \frac{n}{{2\pi }}\int\limits_0^{2\pi } {\frac{{{R^2} - R{x_0}\cos \varphi }}{{{R^2} + x_0^2 - 2R{x_0}\cos \varphi }}} d\varphi = \left\{ \begin{array}{l} n,\;{x_0} < R,\\ n/2,\;{x_0} = R. \end{array} \right.$$
Equation (12) illustrates that shifting the SPP center conserves TC which equals n if the SPP center is within the diaphragm.

If the SPP center is on the diaphragm edge, TC decreases two times immediately. Expression (12) is consistent with the condition from the previous section, which states that for TC conservation a closed contour with nonzero amplitude should exist around the singularity center. If the center of the optical vortex is on the diaphragm edge (x0 = R) then such a contour does not exist and therefore the TC of the vortex changes its value (TC = n/2). Interestingly, OAM of the beam from Eq. (11) decreases continuously till zero when the shift distance x0 increases from 0 to R:

$$\frac{{{J_z}}}{W} = n\left( {1 - \frac{{x_0^2}}{{{R^2}}}} \right). $$
From Eq. (13), if x0 = R, the beam's OAM is zero.

4. TC of an optical vortex with multi-center optical singularities

Below, we analyze a laser Gaussian beam with m embedded simple (TC = +1) phase singularities distributed uniformly on a circle of radius a, i.e. at points defined by the Cartesian coordinates

$$\left\{ \begin{array}{l} x = a\cos {\varphi_p},\\ y = a\sin {\varphi_p}, \end{array} \right.$$
where φp = 2πp/m, p = 0, …, m – 1. The complex amplitude of such an OV at an arbitrary distance from the waist can be shown to be given by
$$E({r,\varphi ,z} )= \frac{1}{\sigma }{\left( {\frac{{\sqrt 2 }}{{{w_0}}}} \right)^m}\exp \left( { - \frac{{{r^2}}}{{\sigma w_0^2}}} \right)\left( {\frac{{{r^m}{e^{im\varphi }}}}{{{\sigma^m}}} - {a^m}} \right), $$
where σ = 1 + iz/z0 and z0 = kw02/2 is the Rayleigh range (k is the wave number). Substituting (15) into (1) yields:
$$TC = \frac{1}{{2\pi }}\mathop {\lim }\limits_{r \to \infty } {\mathop{\rm Im}\nolimits} \left\{ {\int\limits_0^{2\pi } {\frac{{im{\sigma^{ - m}}{r^m}{e^{im\varphi }}}}{{{\sigma^{ - m}}{r^m}{e^{im\varphi }} - {a^m}}}d\varphi } } \right\} = m. $$
Because at r → ∞, the term am in the denominator is negligibly small, TC of the beam (15) turns out to be independent on the distance z passed and the radius a of the circle of the OV centers, instead, being equal to the number of simple OVs in the beam. This result can be extended onto an arbitrary case of m OV centers with multiplicity mp are found at points (rp, φp), where p = 1, 2,…m and the carrier amplitude A(r) is axially symmetric. Such a complex OV is given by the complex amplitude [26,27]:
$${E_m}({r,\varphi ,z = 0} )= A(r )\prod\limits_{p = 1}^m {{{({r{e^{i\varphi }} - {r_p}{e^{i{\varphi_p}}}} )}^{{m_p}}}}. $$
Substituting (17) into (1) yileds:
$$TC = \frac{1}{{2\pi }}\mathop {\lim }\limits_{r \to \infty } {\mathop{\rm Im}\nolimits} \left\{ {\int\limits_0^{2\pi } {ir{e^{i\varphi }}\sum\limits_{p = 1}^m {\frac{{{m_p}}}{{r{e^{i\varphi }} - {r_p}{e^{i{\varphi_p}}}}}} \,d\varphi } } \right\} = \sum\limits_{p = 1}^m {{m_p}}. $$
Relation (18) suggests that in a beam with axially symmetric amplitude and several degenerate simple OVs of Eq. (17), with their centers located at arbitrary points over the beam cross-section, TC equals the sum of multiplicity (degeneracy) values of all constituent vortices.

5. TC of an on-axis combination of optical vortices

Here, we discuss a light field whose complex amplitude is described by a linear combination of a finite number of Laguerre-Gaussian (LG) modes with the numbers (n, 0):

$${E_{N, - M}}({r,\varphi ,z = 0} )= \exp \left( { - \frac{{{r^2}}}{{{w^2}}}} \right){\sum\limits_{n ={-} M}^N {{C_n}\left( {\frac{r}{w}} \right)} ^{|n |}}{e^{in\varphi }}. $$
Substituting (19) into (1) yields a relation for TC:
$$TC = \frac{1}{{2\pi }}\mathop {\lim }\limits_{r \to \infty } {\mathop{\rm Im}\nolimits} \left\{ {\int\limits_0^{2\pi } {i\frac{{{{\sum\limits_{n ={-} M}^N {n{C_n}\left( {\frac{r}{w}} \right)} }^{|n |}}{e^{in\varphi }}}}{{{{\sum\limits_{n ={-} M}^N {{C_n}\left( {\frac{r}{w}} \right)} }^{|n |}}{e^{in\varphi }}}}\,d\varphi } } \right\}. $$
Following a limiting passage r → ∞ under the integral sign in Eq. (20), the numerator and denominator each retain just one highest-power term under the sum sign. If M > N, then TC of the beam in Eq. (19) is TC = –M, if M < N, then TC in (19) equals TC = N. Finally, if M = N, instead of (20), we obtain:
$$TC = \frac{1}{{2\pi }}{\mathop{\rm Im}\nolimits} \left\{ {\int\limits_0^{2\pi } {iN\frac{{({{C_N}{e^{iN\varphi }} - {C_{ - N}}{e^{ - iN\varphi }}} )}}{{({{C_N}{e^{iN\varphi }} + {C_{ - N}}{e^{ - iN\varphi }}} )}}\,d\varphi } } \right\}. $$
Thus, we can infer that if in a linear combination of a finite number of LG modes with different TC, the absolute value of the maximum positive TC is larger than the maximum negative TC, the TC of the entire beam equals the positive TC = N. If the opposite is the case, the resulting TC equals the negative TC = −M. Finally, in the next section we show that for M = N, the integral in Eq. (21) can be taken analytically and, based on Eq. (27), TC = N if |CN| > |CN| or TC = −N if |CN| < |CN|. When |CN| = |CN|, TC of the entire beam equals zero.

