## Anisotropic dynamics of optical vortex-beam propagating in biaxial crystals: a numerical method based on asymptotic expansion |

Optics Express, Vol. 21, Issue 7, pp. 8493-8507 (2013)

http://dx.doi.org/10.1364/OE.21.008493

Acrobat PDF (3340 KB)

### Abstract

One difficulty of angular spectrum representation method in studying optical propagation inside anisotropic crystals is to calculate the double integrals containing highly oscillating functions. In this paper, we introduce an accuracy and numerically cheap method based on asymptotic expansion theory to overcome this difficulty, which therefore allows to compute optical fields with arbitrary incident beam and is not restricted to the paraxial limit. This numerical method is benchmarked against the analytical solutions in uniaxial crystals and excellent agreements between them are obtained. As an application, we adopt it to investigate the propagation of a Gaussian vortex-beam in a biaxial crystal. The general features of anisotropic dynamics and power conversion between field components are revealed. The numerical results is interpreted by making appropriate analytical approximation to the wave equations.

© 2013 OSA

## 1. Introduction

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

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

11. J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. **53**, 20701 (2011) [CrossRef] .

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

4. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A **19**, 1680–1688 (2002) [CrossRef] .

12. C. N. Alexeyev, “Circular array of anisotropic fibers: A discrete analog of a *q* plate,” Phys. Rev. A **86**, 063830 (2012) [CrossRef] .

13. T. A. Fadeyeva, C. N. Alexeyev, P. M. Anischenko, and A. V. Volyar, “Engineering of the space-variant linear polarization of vortex-beams in biaxially induced crystals,” Appl. Opt. **51**, C224–C230 (2012) [CrossRef] [PubMed] .

22. A. Volyar and T. Fadeeva, “Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal,” Opt. Spectros. **101**, 450–457 (2006) [CrossRef] .

*et al.*develop a general scheme to describe the propagation of paraxial beam in uniaxial crystal [20

20. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A **18**, 1656–1661 (2001) [CrossRef] .

*et al.*: the propagation parallel to and orthogonal to the optical axis [5

5. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A **20**, 163–171 (2003) [CrossRef] .

20. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A **18**, 1656–1661 (2001) [CrossRef] .

23. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**, 2163–2171 (2003) [CrossRef] .

4. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A **19**, 1680–1688 (2002) [CrossRef] .

24. A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E **67**, 036618 (2003) [CrossRef] .

26. A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. **231**, 79–92 (2004) [CrossRef] .

20. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A **18**, 1656–1661 (2001) [CrossRef] .

27. A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E **66**, 036614 (2002) [CrossRef] .

32. D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM J. Numer. Anal. **44**, 1026–1048 (2006) [CrossRef] .

29. D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in *Highly Oscillatory Problems* (Cambridge University, 2009), pp. 25–50 [CrossRef] .

27. A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E **66**, 036614 (2002) [CrossRef] .

33. X. Lu and L. Chen, “Spin-orbit interactions of a Gaussian light propagating in biaxial crystals,” Opt. Express **20**, 11753–11766 (2012) [CrossRef] [PubMed] .

*et al.*adopt the quadpack routines (Clenshaw-Curtis method) [34] to compute the oscillating integral and obtain excellent accuracy at small propagating distance, where the oscillation of integrand is not rapidly. However, the calculation becomes slower and also the accuracy decreases as the propagating distance increases. This situation motives us to look for better method to approximate the highly oscillatory integrals in the angular spectrum method, especially at large propagating distance.

35. N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron. **29**, 1020 (1999) [CrossRef] .

43. N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun. **286**, 1–5 (2013) [CrossRef] .

13. T. A. Fadeyeva, C. N. Alexeyev, P. M. Anischenko, and A. V. Volyar, “Engineering of the space-variant linear polarization of vortex-beams in biaxially induced crystals,” Appl. Opt. **51**, C224–C230 (2012) [CrossRef] [PubMed] .

44. M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A: Pure Appl. Opt. **1**, 601 (1999) [CrossRef] .

45. Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. **284**, 3183–3191 (2011) [CrossRef] .

33. X. Lu and L. Chen, “Spin-orbit interactions of a Gaussian light propagating in biaxial crystals,” Opt. Express **20**, 11753–11766 (2012) [CrossRef] [PubMed] .

