Spin-orbit interactions of a Gaussian light propagating in biaxial crystals

doi:10.1364/OE.20.011753

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Spin-orbit interactions of a Gaussian light propagating in biaxial crystals

Xiancong Lu and Lixiang Chen »View Author Affiliations

Optics Express, Vol. 20, Issue 11, pp. 11753-11766 (2012)
http://dx.doi.org/10.1364/OE.20.011753

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– Abstract
1. Introduction
2. Theory
3. Numerical analyses
4. Conclusive remarks
– References and links

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Abstract

Based on the plane-wave angular spectrum representation, we derive a formal expression for any light fields propagating in biaxial crystals, and particularly, present an effective numerical method to investigate the propagation behavior for a Gaussian light beam. Unlike uniaxial crystals, we observe the intriguing formation, repulsion and disappearance of vortex pairs, as the refractive indices deviate slightly and gradually from the uniaxial limit. In the Minkowski angular momentum picture, we also investigate the orbital angular momentum dynamics for both left- and right-handed circularly polarized components. Of further interest is the revelation of nonconservation of the angular momentum within the light field during the spin-orbit interactions, and the optical torque per photon that the light exerts on the biaxial crystal is quantified. We interpret these interesting phenomena by the weakly broken rotational invariance of biaxial crystals. The self-consistency of our theory is confirmed by the balance equation describing the conservation law of total angular momentum of filed and crystal in the Minkowski picture.

1. Introduction

Spin and orbital angular momentum (OAM) are two independent degrees of freedom of photons and each can be explored for quantum information encoding [1

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–204 (2011). [CrossRef]

]. Recently, the spin-orbit interaction has been in the focus of much research, owing to their promising applications for accessing a higher-dimensional quantum space [2

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13(6), 064001 (2011). [CrossRef]

, 3

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011). [CrossRef]

]. It was first theoretically predicted by Ciattoni and associates [4

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003). [CrossRef] [PubMed]

, 5

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3), 036618 (2003). [CrossRef] [PubMed]

] that spin-to-OAM conversion can occur for a circularly polarized beam propagating along the optic axis of uniaxial crystals, and was later experimentally verified by Brasselet et al [6

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(7), 1021–1023 (2009). [CrossRef] [PubMed]

, 7

C. Loussert and E. Brasselet, “Efficient scalar and vectorial singular beam shaping using homogeneous anisotropic media,” Opt. Lett. 35(1), 7–9 (2010). [CrossRef] [PubMed]

]. Such spin-orbit coupling plays an important role in producing polychromatic vortices [8

A. Voylar, V. Shvedov, T. A. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal,” Opt. Express 14(9), 3724–3729 (2006). [CrossRef] [PubMed]

], white-light vortex solitons [9

D. N. Neshev, A. Dreischuh, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of polychromatic vortex solitons,” Opt. Lett. 33(16), 1851–1853 (2008). [CrossRef] [PubMed]

], and spatially engineered polarizations [10

Y. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. Kivshar, “Dynamics of linear polarization conversion in uniaxial crystals,” Opt. Express 17(20), 18196–18208 (2009). [CrossRef] [PubMed]

, 11

A. Desyatnikov, 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(10), 10848–10863 (2010). [CrossRef] [PubMed]

]. Anther elegant device based on Pancharatnam-Berry phase principle, known as q-plates, reverses the spin of photons while transferring the change of spin angular momentum into the orbital kind, and therefore induce the novel spin-to-OAM conversion [12

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

]. Similar to q-plates, an electro-optical device with a pair of opposite spiral phase plates was also proposed to allow a polarization-controlled fractional OAM manipulation [13

L. Chen and W. She, “Electrically tunable and spin-dependent integer or non-integer orbital angular momentum generator,” Opt. Lett. 34(2), 178–180 (2009). [CrossRef] [PubMed]

]. In addition, as the refractive indices are electro-optically tunable, a controllable spin-orbit coupling in unixial crystals was recently demonstrated both in theory and experiment [14

L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. 33(7), 696–698 (2008). [CrossRef] [PubMed]

, 15

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011). [CrossRef]

It is illuminating to attribute all above effects to the rotational symmetry of the associated system [16

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4(2), S7–S16 (2002). [CrossRef]

]. In uniaxial crystals with transverse homogeneity, the rotational invariance around the optic axis holds, so a propagating light with gradual depolarization needs acquire additional OAM to guarantee the conservation of total angular momentum along this direction [17

T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A 27(3), 381–389 (2010). [CrossRef]

]. However, for biaxial crystals, their refractive indices slightly deviates from the uniaxial limit, namely, the rotational invariance is weakly broken. Thus it would be very interesting to see what spin-orbit interaction happens in the biaxial crystals. Here we present an effective numerical method to investigate the behavior of a circularly polarized Gaussian beam propagating in such biaxial crystals. Unlike uniaxial crystals, we observe the intriguing formation, repulsion and disappearance of vortex pairs, as the refractive indices deviates slightly and gradually from the uniaxial limit. Besides, we find that the angular momentum within the light field is not conserved during the process of spin-orbit interactions. However, the total angular momentum of the whole system consisting of light and crystal is conserved, which is manifested quantitatively by the balance equation in the Minkowski picture.