6. TC of the sum of two optical vortices

Now, let us analyze a simple but rather interesting case that produces an unexpected result. Assume a light field with a complex amplitude in the initial plane that describes an axial superposition of two Gaussian OVs with different TC and different amplitudes:

$$E({r,\varphi } )= ({a{e^{in\varphi }} + b{e^{im\varphi }}} ){e^{ - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {{w^2}}}} \right.} {{w^2}}}}}, $$
where w is the Gaussian beam waist radius, n and m are integer topological charges of the OVs, a and b are weight coefficients in the OV superposition, which are generally complex. Substituting (22) into (1) yields a relation for TC:
$$TC = \frac{1}{{2\pi }}\mathop {\lim }\limits_{r \to \infty } {\mathop{\rm Im}\nolimits} \left\{ {\int\limits_0^{2\pi } {\frac{{{{\partial E({r,\varphi } )} \mathord{\left/ {\vphantom {{\partial E({r,\varphi } )} {\partial \varphi }}} \right.} {\partial \varphi }}}}{{E({r,\varphi } )}}d\varphi } } \right\} = \frac{1}{{2\pi }}{\mathop{\rm Re}\nolimits} \left\{ {\int\limits_0^{2\pi } {\frac{{na{e^{in\varphi }} + mb{e^{im\varphi }}}}{{a{e^{in\varphi }} + b{e^{im\varphi }}}}d\varphi } } \right\}. $$
The integral in the right-hand side of Eq. (23) can be reduced to a sum of two integrals:
$$TC = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\left( {\frac{{n + m}}{2} + \frac{{n - m}}{2}\frac{{{{|a |}^2} - {{|b |}^2}}}{{{{|a |}^2} + {{|b |}^2} + 2|a ||b |\cos t}}} \right)dt}. $$
With the first integral in (24) being trivial, the second integral can be rearranged as
$$TC = \frac{{n + m}}{2} + \frac{1}{{2\pi }}\frac{{n - m}}{2}\frac{{{{|a |}^2} - {{|b |}^2}}}{{{{|a |}^2} + {{|b |}^2}}}\int\limits_0^{2\pi } {\frac{{dt}}{{1 + \frac{{2|a ||b |}}{{{{|a |}^2} + {{|b |}^2}}}\cos t}}}. $$
With the coefficient before the cosine function being not larger than unity, this is a reference integral (expression 3.613.1 in [28]):
$$\int\limits_0^\pi {\frac{{\cos ({nx} )dx}}{{1 + a\cos x}}} = \frac{\pi }{{\sqrt {1 - {a^2}} }}{\left( {\frac{{\sqrt {1 - {a^2}} - 1}}{a}} \right)^n}\;\;\;\;[{{a^2} < 1,\;n \ge 0} ]. $$
In the case the integration interval is from zero to 2π, rather than being to π, the expression needs to be multiplied by two. Then, Eq. (23) takes the form:
$$TC = \frac{{n + m}}{2} + \frac{{n - m}}{2}\frac{{{{|a |}^2} - {{|b |}^2}}}{{|{{{|a |}^2} - {{|b |}^2}} |}}.$$
For completeness sake, the normalized OAM of the beam (22) can be given in the form:
$$OAM = \frac{{n{a^2} + m{b^2}}}{{{a^2} + {b^2}}}. $$
From (27) it follows that if |a| > |b|, then TC = n and if |a| < |b|, then TC = m. If m = n, as can be expected, we obtain that TC = n. Thus, TC of the resulting beam (22) equals that of the constituent OV with the larger amplitude. At |a| = |b|, there occurs degeneracy (photon entanglement), with Eq. (27) becoming invalid due to uncertainty 0/0. Because of this, at |a| = |b|, the field in Eq. (22) needs to be rearranged to the form:
$$\begin{array}{l} E({r,\varphi } )= |a |({{e^{in\varphi + i\arg a}} + {e^{im\varphi + i\arg b}}} ){e^{ - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {{w^2}}}} \right.} {{w^2}}}}}\\ = 2|a |\cos \left( {\frac{{n\varphi - m\varphi + \arg a - \arg b}}{2}} \right)\exp \left( { - \frac{{{r^2}}}{{{w^2}}} + i\frac{{n\varphi + m\varphi + \arg a + \arg b}}{2}} \right). \end{array}$$
Substituting (29) into (1) yields
$$TC = \mathop {\lim }\limits_{r \to \infty } \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\frac{\partial }{{\partial \varphi }}\left( {\frac{{n\varphi + m\varphi + \arg a + \arg b}}{2}} \right)d\varphi } = \frac{{n + m}}{2}. $$
In Section 3, an interesting result is obtained. If x0 = R, the topological charge of the optical vortex (11) is equal to TC = n/2 [Eq. (12)], while its OAM is zero [Eq. (13)]. There is nothing strange in it since TC and OAM are different physical properties of an optical vortex. It can be demonstrated by using Eqs. (27) and (28). If b = 2a and n = –4m, Eq. (27) yields TC = m, while Eq. (28) leads to OAM = 0.

As we can infer from (27), superposition of two same-amplitude OVs, with one TC being even and the other odd, produces an OV with a fractional (semi-integer) TC. It should be noted that it is only in the initial plane that TC of a beam can be fractional, whereas during propagation TC needs to be integer for the amplitude to be continuous. It is worth noting that OAM in (28) equals TC in (27) and (30) only when either a = 0, or b = 0, or a = b. At the same time, if the beam is degenerate (a = b), the content of the constituent angular harmonics of the beam cannot be derived from the known TC. For instance, all the below-listed beams carry have the same TC and OAM, which is equal to 4:

$$\begin{array}{l} {E_1}({r,\varphi } )= ({{e^{i\varphi }} + {e^{i7\varphi }}} ){e^{ - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {{w^2}}}} \right.} {{w^2}}}}},\\ {E_2}({r,\varphi } )= ({{e^{i2\varphi }} + {e^{i6\varphi }}} ){e^{ - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {{w^2}}}} \right.} {{w^2}}}}},\\ {E_3}({r,\varphi } )= ({{e^{i3\varphi }} + {e^{i5\varphi }}} ){e^{ - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {{w^2}}}} \right.} {{w^2}}}}},\\ {E_4}({r,\varphi } )= {e^{i4\varphi }}{e^{ - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {{w^2}}}} \right.} {{w^2}}}}}. \end{array}$$
In Fig. 3(a, b) shows the intensity and phase of the superposition of two Gaussian vortices in the initial plane for the following calculation parameters: waist radius w = 1 mm, topological charges n = 12 and m = 7, weight coefficients are unit modulo, but with a random phase: a = e2.9616 i, b = e0.2247 i, computational domain –R ≤ x, y ≤ R (R = 1 mm). The TC calculated numerically by formula (1) is 9.4708, i.e. approximately (12 + 7) / 2. In Fig. 3(c, d) shows the intensity and phase of the same superposition, but at the Fresnel distance (for the wavelength λ = 532 nm) and in a wider calculated region (R = 10 mm). The TC calculated numerically by formula (1) is 11.8167, that is, about 12. In both cases, the TC was calculated by integration over a ring of radius 0.8R. This example corresponds to the situation described by amplitude (27), and when the amplitude moduli are equal |a| = |b| two vortices, the TC of the entire beam will be equal to the larger of the two TC, i.e. 12.