## 2. The expression for electric field inside anisotropic crystals

**18**, 1656–1661 (2001) [CrossRef] .

**E**(

**r**) is the complex amplitude of electric field

**E**(

**r**,

*t*) with a relation

**E**(

**r**,

*t*) =

*Re*[

**E**(

**r**)exp(−

*iωt*)],

*k*

_{0}= 2

*π*/

*λ*with

*λ*being the vacuum wave length of light, and

*ε*is the relative dielectric tensor which has form with

*n*,

_{x}*n*, and

_{y}*n*being the refractive index along crystalline

_{z}*x*,

*y*, and

*z*axis, respectively. For the uniaxial crystals with

*n*=

_{x}*n*≠

_{y}*n*or

_{z}*n*≠

_{x}*n*=

_{y}*n*, the light propagating behaviors have been fully discussed [5

_{z}5. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A **20**, 163–171 (2003) [CrossRef] .

**18**, 1656–1661 (2001) [CrossRef] .

23. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**, 2163–2171 (2003) [CrossRef] .

46. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A **27**, 381–389 (2010) [CrossRef] .

47. T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express **18**, 10848–10863 (2010) [CrossRef] [PubMed] .

*n*≠

_{x}*n*≠

_{y}*n*. We assume that the light is propagating along the

_{z}*z*direction and the crystal fills the whole

*z*> 0 space. In order to achieve the optical field inside the crystals, we first expand it using two-dimensional Fourier transformation [20

**18**, 1656–1661 (2001) [CrossRef] .

23. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**, 2163–2171 (2003) [CrossRef] .

**r**

_{⊥}=

*x*

**ê**

*+*

_{x}*y*

**ê**

*and*

_{y}**k**

_{⊥}=

*k*

_{x}**ê**

*+*

_{x}*k*

_{y}**ê**

*denote the vectors in the transverse planes of real and momentum spaces. By substituting Eq. (3) into the wave equation of Eq. (1), it is straightforward to obtain the exact expression for*

_{y}**Ẽ**(

**k**

_{⊥},

*z*), The coefficients

*λ*

_{1},

*λ*

_{2},

*c*

_{1},

*c*

_{2},

*c*

_{3},

*c*

_{4},

*c*

_{5}, and

*c*

_{6}are given by [33

33. X. Lu and L. Chen, “Spin-orbit interactions of a Gaussian light propagating in biaxial crystals,” Opt. Express **20**, 11753–11766 (2012) [CrossRef] [PubMed] .

**P**and

**Q**are and the vector

**Ẽ**

_{⊥}(0) is the two-dimensional Fourier transformation of the transverse incident field

**E**

_{⊥}(

**r**

_{⊥}, 0) on the plane

*z*= 0, In the above equations, the parameters

*α*,

*β*,

*γ*,

*p*,

*K*,

*L*,

*M*are defined to be

*z*direction in an anisotropic crystal, whose forms are complex. For special input field such as paraxial Gaussian beam, one can simplify these equations to obtain analytical solutions for uniaxial crystal [5

5. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A **20**, 163–171 (2003) [CrossRef] .

**18**, 1656–1661 (2001) [CrossRef] .

*z*is increasing.

## 3. Numerical method

*n*=

_{x}*n*=

_{y}*n*

_{0}and

*n*=

_{z}*n*. The expressions for the coefficients

_{e}*λ*

_{1}(

*λ*

_{2}) and the matrices

**P**(

**Q**) in uniaxial crystals can be simplified as [20

**18**, 1656–1661 (2001) [CrossRef] .

*z*= 0 is where

*s*is the waist of Gaussian beam and

**E**

_{⊥}can be decomposed by using either the Cartesian basis or the circular basis, i.e.,

**E**

_{⊥}=

*E*

_{x}**ê**

*+*

_{x}*E*

_{y}**ê**

*=*

_{y}*E*

_{+}

**ê**

_{+}+

*E*

_{−}

**ê**

_{−}. The relation between different components is The initial field in

*k*space is a Fourier transformation of Eq. (17), which is given by The analytical solutions for the propagation of Gaussian beam of Eq. (17) in uniaxial crystal is derived by Ciattoni

*et al.*in the paraxial limit [5

**20**, 163–171 (2003) [CrossRef] .

24. A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E **67**, 036618 (2003) [CrossRef] .

### 3.1. Asymptotic approximation

48. R. Wong, *Asymptotic Approximations of Integrals* (SIAM, Philadelphia, 2001) [CrossRef] .

*λ*being a large positive parameter and

*D*a bounded domain. In contrast to the conventional quadrature methods, the accuracy of asymptotic expansion improves as the oscillation (or

*λ*) increases. The key point of asymptotic approximation is that the main contributions to the asymptotic expansion of

*I*(

*λ*) come only from regions in the vicinity of certain critical points [15, 48

48. R. Wong, *Asymptotic Approximations of Integrals* (SIAM, Philadelphia, 2001) [CrossRef] .

*f*(

*x*,

*y*) within the domain

*D*at which

*D*. As far as the Eq. (3) is concerned, the boundary of integration goes to infinity and for an inputting Gaussian beam the kernel of integration vanishes on large boundary. So that only the first kind of critical points are important for the integral of Eq. (3) and we do not need to consider the other kinds of critical points.