2. Theory

Our approach is based on the plane-wave angular-spectrum representation developed by Ciattoni and associates [18

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18(7), 1656–1661 (2001). [CrossRef] [PubMed]

]. The basic idea of Ciattoni et al. is to express the optical field in uniaxial crystal as an superposition of an ordinary and extraordinary plane waves, whose propagation independently satisfy two decoupled parabolic equations. Two special cases with the propagation direction parallel to and orthogonal to the optical axis have been extensively discussed, respectively [18

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18(7), 1656–1661 (2001). [CrossRef] [PubMed]

, 19

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(11), 2163–2171 (2003). [CrossRef] [PubMed]

], and many interesting propagating properties are revealed [20

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(9), 1894–1900 (2002). [CrossRef] [PubMed]

, 21

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231(1-6), 79–92 (2004). [CrossRef]

]. Besides, this method has been widely used to study the propagation of various optical beams in uniaxial crystal [22

G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-gauss and bessel-gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 (2002). [CrossRef] [PubMed]

, 23

G. F. Calvo and A. Picón, “Spin-induced angular momentum switching,” Opt. Lett. 32(7), 838–840 (2007). [CrossRef] [PubMed]

], even with an externally applied electric field [14

L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. 33(7), 696–698 (2008). [CrossRef] [PubMed]

, 15

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011). [CrossRef]

]. Along the same line, we present a formal expression for any light fields propagating in biaxial crystals, and particularly, investigate numerically the propagation behavior for an input Gaussian light beam. It is noted that compared with uniaxial crystal, the propagation of optical beam in biaxial crystal is more complicated due to the breaking of rotational invariance, and therefore there are fewer studies on it [24

M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A, Pure Appl. Opt. 1(5), 601–616 (1999). [CrossRef]

, 25

I. Tinkelman and T. Melamed, “Gaussian-beam propagation in generic anisotropic wave-number profiles,” Opt. Lett. 28(13), 1081–1083 (2003). [CrossRef] [PubMed]

]. Besides, the method developed by Ciattoni et al. describes directly the evolutions of each component of a light field, and then provides an effective approach to investigate the dynamics of spin-orbit coupling.

The propagation of monochromatic light with a wavelength

λ

in an anisotropic crystal is described by the following equation [26

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).

\nabla^{2} E - \nabla (\nabla \cdot E) + k_{0}^{2} ε E = 0,

(1)

where

E (r)

is the amplitude of the light field,

k_{0} = 2 π / λ

is the wave number in vacuum, and

ε

is the relative dielectric tensor which has the following form,

\begin{array}{l} ε = [\begin{array}{l} \begin{matrix} n_{x}^{2} & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & n_{y}^{2} & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & n_{z}^{2} \end{matrix} \end{array}], \end{array}

(2)

with

n_{x}

n_{y}

and

n_{z}

being the principle refractive indices along the crystalline x, y and z axis, respectively. For uniaxial crystals with

n_{x} = n_{y} \neq n_{z}

, see Fig. 1(c) , the light propagation behavior and the accompanying angular momentum dynamics have been fully discussed [3

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 064019 (2011). [CrossRef]

–8

, 13

L. Chen and W. She, “Electrically tunable and spin-dependent integer or non-integer orbital angular momentum generator,” Opt. Lett. 34(2), 178–180 (2009). [CrossRef] [PubMed]

–15

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electro-optic Pockels effect in crystals,” Phys. Rev. A 84(4), 043815 (2011). [CrossRef]

, 17

T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A 27(3), 381–389 (2010). [CrossRef]

–22

G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-gauss and bessel-gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19(8), 1680–1688 (2002). [CrossRef] [PubMed]

]. What we will focus on here is the biaxial crystals with

n_{x} \neq n_{y} \neq n_{z}

, see Fig. 1(b). The first key step is to express the light field

E (r) = E (r_{⊥}, z)

using the two-dimensional Fourier transformation,

E (r_{⊥}, z) = \int d^{2} k_{⊥} \exp (i k_{⊥} \cdot r_{⊥}) \tilde{E} (k_{⊥}, z),

(3)

where

r_{⊥} = x {\hat{e}}_{x} + y {\hat{e}}_{y}

and

k_{⊥} = k_{x} {\hat{e}}_{x} + k_{y} {\hat{e}}_{y}

are vectors in the x-y plane of the real space and

k_{x} - k_{y}

plane of the momentum space, respectively. Thus the initial conditions can also be given in the momentum space by doing an inverse Fourier transformation on the incident field

Δ n

, namely,

\tilde{E} (0) = \frac{1}{{(2 π)}^{2}} \int d^{2} r_{⊥} \exp (- i k_{⊥} \cdot r_{⊥}) E (r_{⊥}, 0)

, and accordingly, its x and y components are labeled as

{\tilde{E}}_{x} (0)

and

{\tilde{E}}_{y} (0)

, respectively. By substituting Eq. (3) into Eq. (1) and after some lengthy but straightforward algebra, we can obtain the exact expression,

\tilde{E} (k_{⊥}, z) = (\begin{array}{l} c_{1} \\ c_{3} \\ c_{5} \end{array}) \exp (\sqrt{λ_{1}} z) + (\begin{array}{l} c_{2} \\ c_{4} \\ c_{5} \end{array}) \exp (\sqrt{λ_{2}} z),

(4)

where

λ_{1}, λ_{2} = \frac{1}{2} (- K \pm \sqrt{L}),

(5)

\begin{array}{l} c_{1} = - \frac{1}{\sqrt{L}} [(1 - \frac{β}{γ}) k_{x} k_{y} {\tilde{E}}_{y} (0) + (M - \sqrt{L}) \frac{{\tilde{E}}_{x} (0)}{2}], \\ c_{2} = \frac{1}{\sqrt{L}} [(1 - \frac{β}{γ}) k_{x} k_{y} {\tilde{E}}_{y} (0) + (M + \sqrt{L}) \frac{{\tilde{E}}_{x} (0)}{2}], \end{array}