 

Fig. 3. The intensity (a, c) and phase (b, d) of the axial superposition of two Gaussian OVs with TC 12 and 7, but with the same weight amplitudes (in (22)) in the initial plane (a, c) and at the Rayleigh distance (c, d). The red (dashed) rings on the phase distributions denote the radius of the ring by which the topological charge was calculated by the formula (1).

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7. Topological charge in an arbitrary plane

In this section, we shall demonstrate that a combination of two same-amplitude (a = b) Gaussian OVs of Eq. (22) with different TC produces an OV with half-integer TC (if m + n is odd) of Eq. (27) in the initial plane, generating an OV with integer TC as it propagates. Actually, if in the initial plane there is a Gaussian OV

$$E({r,\varphi } )= {e^{ - {{{r^2}} \mathord{\left/ {\vphantom {{{r^2}} {{w^2}}}} \right.} {{w^2}}} + in\varphi }}, $$
following the propagation through an ABCD-system, its complex amplitude is given by
$${E_z}({\rho ,\theta } )= {({ - i} )^{n + 1}}\sqrt {\frac{\pi }{2}} \frac{{{z_0}}}{{B{q_1}}}\sqrt \xi \exp \left( {\frac{{ikD{\rho^2}}}{{2B}} + in\theta - \xi } \right)\left[ {{I_{\frac{{n - 1}}{2}}}(\xi )- {I_{\frac{{n + 1}}{2}}}(\xi )} \right],$$
where
$$\xi = {{{{({{{{z_0}} \mathord{\left/ {\vphantom {{{z_0}} B}} \right.} B}} )}^2}{{({{\rho \mathord{\left/ {\vphantom {\rho w}} \right.} w}} )}^2}} \mathord{\left/ {\vphantom {{{{({{{{z_0}} \mathord{\left/ {\vphantom {{{z_0}} B}} \right.} B}} )}^2}{{({{\rho \mathord{\left/ {\vphantom {\rho w}} \right.} w}} )}^2}} {({2{q_1}} )}}} \right.} {({2{q_1}} )}},\;{q_1} = 1 - i({{A \mathord{\left/ {\vphantom {A B}} \right.} B}} ){z_0}.$$
Since in Eq. (33), Iν(x) is a modified Bessel function, then for the superposition in (22), the complex amplitude is
$$\begin{array}{l} {E_z}({\rho ,\theta } )={-} i\sqrt {\frac{\pi }{2}} \frac{{{z_0}}}{{B{q_1}}}\exp \left( {\frac{{ikD{\rho^2}}}{{2B}}} \right)\sqrt \xi \exp ({ - \xi } )\\ \;\;\;\; \times \left\{ {a{{({ - i} )}^n}{e^{in\theta }}\left[ {{I_{\frac{{n - 1}}{2}}}(\xi )- {I_{\frac{{n + 1}}{2}}}(\xi )} \right]} \right. + \left. {b{{({ - i} )}^m}{e^{im\theta }}\left[ {{I_{\frac{{m - 1}}{2}}}(\xi )- {I_{\frac{{m + 1}}{2}}}(\xi )} \right]} \right\}. \end{array}$$
Retaining in the asymptotic expansion of the modified Bessel function just first two terms yields a relationship to describe the difference of two modified Bessel functions of adjacent orders at large values of the argument:
$${I_{\frac{{n - 1}}{2}}}(\xi )- {I_{\frac{{n + 1}}{2}}}(\xi )\sim \frac{{{e^\xi }}}{{\sqrt {2\pi \xi } }}\left\{ {\left[ {1 - \frac{{{{({n - 1} )}^2} - 1}}{{8\xi }}} \right] - \left[ {1 - \frac{{{{({n + 1} )}^2} - 1}}{{8\xi }}} \right]} \right\} = \frac{{n{e^\xi }}}{{2\xi \sqrt {2\pi \xi } }}.$$
Then, at large values of ρ, Eq. (35) takes the form:
$$\begin{aligned} {E_z}({\rho ,\theta } )&={-} i\sqrt {\frac{\pi }{2}} \frac{{{z_0}}}{{B{q_1}}}\exp \left( {\frac{{ikD{\rho^2}}}{{2B}}} \right)\sqrt \xi \exp ({ - \xi } )\\ &\times \left[ {a{{({ - i} )}^n}\exp ({in\theta } )\frac{{n{e^\xi }}}{{2\xi \sqrt {2\pi \xi } }} + b{{({ - i} )}^m}\exp ({im\theta } )\frac{{m{e^\xi }}}{{2\xi \sqrt {2\pi \xi } }}} \right]\\ &= \frac{{ - i{z_0}}}{{4B{q_1}\xi }}\exp \left( {\frac{{ikD{\rho^2}}}{{2B}}} \right)[{an{{({ - i} )}^n}\exp ({in\theta } )+ bm{{({ - i} )}^m}\exp ({im\theta } )} ]. \end{aligned}$$
In view of Eqs. (23) and (27), Eq. (37) suggests that at |a| = |b|, in the initial plane TC = (m + n)/2. At the same time, in any other plane, modules of the coefficients in front of einθ and eimθ are proportional to |n| and |m|, being no more equal to each other (at nm), hence, according to Eq. (27), TC = max(n, m). Note, however, that if in Eq. (37) |an| = |bm|, we again find ourselves in the situation of degeneracy, because in view of Eq. (27) and at z>0, TC of two OVs of Eq. (22) equals the arithmetic mean of Eq. (30): TC = (n + m)/2. This situation can be addressed as follows: with the equality |an| = |bm| meaning that |a| ≠ |b|, Eq. (27) suggests that the total TC of the field in the initial plane equals that of the OV with the larger amplitude (respectively, |a| or |b|). In the meantime, the integer TC in the initial plane conserves upon propagation.

