Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams

Guo Liang; Yuqing Wang; Qi Guo; Huicong Zhang

doi:10.1364/OE.26.008084

1. Introduction

The diffraction phenomenon of light is described as the interference of electromagnetic waves according to the Huygens-Fresnel principle, which is a very old story back to the 15th century [1]. Due to the diffraction, the (1+2)-dimensional paraxial optical beam expands its cross-sectional shape uniformly along the two transverse directions when it propagates in an isotropic bulk medium [2, 3]. The situation in anisotropic media, either a uniaxial one [4–9] or a biaxial one [10,11], however, is a little bit complicated. This circumstance entails that the diffraction is anisotropic and does not exhibit the well-know circularly symmetric behavior typical of the isotropic one [9]. The beam loses its initial cylindrical symmetry during the propagation [6], and the fundamental (basic) mode among all eigenmodes [2, 3] is the Gaussian beam with an elliptical cross section since only an elliptical beam can preserve its cross-sectional shape on the propagation [8]. There are two theoretical descriptions of the propagation in the anisotropic media. One of them is an integration of all of plane-wave angular spectrums of the electrical field [6, 7, 9], and the other is to solve the paraxial (differential) equation governing the beam propagation [4,5,7,10] [see, Eq. (3) in Sec. 2], in which there exists the anisotropic diffraction (AD) [12], that is, the two coefficients (the Fresnel diffraction coefficients [10]), δ_xx and δ_yy, of the second derivative terms are different.

Different from a long history of diffraction, it was 1992 when Allen and coworkers [13] recognised that optical beams can carry a discrete orbital angular momentum (OAM). The concept of the OAM is now leading to new understanding of a wide range of phenomena [14], including fundamental processes in Bose-Einstein condensates [15] and quantum information [16], while the associated technologies have led to new applications in optical manipulation [17,18]. The optical beams carrying the OAM are usually associated with optical vortices and related ring-shaped beams with the phase-singularity, such as the Laguerre-Gauss beams [13] and the Bessel beams [19], as well as the other hollow beams [20]. At the same time, the vortex-free beam with nonzero OAM and without phase-singularity was also found [21], and is an elliptical Gaussian beam with the cross-phase [22, 23], which the discoverer of the beam [21] and the other [24] called as an “astigmatic (elliptical Gaussian) beam” [25]. It has been shown that such elliptical Gaussian beams with nonzero OAM can survive in the form of the stably spiraling elliptic solitons in the nonlinear media with both linear and nonlinear isotropy [22, 23]. To explain the physical mechanism of such a phenomenon, we (two of the authors of this paper) guessed that [23] the OAM could result in effective anisotropic diffraction, and went to explain qualitatively why the elliptic mode can survive in the medium with both linear and nonlinear isotropy where only the mode with cylindrical-symmetry is supposed to exist. This open question will be addressed in this paper.

The diffraction and the OAM are considered to be two essential and intrinsic properties of the many properties that optical beams possess. Their interaction, that is the diffraction of the optical beam with the OAM, has been extensively exploited, and most of them focused on optical vortice diffraction in presence of obstacles, including a knife edge [26], a single slit [27], forked [28] and gradually-changing-period [29] gratings, triangular and square apertures [30,31], an annular aperture [32] and so on. The propagation properties of optical vortices in bulk linear media have also been discussed [33]. But so far, there are few investigations, to our knowledge at least, that the optical beam with the OAM and without phase-singularity, for example the elliptical Gaussian beam with the cross-phase, propagates in the bulk linear materials without obstacles, except the measurement of the orbital angular momentum of such a beams published just now [24].

In this paper, we discuss the linear propagations of the elliptic beam without the OAM in the anisotropic bulk medium and that with the nonzero OAM in the isotropic one, respectively. The AD induced by the OAM is analytically obtained by the comparison of two different kinds of propagations. Besides, it is also found that the elliptic beam with the OAM will rotate during the propagation, and the rotation property is discussed in detail.