(6)

\begin{array}{l} c_{3} = - \frac{1}{\sqrt{L}} [(1 - \frac{α}{γ}) k_{x} k_{y} {\tilde{E}}_{x} (0) - (M + \sqrt{L}) \frac{{\tilde{E}}_{y} (0)}{2}], \\ c_{4} = \frac{1}{\sqrt{L}} [(1 - \frac{α}{γ}) k_{x} k_{y} {\tilde{E}}_{x} (0) - (M - \sqrt{L}) \frac{{\tilde{E}}_{y} (0)}{2}], \end{array}

(7)

\begin{array}{l} c_{5} = - \frac{\sqrt{- λ_{1}}}{p} (k_{x} c_{1} + k_{y} c_{3}), \\ c_{6} = - \frac{\sqrt{- λ_{2}}}{p} (k_{x} c_{2} + k_{y} c_{4}), \end{array}

(8)

and

α

β

γ

p

K

L

M

are defined as follows:

Fig. 1 (a) The schematic diagram of our demonstration. (b) For biaxial crystals, the rotational invariance is weakly broken, while (c) for uniaxial crystals, it holds.

α = k_{0}^{2} n_{x}^{2}, β = k_{0}^{2} n_{y}^{2}, γ = k_{0}^{2} n_{z}^{2}, p = γ - (k_{x}^{2} + k_{y}^{2}),

(9)

K = (α + β) - (1 + \frac{α}{γ}) k_{x}^{2} - (1 + \frac{β}{γ}) k_{y}^{2},

(10)

\begin{array}{l} L = (^{α - β) 2} + {(1 - \frac{α}{γ})}^{2} k_{x}^{4} + {(1 - \frac{β}{γ})}^{2} k_{y}^{4} + 2 (α - β) (1 - \frac{α}{γ}) k_{x}^{2} \\ + 2 (β - α) (1 - \frac{β}{γ}) k_{y}^{2} + 2 (1 - \frac{α}{γ}) (1 - \frac{β}{γ}) k_{x}^{2} k_{y}^{2}, \end{array}

(11)

M = (α - β) + (1 - \frac{α}{γ}) k_{x}^{2} - (1 - \frac{β}{γ}) k_{y}^{2} .

(12)

The analytic solutions of Eqs. (3) and (4) consist of a complete description of optical propagation along z direction in any anisotropic crystals, including uniaxial and biaxial ones. For uniaxial crystals and paraxial limit, one can simplify these equations to obtain analytical results for some special input fields [4

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003). [CrossRef] [PubMed]

, 5

]. Besides, for a biaxial crystal if the light propagates along the optic axis or in the vicinity of binormals, it is also possible to allow an analytic treatment as that in [27

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(11), 1020–1024 (1999). [CrossRef]

, 28

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001). [CrossRef]

]. However, it is difficult to do this for biaxial crystals when the light propagates along one crystalline axis (e.g., z axis) due to the lack of rotational symmetry. We therefore in this paper use an accurate numerical method to effectively perform the oscillating Fourier integral of the complex form of

\tilde{E} (k_{⊥}, z)

in Eq. (3) and obtain the numerical solutions.

Assume that a left-handed circularly polarized Gaussian beam is incident on a biaxial crystal, where the initial field at z = 0 is chosen to be

E (r_{⊥}, 0) = \exp (- \frac{r_{⊥}^{2}}{2 s^{2}}) {\hat{e}}_{+},

(13)

where s is the waist of Gaussian beam, while

{\hat{e}}_{+} = ({\hat{e}}_{x} + i {\hat{e}}_{y}) / \sqrt{2}

as well as its partner

{\hat{e}}_{-} = ({\hat{e}}_{x} - i {\hat{e}}_{y}) / \sqrt{2}

forms the basis of circular polarizations. Generally, the transverse part of electric field

E_{⊥}

can be decomposed by using either the Cartesian basis or the circular basis:

E_{⊥} = E_{x} {\hat{e}}_{x} + E_{y} {\hat{e}}_{y} = E_{+} {\hat{e}}_{+} + E_{-} {\hat{e}}_{-}

, and the following relation holds,

(\begin{array}{l} E_{+} \\ E_{-} \end{array}) = \frac{1}{\sqrt{2}} [\begin{array}{l} \begin{matrix} 1 & - i \end{matrix} \\ \begin{matrix} 1 & i \end{matrix} \end{array}] (\begin{array}{l} E_{x} \\ E_{y} \end{array}) .

(14)

After making a Fourier transformation toward Eq. (13), we get the initial field in the k space,

\tilde{E} (0) = \frac{s^{2}}{2 π} \exp (- \frac{k_{⊥}^{2} s^{2}}{2}) {\hat{e}}_{+} .

(15)

By substituting Eq. (15) into the coefficients

c_{1} ~ c_{6}

of Eqs. (6)–(8), one can formally write out the k space propagating field

\tilde{E} (k_{⊥}, z)

in Eq. (4). Then the real space field

E (r_{⊥}, z)

is obtained after the Fourier integral transformation of Eq. (3). Here we adopt the QUADPACK library to compute this highly oscillating integral [29

R. Piessens, E. deDoncker-Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983), http://www.netlib.org/quadpack/.

]. Before proceeding to discuss the biaxial crystals, we firstly benchmark our numerical method by means of the analytical solutions in uniaxial crystal [4

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003). [CrossRef] [PubMed]

], as shown in Fig. 2 . The values of parameters in the calculation are chosen as follows: the refractive index

n_{x} = n_{y} = 1.656

n_{z} = 1.458

, propagation distance

z = 8000 μ m

, wave length

λ = 0.633 μ m

, and the waist of initial Gaussian beam

s = 4.59 μ m

]. In Figs. 2(a) and 2(b), both numerical and analytical results for the moduli of

E_{+}

and

E_{-}

in a uniaxial crystal are shown as a function of transverse radius

r

. The numerical results agree very well with the analytical solutions, even for a quite large propagating distance

z = 8000 μ m

here. This indicates that the accuracy of present numerical method is excellent, e.g., the absolute errors shown in Figs. 2(c) and 2(d) are smaller than

4 \times 10^{- 5}

. Note that, as the distance z increases, the calculation becomes slower and also the accuracy decreases.