*E*, where For large distance

_{x}*z*,

*E*consists of two typical highly oscillatory integral, which can be approximated by asymptotic expansion. It’s easy to show that both functions

_{x}*f*

_{1}and

*f*

_{2}have only one stationary point, which in the case of uniaxial crystals are at where

*r*= (

_{o}*x*

^{2}+

*y*

^{2}+

*z*

^{2})

^{1/2},

*E*come from the critical points of first kind, one can expand the functions

_{x}*f*

_{1}and

*f*

_{2}on the critical points (

*k*,

_{xo}*k*) and (

_{yo}*k*,

_{xe}*k*), respectively. For example, function

_{ye}*f*

_{1}can be expanded as where the coefficients are

*f*

_{2},

*f*

_{1}and

*f*

_{2}to second order and substituting them into Eq. (21), we can derive the asymptotic approximation to

*E*, Note that since the leading contributions to the integral Eq. (21) come from the stationary points we have approximated the functions

_{x}*c*

_{1}(

*k*,

_{x}*k*) and

_{y}*c*

_{2}(

*k*,

_{x}*k*) in the integral by constants

_{y}*c*

_{1}(

*k*,

_{xo}*k*) and

_{yo}*c*

_{2}(

*k*,

_{xe}*k*), which can be evaluated by substituting (

_{ye}*k*,

_{xo}*k*) and (

_{yo}*k*,

_{xe}*k*) into Eq. (6). However, this approximation is valid only when the functions

_{ye}*c*

_{1}and

*c*

_{2}are differentiable [48

48. R. Wong, *Asymptotic Approximations of Integrals* (SIAM, Philadelphia, 2001) [CrossRef] .

*E*and

_{y}*E*, can be obtained in a similar way, by replacing

_{z}*c*

_{1},

*c*

_{2}with

*c*

_{3},

*c*

_{4}and

*c*

_{5},

*c*

_{6}, respectively. After some straightforward algebra, the electric field

**E**, in the case of uniaxial crystal, can be simplified as This is one of the main results of present paper.

*c*(

_{i}*i*= 1, ⋯ ,6) at respective stationary points with a given incident field

**Ẽ**

_{⊥}(0), it’s straightforward to obtain the asymptotic expansion for

**E**(

**r**

_{⊥},

*z*) by using Eq. (27). In the following, we will qualify this approximation of asymptotic expansion by comparing it with the analytical solutions in uniaxial crystal [5

**20**, 163–171 (2003) [CrossRef] .

24. A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E **67**, 036618 (2003) [CrossRef] .

*n*= 1.656,

_{o}*n*= 1.458, the wave length

_{e}*λ*= 0.633

*μm*, and the waist of initial Gaussian beam

*s*= 4.59

*μm*[33

**20**, 11753–11766 (2012) [CrossRef] [PubMed] .

49. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. **34**, 1021–1023 (2009) [CrossRef] [PubMed] .

*z*= 100000

*μm*. In Figs. 1(a) and 1(b), both asymptotic and analytical results for the moduli of

*E*

_{+}and

*E*

_{−}, the circularly polarized components of propagating beam, are shown as a function of transverse radius

*r*

_{⊥}. The results of asymptotic expansion agree very well with the analytical solutions, except for a small region close to the

*r*

_{⊥}= 0 point. For clarity, we show in Figs. 1(c) and 1(d) the absolute errors of asymptotic approximation with respect to the analytical solutions. One can see that the errors are smaller than 1 × 10

^{−5}for

*E*

_{+}and most region of

*E*

_{−}(e.g.,

*r*

_{⊥}> 600

*μm*). In the following, we will discuss why the errors of

*E*

_{−}are large in the region close to

*r*

_{⊥}= 0.