2. Paraxial equation governing the beam propagation and its eigenmodes

The paraxial optical beams propagating in the anisotropic bulk media is in general modeled by [10]

i (\frac{\partial ϕ}{\partial ζ} - δ_{ξ} \frac{\partial ϕ}{\partial ξ} - δ_{η} \frac{\partial ϕ}{\partial η}) + \frac{1}{2 k_{0}} (δ_{ξ ξ} \frac{\partial^{2} ϕ}{\partial ξ^{2}} + δ_{ξ η} \frac{\partial^{2} ϕ}{\partial ξ \partial η} + δ_{η η} \frac{\partial^{2} ϕ}{\partial η^{2}}) = 0,

where ϕ is the slowly varying amplitude for the paraxial beam, ζ is the longitudinal (the propagation direction of the beams) coordinate, ξ and η are the transverse coordinates normal to ζ, k₀ = k_ζ (k_ξ, k_η)|_{k_ξ=k_η=0}, δ_i = ∂_{k_i}k_ζ (k_ξ, k_η)|_{k_ξ=k_η=0} and

{δ_{i l} = \partial_{k_{i} k_{l}}^{2} k_{ζ} (k_{ξ}, k_{η}) |}_{k_{ξ} = k_{η} = 0}

(i, l means ξ or η). The function k_ζ (k_ξ, k_η) can be derived from the dispersion equation to determine the plane wave modes with the form of exp [i(k_ζζ + k_ξξ + k_ηη − ωt)] existing in the anisotropic media [10, 34]. δ_ξ and δ_η describe the walk-off of the beam center in ξ and η directions, and δ_ξξ, δ_ξη and δ_ηη are the Fresnel diffraction coefficients [10]. All of δ-coefficients depend upon the direction of the propagation. For the uniaxial crystals, the paraxial equation for the extraordinary light is simplified to [4,10,34]

i (\frac{\partial ϕ}{\partial ζ} - δ \frac{\partial ϕ}{\partial ξ}) + \frac{1}{2 k_{0}} (δ_{ξ ξ} \frac{\partial^{2} ϕ}{\partial ξ^{2}} + δ_{η η} \frac{\partial^{2} ϕ}{\partial η^{2}}) = 0 .

In fact, the mixed derivative ∂²ϕ/∂ξ∂η in Eq. (1) can be made vanished by the rotation of coordinates in the ξ-η plane [10], and then the paraxial equation for the optical beams in the biaxial crystals [Eq. (1)] can also be reduced into the similiar form with Eq. (2). Using the transformation x = (δζ + ξ)/w₀, y = η/w₀ and

z = ζ / k_{0} w_{0}^{2}

, the wave equation (2) becomes its dimensionless form

i \frac{\partial ϕ}{\partial z} + \frac{1}{2} (δ_{x x} \frac{\partial^{2} ϕ}{\partial x^{2}} + δ_{y y} \frac{\partial^{2} ϕ}{\partial y^{2}}) = 0,

where δ_xx = δ_ξξ and δ_yy = δ_ηη. The last two terms of Eq. (3) are diffraction terms, and determine the expansion of the optical beams. That δ_xx ≠ δ_yy is general case and is called the anisotropic diffraction [12]. Otherwise, that δ_xx = δ_yy(= 1) is special case for the ordinary light in the uniaxial crystal or light in the isotropic medium, and the diffraction is isotropic. By application of an order-of-magnitude analysis method [35], we have the order of the diffraction terms, respectively

| δ_{x x} \frac{\partial^{2} ϕ}{\partial x^{2}} | ~ \frac{δ_{x x}}{w_{x}^{2}} | ϕ |, | δ_{y y} \frac{\partial^{2} ϕ}{\partial y^{2}} | ~ \frac{δ_{y y}}{w_{y}^{2}} | ϕ |,

where w_x and w_y are the (dimensionless) beam-width of the optical beam in x and y directions, respectively. Obviously, the beam-expansion will be determined by two factors: a linear relation with the coefficients δ_xx and δ_yy that result from the medium anisotropy and an inverse-square relation with the beam-widths that are determined by the specific profile of the beam itself propagating in the medium. We define the radio δ_xx/δ_yy as an anisotropy parameter of the diffraction and assume that δ_xx > δ_yy without loss of generality.