Fig. 2 A comparison of numerical and analytical results. (a) and (b) show the moduli of E₊ and E₋, the circularly polarized components of Gaussian beam propagating in uniaxial crystal, as a function of transverse radius r. (c) and (d) show the absolute errors of our numerical method for |E₊| and |E₋|, respectively.

3. Numerical analyses

3.1 Light propagation behaviors in biaxial crystals

In this part, we transfer our attention to the case of biaxial crystals with different deviations from the uniaxial limit. Without losing generality, we fix

n_{x} = 1.656

and

n_{z} = 1.458

, while let

n_{y}

vary in the range between

n_{x}

and

n_{z}

(A practical method is to use the linear electro-optic effect to tune

n_{y}

[30

W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195(1-4), 303–311 (2001). [CrossRef]

]). Correspondingly, we introduce

Δ n = n_{x} - n_{y}

to describe the degree of the deviation. The incident left-handed circularly polarized Gaussian beam is assumed the same as Eq. (13) (The beam parameters are still chosen as:

λ = 0.633 μ m

s = 4.59 μ m

) and the propagation distance is fixed at

z = 2000 μ m

. As the spin eigenstates, a left- and right-handed circularly polarized photon possesses a spin angular momentum of

\pm ℏ

, respectively. We therefore investigate the evolutions of

E_{+}

and

E_{-}

with different deviations

Δ n

. The results of numerical simulations are presented in Fig. 3 .

Fig. 3 The moduli of E₊ (upper row) and E₋ (lower row) at a fixed propagating distance z = 2000μm for various deviations of Δn. (a) and (f): Δn = 0, (b) and (g): Δn = 0.00001, (c) and (h): Δn = 0.00005, (d) and (i): Δn = 0.0001, (e) and (j): Δn = 0.01. The x and y coordinates are in unit of μm. We observe the intriguing formation, repulsion and disappearance of vortex pairs, as shown in the lower row. (Media 1).

As can be seen in Figs. 3(a) and 3(f) with the uniaxial limit

Δ n = 0

, the input left-handed polarized Gaussian beam without vortex is partially converted to the right-handed one carrying a vortex with topological charge 2, and therefore, a spin-to-OAM conversion is clearly observed [5

, 6

]. Of interest is when

n_{y}

slightly deviates from

n_{x}

, for example,

Δ n = 0.00001

in Figs. 3(b) and 3(g), the vortex core in

E_{-}

is split into two cores, and then a vortex pair forms, lying along the x axis. As

Δ n

increases, the nonzero pattern in the center becomes more and more significant, therefore, the vortex pairs is pushed far away and disappears gradually, see Figs. 3(h) and 3(i). From Figs. 3(a) to 3(j), we also see that the main part of energy power is transferred from

E_{+}

E_{-}

. This is a polarization transfer between opposite spin angular momentum. Another feature of the propagation is the anisotropy in

E_{+}

and

E_{-}

, as could be clearly seen in Figs. 3(c) and 3(h) for a quite small

Δ n = 0.00005

. The larger

Δ n

, the more profound anisotropy in

E_{+}

and

E_{-}

. The reason underlying can be interpreted by the angular momentum dynamics of spin-orbit interactions, as we will reveal below.

3.2 Spin-orbit dynamics of the light field emerging from biaxial crystals

Unlike photon spin associated with polarization, OAM is related to the helical phase structure

\exp (i l ϕ)

of light, which carries an eigenvalue of

l ℏ

per photon (l is an integer) [31

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(11), 8185–8189 (1992). [CrossRef] [PubMed]

]. For photon spin, it is well known that a left- and right-handed circularly polarized photons possess spin angular momentum of

\pm ℏ

, respectively. For optical OAM, it could be well defined in the isotropic media with (and beyond) paraxial approximation or in the uniaxial media with propagation direction coinciding with the optic axis. In uniaxial crystals, the spin-orbit interaction has been reported in detail [4

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003). [CrossRef] [PubMed]

–6

]. In their situations, as the light propagates, the left-handed circularly polarized component with initial zero OAM is gradually and monotonously converted into the right-handed component with OAM of

2 ℏ

per photon, see also the dashed lines (blue) in Fig. 4(a) . The total power is equally distributed between these two angular momentum components in the limit of

z \to \infty

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003). [CrossRef] [PubMed]

, 5

]. Besides, the extreme spin-orbit coupling is also recently demonstrated for Bessel-Gaussian beams propagating through a uniaxial crystal [17

T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A 27(3), 381–389 (2010). [CrossRef]

Fig. 4 Spiral spectra of the emergent light field carried by (a) left-handed component,

W_{l}^{+}

, and (b) right-handed one,

W_{l}^{-}

, as a function of the crystal length Z, where l denotes the OAM number. The deviation is fixed at Δn = 0.0001 and the other parameters used for calculation are the same as those in Fig. 3. The dashed lines in the uniaxial case are also plotted for comparison.

