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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 11753–11766
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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

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.

© 2012 OSA

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

, 25

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

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

]:
2E(E)+k02εE=0,
(1)
where E(r) is the amplitude of the light field, k0=2π/λ is the wave number in vacuum, and ε is the relative dielectric tensor which has the following form,
ε=[nx2000ny2000nz2],
(2)
with nx, ny and nz being the principle refractive indices along the crystalline x, y and z axis, respectively. For uniaxial crystals with nx=nynz, see Fig. 1(c)
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.
, the light propagation behavior and the accompanying angular momentum dynamics have been fully discussed [3

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

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]

, 13

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

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

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

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 nxnynz, 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)=d2kexp(ikr)E˜(k,z),
(3)
where r=xe^x+ye^y and k=kxe^x+kye^y are vectors in the x-y plane of the real space and kxky 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 Δn, namely, E˜(0)=1(2π)2d2rexp(ikr)E(r,0), and accordingly, its x and y components are labeled as E˜x(0) and E˜y(0), respectively. By substituting Eq. (3) into Eq. (1) and after some lengthy but straightforward algebra, we can obtain the exact expression,
E˜(k,z)=(c1c3c5)exp(λ1z)+(c2c4c5)exp(λ2z),
(4)
where
λ1,λ2=12(K±L),
(5)
c1=1L[(1βγ)kxkyE˜y(0)+(ML)E˜x(0)2],c2=1L[(1βγ)kxkyE˜y(0)+(M+L)E˜x(0)2],
(6)
c3=1L[(1αγ)kxkyE˜x(0)(M+L)E˜y(0)2],c4=1L[(1αγ)kxkyE˜x(0)(ML)E˜y(0)2],
(7)
c5=λ1p(kxc1+kyc3),c6=λ2p(kxc2+kyc4),
(8)
and α, β, γ, p, K, L, Mare defined as follows:

α=k02nx2,β=k02ny2,γ=k02nz2,p=γ(kx2+ky2),
(9)
K=(α+β)(1+αγ)kx2(1+βγ)ky2,
(10)
L=(αβ)2+(1αγ)2kx4+(1βγ)2ky4+2(αβ)(1αγ)kx2+2(βα)(1βγ)ky2+2(1αγ)(1βγ)kx2ky2,
(11)
M=(αβ)+(1αγ)kx2(1βγ)ky2.
(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

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

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]

]. 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

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

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 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(r22s2)e^+,
(13)
where s is the waist of Gaussian beam, while e^+=(e^x+ie^y)/2 as well as its partner e^=(e^xie^y)/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=Exe^x+Eye^y=E+e^++Ee^, and the following relation holds,
(E+E)=12[1i1i](ExEy).
(14)
After making a Fourier transformation toward Eq. (13), we get the initial field in the k space,

E˜(0)=s22πexp(k2s22)e^+.
(15)

By substituting Eq. (15) into the coefficients c1~c6 of Eqs. (6)(8), one can formally write out the k space propagating field 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

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

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
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.
. The values of parameters in the calculation are chosen as follows: the refractive index nx=ny=1.656, nz=1.458, propagation distance z=8000μm, wave length λ=0.633μm, and the waist of initial Gaussian beam s=4.59μm [6

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]

]. 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=8000μm 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 4×105. Note that, as the distance z increases, the calculation becomes slower and also the accuracy decreases.

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 nx=1.656 and nz=1.458, while let ny vary in the range between nx and nz (A practical method is to use the linear electro-optic effect to tune ny [30

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 Δn=nxny 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=0.633μm, s=4.59μm) and the propagation distance is fixed at z=2000μm. As the spin eigenstates, a left- and right-handed circularly polarized photon possesses a spin angular momentum of ±, respectively. We therefore investigate the evolutions of E+ and E with different deviations Δn. 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 Δn=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

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]

, 6

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]

]. Of interest is when ny slightly deviates from nx, for example, Δn=0.00001in 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 Δn 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+ to 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 Δn=0.00005. The larger Δn, 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

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

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

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±(Z+)=1Wz00rdr02πdϕ|E±(r,ϕ,Z+)|2,
(16)
which is normalized by the input total power Wz0 at z=0. As has been well-defined, the amount of spin angular momentum per photon within the light field is defined as, S(Z+)=[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(ilϕ), where l is the winding number. We thus have

E±(r,ϕ,Z+)=l=+El±(r,Z+)eilϕ,
(17)
El±(r,Z+)=12π02πE±(r,ϕ,Z+)eilϕdϕ.
(18)

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

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(ilϕ)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

Wl±(Z+)=1Wz002πr|El±(r,Z+)|2dr.
(19)