On the propagation for the input beam with a circular shape, the initial beam-expandings are different along the two transverse directions due to the AD, and the beam-expanding in the x-direction is stronger than that in the y-direction. Thus, the beam will adjust continually its ellipticity during the propagation, and eventually evolves towards an elliptic beam with a specific ellipticity, as discussed for the case along the optical axis of a uniaxial crystal [6]. On the other hand, the eigenmodes [36] in the anisotropic media can be directly obtained from the ones in the isotropic medium [2,3] by the scale transform $x / δ_{x x}^{1 / 2}$ and $y / δ_{y y}^{1 / 2}$ in Eq. (3). The eigenmodes of Eq. (3) in the Cartesian coordinates are Hermite-Gaussian modes

\begin{array}{l} ϕ_{m n} (x, y, z) & = & \frac{C_{m n}}{w} H_{m} (\frac{x}{w_{x h g}}) H_{n} (\frac{y}{w_{y h g}}) exp [- (\frac{x^{2}}{2 w_{x h g}^{2}} + \frac{y^{2}}{2 w_{y h g}^{2}})] \\ \times & exp [- i \frac{1}{2 R} (\frac{x^{2}}{δ_{x x}} + \frac{y^{2}}{δ_{y y}}) + i (m + n + 1) ψ], \end{array}

where H_m(x) is Hermite polynomials,

w_{x h g} = δ_{x x}^{1 / 2} w (z)

,

w_{y h g} = δ_{y y}^{1 / 2} w (z)

, w(z), R(z), and ψ(z) are given by

w (z) = \sqrt{1 + z^{2}}

, R(z) = (1 + z²)/z, and ψ(z) = arctan (z), respectively, and

C_{m n} [= {(2^{m + n} π m! n! δ_{x x}^{1 / 2} δ_{y y}^{1 / 2})}^{- 1 / 2}]

is the normalization constant which is chosen so that ∫|ϕ|²dxdy = 1. The eigenmodes in the cylindrical coordinates are Laguerre-Gaussian modes, which are like those in the isotropic medium [2] and can be easily written out in the same way above. It can be observed that the fundamental (basic) eigenmode (m = n = 0) in anisotropic medium exhibits transverse asymmetrical structure,

| ϕ_{00} (x, y, z) | = \frac{1}{\sqrt{π δ_{x x}^{1 / 2} δ_{y y}^{1 / 2} w}} exp (- \frac{x^{2}}{2 w_{x h g}^{2}} - \frac{y^{2}}{2 w_{y h g}^{2}}),

as was discussed [8], while the fundamental mode in the isotropic medium must be transversely circular [2,3].

3. Two different propagation cases

If the optical beams carry the OAM, the elliptically shaped modes even can exist in the media with both linear and nonlinear isotropy [22,23]. We have guessed that [23] the OAM can result in the effective AD, that is, the OAM enhances the expansion of the beams in the major-axis directions and weakens that in the minor-axis direction. In the following, we will quantitatively investigate such an AD induced by the OAM.

In order to find the quantitative relation between the OAM and its induced AD, we consider the following two cases. Case 1: the propagation of an elliptic beam without the OAM in the anisotropic media. Case 2: the propagation of an elliptic beam carrying the OAM in the isotropic media. By comparing the two kinds of propagations, we come to a conclusion about the AD induced by the OAM.

We assume the optical beam at z = 0 is of the form

ϕ (x, y, 0) = \sqrt{\frac{P_{0}}{π w_{0 x} w_{0 y}}} exp (- \frac{x^{2}}{2 w_{0 x}^{2}} - \frac{y^{2}}{2 w_{0 y}^{2}}) exp (i Θ x y),

where w_0x and w_0y are two semi-axes, P₀ = ∬|ϕ(x, y, z)|²dxdy is the optical power, and Θ is the cross phase coefficient, which determines the OAM carried by the optical beam [22,23]

M = Im \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ϕ^{*} (x \frac{\partial ϕ}{\partial y} - y \frac{\partial ϕ}{\partial x}) d x d y = \frac{P_{0}}{2} (w_{0 x}^{2} - w_{0 y}^{2}) Θ .

The solution ϕ(x, y, z) of Eq. (3) with the initial condition (7) can be derived by the Fresnel diffraction integral [3] with its anisotropic form