It is worth emphasizing that the definition of optical OAM is still a tricky question in the anisotropic medium without rotational symmetry [5

, 32

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005). [CrossRef]

], as in our case of biaxial crystals. In order to elucidate the spin-orbit interaction of the light filed within the conventional frame of angular momentum dynamics, here we follow Berry's way [32

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005). [CrossRef]

] to instead consider the light fields at the exit interface outside the biaxial crystal. In the next section, we will employ the balance equation for total angular momentum inside biaxial crystal to manifest the angular momentum coupling that occurs between the light field and the crystal. Of importance is that we find our analysis and calculation are self-consistent and therefore valid in the Minkowski picture.

We consider a stripe of crystal with a thickness of Z, see Fig. 1(a), so that we have

0 < z < Z

inside a crystal. Since the light is propagating along the z axis, we are interested in the total amount of z component angular momentum of light field flowing out of the crystal at

z = Z

. For clarity, we distinguish the two sides of the exit interface by

z = Z^{-}

and

z = Z^{+}

, which are defined inside and outside the crystal, respectively. According to the boundary conditions of Maxwell's equations, the tangential component of the electric field vector is continuous at the boundary surface [33

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

]. Therefore, we have the following important relation in our case:

E_{⊥} (r, ϕ, Z^{-}) = E_{⊥} (r, ϕ, Z^{+}) = E_{⊥} (r, ϕ, Z)

. The key point in such consideration is that both the spin and orbital components of angular momentum outside the crystal are well-defined and can be calculated by their conventional definitions [32

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005). [CrossRef]

]. This thus allows us to study the spin-orbit interactions of the emergent light field from the biaxial crystal, without the need to touch the rigorous OAM definition inside the biaxial crystal.

Generally, the angular momentum within a light field outside the crystal

(z = Z^{+})

can be formally divided into the spin and orbital parts, namely,

J (Z^{+}) = S (Z^{+}) + L (Z^{+})

. For the spin part, we firstly consider the normalized energy power carried by the left- and right-handed components,

W_{\pm} (Z^{+}) = \frac{1}{W_{z 0}} \int_{0}^{\infty} r d r \int_{0}^{2 π} d ϕ {| E_{\pm} (r, ϕ, Z^{+}) |}^{2},

(16)

which is normalized by the input total power

W_{z 0}

z = 0

. As has been well-defined, the amount of spin angular momentum per photon within the light field is defined as,

S (Z^{+}) = ℏ [W_{+} (Z^{+}) - W_{-} (Z^{+})]

. For the orbital part, it is valid for us to compute the projection of both left- and right-handed components into the discrete spiral harmonics

\exp (i l ϕ)

, where l is the winding number. We thus have

E_{\pm} (r, ϕ, Z^{+}) = \sum_{l = - \infty}^{+ \infty} E_{l}^{\pm} (r, Z^{+}) e^{i l ϕ},

(17)

E_{l}^{\pm} (r, Z^{+}) = \frac{1}{2 π} \int_{0}^{2 π} E_{\pm} (r, ϕ, Z^{+}) e^{- i l ϕ} d ϕ .

(18)

We emphasize again here we have followed Berry's way to consider the light field outside the biaxial crystal [32

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005). [CrossRef]

], such that the spiral harmonics

\exp (i l ϕ)

in Eq. (17) can be validly recognized as the OAM eigenstates. As a result, the weight of each OAM component of the light field at the exit interface outside the crystal can be given by

W_{l}^{\pm} (Z^{+}) = \frac{1}{W_{z 0}} \int_{0}^{\infty} 2 π r {| E_{l}^{\pm} (r, Z^{+}) |}^{2} d r .

(19)

The above OAM quantities can be computed by just using the light field

E_{\pm} (r, ϕ, Z)

owing to the continuous boundary condition at the interface, namely,

E_{\pm} (r, ϕ, Z^{-})

= E_{\pm} (r, ϕ, Z^{+}) = E_{\pm} (r, ϕ, Z)

, as mentioned above [33

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

]. The numerical results are illustrated in Fig. 4. Different from uniaxial crystals, one can see that the field spiral spectrum outside the biaxial crystal with

Δ n = 0.0001

is spread here. The incident beam is a Gaussian one with zero OAM, however, both left- and right-handed components of emergent light field distribute among the even OAM numbers of

l = 0, \pm 2, \pm 4, \cdot \cdot \cdot

, in contrast to only a single OAM in uniaxial cases. For a relatively short crystal thickness Z, the small OAM numbers are dominant in the spectrum. But for a large Z, some of their weights are not always changing monotonously, instead, in an oscillating format. As mentioned above, as a light propagates in a biaxial crystal, it accompanies the formation of a vortex pair. The reason can be further revealed here. We focus on the spiral spectrum of the right-handed component. As can be seen in Fig. 4(b), when

z = 2000 μ m

there are two dominant OAM components, namely,

l = 0

and

l = 2

. It is just the superposition of these two components that accounts for the formation of the vortex pair, see Fig. 3(i). While for uniaxial crystals

(Δ n = 0)

, only one OAM component

l = 2

exists such that only a single vortex emerges in the center of the beam [4

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003). [CrossRef] [PubMed]

, 5

], see also Figs. 3(a) and 3(f). Along the same line, one can conclude that, as the deviation

Δ n

becomes larger, the contribution of

l = 2

component become smaller while that of

l = 0

component becomes larger, such that the vortex pair is pushed away, and finally disappears due to the zero contribution of

l = 2

component, as clearly shown by Figs. 3(i)–3(j).