8. Topological charge for an optical vortex with an initial fractional charge

For an OV with fractional TC = µ (µ is an arbitrary real number), a relation to describe the corresponding fractional TC has been derived [29]. The mutual transformations between beams with fractional-order and integer-order vortices was in detail considered in [30]. An OV with fractional TC, which is possible only in the initial plane, can be decomposed in terms of OVs with integer TC n (µ is an arbitrary real number) as follows:

$${E_\mu }(r,\varphi ,z) = \exp (i\mu \varphi )\Psi (r,z) = \frac{{{e^{i\pi \mu }}\sin \pi \mu }}{\pi }\Psi (r,z)\sum\limits_{n ={-} \infty }^\infty {\frac{{{e^{in\varphi }}}}{{\mu - n}}} .$$
In (38), the function Ψ(r, z) is real. Substituting the right-hand side of Eq. (38) into a general relation for OAM
$${J_z} = {\mathop{\rm Im}\nolimits} \int\limits_0^\infty {\int\limits_0^{2\pi } {\bar{E}({r,\varphi ,z} )\left( {\frac{{\partial E({r,\varphi ,z} )}}{{\partial \varphi }}} \right)rdrd\varphi } }$$
yields:
$${J_z} = W\frac{{{{\sin }^2}(\pi \mu )}}{{{\pi ^2}}}\sum\limits_{n ={-} \infty }^\infty {\frac{n}{{{{({\mu - n} )}^2}}}}, $$
where W is the energy (power) of the beam:
$$W = \int\limits_0^\infty {\int\limits_0^{2\pi } {E(r,\varphi ,z)\bar{E}(r,\varphi ,z)rdrd\varphi } }. $$
The series in the right-hand side of Eq. (40) can be reduced to a reference series [26]:
$$\sum\limits_{n = 1}^\infty {\frac{{{n^2}}}{{{{({{n^2} \pm {a^2}} )}^2}}}} = \frac{\pi }{{4a}}\left[ { \pm \left\{ {\begin{array}{c} {\coth \pi a}\\ {\cot \pi a} \end{array}} \right\} \mp a\left\{ {\begin{array}{c} {{{{\mathop{\rm cosech}\nolimits} }^2}\pi a}\\ {{{{\mathop{\rm cosec}\nolimits} }^2}\pi a} \end{array}} \right\}} \right], $$
using which the final relation for the normalized OAM of the field in Eq. (38) is rearranged to
$$\frac{{{J_z}}}{W} = \mu - \frac{{\sin 2\pi \mu }}{{2\pi }}. $$
From (43) it follows that OAM equals TC = µ only if µ is integer and half integer. This conclusion is in agreement with Eqs. (35) and (37) for the linear combination composed of two angular harmonics.

We obtain the expression for the TC of the optical vortex in the Fresnel diffraction zone for the initial field with a fractional topological charge (38), but for definiteness we choose the amplitude function in the form of a Gaussian one. Then instead of (38) we get:

$${E_\mu }(r,\varphi ,z = 0) = \exp \left(i\mu \varphi - {\left( {\frac{r}{w}} \right)^2}\right) = \frac{{{e^{i\pi \mu }}\sin \pi \mu }}{\pi }\sum\limits_{n ={-} \infty }^\infty {\frac{{{e^{in\varphi - {r^2}/{w^2}}}}}{{\mu - n}}} .$$
In view of (33), the amplitude of the optical vortex (44) for any z will be equal to (B = z, A = D = 1):
$$\begin{array}{l} {E_2}({\rho ,\theta } )= \frac{1}{{\sqrt {2\pi } }}\left( {\frac{{ - i{z_0}}}{{{q_1}z}}} \right)\exp \left( {\frac{{ik{\rho^2}}}{{2z}} + i\pi \mu } \right)\sin ({\pi \mu } )\sqrt x \exp ({ - x} )\times \\ \times \sum\limits_{m ={-} \infty }^\infty {{{({ - i} )}^{|m |}}\frac{{\exp ({im\theta } )}}{{\mu - m}}\left[ {{I_{\frac{{|m |- 1}}{2}}}(x )- {I_{\frac{{|m |+ 1}}{2}}}(x )} \right]} \end{array}$$
We substitute (45) in (1) and, when passing to the limit in (1), we take into account the asymptotic behavior (36), then we obtain the expression for calculating the TC of the optical vortex (44):
$$TC = \frac{{{\mathop{\rm Re}\nolimits} }}{{2\pi }}\left\{ {\int\limits_0^{2\pi } {\left[ {\sum\limits_{n ={-} \infty }^\infty {\frac{{{{( - i)}^{|n |}}n|n |{e^{in\varphi }}}}{{\mu - n}}} } \right] \cdot {{\left[ {\sum\limits_{n ={-} \infty }^\infty {\frac{{{{( - i)}^{|n |}}|n |{e^{in\varphi }}}}{{\mu - n}}} } \right]}^{ - 1}}\,d\varphi } } \right\}. $$
In the near field (z << z0), Eq. (46) should be replaced by the following expression, which follows immediately from Eq. (38):
$$TC = \frac{{{\mathop{\rm Re}\nolimits} }}{{2\pi }}\left\{ {\int\limits_0^{2\pi } {\left[ {\sum\limits_{n ={-} \infty }^\infty {\frac{{n{e^{in\varphi }}}}{{\mu - n}}} } \right] \cdot {{\left[ {\sum\limits_{n ={-} \infty }^\infty {\frac{{{e^{in\varphi }}}}{{\mu - n}}} } \right]}^{ - 1}}\,d\varphi } } \right\}. $$
Expression (47) is remarkable in that the answer is known, which was numerically obtained in [16], but has not yet been obtained analytically. Calculation (47) can be called the Berry problem [16]. The right-hand side of (47) should give only whole TC, closest to µ:
$$TC = \sum\limits_{n ={-} \infty }^\infty {n{\mathop{\rm rect}\nolimits} } ({\mu - n} )\,,\quad {\mathop{\rm rect}\nolimits} (x) = \left\{ \begin{array}{l} 1,\;|x |\le 1/2,\\ 0,\;|x |> 1/2 \end{array} \right.$$
From a comparison of Eqs. (47) and (48), it can be said that the TC (47) is equal to the TC of the angular harmonic in the series in the numerator and the denominators for which the weight coefficient is greater in absolute value. This is also consistent with the results for a complete linear combination of LG modes (19) and for the sum of two angular harmonics (22).