ϕ (x, y, z) = - i \frac{exp (i z)}{2 π z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ϕ (x^{'}, y^{'}, 0) exp [i \frac{{(x - x^{'})}^{2} / δ_{x x} + {(y - y^{'})}^{2} / δ_{y y}}{2 z}] d x^{'} d y^{'},

and it is obtained as follows after some complicated calculations

ϕ (x, y, z) = ϕ_{0} exp (- \frac{x^{2}}{2 w_{x}^{2}} - \frac{y^{2}}{2 w_{y}^{2}} - \frac{x y}{2 w_{x y}} + i φ),

where

ϕ_{0} = \frac{w_{0 y} \sqrt{P_{0} w_{0 x} w_{0 y}}}{\sqrt{π α_{r}}}, φ = c_{x y} x y + c_{x} x^{2} + c_{y} y^{2} + \frac{arctan κ - π}{2},

w_{x} = \frac{δ_{x x} δ_{y y}}{w_{0 x}} \sqrt{\frac{α_{r}}{w_{0 y}^{4} + δ_{y y}^{2} (w_{0 x}^{2} + w_{0 y}^{2} Θ^{2} + 1) z^{2}}}, w_{y} = \frac{δ_{x x} δ_{y y}}{w_{0 y}} \sqrt{\frac{α_{r}}{w_{0 x}^{4} + δ_{x x}^{2} (w_{0 x}^{2} + w_{0 y}^{2} Θ^{2} + 1) z^{2}}},

w_{x y} = - \frac{α_{r} δ_{x x}^{2} δ_{y y}^{2}}{2 w_{0 x}^{2} w_{0 y}^{2} (w_{0 x}^{2} δ_{y y} + w_{0 y}^{2} δ_{x x}) Θ z}, κ = \frac{(δ_{y y} w_{0 x}^{2} + δ_{x x} w_{0 y}^{2}) z}{w_{0 x}^{2} w_{0 y}^{2} (1 - δ_{x x} δ_{y y} Θ^{2} z^{2}) - δ_{x x} δ_{y y} z^{2}},

c_{x} = \frac{z [δ_{x x} δ_{y y}^{2} z^{2} (2 w_{0 x}^{2} w_{0 y}^{2} Θ^{2} + w_{0 x}^{4} w_{0 y}^{4} Θ^{4}) + w_{0 y}^{4} (δ_{x x} - δ_{y y} w_{0 x}^{4} Θ^{2})]}{2 δ_{x x}^{2} δ_{y y}^{2} α_{r}},

c_{y} = \frac{z [δ_{x x}^{2} δ_{y y} z^{2} (2 w_{0 x}^{2} w_{0 y}^{2} Θ^{2} + w_{0 x}^{4} w_{0 y}^{4} Θ^{4}) + w_{0 x}^{4} (δ_{y y} - δ_{x x} w_{0 y}^{4} Θ^{2})]}{2 δ_{x x}^{2} δ_{y y}^{2} α_{r}},

c_{x y} = \frac{w_{0 x}^{2} w_{0 y}^{2} Θ [w_{0 x}^{2} w_{0 y}^{2} - δ_{x x} δ_{y y} z^{2} (w_{0 x}^{2} w_{0 y}^{2} Θ^{2} + 1)]}{α_{r} δ_{x x}^{2} δ_{y y}^{2}},

and

α_{r} = (2 w_{0 x}^{2} w_{0 y}^{2} Θ^{2} + 1) z^{4} + (\frac{w_{0 x}^{4}}{δ_{x x}^{2}} + \frac{w_{0 y}^{4}}{δ_{y y}^{2}}) z^{2} + w_{0 x}^{4} w_{0 y}^{4} {(\frac{1}{δ_{x x} δ_{y y}} - Θ^{2} z^{2})}^{2} .

3.1. Case one: elliptic beam without the OAM in anisotropic medium

To investigate the evolution of the elliptic beam without the OAM in the anisotropic medium, we set Θ = 0, then the solution (10) becomes

| ϕ (x, y, z) | = ϕ_{0} exp (- \frac{x^{2}}{2 w_{x a}^{2}} - \frac{y^{2}}{2 w_{y a}^{2}}),

where the two semi-axes are

w_{x a}^{2} = w_{0 x}^{2} + {(\frac{δ_{x x}}{w_{0 x}})}^{2} z^{2}, w_{y a}^{2} = w_{0 y}^{2} + {(\frac{δ_{y y}}{w_{0 y}})}^{2} z^{2} .