3.3 Nonconservation of the angular momentum within the light field in biaxial crystals

Another interesting phenomenon is the observation of nonconservation of the angular momentum within the light field in the biaxial crystal, different from that in uniaxial case. We employ the balance equation for the angular momentum of the light field and biaxial crystal to manifest the angular momentum coupling that occurs between the light field and crystal, which conserves the total angular momentum of the whole system (field + crystal). Of importance is that we find present analysis and calculation are self-consistent in the Minkowski picture.

Generally, we have the following balance equation for the total angular momentum in any anisotropic media without charges [5

, 16

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4(2), S7–S16 (2002). [CrossRef]

\frac{d L_{f}}{d t} = \int_{S} F \hat{n} d S + \int_{V} g d r,

(20)

where

L_{f} = \int_{V} r \times (D \times B) d r

; S is the surface boundary enclosing the volume V;

\hat{n}

is the unit vector pointing outward from S; the tensor F and vector g are given by

F_{i j} = ε_{i m n} x_{m} T_{n j}, g_{i} = ε_{i m n} T_{m n},

(21)

respectively, with

ε_{i m n}

being the Levi-Cività symbol and

T_{i j}

being

T_{i j} = ε_{0} E_{i} {(ε E)}_{_{j}} + \frac{1}{μ_{0}} B_{i} B_{j} - \frac{1}{2} (ε_{0} E \cdot ε E + \frac{1}{μ_{0}} B \cdot B) δ_{i j} .

(22)

Specifically, for the biaxial crystals we have

g = [\begin{array}{l} T_{y z} - T_{z y} \\ T_{z x} - T_{x z} \\ T_{x y} - T_{y x}^{} \end{array}] = ε_{0} [\begin{array}{l} E_{y} E_{z} (n_{z}^{2} - n_{y}^{2}) \\ E_{z} E_{x} (n_{x}^{2} - n_{z}^{2}) \\ E_{x} E_{y} (n_{y}^{2} - n_{x}^{2}) \end{array}],

(23)

Along the same line in Ref [5

], we choose the volume V to be a stripe 0<z<Z. Remember that in subsection 3.2, we have distinguished the two sides of the interface inside and outside the biaxial crystal by

z = Z^{-}

and

z = Z^{+}

. In the strict sense, Eq. (20) is established inside the crystal with the stripe in the bound from

z = 0^{+}

z = Z^{-}

, as shown in Fig. 1(a). But the spin-orbit dynamics described by Fig. 4 is defined well only outside the crystal. A delicate treatment is therefore required to consider the angular momentum exchange at the interfaces between inside and outside crystal. Obviously, such consideration is associated with the controversy over the momentum (also angular momentum) of light inside a dielectric, namely, Minkowski formulation

D \times B

, vs. Abraham formulation,

E \times H / c^{2}

[34

U. Leonhardt, “Optics: momentum in an uncertain light,” Nature 444(7121), 823–824 (2006). [CrossRef] [PubMed]

]. Of importance is then we can interpret well that the formulation,

L_{f} = \int_{V} r \times (D \times B) d r

defined in the left-handed side of Eq. (20), is just the Minkowski angular momentum of light field stored in a volume V, rather than the Abraham one. Along this line, we consider the z component of Eq. (20) and take the time average, that is to say,

〈 \frac{d L_{f z}}{d t} 〉 = Φ (0^{+}) - Φ (Z^{-}) + \int_{V} 〈 g_{z} 〉 d r,

(24)

where

Φ (z) = - \int_{S} d^{2} r_{⊥} 〈 F_{z z} (r_{⊥}, z, t) 〉

describes the time-average angular momentum flux flowing through any z plane from left to right. Of further interest is that in the frame of the Minkowski picture the angular momentum flux is also continuous at the interface inside and outside the crystal, namely,

Φ (Z^{-}) = Φ (Z^{+})

, as we show below. In this perspective, it is therefore convenient for us to know the amount of spin and OAM inside the crystal, but avoid the need to touch the rigorous OAM definition in the biaxial crystal.

As has been well-known, photon spin angular momentum is in essence associated with the polarization of light, so we have the relation at the exit interface:

S (Z^{-}) = S (Z^{+})

= ℏ [W_{+} (Z^{+}) - W_{-} (Z^{+})]

. While for optical OAM, it is found that the OAM content of a light field could be the same inside and outside the crystal in the Minkowski picture. In principle, we can express the total light field in terms of linear polarization as (irrespective of its spin),

E_{⊥} (r, ϕ, z) = \sum_{l = - \infty}^{+ \infty} (E_{x, l} (r, z) {\hat{e}}_{x} + E_{y, l} (r, z) {\hat{e}}_{y}) \exp (i l ϕ),

(25)

where

E_{x, l} (r, z) = [E_{l}^{+} (r, z) + E_{l}^{-} (r, z)] / \sqrt{2}

and

E_{y, l} (r, z) = i [E_{l}^{+} (r, z) - E_{l}^{-} (r, z)] / \sqrt{2}

. Recall that this expression of

E_{⊥} (r, ϕ, z)

is applicable to the light fields both inside and outside crystal, as it is continuous at

z = Z^{-}

and

z = Z^{+}

. Besides, we can reckon that the x and y components,

E_{x, l} (r, z) \exp (i l ϕ)

and

E_{y, l} (r, z) \exp (i l ϕ)

in Eq. (25), inside the crystal experience mainly the refractive indices

n_{x}

and

n_{y}

, respectively. Thus, their rays inside the crystal have a skew angle with respect to the beam axis,

θ_{x} = 1 / (n_{x} k_{0} r)

and

θ_{y} = 1 / (n_{y} k_{0} r)

, respectively, as demonstrated by Padgett et al. [35

M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 10, 1555–1562 (2003).