*c*(

_{i}*k*,

_{x}*k*) (

_{y}*i*= 1, ⋯,6) are smooth and differentiable [48

48. R. Wong, *Asymptotic Approximations of Integrals* (SIAM, Philadelphia, 2001) [CrossRef] .

*E*

_{±}, one should look at the behavior of functions

*u*

_{±}=

*c*

_{1}±

*ic*

_{3}and

*v*

_{±}=

*c*

_{2}±

*ic*

_{4}, respectively. It is interesting to observe that although the functions

*u*

_{+}and

*v*

_{+}corresponding to

*E*

_{+}are smooth everywhere, the functions

*u*

_{−}and

*v*

_{−}for

*E*

_{−}are singular in the momentum space. As an example, we plot the real parts of functions

*u*

_{+}and

*u*

_{−}in Fig. 2, from which we can see a singularity at the point (0, 0) for the function

*u*

_{−}. For small values of

*r*

_{⊥}and large distance

*z*, the stationary points (

*k*,

_{xo}*k*) and (

_{yo}*k*,

_{xe}*k*) are very close to the singular point (0, 0) in

_{ye}*u*

_{−}and

*v*

_{−}, also see Eqs. (23) and (24). Therefore, the asymptotic approximation of Eq. (27), which does not consider the effect of singularity, is not good for

*E*

_{−}in this case. But the effect of singularity is negligible when the radius

*r*

_{⊥}is large, for the stationary points are far from the singular point. This results in excellent accuracy for

*E*

_{−}in large

*r*

_{⊥}region, as can be seen in Fig. 1(b).

*c*are smooth, the accuracy of asymptotic approximation is great for large propagating distance

_{i}*z*. Note that the functions

*c*depend on the incident beams, which often contains singularities. In the following, we propose a method to take the effect of singularity into account.

_{i}### 3.2. Method of integrating near important points

*f*

_{1}and

*f*

_{2}, and also the singular points in functions

*c*. Thus, a straightforward idea of approximation is to numerically integrate over small regions near such important points and throw away the contributions from the other more oscillatory and smooth regions [29

_{i}29. D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in *Highly Oscillatory Problems* (Cambridge University, 2009), pp. 25–50 [CrossRef] .

29. D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in *Highly Oscillatory Problems* (Cambridge University, 2009), pp. 25–50 [CrossRef] .

48. R. Wong, *Asymptotic Approximations of Integrals* (SIAM, Philadelphia, 2001) [CrossRef] .

*z*is, the smaller region is needed to be integrated over in order to obtain a fixed accuracy. The integrand close to the stationary points does not oscillate rapidly, therefore the integration over a nearby region can be performed by using the classic quadrature methods such as Clenshaw-Curtis and Gauss-Kronrod [33

**20**, 11753–11766 (2012) [CrossRef] [PubMed] .

*k*,

_{xo}*k*) and (

_{yo}*k*,

_{xe}*k*). How to choose the domain of integration is a little bit tricky. It is not a good idea to choose a circular domain around the stationary point, for the value of integral will oscillate as the radius of domain is increasing. It turns out that a square domain is a good choice (see the inset in Fig. 3(b) for the shape and position). The oscillating part of the integrand on the four corners of square can partially cancel each other, making the result of integration much more smooth.

_{ye}*E*

_{+}and

*E*

_{−}are plotted as a function of the width

*d*of the square domain. All parameters used in the calculations are the same as those in Fig. 1. One can see that the errors of both

_{k}*E*

_{+}and

*E*

_{−}decrease as the width

*d*is increasing. For a small domain with

_{k}*d*= 0.5, we already obtain an excellent accuracy, i.e., the errors for all values of transverse radius

_{k}*r*

_{⊥}are smaller than 2.0×10

^{−5}for

*E*

_{+}and smaller than 4.0 × 10

^{−5}for

*E*

_{−}. The relative large errors of

*E*

_{−}compared with

*E*

_{+}may come from the singularities in functions

*u*

_{−}and

*v*

_{−}, which make the numerical integration difficult. We emphasize that in contrast with the asymptotic expansion in the above section, the errors of

*E*

_{−}in small

*r*

_{⊥}region are very small even for a quite small integration width

*d*; see Fig. 3(b) for details. This is basically due to the fact that the singularities in

_{k}*u*

_{−}and

*v*

_{−}have been included into the square domain of integration.