As mentioned above [see, Eq. (4)], the beam-expansion is determined by both the medium anisotropy and the beam structure. Here, we assume that δ_xx > δ_yy (without loss of generality), and discuss the beam-expanding under two initial conditions that w_0x < w_0y and w_0x > w_0y, which can be featured by the ellipticity function ρ(z) that is defined as the ratio of the two semi-axes of the elliptical beam and ρ(z) = w_xa(z)/w_ya(z) here for this case. Figure 1 gives the evolution of ρ(z) for different initial ellipticities ρ₀[= ρ(0)]. In the case that w_0x < w_0y (ρ₀ < 1), we will have

δ_{x x} / w_{0 x}^{2} > δ_{y y} / w_{0 y}^{2}

and ∂_z ρ(z) > 0. This means that the stronger initial-beam-expanding in the x direction will make the beam-expanding always be stronger in that direction such that ρ(z) will keep increase. Thus, the elliptic beam will evolve towards a circular beam first, and again do an elliptic beam but with an ellipticity ρ(z) > 1 till at a very large distance (z → ∞), ρ(∞) = δ_xx/δ_yy ρ₀, as shown by the curve of ρ₀ = 1/1.5 in Fig. 1. In the case that w_0x > w_0y (ρ₀ > 1), the elliptic beam will exhibit two different evolutions depending on the relative magnitudes between

δ_{x x} / w_{0 x}^{2}

and

δ_{y y} / w_{0 y}^{2}

. That

δ_{x x} / w_{0 x}^{2} > δ_{y y} / w_{0 y}^{2}

(w_0x > w_0y and δ_xx > δ_yy) will also yield ∂_z ρ(z) > 0, the ellipticity ρ(z) will also continuously increase till approach the saturated value ρ(∞), as can be seen from the curve of ρ₀ = 1.03 in Fig. 1; while if

δ_{x x} / w_{0 x}^{2} < δ_{y y} / w_{0 y}^{2}

, that ∂_z ρ(z) < 0 can be obtained, thus the beam-expanding will always be weaker in the x direction and ρ(z) will keep decrease, contrary to the case that w_0x < w_0y. The elliptic beam with ρ > 1 will evolve towards a circular beam in the initial phase, and again do an elliptic beam with ρ < 1, as shown by the curve of ρ₀ = 1.5 in Fig. 1. Specially, when

δ_{x x} / w_{0 x}^{2} = δ_{y y} / w_{0 y}^{2}

, that is

ρ_{0} = \sqrt{\frac{δ_{x x}}{δ_{y y}}},

the total contributions of the beam-expansions due to the medium anisotropy and due to the beam-width are the same in both x and y directions, thus we can have ∂_zρ(z) = 0. In this case, the ellipticity ρ(z) will be constant shown by the curve of

ρ_{0} = \sqrt{1.2}

in Fig. 1, and Eq. (11) will be deduced to the fundamental mode, given by Eq. (6), of Eq. (3).

Fig. 1 Ellipticity functions ρ(z) of elliptic beams propagating in anisotropic media for four different initial ρ₀s: w_0x = 1.5 and w_0y = 1, $w_{0 x} = \sqrt{1.2}$ and w_0y = 1, w_0x = 1.03 and w_0y = 1 (ρ₀ > 1), and w_0x = 1 and w_0y = 1.5 (ρ₀ < 1). Parameters are taken to be δ_xx = 1.2, and δ_yy = 1.

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3.2. Case two: elliptic beam with the OAM in isotropic medium

The solution for the evolution of the beam carrying the OAM in the isotropic medium can be obtained from Eq. (10) by setting δ_xx = δ_yy = 1. Without loss of generality, it is convenient to assume that w_0x > w_0y in this case. Thus, Θ can be either positive or negative. The cross-sectional shape of the beam will be an oblique ellipse in the x–y coordinates for the nonzero OAM. Moreover, the expression of the standard ellipse can be derived by a coordinate rotation X = x cos θ + y sin θ, Y = −x sin θ + y cos θ, Z = z, and the rotational angle θ is determined by

tan (2 θ) = \frac{2 w_{0 x}^{2} w_{0 y}^{2} (w_{0 x}^{2} + w_{0 y}^{2}) Θ z}{(w_{0 x}^{2} - w_{0 y}^{2}) (w_{0 x}^{2} w_{0 y}^{2} - w_{0 x}^{2} w_{0 y}^{2} Θ^{2} z^{2} - z^{2})} .

