## Abstract

The analytical expression for the propagation of guided optical vortices through free space is derived and used to study the dynamic evolution of guided optical vortices after passing through the free space, and the dependence of guided optical vortices on the control parameters where the effect of propagation distance is stressed. It is shown that the motion, pair creation and annihilation of guided optical vortices may take place. In particular, the creation and annihilation of a pair of guided optical vortices do not take place by varying fiber length.

©2012 Optical Society of America

## 1. Introduction

Singular optics is recognized as a rapidly emerging branch of modern optics that deals with singularities in optical fields such as caustics in geometrical optics, screw/edge phase dislocations in scalar wave optics, and *C*-points/disclinations in vector wave optics [1–4]. Among these, optical vortices (screw/edge phase dislocations) have recently attracted much interest because of their importance in the theoretical aspect and potential applications in optical communications, high-resolution metrology, optoelectronics, optical manipulation, atom trapping and micromachining etc. [3, 5, 6]. Optical vortices are known to profoundly affect the physical properties of systems thus research into the vortex dynamical phenomena can provide surprising and potentially useful new optical phenomena. Theoretically, optical fibers can be the source of optical vortices with proper parameters, in which optical vortices can appear and propagate [7–9]. However, it is difficult to obtain a single optical vortex in real fibers unless some defects exist in the fiber [10]. Lim and Lee discussed the formation, structural characteristics and properties of optical vortices in optical fibers theoretically and experimentally, and also found that all the mixed modes except for fundamental linearly polarized modes displayed phase singularities in the transverse plane [11]. References [12–15] showed the processes of the formation of directed optical vortices of a low-mode fiber, where the optical vortices is formed of the combinations of CP_{11} (two orthogonal linearly polarized (LP) modes with a relative phase shift π/2 and the same parity) modes. Above all, circular vortices (CV) and instability vortices (IV) were both introduced. The interconversion of dislocations and disclinations of the field of a few-mode optical fiber is studied in detail in [16–19], which particularly made an experimental and theoretical study of the physical mechanisms responsible for the formation of an optical vortex. It was found that the propagation of a circular polarized CV vortex can be represented as a screwing of a helical wavefront into the fiber core and the propagation of a linearly polarized vortex in free space is characterized by the translational displacement of a helical wavefront [20]. An optical fiber with axial losses not only can retain a guided optical vortex but also operates as a mode filter for the exciting field [21]. Egorov et al. showed that the birth and death events of a pair of optical vortices after emitting from an optical fiber took place, and also a single vortex may appear separately [22]. The simultaneous generation, propagation and detection of optical vortices using all fiber-optic system were made by Kumar, and the method might find applications for efficient optical trapping of micro-particles, optical communications, for studying the orbital angular momentum state light and for the sensing of smallest amount of coherent photonic forces [23]. A two-mode fiber can convert the input Gaussian beam into Laguerre-Gaussian with single helical charge or coherent linear combinations of the different vector modes guided in the fiber [24]. It is suggested that annular core optical fibers (ACFs) where the annulus between the two non-guiding regions serves the purpose of core [25–29] may be used in optical sensors, optical sensitivity, optical communications, and so on. However, to the best of our knowledge, up to now, the free-space propagation dynamics of guided optical vortices excited in an annular core fiber has not been found in the literature. This paper is devoted to the study of the dynamic evolution behavior of guided optical vortices in free-space propagation.

## 2. Theoretical formulation

Figure 1
shows the refractive index profile of the annular core fiber with two layer boundaries *a* and *b*, the annulus being the guiding region having refractive index *n*_{1} covered with two concentric claddings of the same *n*_{2}. Assume that the index difference isvery small, so the weak guidance approximation [30] can be well applicable to the case. The guided mode in the annular core fiber is given in cylindrical coordinates by its transverse component [25–29, 31]

*J*and

_{m}*Y*(

_{m}*I*and

_{m}*K*) are the (modified) Bessel functions of the first and the second kind of order

_{m}*m*, respectively, ${u}^{2}={n}_{1}^{2}{k}^{2}-{\beta}_{m}^{2}$, ${w}^{2}={\beta}_{m}^{2}-{n}_{2}^{2}{k}^{2}$, and the wave number

*k*is related to the wave length

*λ*by

*k*= 2

*π*/

*λ*. In Eq. (1),

*C*

_{1},

*C*

_{2},

*C*

_{3}and

*C*

_{4}are arbitrary constants which can be determined by the boundary conditions, according to which fields and their first derivatives must have a smooth match at different interfaces. We derive the characteristic eigenvalue equation in

*β*by matching

*F*and δ

*F*/δ

*r*at the different layer interfaces between annulus, inner cladding, and outer cladding. In the limit of a weakly guiding fiber, this continuity is equivalent to the continuity of the transverse components. This leads to a (4 × 4) matrix which determinant must be equal to zero to ensure a nontrivial solution. The dispersion equation is expressed as

*a*→0) so that the annular core fiber tends towards the standard central core step index fiber, and the dispersion equation takes the following limit form

_{11}mode combination can exist (

*m*≤1) should be chosen. In the following Sec. 3 the two-mode annular core optical fiber with two layer boundaries