TC of an optical vortex can be measured using a cylindrical lens by the method described in [31]. In Fig. 4 shows the intensity distributions at a double focal length from a cylindrical lens for optical vortices with an initial fractional TC (38). It can be seen that on the line at an angle of –45 degrees in the center of the picture are two zeros (two dark lines) (Fig. 4(a)) for µ < 2.5 and three zeros (three dark lines) for µ> 2.5 (Fig. 4(b), (c), d). As shown in Fig. 4, for arbitrary initial fractional TC between 2 and 2.5, TC of the optical vortex equals 2, and for arbitrary initial fractional TC [Eq. (38)] higher than 2.5 and lower than 3, TC of the beam equals 3. The experiment in Fig. 4 confirms the numerical result (48).

 

Fig. 4. Intensity distributions measured at a distance z = 200 mm (at a double focal length from a cylindrical lens) from a spiral phase plate with fractional order µ: (a) 2.3, (b) 2.5, (c) 2.7, (g) 2.9. The sizes of pictures are 4000 by 4000 microns.

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Appendix contains results of the numerical TC calculation by using different expressions, which yield different results. Computation of TC by Eq. (47) (z << z0) confirms that Eq. (48) is correct and coincides with calculations in [16], i.e. the jumps between the adjacent TC values are located at the half-integer values $\mu = n + 1/2$. Computation by Eq. (46) shows that these jumps are near every integer number, when $\mu \approx n + 0.1$. This result is close to that obtained in [32]. Computation by Eq. (45) (at z >> z0) and by Eq. (1) yields the result obtained in [33], i.e. the jumps between the adjacent integer TC values happen at the even integer values µ = 2p, p = 1,2,3… It is worth noting that all these computations are confirmed by experiments in [32], [33] and in Fig. 4. In all three cases, TC is always integer and at integer vortex order µ = n, TC always equals this order, i.e. TC = n. The results in papers [16], [32] and [33] are different only in that TC jumps (i.e. birth of new vortex) happen at different values of µ. Detailed comparison of works [16], [32] and [33] requires additional investigations.

9. Topological charge of an elliptic optical vortex embedded in a Gaussian beam

Let us analyze a simple example of an OV with introduced phase distortion by making it ellipse-shaped. While for a conventional OV the complex amplitude in the initial plane is given by

$$E({r,\varphi } )= A(r )\exp ({in\varphi } ), $$
for an elliptic vortex imbedded, say, into a Gaussian beam (or any other radially symmetric beam) it takes the form:
$$\begin{array}{l} {E_e}(x,y) = A(\sqrt {{x^2} + {y^2}} ){({x + i\alpha y} )^n} = \\ = A(\sqrt {{x^2} + {y^2}} ){({{x^2} + {\alpha^2}{y^2}} )^{n/2}}\exp \left( {in\arctan \left( {\frac{{\alpha y}}{x}} \right)} \right). \end{array}$$
Substituting (50) into (1) yields:
$$\begin{array}{l} TC = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {d\varphi \frac{\partial }{{\partial \varphi }}} \arg {E_e}(r,\varphi ) = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {d\varphi \frac{\partial }{{\partial \varphi }}} ({n\arctan ({\alpha \tan \varphi } )} )= \\ = \left( {\frac{{n\alpha }}{{2\pi }}} \right)\int\limits_0^{2\pi } {\frac{{d\varphi }}{{{{\cos }^2}\varphi + {\alpha ^2}{{\sin }^2}\varphi }}} = n. \end{array}$$
Note that a result similar to (51), but only for n = 1, was previously obtained in [34]. From (51) it follows that the fact that an optical vortex or SPP is ellipse-shaped does not change TC of the original simple OV in Eq. (49). At any degree of ellipticity (any α), an elliptic OV has TC = n. In the meantime, OAM of an elliptic OV is always lower than n, being equal to
$$\frac{{{J_z}}}{W} = \frac{{n{P_{n - 1}}(y )}}{{{P_n}(y )}} < n,$$
where y = (1 + α2)/(2α) > 1 and Pn(y) is Legendre polynomial. Figure 5 shows the intensity and phase distributions of a Gaussian beam with an elliptical vortex in the initial plane and after propagation in space for different ellipticities. The complex amplitude in the initial plane is ${E_e}({x,y} )= \exp [{ - {{({{x^2} + {y^2}} )} \mathord{\left/ {\vphantom {{({{x^2} + {y^2}} )} {{w^2}}}} \right.} {{w^2}}}} ]{({x + i\alpha y} )^n}$, where w is the waist radius of the Gaussian beam, n and α are the topological charge and ellipticity of the optical vortex, respectively. The following calculation parameters were used: w = 1 mm, n = 7, α = 1.1 [Fig. 5(a,b,c,d)], α = 1.5 [Fig. 5(e,f,g,h)], α = 3 [Fig. 5(i,j,r,l)], the distance of propagation in space z = z0/2, the computational domain is –R ≤ x, y ≤ R (R = 5 mm). The TC in the initial plane, calculated numerically by formula (1) (along a ring of radius 0.8R), is 6.9997 at α = 1.1, 6.9996 at α = 1.5, 6.9987 at α = 3, that is, in all cases about 7. At a distance z = z0/2 TC is 6.9989, 6.9988, and 6.9979, respectively. That is, it is also approximately equal to 7.

 

Fig. 5. Distributions of intensity (a, c, e, g, i, k) and phase (b, d, f, h, j, l) of a Gaussian beam with an elliptical vortex in the initial plane (a, b, e, f, i, j) and after propagation in space z = z0/2 (c, d, g, h, k, l) for different ellipticities. The red (dashed) rings on the phase distributions denote the radius of the ring by which the TC was calculated by the formula (1).