Then a major-axis w_b and a minor-axis w_c of the ellipse are found to be

\begin{array}{l} w_{b}^{2} = \frac{2 w_{x m}^{2} w_{y m}^{2}}{w_{x m}^{2} + w_{y m}^{2} - \sqrt{{(w_{x m}^{2} - w_{y m}^{2})}^{2} + w_{x m}^{2} w_{y m}^{2} / w_{x y m}^{2}}}, \\ w_{c}^{2} = \frac{2 w_{x m}^{2} w_{y m}^{2}}{w_{x m}^{2} + w_{y m}^{2} + \sqrt{{(w_{x m}^{2} - w_{y m}^{2})}^{2} + w_{x m}^{2} w_{y m}^{2} / w_{x y m}^{2}}}, \end{array}

where w_xm = w_x|_{δ_xx=δ_yy=1}, w_ym = w_y|_{δ_xx=δ_yy=1} and w_xym = w_xy|_{δ_xx=δ_yy=1}, and all of them are functions of the propagation distance z. It can also be observed from Eq. (14) that the elliptic beam will rotate because of the OAM it carries, the detail of which will be discussed later.

Comparing Eq. (15) with Eq. (12), we can obtain the effective Fresnel diffraction coefficients

δ_{x x}^{eff} = \frac{w_{0 x} \sqrt{w_{b}^{2} - w_{0 x}^{2}}}{z}, δ_{y y}^{eff} = \frac{w_{0 y} \sqrt{w_{c}^{2} - w_{0 y}^{2}}}{z},

both of which are generally dependent on the propagation distance z. In order to understand what is the physical picture of

δ_{x x}^{eff}

and

δ_{y y}^{eff}

, we inspect the incipient effect of the OAM on the beam propagation. For this purpose, we expand

w_{b}^{2}

and

w_{c}^{2}

in Eq. (15) with respect to z in a Taylor series about z = 0 to second order:

\begin{array}{l} w_{b}^{2} \approx w_{0 x}^{2} + \frac{1}{w_{0 x}^{2}} [1 + \frac{w_{0 x}^{4} (w_{0 x}^{2} + 3 w_{0 y}^{2}) Θ^{2}}{w_{0 x}^{2} - w_{0 y}^{2}}] z^{2}, \\ w_{c}^{2} \approx w_{0 y}^{2} + \frac{1}{w_{0 y}^{2}} [1 - \frac{w_{0 y}^{4} (w_{0 y}^{2} + 3 w_{0 x}^{2}) Θ^{2}}{w_{0 x}^{2} - w_{0 y}^{2}}] z^{2}, \end{array}

and their substitution into Eq. (16) yields

δ_{x x}^{eff} \approx \sqrt{1 + \frac{w_{0 x}^{4} (w_{0 x}^{2} + 3 w_{0 y}^{2}) Θ^{2}}{w_{0 x}^{2} - w_{0 y}^{2}}}, δ_{y y}^{eff} \approx \sqrt{1 - \frac{w_{0 y}^{4} (w_{0 y}^{2} + 3 w_{0 x}^{2}) Θ^{2}}{w_{0 x}^{2} - w_{0 y}^{2}}} .

The comparison between the behavior of the elliptic beam with the OAM in the isotropic medium [Eq. (17)] and that without the OAM in the anisotropic medium [Eq. (12)] tells a story that the former acts, watched in the X–Y coordinate system, as if it propagated in the medium with the Fresnel diffraction coefficients

δ_{x x}^{eff}

and

δ_{y y}^{eff}

. Obviously the two coefficients given by Eq. (18), or more generally Eq. (16), are not equal as long as Θ ≠ 0.

δ_{x x}^{eff}

and

δ_{y y}^{eff}

, therefore, are the anisotropic diffraction induced by the orbital angular momentum.

Furthermore, the AD induced by the OAM can be better understood by making a comparison of the propagations of the elliptic beams with and without the OAM in the isotropic medium, as shown in Fig. 2. For the isotropic medium, the beam-expanding is only inversely proportional to square of the beam width, as discussed in §2. Thus, the elliptic beam without the OAM spreads more quickly in y (minor-axis) direction than the x (major-axis) direction, resulting in the decrease of its ellipticity; on the contrary, the ellipticity increases for the elliptic beam with the OAM, as shown in Fig. 2 (e). The reason for such a difference is that the AD induced by the OAM makes the expanding in the X (major-axis) direction be stronger than that in the x direction for the case without OAM and one in the Y (minor-axis) direction be weaker than that in y direction, as shown in Fig. 2 (c–d) and also observed in Eq. (18).