]. Then the amount of OAM per photon can be given by multiplying the skew angles with the axial momenta (for the Minkowski's formations

n_{x} ℏ k_{0}

and

n_{y} ℏ k_{0}

) and the radius vector, which are

L_{x} = θ_{x} \times n_{x} ℏ k_{0} \times r = l ℏ, L_{y} = θ_{y} \times n_{y} ℏ k_{0} \times r = l ℏ,

(26)

which is evidently equal to that of a light field with the same helical phase

\exp (i l ϕ)

outside the crystal, although it is still difficult to affirm that whether Eq. (17) represents the OAM eigenmodes in the biaxial crystal. Then we know the total OAM amount per photon within a light field inside the crystal is just equal to that outside the crystal, namely,

L (Z^{-}) = L (Z^{+}) = \sum_{l} (W_{l}^{+} + W_{l}^{-}) l = \sum_{l} (W_{x, l}^{} + W_{y, l}^{}) l

, where

W_{l}^{\pm}

is given by Eq. (19) and similarly,

W_{i, l} = \frac{1}{W_{z 0}} \int_{0}^{\infty} 2 π r {| E_{i, l} (r, Z) |}^{2} d r

(i = x, y)

. As a result, the total amount of angular momentum within the light field remains unchanged at the exit interface when the light emerges from the biaxial crystal, namely,

Φ (Z^{-}) = Φ (Z^{+}) = Φ (Z)

. In the similar manner, we also have

Φ (0^{-}) = Φ (0^{+}) = Φ (0)

at the entrance interface. Therefore, the superscripts ' + ' and '-' at the right-handed side of Eq. (24) can be ignored. Besides, it thus allows us to investigate the spin-orbit interaction of the light field in the biaxial crystal straightforwardly based on Eqs. (17)–(19).

Following Eq. (19), the OAM value per photon carried by left- and right-handed circularly polarized components in the biaxial crystal are calculated, respectively, as follows:

L_{+} (z) = \frac{\sum_{l} l W_{l}^{+}}{\sum_{l} W_{l}^{+}},

(27)

L_{-} (z) = \frac{\sum_{l} l W_{l}^{-}}{\sum_{l} W_{l}^{-}} .

(28)

The numerical results of Eqs. (27) and (28) are shown in Fig. 5(a) , where

L_{+} (z)

and

L_{-} (z)

change independently and slightly with the propagation distance z. While the field angular momentum per photon, as a sum of spin and orbital angular momentum, can be obtained as

Fig. 5 (a) Spin-orbit angular momentum dynamics within the light field emerging from the biaxial crystal; (b) Conservation law of total angular momentum of the light field and biaxial crystal. All parameters used in calculation are the same as those in Fig. 4.

J_{f} (z) = \sum_{l} (l + 1) W_{l}^{+} (z) + \sum_{l} (l - 1) W_{l}^{-} (z) .

(29)

Of interest is the further observation of nonconservation of the angular momentum within the light field, in contrast with the uniaxial case where the sum of spin and orbital angular momentum remains always conserved [5

, 6

]. As shown by the black line in Fig. 5(b),

J_{f} (z)

is different from that of the incident light, which is one

ℏ

per photon in our case. The reason underlying can be well interpreted by the weakly broken rotation invariance of a biaxial crystal. For a uniaxial crystal, the left-handed component with initial zero OAM is converted into the right-handed one with OAM of

2 ℏ

per photon, therefore conserving the total angular momentum. This is related to the transverse homogeneity of uniaxial crystals,

n_{x} = n_{y} \neq n_{z}

, namely, the rotational invariance holds along the z axis, see Fig. 1(c). In contrast, inside a biaxial crystal,

n_{y}

is varied to depart slightly from

n_{x}

such that the rotational invariance in the transverse plane is weakly broken, see Fig. 1(b). Therefore, the total angular momentum within the light field cannot be conserved any longer as propagating.

We can also illustrate quantitatively the nonconservation of field angular momentum in the frame of the balance equation of Eq. (24). For uniaxial crystals, it has been proved that the z component of g vanishes as a consequence of the rotational invariance around the z axis of the crystal, namely,

g_{z} = 0

], see also Eq. (23) with

n_{x} = n_{y}

. In contrast, due to the lack of rotational variance in the biaxial crystal with

n_{x} \neq n_{y}

, we have

g_{z} = E_{x} E_{y} (n_{y}^{2} - n_{x}^{2}) \neq 0.

(30)

The nonzero

g_{z}

therefore will play an important role in the angular momentum coupling that occurs between light and crystal. Since the time average of a time derivative at the left-handed side of Eq. (24) vanishes, namely,

〈 d L_{f z} / d t 〉 = 0

, it is reasonable for us to anticipate that the total angular momentum of the system consisting of the light field and the biaxial crystal would be conserved, although the field angular momentum is not a conserved one. From this angle, we can interpret the third term in the right-handed side of Eq. (24) as the angular momentum transferred from the light field and stored in the biaxial crystal [36

W. Gough, “The angular momentum of radiatlon,” Eur. J. Phys. 7(2), 81–87 (1986). [CrossRef]

]. After normalized by the total photon numbers, Eq. (24) can be at last rewritten as,

J_{f} (z) + J_{c} (z) = J_{f} (0),

(31)

where

J_{f} (z)

, as given by Eq. (29), denotes the angular momentum per photon within the light field at a propagation distance z, and

J_{f} (0)

denotes the angular momentum per photon of the incident light field at

z = 0

. While

J_{c} (z)

denotes the angular momentum transferred and stored in the biaxial crystal per photon, which is a normalized quantity defined as,

J_{c} (z) = - \frac{1}{η} \int_{V} 〈 g_{z} 〉 d r = \frac{1}{η} ε_{0} (n_{x}^{2} - n_{y}^{2}) \int_{0}^{\infty} \int_{0}^{2 π} \int_{0}^{z} \frac{1}{2} Re (E_{x} E_{y}^{*}) d r d ϕ d z,

(32)

with

η

being the normalized constant. In the other word,

J_{c} (z)

is just the optical torque exerting on the biaxial crystal contributed by per photon, which is induced by the angular momentum coupling from light to matter.