*a*= 20 µm and

*b*= 26 µm, refractive indices

*n*

_{1}= 1.45213,

*n*

_{2}= 1.45156,

*λ*= 1.55 µm is taken as the object studied. From Eq. (2), we obtain propagation constant

*β*

_{0}= 5.8846,

*C*

_{1}= 0.4578,

*C*

_{2}= −1.4823,

*C*

_{3}= 1,

*C*

_{4}= 5.3150 when

*m*= 0 and

*β*

_{1}= 5.8844,

*C*

_{1}= 0.6460,

*C*

_{2}= 0.1530,

*C*

_{3}= 1,

*C*

_{4}= 1.1684 when

*m*= 1. Thus, there are LP

_{01}mode and LP

_{11}mode in the annular core optical fiber in scalar approximation.

We consider the field of a stable single-charged CV vortex in a two-mode annular core fiber when it is perturbed by the circularly polarized field of the HE_{11} mode. The field of this mode perturbs the field of the vortex and alters the location of the pure screw dislocations. The transverse electric field containing a single-charged CV vortex and the HE_{11} mode which are both right-circularly polarized excited in an annular optical fiber reads as [20]

**e**

^{+1}and (

*r*

_{0},

*θ*

_{0},

*z*

_{0}) denote the unit vector of the right-circular polarization and the cylindrical polar coordinate system of the annular core fiber, respectively,

*z*

_{0}is the fiber length,

*η*is amplitude factor, and ${\beta}_{1}^{1}={\beta}_{1}+\Delta \beta $ and ${\beta}_{0}$are the propagation constants of the CV and the HE

_{11}mode fields, respectively. In addition, $\Delta \beta =0$, ${\beta}_{1}^{1}$ is degenerated with respect to the propagation constant in scalar approximation.

*F*

_{1}(

*r*

_{0}) and

*F*

_{0}(

*r*

_{0}) are radial functions of the transverse fields of the annular core fiber in Eq. (1) for

*m*= 1 and 0 [31].

Based on the framework of the paraxial approximation and Fresnel diffraction integral, after tedious integral calculations, the output fields after emitting from an annular core fiber in the *z* plane in free space are given by

*x*,

*y*,

*z*) is the Cartesian coordinate system of the field in free space. The positions of guided optical vortices in the field are determined by [1]

*e*(

_{t}*x*,

*y*,

*z*). From Eqs. (1) and (5), we see that the evolution of guided optical vortices in free space depend on the propagation distance

*z*, amplitude factor

*η*, fiber length

*z*

_{0}, and on the fiber parameters like refractive indices

*n*

_{1},

*n*

_{2}, two layer boundaries

*a*and

*b*. Therefore, the transverse position and number of guided optical vortices can be controlled by varying a control parameter

*z*,

*η*,

*z*

_{0}or a fiber parameter.

## 3. Propagation of guided optical vortices in free space

The dynamic evolution behavior of guided optical vortices in free space is illustrated numerically by using Eqs. (1), (4)–(6), and in the following numerical calculations *n*_{1}, *n*_{2}, *a*, *b* and *λ* are fixed which can guarantee the two-mode optical fiber.

Figures 2(a)
–2(j) give the contour lines of phase and intensity patterns in the *z* plane, where the calculation parameters are *η* = 0.1, *z*_{0} = 5 × 10^{5} µm, *z* = 80 µm in Figs. 2(a) and 2(b), *z* = 965 µm in Figs. 2(c) and 2(d), *z* = 1040 µm in Figs. 2(e) and 2(f), *z* = 1060 µm in Figs. 2(g) and 2(h), *z* = 10^{4} µm in Figs. 2(i) and 2(j). Figure 3
represents the distance Δ/*λ* between a pair of guided optical vortices B and C versus the propagation distance *z*, where the other calculation parameters are the same as in Fig. 2. From Figs. 2(a) and 2(b) we see that near the output of the annular core fiber there is only one optical vortex who is marked A located at (−0.86 µm, −2.61 µm) in free space. The topological charge of the optical vortex A is + 1 by analyzing vorticity of phase contours around the vortices [3]. The results are similar to the fields of a few-mode fiber in Ref [20]. However, A further increase of z = 965 µm in Figs. 2(c) and 2(d) leads to the creation of a pair of oppositely charged vortices B with topological charge + 1 and C with topological charge −1 located at (31.94 µm, 61.73 µm) and (−53.48 µm, 42.66 µm), respectively. With increasing the propagation distance from z = 965 µm to z = 1040 µm in Figs. 2(e) and 2(f), the oppositely charged vortices B, C approach each other. However, when the propagation distance gradually increases, say *z* = 1044.338 µm and 1044.3387 µm in Fig. 3, the vortices B and C approach to very small distance Δ = 0.6*λ* and Δ = 0.3*λ*, respectively, leading to subwavelength structures [32]. As the propagation distance increases to *z* = 1060 µm in Figs. 2(g) and 2(h), the collision and annihilation of the vortices B and C appear. Finally, if *z* = 10^{4} µm in Fig. 2(i) and 2(j), the optical vortex A still exists which moves from (−4.32 µm, −0.80 µm) in Figs. 2(g) and 2(h) to (−22.90 µm, 1.83 µm), and the creation of a pair of oppositely charged optical vortices such as the vortices D and E also appear. In addition to D and E, there are other pairs of optical vortices outside the region in Figs. 2(i) and 2(j) which are not shown because the intensity of which is almost equal to zero. It means that when *z*>10^{4} µm the longer the propagation distance, the larger the number of guided optical vortices. Therefore, the motion, pair creation and annihilation of guided optical vortices may take place, when guided optical vortices propagate in free space. It can be seen that the total topological charge is equal to + 1 and remains unchanged upon propagation.