Fig. 2 Evolutions of the elliptic beams with and without the OAM in the isotropic medium. (a1–a3): zero OAM Θ = 0, (b1–b3): with OAM Θ = 1/2, (c–d): evolutions of two semi-axes, and (e): evolutions of the ellipticities. $w_{0 x} = \sqrt{2}$ and w_0y = 1. Red dashed line: with the OAM, and green solid line: without the OAM.

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Setting ∂_z ρ(z) = 0 where ρ(z) = w_b(z)/w_c(z) that are given by Eq. (15) yields the critical cross phase coefficient

Θ_{c} = \pm \frac{w_{0 x}^{2} - w_{0 y}^{2}}{2 w_{0 x}^{2} w_{0 y}^{2}} .

When Θ = Θ_c, the elliptic beam carrying OAM will evolve with an invariant ellipticity, and the beam profile in the X–Y (rotating) coordinate system is the fundamental mode given by Eq. (6) with two semi-axes

{(w_{b}^{cri})}^{2} = w_{0 x}^{2} + {(\frac{δ_{x x c}^{eff}}{w_{0 x}})}^{2} z^{2}, {(w_{c}^{cri})}^{2} = w_{0 y}^{2} + {(\frac{δ_{y y c}^{eff}}{w_{0 y}})}^{2} z^{2},

and the AD,

δ_{x x c}^{eff}

and

δ_{y y c}^{eff}

, induced by the OAM

δ_{x x c}^{eff} = | \tilde{M} | + 1 + \sqrt{| \tilde{M} | (| \tilde{M} | + 1)}, δ_{y y c}^{eff} = | \tilde{M} | + 1 - \sqrt{| \tilde{M} | (| \tilde{M} | + 1)},

where M̃ = M_c/P₀ is the critical OAM per unit power. Eq. (20) is obtained from Eq. (15) for Θ = Θ_c and Eq. (21) is from Eq. (16) for Θ = Θ_c and a relation [see, Eq. (8)]

\frac{M_{c}}{P_{0}} = \frac{1}{2} (w_{0 x}^{2} - w_{0 y}^{2}) Θ_{c} = \pm \frac{1}{4} \frac{{(w_{0 x}^{2} - w_{0 y}^{2})}^{2}}{w_{0 x}^{2} w_{0 y}^{2}} .

It can also be found that the two semi-axes given by Eq. (20) satisfy

\frac{w_{b}^{cri}}{w_{c}^{cri}} = {(\frac{δ_{x x c}^{eff}}{δ_{y y c}^{eff}})}^{1 / 2} = \sqrt{| \tilde{M} | + 1} + \sqrt{| \tilde{M} |} .

The ratio

δ_{x x c}^{eff} / δ_{y y c}^{eff}

above gives the anisotropic parameter of the AD induced by the OAM. It tells that an inequation

δ_{x x c}^{eff} > δ_{y y c}^{eff}

holds always no matter the sign of the OAM, which means the OAM always enhances the beam-expanding in the major-axis direction and weakens that in the minor-axis direction, and also that the larger the OAM, the stronger the AD induced by it.

Now we can draw a conclusion that observed in the rotating (X–Y) coordinate system, the elliptic beam in the isotropic medium when its carrying OAM equals the critical value given by Eq. (22) behaves exactly same as that with the same widthes in the anisotropic medium with the two coefficients $δ_{x x} = δ_{x x c}^{eff}$ and $δ_{y y} = δ_{y y c}^{eff}$ , no matter how long distance they propagate. Here we give an example to show such a similarity. The input elliptic beam are assumed with $w_{0 x} = \sqrt{2}$ and w_0y = 1 and carrying the critical OAM M̃ = 1/8. Because its carrying OAM induce the AD $δ_{x x c}^{eff} = 3 / 2$ and $δ_{y y c}^{eff} = 3 / 4$ , the elliptic beam can evolve with an invariant ellipticity $\sqrt{2}$ in the isotropic medium, as shown in Fig. 3 (a1–a3). The fundamental-mode-propagation of the elliptic beam with the same structure but without the OAM in the fictitious anisotropic medium with δ_xx = 3/2 and δ_yy = 3/4 also has an invariant ellipticity $\sqrt{2}$ , as displayed in Fig. 3 (b1–b3). It can be found the two evolutions are completely equivalent without taking the rotation from the OAM into consideration.