Our simulation results are shown by Fig. 5(b), which do support our prediction. As can be seen, both

J_{f} (z)

and

J_{c} (z)

are changing with respect to the propagation distance z; however, they do show an opposite behavior such that their sum,

J_{t} (z) = J_{f} (z) + J_{c} (z)

, remains almost unchanged. It is noted that

J_{t} (z)

is not ideally a constant one. We think this may arise from the finite numbers of points we used for calculations as well as the numerical error, where the deviation may be accumulated as

J_{c} (z)

is an integral of

g_{z} (z)

. But we do observe the evident nonconservation of the angular momentum within the light field, for example, the change amount of angular momentum from light to matter can reach

1.5 ℏ

per photon at

z = 3000 μ m

. Besides, it can be concluded from Eq. (32) that, the spin and orbital contributions to the optical torque are indistinguishable, as

g_{z}

cannot be separated well into two parts that describe independently the evolutions of the polarization state and the helical phase structure. Finally, it is worth pointing out again that the angular momentum we consider here is the Minkowski one, as shown in Eq. (20); and the good agreement between the theoretical predictions and numerical simulations in Fig. 5 does imply our treatment above describing the spin-orbit interaction of light in the biaxial crystals is self-consistent and therefore valid in the Minkowski picture.

4. Conclusive remarks

We have presented an effective numerical method to investigate the behavior of a focused and circularly polarized Gaussian light beam propagating in biaxial crystals. Different from uniaxial crystals, we have observed the intriguing formation, repulsion and disappearance of vortex pairs, as the refractive indices deviates slightly and gradually from the uniaxial limit. Besides, we investigate the OAM dynamic for both left- and right-handed circularly polarized components in the light field. Of further interest is the revelation of nonconservation of the angular momentum within the light field, the reason underlying is well interpreted by the weakly broken rotational invariance of the biaxial crystals. We emphasize again that in our simulations, we have followed Berry's way to perform the calculation on the light field outside the crystal, and therefore avoid the need to touch the rigorous definition of optical OAM in the biaxial crystals. The self-consistency and validity of our treatment is then confirmed quantitatively by the balance equation describing the conservation law of total angular momentum of the whole system (field + crystal) in the Minkowski picture. It is worth noting that our work is related to two basic but ambiguous questions, namely, the Minkowski and Abraham (angular) momentum in the transparent dielectric, and the rigorous definition of optical OAM eigenstates in the anisotropic media with the lack of rotation symmetry, both of which deserve our further research interest in the near future.

Acknowledgments

The authors acknowledge gratefully the financial support from the National Natural Science Foundation of China (NSFC) (Grant No. 11004164 and 11104233), the Doctoral Fund of Ministry of Education of China (Grant No. 2011012112003), the Fundamental Research Funds for the Central Universities (Grant No. 2011121043, 2012121015) and the Natural Science Foundation of Fujian Province of China (Grant No. 2011J05010).

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26.	M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).
27.	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(11), 1020–1024 (1999). [CrossRef]
28.	V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE 4403, 229–240 (2001). [CrossRef]
29.	R. Piessens, E. deDoncker-Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983), http://www.netlib.org/quadpack/.
30.	W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun. 195(1-4), 303–311 (2001). [CrossRef]
31.	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(11), 8185–8189 (1992). [CrossRef] [PubMed]
32.	M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005). [CrossRef]
33.	A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
34.	U. Leonhardt, “Optics: momentum in an uncertain light,” Nature 444(7121), 823–824 (2006). [CrossRef] [PubMed]
35.	M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 10, 1555–1562 (2003).
36.	W. Gough, “The angular momentum of radiatlon,” Eur. J. Phys. 7(2), 81–87 (1986). [CrossRef]

OCIS Codes
(260.1180) Physical optics : Crystal optics
(260.1960) Physical optics : Diffraction theory
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Physical Optics

History
Original Manuscript: February 21, 2012
Revised Manuscript: April 23, 2012
Manuscript Accepted: April 23, 2012
Published: May 9, 2012

Citation
Xiancong Lu and Lixiang Chen, "Spin-orbit interactions of a Gaussian light propagating in biaxial crystals," Opt. Express 20, 11753-11766 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-11753

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References

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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(11), 1020–1024 (1999). [CrossRef]
V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khjlo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion ofBessel optical vortices,” Proc. SPIE4403, 229–240 (2001). [CrossRef]
R. Piessens, E. deDoncker-Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983), http://www.netlib.org/quadpack/ .
W. She and W. Lee, “Wave coupling theory of linear electrooptic effect,” Opt. Commun.195(1-4), 303–311 (2001). [CrossRef]
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(11), 8185–8189 (1992). [CrossRef] [PubMed]
M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt.7(11), 685–690 (2005). [CrossRef]
A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
U. Leonhardt, “Optics: momentum in an uncertain light,” Nature444(7121), 823–824 (2006). [CrossRef] [PubMed]
M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt.10, 1555–1562 (2003).
W. Gough, “The angular momentum of radiatlon,” Eur. J. Phys.7(2), 81–87 (1986). [CrossRef]

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