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10. Conclusion

Summing up, it has been theoretically shown that OVs conserve the integer TC when passing through an arbitrary aperture or shifted from the optical axis of an arbitrary axisymmetric carrier beam. If the beam contains a finite number of off-axis optical vortices with different-value same-sign TC, the total TC of the resulting beam has been shown to be equal to the sum of all constituent TCs. In this case, there is no topological competition, because it takes place as a result of on-axis superposition of OVs. By way of illustration, if an on-axis superposition is composed of a finite number of Laguerre-Gaussian modes (n, 0), the resulting TC equals that of the constituent mode with the highest TC (including sign). If the highest positive and negative TCs of the constituent modes are equal in magnitude, the “winning” TC is the one with the larger absolute value of the weight coefficient. If the constituent modes have the same weight coefficients, the resulting TC equals zero. If the beam is composed of two on-axis different-amplitude Gaussian vortices with different TC, the resulting TC equals that of the constituent vortex with the larger absolute value of the weight coefficient amplitude, irrespective of the correlation between the individual TCs. If the constituent beams have equal weight coefficients, there occurs degeneracy, with the resulting TC being equal to the mean arithmetic of the constituent Gaussian OVs. If in the superposition of two Gaussian OVs, one TC is odd and the other is even, the resulting TC in the initial plane is half-integer. As the beam propagates, degeneracy is eliminated, with the resulting TC becoming equal to the larger (positive) integer constituent TC. This effect has been given the name “topological competition of optical vortices”. Theoretical predictions have been corroborates by the numerical simulation and experiment.

Appendix. Numerical TC calculation by different expressions

Figure 6(a) shows TC of a Gaussian beam with the fractional-order vortex in the near field (z << z0) computed by Eq. (47) for –7 ≤ µ ≤ 7 with the step of 0.05. As seen in Fig. 6(a), TC jumps are located at the half-integer values, when µn + 0.5, i.e. like is described in [16].

 

Fig. 6. TC of a Gaussian beam with the fractional-order vortex computed by Eq. (47) (a), by Eq. (46) (b), and by Eqs. (45) and (1) (c).

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Figure 6(b) shows TC of a Gaussian beam with the fractional-order vortex computed by Eq. (46) for 5 ≤ µ ≤ 10 with the step of 0.05. It is seen in Fig. 6(b) that TC jumps are near every integer number, when µn + 0.1, like is described in [32].

Figure 6(c) shows TC of a Gaussian beam with the fractional-order vortex in the far field computed by Eqs. (45) (A = 0, B = f, C = –1/f, D = 0) and (1) for –5 ≤ µ ≤ 5 with the step of 0.05. According to Fig. 6(c), the jumps between the adjacent integer TC values happen at the even integer values, like is described in [33].

Funding

Russian Foundation for Basic Research (18-07-01129, 18-29-20003, 19-29-01233); Russian Science Foundation (17-19-01186); Ministry of Education and Science of the Russian Federation (007-ГЗ/Ч3363/26).

Acknowledgments

This work was supported by the Russian Foundation for Basic Research grants 18-29-20003 (in the parts “Topological charge of an optical vortex passing through an amplitude mask”, “Topological charge of an axial linear combination of optical vortices”, “Topological charge outside an axial optical vortex”), 18-07-01129 (in the part “Topological charge of an optical vortex with several centers of phase singularity”) and 19-29-01233 (in the part “The topological charge of an elliptic vortex in a Gaussian beam”), by the Russian Science Foundation grant 17-19-01186 (in the parts “Topological charge in an arbitrary plane” and “Topological charge of the sum of two Gaussian optical vortices”), as well as by the Ministry of Science and Higher Education of the Russian Federation in the framework of work on the State assignment of the Federal Research Center for Crystallography and Photonics of the Russian Academy of Sciences (in the part “Topological charge for an optical vortex with an initial fractional charge”).

Disclosures

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

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19. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Orbital angular momentum and informational entropy in perturbed vortex beams,” Opt. Lett. 44(23), 5687–5690 (2019). [CrossRef]  

20. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge,” Comput. Opt. 43(5), 723–734 (2019). [CrossRef]  

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

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

23. D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018). [CrossRef]  

24. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Asymmetric Gaussian optical vortex,” Opt. Lett. 42(1), 139–142 (2017). [CrossRef]  

25. V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. G. Abramochkin, “Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex,” Comput. Opt. 41(1), 22–29 (2017). [CrossRef]  

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

27. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993). [CrossRef]  

28. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

29. J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007). [CrossRef]  

30. C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017). [CrossRef]  

31. V. V. Kotlyar, A. A. Kovalev, and A. P. Pofirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017). [CrossRef]  

32. A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Study of the birth of a vortex at Fraunhofer zone,” Opt. Lett. 37(21), 4552–4554 (2012). [CrossRef]  

33. J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019). [CrossRef]  

34. G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020). [CrossRef]  

References

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  1. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  2. A. Volyar, M. Bretsko, Y. Akimova, and , and Y. Egorov, “Vortex avalanche in the perturbed singular beams,” J. Opt. Soc. Am. A 36(6), 1064–1071 (2019).
    [Crossref]
  3. Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
    [Crossref]
  4. H. Zhang, X. Li, H. Ma, M. Tang, H. Li, J. Tang, and Y. Cai, “Grafted optical vortex with controllable orbital angular momentum distribution,” Opt. Express 27(16), 22930–22938 (2019).
    [Crossref]
  5. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019).
    [Crossref]
  6. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27(8), 11236–11251 (2019).
    [Crossref]
  7. V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. S. Kozlova, “Orbital angular momentum of a laser beam behind an off-axis spiral phase plate,” Opt. Lett. 44(15), 3673–3676 (2019).
    [Crossref]
  8. S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
    [Crossref]
  9. A. E. Siegman, Lasers (University Science, 1986).
  10. J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [Crossref]
  11. F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [Crossref]
  12. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007).
    [Crossref]
  13. M. A. Bandres and J. C. Gutierrez-Vega, “Circular beams,” Opt. Lett. 33(2), 177–179 (2008).
    [Crossref]
  14. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Asymmetric Bessel modes,” Opt. Lett. 39(8), 2395–2398 (2014).
    [Crossref]
  15. A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Asymmetric Laguerre- Gaussian beams,” Phys. Rev. A 93(6), 063858 (2016).
    [Crossref]
  16. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
    [Crossref]
  17. G. Gbur and R. K. Tuson, “Vortex beam propagation through atmodspheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008).
    [Crossref]
  18. M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
    [Crossref]
  19. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Orbital angular momentum and informational entropy in perturbed vortex beams,” Opt. Lett. 44(23), 5687–5690 (2019).
    [Crossref]
  20. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge,” Comput. Opt. 43(5), 723–734 (2019).
    [Crossref]
  21. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).
    [Crossref]
  22. M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carring optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
    [Crossref]
  23. D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
    [Crossref]
  24. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Asymmetric Gaussian optical vortex,” Opt. Lett. 42(1), 139–142 (2017).
    [Crossref]
  25. V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. G. Abramochkin, “Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex,” Comput. Opt. 41(1), 22–29 (2017).
    [Crossref]
  26. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
    [Crossref]
  27. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
    [Crossref]
  28. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  29. J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
    [Crossref]
  30. C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
    [Crossref]
  31. V. V. Kotlyar, A. A. Kovalev, and A. P. Pofirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
    [Crossref]
  32. A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Study of the birth of a vortex at Fraunhofer zone,” Opt. Lett. 37(21), 4552–4554 (2012).
    [Crossref]
  33. J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
    [Crossref]
  34. G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
    [Crossref]