*d*is increasing, we can do a coarse-grained average in one oscillating period of the integrand. For example, we can choose ten values of

_{k}*d*in a period, e.g., between the two squares in the inset of Fig. 4(b), calculate the integral for each

_{k}*d*, and treat the average as the final result. In this way, the highly oscillating part far away from the stationary point is canceled, leaving the most important contribution from the small area close to the stationary point. The calculated results for a Gaussian incident beam are shown in Fig. 4. One can see that the accuracy is improved greatly comparing with the Fig. 3, e.g., for a small width

_{k}*d*= 0.4, the errors for both

_{k}*E*

_{+}and

*E*

_{−}are smaller than 1.0×10

^{−5}for all radii

*r*

_{⊥}. Note that for even smaller width such as

*d*= 0.2 the accuracy is still good.

_{k}## 4. The anisotropic propagating dynamics in biaxial crystals

*f*

_{1}and

*f*

_{2}in Eq. (22) are complicated. Nevertheless, one can numerically show that

*f*

_{1}and

*f*

_{2}still have only one stationary point, whose position can also be obtained numerically. With these points, the method of integrating near important points can be straightforward applied to compute the optical field in biaxial crystals. The only difference from uniaxial crystals is that the domain of integration is no longer square, but is rectangle due to the breaking symmetry in the transverse plane. In the following, we adopt this method to investigate the propagation of vortex beam in the biaxial crystals.

*z*= 0 plane, where the two-dimensional complex vector

**d̂**

*=*

_{in}**ê**

*cos(*

_{x}*θ*)+

**ê**

*sin(*

_{y}*θ*)

*e*representing the state of polarization. The Fourier transformation of incident field Eq. (28) is In order to study the properties of propagation in biaxial crystals, we fix the values

^{iϕ}*n*= 1.656 and

_{x}*n*= 1.458, while let

_{z}*n*vary in the range between

_{y}*n*and

_{x}*n*. Correspondingly, we introduce parameters Δ

_{z}*n*=

*n*−

_{x}*n*and

_{y}*η*= Δ

*n*/(

*n*−

_{x}*n*) to describe the degree of deviation from the uniaxial crystals. The cases of

_{z}*η*= 0 and

*η*= 1 correspond to two limits of uniaxial crystal with propagation direction parallel to and orthogonal to the optical axis respectively (

*n*=

_{x}*n*≠

_{y}*n*and

_{z}*n*≠

_{x}*n*=

_{y}*n*). In the past, the propagating behaviors of vortex beam of Eq. (28) in these two uniaxial limits have been extensively studied [4

_{z}4. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A **19**, 1680–1688 (2002) [CrossRef] .

11. J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. **53**, 20701 (2011) [CrossRef] .

**d̂**

*=*

_{in}**ê**

_{+}. The beam parameters are still chosen as:

*λ*= 0.633

*μm*and

*s*= 4.59

*μm*, so that the paraxial condition

*k*

_{0}

*s*≫ 1 is satisfied. We investigate the evolutions of Cartesian components

*E*and

_{x}*E*with different values of deviations

_{y}*η*at a fixed propagating distance

*z*= 5000

*μm*. The results of numerical simulations are presented in Fig. 5.

*E*and

_{x}*E*components for biaxial crystals exhibit anisotropic propagating behaviors due to the broken symmetry in the transverse plane. However, the anisotropic characters of the

_{y}*E*and

_{x}*E*components are quite different. The profile of

_{y}*E*is extended in the

_{x}*x*direction, nevertheless the profile of

*E*is extended in the

_{y}*y*direction. In the region where the deviation

*η*is very small (e.g., the panels (b) and (g), (c) and (h)), the regular pattern formed in the uniaxial limit with

*η*= 0 [5

**20**, 163–171 (2003) [CrossRef] .

49. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. **34**, 1021–1023 (2009) [CrossRef] [PubMed] .

*η*is increasing. The propagation and angular momentum dynamics in this region are studied in detail before [33

**20**, 11753–11766 (2012) [CrossRef] [PubMed] .

*η*, the propagating behaviors of

*E*and

_{x}*E*components can be well understood by an analytical approximation.

_{y}*s*is much greater than the vacuum wave length

*λ*. Thus, the incident field

**Ẽ**

_{⊥}(0) in the Fourier space is nonvanishing only in a very small region with |

**k**

_{⊥}| ≪

*k*

_{0}. For biaxial crystals with strong birefringence, e.g.