The contour lines of phase and intensity patterns in the plane z = 1040 µm for selected values of amplitude factor *η* are depicted in Figs. 4(a)
–4(f) and Figs. 2(e) and 2(f), where (a) and (b) *η* = 0, (c) and (d) *η* = 0.21, (e) and (f) *η* = 0.22, and z_{0} = 5 × 10^{5} µm. For *η* = 0 in Figs. 4(a) and 4(b), namely, the perturbing field is completely absent, the guided optical vortex A with topological charge + 1 still exists in the location of (0, 0) in the plane *z* = 1040 µm. When the amplitude factor *η* is increased to *η* = 0.1 in Figs. 2(e) and 2(f) a pair of optical vortices B and C with topological charge + 1, −1 are present at (53.81 µm, −46.16 µm), (4.05 µm, −70.01 µm), respectively. The guided optical vortex A shifts to (−4.25 µm, −0.79 µm), whose topological charge remains unchanged. With increasing the amplitude factor from *η* = 0.1 to *η* = 0.21 in Figs. 4(c) and 4(d) vortices B and C move away from each other and are located at (68.59 µm, −22.79 µm) and (−28.93 µm, −62.55 µm), respectively. With a further increase of the amplitude factor to *η* = 0.22 in Figs. 4(e) and 4(f) the creation of a pair of vortices D and E with opposite topological charge + 1, −1 takes place. Vortices A, B and C shift to (−8.94 µm, −1.98 µm), (68.60 µm, −23.12 µm), and (−29.51 µm, −62.18 µm), respectively. Similar effect may appear by increasing the amplitude factor *η*, and is omitted here. It is seen that the larger amplitude factor *η* leads to the greater number of the guided optical vortices. However, the total topological charge is conserved in the variation of *η*.

Figures 5(a)
–5(h) show the intensity patterns and contour lines of phase in the plane z = 1040 µm for different fiber length (a) and (b) *z*_{0} = 1 × 10^{5} µm, (c) and (d) *z*_{0} = 4.89 × 10^{5} µm, (e) and (f) *z*_{0} = 4.95 × 10^{5} µm, (g) and (h) *z*_{0} = 6 × 10^{5} µm, where *z* = 1040 µm, *η* = 0.1. When the fiber length *z*_{0} is 1 × 10^{5} µm in Figs. 5(a) and 5(b), there are three guided vortices A, B and C with topological charge + 1, + 1, −1 in the plane *z* = 1040 µm which are located at (4.28 µm, −0.66 µm), (−35.42 µm, 61.42 µm), and (19.44 µm, 67.38 µm), respectively. As *z*_{0} is increased to *z*_{0} = 4.89 × 10^{5} µm in Figs. 5(c) and 5(d), vortices A, B and C shift to (2.98 µm, 3.14 µm), (−70.66 µm, 5.82 µm), and (−44.35 µm, 54.33 µm), respectively. With increasing fiber length to *z*_{0} = 4.95 × 10^{5} µm and *z*_{0} **=** 6 × 10^{5} µm in Figs. 5(e)–5(h), the motion of vortices A, B and C is observed, e.g., the position of vortex A shifts from (−2.49 µm, 3.54 µm) in Figs. 5(e) and 5(f) to (2.14 µm, 3.76 µm) in Figs. 5(g) and 5(h). It can be seen that the number of guided optical vortices is always three, which is independent of the fiber length in the free space propagation. Therefore, by varying fiber length the motion of guided vortices and the conservation of topological charge in free-space propagation may take place.

## 4. Concluding remarks

In conclusion, to the best of our knowledge, we have found for the first time the free-space propagation of guided optical vortices excited in an annular core fiber. The analytical expression for the propagation of guided optical vortices through free space has been derived, and used to study the dynamic evolution of guided optical vortices after propagating through the free space, and the dependence of guided optical vortices on the control parameters, such as the propagation distance, amplitude factor, fiber length, refractive indices, and the two layer boundaries *a* and *b*, where the effect of propagation distance has been stressed. It has been shown that, the motion, pair creation and annihilation may take place after the propagation in free space. In particular, the creation and annihilation of a pair of guided optical vortices do not take place by varying fiber length. Thus, the conservation of topological charge of guided optical vortices in free-space propagation is valid. The results obtained in this paper are beneficial to understanding the dynamic behavior of guided optical vortices propagating in free space and to their controlling.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China (NSFC) under grant No. 61138007.

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