2020 (1)

G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
[Crossref]

2019 (10)

J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Orbital angular momentum and informational entropy in perturbed vortex beams,” Opt. Lett. 44(23), 5687–5690 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge,” Comput. Opt. 43(5), 723–734 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and , and Y. Egorov, “Vortex avalanche in the perturbed singular beams,” J. Opt. Soc. Am. A 36(6), 1064–1071 (2019).
[Crossref]

Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
[Crossref]

H. Zhang, X. Li, H. Ma, M. Tang, H. Li, J. Tang, and Y. Cai, “Grafted optical vortex with controllable orbital angular momentum distribution,” Opt. Express 27(16), 22930–22938 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27(8), 11236–11251 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. S. Kozlova, “Orbital angular momentum of a laser beam behind an off-axis spiral phase plate,” Opt. Lett. 44(15), 3673–3676 (2019).
[Crossref]

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[Crossref]

2018 (1)

D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
[Crossref]

2017 (4)

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Asymmetric Gaussian optical vortex,” Opt. Lett. 42(1), 139–142 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. G. Abramochkin, “Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex,” Comput. Opt. 41(1), 22–29 (2017).
[Crossref]

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Pofirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
[Crossref]

2016 (1)

A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Asymmetric Laguerre- Gaussian beams,” Phys. Rev. A 93(6), 063858 (2016).
[Crossref]

2014 (1)

2012 (1)

2009 (1)

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

2008 (2)

2007 (2)

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007).
[Crossref]

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

2004 (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

1997 (1)

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

1993 (2)

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

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[Crossref]

1992 (1)

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

1987 (2)

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

1974 (1)

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

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[Crossref]

Abramochkin, E. G.

V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. G. Abramochkin, “Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex,” Comput. Opt. 41(1), 22–29 (2017).
[Crossref]

Akimova, Y.

Alexeyev, C. N.

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[Crossref]

Allen, L.

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

Bandres, M. A.

Barnett, S. M.

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. 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. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

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

Bretsko, M.

Brundavanam, M. M.

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[Crossref]

Cai, Y.

Ceballos-Herrera, D. E.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Cheng, W.

G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
[Crossref]

Durnin, J.

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Egorov, Y.

Egorov, Y. A.

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[Crossref]

Ferrando, A.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Fonseca, E. J. S.

Franke-Arnold, S.

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Gao, J.

Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
[Crossref]

Garcia-March, M. A.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Gbur, G.

Gori, F.

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Gorshkov, V. N.

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

Gotte, J. B.

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Guattary, G.

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Gutierrez-Vega, J. C.

Heckenberg, N. R.

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

Herbi, D.

D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
[Crossref]

Hickmann, J. M.

Indebetouw, G.

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

Jacob, P.

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[Crossref]

Jesus-Silva, A. J.

Khonina, S. N.

Kotlyar, V. V.

Kovalev, A. A.

Kozlova, E. S.

Li, H.

Li, X.

Liang, G.

G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
[Crossref]

Ma, H.

Maji, S.

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[Crossref]

Malos, J. T.

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

Micely, J. J.

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Nye, J. F.

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

Padovani, C.

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Pofirev, A. P.

Porfirev, A. P.

Rasouli, S.

D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Sahu, S.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Skidanov, R. V.

Soifer, V. A.

Soskin, M. S.

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

Spreeuw, R.

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

Tang, J.

Tang, M.

Tuson, R. K.

Vastnetsov, M. V.

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

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[Crossref]

Volyar, A.

Volyar, A. V.

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[Crossref]

Wang, L.

Wen, J.

Woerdman, J.

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

Yang, X.

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Figures (6)

Fig. 1.
Fig. 1. Distributions of intensity (a,c,e,g) and phase (b,d,f,h) of a Gaussian optical vortex bounded by a sector-shape diaphragm in the initial plane z = 0 (a-d) and after propagation in free space (e-h) for two different angles of the sector aperture α = π/6 (a,b,e,f) and α = π/4 (c,d,g,h). Red rings (f,h) show the circle over which the TC was calculated. Yellow text (e,g) shows the TC.
Fig. 2.
Fig. 2. Distributions of intensity (a, c, e, g, i, k) and phase (b, d, f, h, j, l) of a Gaussian beam with an off-axis optical vortex in the initial plane (a, b, e, f, i, j) and after propagation in space (c, d, g, h, k, l) for different lateral displacements of the vortex from the optical axis. Calculation parameters: waist radius w = 1 mm, TC is n = 7, displacement r0 = w0/4 (a-d), r0 = w0/2 (e-h), r0 = 2w0 (i-l); φ0 = 0 in all figures, the propagation distance in space is z = z0/2 (z0 is the Rayleigh distance). The red (dashed) rings on the phase distributions denote the radius of the ring by which the TC was calculated by formula (1).
Fig. 3.
Fig. 3. The intensity (a, c) and phase (b, d) of the axial superposition of two Gaussian OVs with TC 12 and 7, but with the same weight amplitudes (in (22)) in the initial plane (a, c) and at the Rayleigh distance (c, d). The red (dashed) rings on the phase distributions denote the radius of the ring by which the topological charge was calculated by the formula (1).
Fig. 4.
Fig. 4. Intensity distributions measured at a distance z = 200 mm (at a double focal length from a cylindrical lens) from a spiral phase plate with fractional order µ: (a) 2.3, (b) 2.5, (c) 2.7, (g) 2.9. The sizes of pictures are 4000 by 4000 microns.
Fig. 5.
Fig. 5. Distributions of intensity (a, c, e, g, i, k) and phase (b, d, f, h, j, l) of a Gaussian beam with an elliptical vortex in the initial plane (a, b, e, f, i, j) and after propagation in space z = z0/2 (c, d, g, h, k, l) for different ellipticities. The red (dashed) rings on the phase distributions denote the radius of the ring by which the TC was calculated by the formula (1).
Fig. 6.
Fig. 6. TC of a Gaussian beam with the fractional-order vortex computed by Eq. (47) (a), by Eq. (46) (b), and by Eqs. (45) and (1) (c).