*η*> 0.05 as in our calculations, we expect the following relation is satisfied for a paraxial beam, In this limit, the complicated optical field inside biaxial crystals can be greatly simplified. We first look at the expression of

*L*in Eq. (13). Since the dominate term of

*L*is (

*α*−

*β*)

^{2}when the value of

*η*is not very small, it’s reasonable to change the form of

*L*a term proportional to

*η*= 1, for the added term is zero when

*β*=

*γ*. After the treatment, the expression of

*L*in Eq. (13) can be written as which is actually the square of

*M*in Eq. (14),

*L*=

*M*

^{2}. Then the eigenvalues

*λ*

_{1},

*λ*

_{2}and the matrix

**P**,

**Q**can be simplified as For the case

*n*≠

_{x}*n*=

_{y}*n*(i.e.,

_{z}*β*=

*γ*and

*η*= 1), the matrix

**P**and

**Q**reduce to be which is exactly the same as those in Ref. [23

**20**, 2163–2171 (2003) [CrossRef] .

**18**, 1656–1661 (2001) [CrossRef] .

**20**, 2163–2171 (2003) [CrossRef] .

*k*/

_{x}*k*

_{0}and

*k*/

_{y}*k*

_{0}. Finally, we get The optical field in Eqs. (35) and (36) is a generalization of the results in Ref. [23

**20**, 2163–2171 (2003) [CrossRef] .

*η*= 1 is discussed. Many features of the beam propagation in Fig. 5 can be explained by employing these equations.

*E*and

_{x}*E*are uncoupled, that is, the

_{y}*E*and

_{x}*E*depend on the

_{y}*x*and

*y*components of the incident field respectively, and the polarization conversion between them is forbidden during propagation.

*η*> 0.05, the total power of

*E*and

_{x}*E*is almost equal and do not change as propagating, which is in agreement with the predictions from Eqs. (35) and (36).

_{y}*E*component comes from the exponential factor

_{x}*Ẽ*(0) and

_{x}*Ẽ*(0) are isotropic in the transverse plane. As in our calculations, the value of

_{y}*n*= 1.656 is larger than that of

_{x}*n*= 1.458, so that the profile of

_{z}*E*will extend in the

_{x}*x*direction. Another interesting feature of

*E*in Eq. (35) is that it does not depend on the value of

_{x}*n*, which implies that its profile will not change as the value of

_{y}*η*changes. This feature can be seen from the panels (d) and (e) in Fig. 5, where the value of

*η*is large enough. The component

*E*exhibits a different anisotropic behavior from that of

_{y}*E*. The anisotropy is caused by a different factor

_{x}*y*direction in case when the value

*n*>

_{y}*n*. Moreover, the anisotropy in the profile of

_{z}*E*loses gradually as the value of

_{y}*n*approaches to that of

_{y}*n*(

_{z}*η*approaches to 1). At the uniaxial limit of

*η*= 1,

*E*turns out to be isotropic in the transverse plane, see the panel (j) in Fig. 5.

_{y}*z*= 5000

*μm*, i.e.,

**d̂**

*in Eq. (28) is chosen to be*

_{in}**ê**

*. For*

_{x}*η*= 0, a significant amount of power is transferred from

*E*to

_{x}*E*; see Fig. 6(a) and (f). The power conversion in this limit has been well studied both experimentally and theoretically in Refs. [5

_{y}**20**, 163–171 (2003) [CrossRef] .

25. A. Ciattoni, G. Cincotti, C. Palma, and H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A **19**, 1894–1900 (2002) [CrossRef] .

49. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. **34**, 1021–1023 (2009) [CrossRef] [PubMed] .

50. Y. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. Kivshar, “Dynamics of linear polarization conversion in uniaxial crystals,” Opt. Express **17**, 18196–18208 (2009) [CrossRef] [PubMed] .

*η*is increasing, the amount of transferred power decreases sharply, and for large

*η*the intensity of

*E*is much smaller than that of

_{y}*E*, implying that the components

_{x}*E*and

_{x}*E*are almost decoupled. This agrees with our previous analysis on Eqs. (35) and (36). Note that the colorbar on the right side of the second row is only for subfigures (i) and (j). The colorbar for subfigures (f), (g), and (h) is the same as that of subfigures in the first row, which is on the right side of it. For completeness, the total intensity of the transverse fields

_{y}*I*= |

*E*|

_{x}^{2}+ |

*E*|

_{y}^{2}are shown in the third rows in Fig. 5 and Fig. 6 for circularly and linearly polarized beam, respectively. Regular pattern forms when

*η*= 0, which is destroyed quickly as the value of

*η*is increasing. For large

*η*, the intensity

*I*basically forms a elliptical profile and does not change a lot as

*η*varies.