Fig. 3 Evolutions of the elliptic beams for two cases. (a1–a3): with the OAM M̃ = 1/8 in the isotropic medium, (b1–b3): without the OAM in the fictitious anisotropic medium of δ_xx = 3/2 and δ_yy = 3/4. The beam widthes are $w_{0 x} = \sqrt{2}$ and w_0y = 1.

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As mentioned above, the OAM can not only induce the AD, but also make the elliptic beam rotate, as shown in Figs. (2) and (3). The rotation angle θ of the elliptic beam can be obtained from Eq. (14) as

θ = {\begin{matrix} - \frac{1}{2} arctan [\frac{2 w_{0 x}^{2} w_{0 y}^{2} (w_{0 x}^{2} + w_{0 y}^{2}) Θ z}{(w_{0 x}^{2} + w_{0 y}^{2}) [z^{2} + w_{0 x}^{2} w_{0 y}^{2} (- 1 + z^{2} Θ^{2})]}], & z < \frac{w_{0 x} w_{0 y}}{\sqrt{w_{0 x}^{2} w_{0 y}^{2} Θ^{2} + 1}} \\ - \frac{1}{2} arctan [\frac{2 w_{0 x}^{2} w_{0 y}^{2} (w_{0 x}^{2} + w_{0 y}^{2}) Θ z}{(w_{0 x}^{2} + w_{0 y}^{2}) [z^{2} + w_{0 x}^{2} w_{0 y}^{2} (- 1 + z^{2} Θ^{2})]}] + \frac{π}{2}, & z > \frac{w_{0 x} w_{0 y}}{\sqrt{w_{0 x}^{2} w_{0 y}^{2} Θ^{2} + 1}} . \end{matrix}

The positive OAM (Θ > 0) makes the elliptic beam rotate anticlockwise, otherwise clockwise. When z → +∞, rotation angle approaches to ±π/2 depending up the sign of the OAM, as shown in Fig. 4 (a). And the angular velocity can be obtained from Eq. (14)

ω \equiv \frac{d θ}{d z} = \frac{1}{2 {(1 + tan 2 θ)}^{2}} \frac{d}{d z} [\frac{w_{x m}^{2} w_{y m}^{2}}{w_{x y m} (w_{y m}^{2} - w_{x m}^{2})}] .

It is known that the OAM is a conservative quantity in isotropic media [22,23]. However, because of the diffraction, the beam width becomes larger as the propagation distance increases, it leads to slowing down of the rotation [37], which is shown in Fig. 4 (b).

Fig. 4 Rotation angle θ (a) and angular velocity ω (b) of elliptic beams with OAM propagating in isotropic media. Parameters are taken to be $w_{0 x} = \sqrt{2}$ , w_0y = 1 and Θ = 1/4.

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4. Conclusion

It is found that the orbital angular momentum carried by the elliptic beam can induce the anisotropic diffraction. By discussing the propagations of the elliptic beam without the OAM in the anisotropic medium and the elliptic beam carrying the OAM in the isotropic medium, we analytically obtain the quantitative relation between the OAM and its induced AD. The OAM always enhances the beam-expanding in the major-axis direction and weakens that in the minor-axis direction no matter the sign of the OAM, and the larger the OAM, the stronger the AD induced by it. When the elliptic beam without the OAM propagate in the anisotropic medium, it evolves as the fundamental mode if its ellipticity is the square root of anisotropic parameter δ_xx/δ_yy. And in the isotropic medium, the beam profile in the X–Y (rotating) coordinate system is the fundamental mode if the OAM carried by optical beam equals to its critical value M_c(Θ_c). The rotation resulting form the OAM is also discussed, and is found that the absolute value of the rotation angle is no larger than π/2 and the angular velocity slows down during the propagation because of the diffraction effect.

Our results give the answer why the elliptic mode can survive in the medium with both linear and nonlinear isotropy where only the mode with cylindrical-symmetry is supposed to exist.

Acknowledgments

National Natural Science Foundation of China (NSFC) (11474109, 11604199).

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Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams

Abstract

1. Introduction

2. Paraxial equation governing the beam propagation and its eigenmodes

3. Two different propagation cases

3.1. Case one: elliptic beam without the OAM in anisotropic medium

3.2. Case two: elliptic beam with the OAM in isotropic medium

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (32)

Optics Express