Equations (52)

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TC=limr12π02πdφφargE(r,φ)=12πlimrIm02πdφE(r,φ)/φE(r,φ).
E(r,φ)=A(r)exp(inφ)f(φ),
f(φ)={1,α<φ<α,δ1,otherwise.
TC=limrIm2π02πdφinE(r,φ)+A(r)einφf(φ)φE(r,φ)=Im2πlimr02πdφ(in+f(φ)φ1f(φ))=n.
f(r,φ)={1,(r,φ)Ω,δ1,(r,φ)Ω,
TC=Im2πlimr02πdφ(in+f(r,φ)φ1f(r,φ))=n.
Jz=Im002πE¯(r,φ,z)(E(r,φ,z)dφ)rdrdφ=Im0ααA(r)einφ(inA¯(r)einφ)rdrdφ=2αn0|A(r)|2rdr,JzW=n,W=2α0|A(r)|2rdr.
E(x,y,0)=exp[x2+y2w2+inarg(x+iy)]rect{arg[(xx0)+i(yy0)]2α},
En(r,φ)=(reiφr0eiφ0w)nA(r).
TC=limrIm2π02πdφinreiφreiφr0eiφ0=12πImlimr02πdφinreiφreiφr0eiφ0=n.
E(r,φ)=circ(rR)exp[inarctan(rsinφrcosφx0)],
TC=n2π02πr2rx0cosφR2+x022rx0cosφdφ={n,x0<R,n/2,x0=R.
JzW=n(1x02R2).
{x=acosφp,y=asinφp,
E(r,φ,z)=1σ(2w0)mexp(r2σw02)(rmeimφσmam),
TC=12πlimrIm{02πimσmrmeimφσmrmeimφamdφ}=m.
Em(r,φ,z=0)=A(r)p=1m(reiφrpeiφp)mp.
TC=12πlimrIm{02πireiφp=1mmpreiφrpeiφpdφ}=p=1mmp.
EN,M(r,φ,z=0)=exp(r2w2)n=MNCn(rw)|n|einφ.
TC=12πlimrIm{02πin=MNnCn(rw)|n|einφn=MNCn(rw)|n|einφdφ}.
TC=12πIm{02πiN(CNeiNφCNeiNφ)(CNeiNφ+CNeiNφ)dφ}.
E(r,φ)=(aeinφ+beimφ)er2/r2w2w2,
TC=12πlimrIm{02πE(r,φ)/E(r,φ)φφE(r,φ)dφ}=12πRe{02πnaeinφ+mbeimφaeinφ+beimφdφ}.
TC=12π02π(n+m2+nm2|a|2|b|2|a|2+|b|2+2|a||b|cost)dt.
TC=n+m2+12πnm2|a|2|b|2|a|2+|b|202πdt1+2|a||b||a|2+|b|2cost.
0πcos(nx)dx1+acosx=π1a2(1a21a)n[a2<1,n0].
TC=n+m2+nm2|a|2|b|2||a|2|b|2|.
OAM=na2+mb2a2+b2.
E(r,φ)=|a|(einφ+iarga+eimφ+iargb)er2/r2w2w2=2|a|cos(nφmφ+argaargb2)exp(r2w2+inφ+mφ+arga+argb2).
TC=limr12π02πφ(nφ+mφ+arga+argb2)dφ=n+m2.
E1(r,φ)=(eiφ+ei7φ)er2/r2w2w2,E2(r,φ)=(ei2φ+ei6φ)er2/r2w2w2,E3(r,φ)=(ei3φ+ei5φ)er2/r2w2w2,E4(r,φ)=ei4φer2/r2w2w2.
E(r,φ)=er2/r2w2w2+inφ,
Ez(ρ,θ)=(i)n+1π2z0Bq1ξexp(ikDρ22B+inθξ)[In12(ξ)In+12(ξ)],
ξ=(z0/z0BB)2(ρ/ρww)2/(z0/z0BB)2(ρ/ρww)2(2q1)(2q1),q1=1i(A/ABB)z0.
Ez(ρ,θ)=iπ2z0Bq1exp(ikDρ22B)ξexp(ξ)×{a(i)neinθ[In12(ξ)In+12(ξ)]+b(i)meimθ[Im12(ξ)Im+12(ξ)]}.
In12(ξ)In+12(ξ)eξ2πξ{[1(n1)218ξ][1(n+1)218ξ]}=neξ2ξ2πξ.
Ez(ρ,θ)=iπ2z0Bq1exp(ikDρ22B)ξexp(ξ)×[a(i)nexp(inθ)neξ2ξ2πξ+b(i)mexp(imθ)meξ2ξ2πξ]=iz04Bq1ξexp(ikDρ22B)[an(i)nexp(inθ)+bm(i)mexp(imθ)].
Eμ(r,φ,z)=exp(iμφ)Ψ(r,z)=eiπμsinπμπΨ(r,z)n=einφμn.
Jz=Im002πE¯(r,φ,z)(E(r,φ,z)φ)rdrdφ
Jz=Wsin2(πμ)π2n=n(μn)2,
W=002πE(r,φ,z)E¯(r,φ,z)rdrdφ.
n=1n2(n2±a2)2=π4a[±{cothπacotπa}a{cosech2πacosec2πa}],
JzW=μsin2πμ2π.
Eμ(r,φ,z=0)=exp(iμφ(rw)2)=eiπμsinπμπn=einφr2/w2μn.
E2(ρ,θ)=12π(iz0q1z)exp(ikρ22z+iπμ)sin(πμ)xexp(x)××m=(i)|m|exp(imθ)μm[I|m|12(x)I|m|+12(x)]
TC=Re2π{02π[n=(i)|n|n|n|einφμn][n=(i)|n||n|einφμn]1dφ}.
TC=Re2π{02π[n=neinφμn][n=einφμn]1dφ}.
TC=n=nrect(μn),rect(x)={1,|x|1/2,0,|x|>1/2
E(r,φ)=A(r)exp(inφ),
Ee(x,y)=A(x2+y2)(x+iαy)n==A(x2+y2)(x2+α2y2)n/2exp(inarctan(αyx)).
TC=12π02πdφφargEe(r,φ)=12π02πdφφ(narctan(αtanφ))==(nα2π)02πdφcos2φ+α2sin2φ=n.
JzW=nPn1(y)Pn(y)<n,

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