## 5. Conclutions

*η*= 0 and

*η*= 1. We found that for the paraxial limit and a large value of

*η*(the condition

*E*and

_{x}*E*are uncoupled and the power exchange between them is very small. The general features of anisotropic propagation are identified and explained by an analytical approximation.

_{y}44. M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A: Pure Appl. Opt. **1**, 601 (1999) [CrossRef] .

*θ*,

*ϕ*) in the crystalline coordinates, where

*θ*is the polar angle and

*ϕ*the azimuthal angle. Then the transformation matrix can be given as, In this perspective, the diagonal dielectric tensor of Eq. (2) will become

*ε*′ =

*TεT*in the experimental frame. It thus allows one to make our numerical description to the well-known phenomena of conical diffraction when the light propagates along the optic axis or in vicinity of “the diabolic points” [37

^{t}37. V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE **4403**, 229–240 (2001) [CrossRef] .

43. N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun. **286**, 1–5 (2013) [CrossRef] .

## Acknowledgments

## References and links

1. | J. F. Nye, |

2. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

3. | M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A |

4. | G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A |

5. | A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A |

6. | F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. |

7. | T. Fadeyeva, A. Rubass, Y. Egorov, A. Volyar, and J. Grover Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A |

8. | T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A |

9. | T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt. |

10. | T. Fadeyeva, C. Alexeyev, A. Rubass, A. Zinov’ev, V. Konovalenko, and A. Volyar, “Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal,” Opt. Lett. |

11. | J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. |

12. | C. N. Alexeyev, “Circular array of anisotropic fibers: A discrete analog of a |

13. | T. A. Fadeyeva, C. N. Alexeyev, P. M. Anischenko, and A. V. Volyar, “Engineering of the space-variant linear polarization of vortex-beams in biaxially induced crystals,” Appl. Opt. |

14. | T. Fadeyeva, C. Alexeyev, P. Anischenko, and A. Volyar, “The fine structure of the vortex-beams in the biaxial and biaxially-induced birefringent media caused by the conical diffraction,” arXiv:1107.5775 (2011). |

15. | M. Born and E. Wolf, |

16. | H. C. Chen, |

17. | J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. |

18. | J. J. A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. |

19. | M. Trippenbach and Y. B. Band, “Propagation of light pulses in nonisotropic media,” J. Opt. Soc. Am. B |

20. | A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A |

21. | A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Optics and Spectroscopy |

22. | A. Volyar and T. Fadeeva, “Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal,” Opt. Spectros. |

23. | A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A |

24. | A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E |

25. | A. Ciattoni, G. Cincotti, C. Palma, and H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A |

26. | A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. |

27. | A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E |

28. | S. Olver, Numerical Approximation of Highly Oscillatory Integrals (PhD thesis, 2008). |

29. | D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in |

30. | A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proc. R. Soc. A |

31. | S. Olver, “Moment-free numerical integration of highly oscillatory functions,” IMA J. Numer. Anal. |

32. | D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM J. Numer. Anal. |

33. | X. Lu and L. Chen, “Spin-orbit interactions of a Gaussian light propagating in biaxial crystals,” Opt. Express |

34. | R. Piessens, E. deDoncker Kapenga, C. Uberhuber, and D. Kahaner, |

35. | N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron. |

36. | A. Belsky and M. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. |

37. | V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE |

38. | M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. |

39. | M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A: Pure Appl. Opt. |

40. | M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” in |

41. | M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. |

42. | D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “The creation and annihilation of optical vortices using cascade conical diffraction,” Opt. Express |

43. | N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun. |

44. | M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A: Pure Appl. Opt. |

45. | Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. |

46. | T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A |

47. | T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express |

48. | R. Wong, |

49. | E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. |

50. | Y. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. Kivshar, “Dynamics of linear polarization conversion in uniaxial crystals,” Opt. Express |

**OCIS Codes**

(260.1180) Physical optics : Crystal optics

(260.1960) Physical optics : Diffraction theory

(050.4865) Diffraction and gratings : Optical vortices

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 4, 2013

Revised Manuscript: March 20, 2013

Manuscript Accepted: March 22, 2013

Published: April 1, 2013

**Citation**

Xiancong Lu and Lixiang Chen, "Anisotropic dynamics of optical vortex-beam propagating in biaxial crystals: a numerical method based on asymptotic expansion," Opt. Express **21**, 8493-8507 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8493

Sort: Year | Journal | Reset

### References

- J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, Bristol, 1999).
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45, 8185–8189 (1992). [CrossRef] [PubMed]
- M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997). [CrossRef]
- G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A19, 1680–1688 (2002). [CrossRef]
- A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A20, 163–171 (2003). [CrossRef]
- F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett.95, 253901 (2005). [CrossRef] [PubMed]
- T. Fadeyeva, A. Rubass, Y. Egorov, A. Volyar, and J. Grover Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A25, 1634–1641 (2008). [CrossRef]
- T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A79, 053815 (2009). [CrossRef]
- T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011). [CrossRef]
- T. Fadeyeva, C. Alexeyev, A. Rubass, A. Zinov’ev, V. Konovalenko, and A. Volyar, “Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal,” Opt. Lett.36, 4215–4217 (2011). [CrossRef] [PubMed]
- J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011). [CrossRef]
- C. N. Alexeyev, “Circular array of anisotropic fibers: A discrete analog of a q plate,” Phys. Rev. A86, 063830 (2012). [CrossRef]
- T. A. Fadeyeva, C. N. Alexeyev, P. M. Anischenko, and A. V. Volyar, “Engineering of the space-variant linear polarization of vortex-beams in biaxially induced crystals,” Appl. Opt.51, C224–C230 (2012). [CrossRef] [PubMed]
- T. Fadeyeva, C. Alexeyev, P. Anischenko, and A. Volyar, “The fine structure of the vortex-beams in the biaxial and biaxially-induced birefringent media caused by the conical diffraction,” arXiv:1107.5775 (2011).
- M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).
- H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).
- J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am.66, 780–788 (1976). [CrossRef]
- J. J. A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am.73, 920–926 (1983). [CrossRef]
- M. Trippenbach and Y. B. Band, “Propagation of light pulses in nonisotropic media,” J. Opt. Soc. Am. B13, 1403–1411 (1996). [CrossRef]
- A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A18, 1656–1661 (2001). [CrossRef]
- A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Optics and Spectroscopy94, 235–244 (2003). [CrossRef]
- A. Volyar and T. Fadeeva, “Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal,” Opt. Spectros.101, 450–457 (2006). [CrossRef]
- A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A20, 2163–2171 (2003). [CrossRef]
- A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E67, 036618 (2003). [CrossRef]
- A. Ciattoni, G. Cincotti, C. Palma, and H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A19, 1894–1900 (2002). [CrossRef]
- A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun.231, 79–92 (2004). [CrossRef]
- A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E66, 036614 (2002). [CrossRef]
- S. Olver, Numerical Approximation of Highly Oscillatory Integrals (PhD thesis, 2008).
- D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in Highly Oscillatory Problems (Cambridge University, 2009), pp. 25–50. [CrossRef]
- A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proc. R. Soc. A461, 1383–1399 (2005). [CrossRef]
- S. Olver, “Moment-free numerical integration of highly oscillatory functions,” IMA J. Numer. Anal.26, 213–227 (2006). [CrossRef]
- D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM J. Numer. Anal.44, 1026–1048 (2006). [CrossRef]
- X. Lu and L. Chen, “Spin-orbit interactions of a Gaussian light propagating in biaxial crystals,” Opt. Express20, 11753–11766 (2012). [CrossRef] [PubMed]
- R. Piessens, E. deDoncker Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983).
- N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron.29, 1020 (1999). [CrossRef]
- A. Belsky and M. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun.167, 1–5 (1999). [CrossRef]
- V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001). [CrossRef]
- M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt.6, 289 (2004). [CrossRef]
- M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A: Pure Appl. Opt.7, 685 (2005). [CrossRef]
- M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2007), pp. 13–50. [CrossRef]
- M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt.12, 075704 (2010). [CrossRef]
- D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “The creation and annihilation of optical vortices using cascade conical diffraction,” Opt. Express19, 2580–2588 (2011). [CrossRef]
- N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun.286, 1–5 (2013). [CrossRef]
- M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A: Pure Appl. Opt.1, 601 (1999). [CrossRef]
- Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun.284, 3183–3191 (2011). [CrossRef]
- T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A27, 381–389 (2010). [CrossRef]
- T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express18, 10848–10863 (2010). [CrossRef] [PubMed]
- R. Wong, Asymptotic Approximations of Integrals (SIAM, Philadelphia, 2001). [CrossRef]
- E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett.34, 1021–1023 (2009). [CrossRef] [PubMed]
- Y. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. Kivshar, “Dynamics of linear polarization conversion in uniaxial crystals,” Opt. Express17, 18196–18208 (2009). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.