The above OAM quantities can be computed by just using the light field E±(r,ϕ,Z) owing to the continuous boundary condition at the interface, namely, E±(r,ϕ,Z) =E±(r,ϕ,Z+)=E±(r,ϕ,Z), as mentioned above [33

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 Δn=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,±2,±4,, 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=2000μm 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 (Δn=0), only one OAM component l=2 exists such that only a single vortex emerges in the center of the beam [4

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

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]

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

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]

, 16

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

],
dLfdt=SFn^dS+Vgdr,
(20)
where Lf=Vr×(D×B)dr; S is the surface boundary enclosing the volume V; n^ is the unit vector pointing outward from S; the tensor F and vector g are given by
Fij=εimnxmTnj,gi=εimnTmn,
(21)
respectively, with εimn being the Levi-Cività symbol and Tij being
Tij=ε0Ei(εE)j+1μ0BiBj12(ε0EεE+1μ0BB)δij.
(22)
Specifically, for the biaxial crystals we have

g=[TyzTzyTzxTxzTxyTyx]=ε0[EyEz(nz2ny2)EzEx(nx2nz2)ExEy(ny2nx2)],
(23)

Along the same line in Ref [5

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]

], 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+ to 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×B, vs. Abraham formulation, E×H/c2 [34

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, Lf=Vr×(D×B)dr 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,
dLfzdt=Φ(0+)Φ(Z)+Vgzdr,
(24)
where Φ(z)=Sd2rFzz(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+) =[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)=l=+(Ex,l(r,z)e^x+Ey,l(r,z)e^y)exp(ilϕ),
(25)
whereEx,l(r,z)=[El+(r,z)+El(r,z)]/2andEy,l(r,z)=i[El+(r,z)El(r,z)]/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, Ex,l(r,z)exp(ilϕ) and Ey,l(r,z)exp(ilϕ) in Eq. (25), inside the crystal experience mainly the refractive indices nx and ny, respectively. Thus, their rays inside the crystal have a skew angle with respect to the beam axis, θx=1/(nxk0r) and θy=1/(nyk0r), respectively, as demonstrated by Padgett et al. [35

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 nxk0 and nyk0) and the radius vector, which are
Lx=θx×nxk0×r=l,Ly=θy×nyk0×r=l,
(26)
which is evidently equal to that of a light field with the same helical phase exp(ilϕ) 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+)=l(Wl++Wl)l=l(Wx,l+Wy,l)l, where Wl± is given by Eq. (19) and similarly, Wi,l=1Wz002πr|Ei,l(r,Z)|2dr(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)=llWl+lWl+,
(27)
L(z)=llWllWl.
(28)
The numerical results of Eqs. (27) and (28) are shown in Fig. 5(a)
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.
, 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

Jf(z)=l(l+1)Wl+(z)+l(l1)Wl(z).
(29)

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, gz=0 [5

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]

], see also Eq. (23) with nx=ny. In contrast, due to the lack of rotational variance in the biaxial crystal with nxny, we have

gz=ExEy(ny2nx2)0.
(30)

The nonzero gz 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, dLfz/dt=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

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,
Jf(z)+Jc(z)=Jf(0),
(31)
where Jf(z), as given by Eq. (29), denotes the angular momentum per photon within the light field at a propagation distance z, and Jf(0) denotes the angular momentum per photon of the incident light field at z=0. While Jc(z) denotes the angular momentum transferred and stored in the biaxial crystal per photon, which is a normalized quantity defined as,
Jc(z)=1ηVgzdr=1ηε0(nx2ny2)002π0z12Re(ExEy*)drdϕdz,
(32)
with η being the normalized constant. In the other word, Jc(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 Jf(z) and Jc(z) are changing with respect to the propagation distance z; however, they do show an opposite behavior such that their sum, Jt(z)=Jf(z)+Jc(z), remains almost unchanged. It is noted that Jt(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 Jc(z) is an integral of gz(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 per photon at z=3000μm. Besides, it can be concluded from Eq. (32) that, the spin and orbital contributions to the optical torque are indistinguishable, as gz 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

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).

References and links

1.

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

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]

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]

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]

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]

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]

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]

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]

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]

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]

16.

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

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]

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]

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]

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]

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

  1. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3(2), 161–204 (2011). [CrossRef]
  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]
  4. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A20(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]
  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]
  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. Express14(9), 3724–3729 (2006). [CrossRef] [PubMed]
  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]
  10. Y. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. Kivshar, “Dynamics of linear polarization conversion in uniaxial crystals,” Opt. Express17(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. Express18(10), 10848–10863 (2010). [CrossRef] [PubMed]
  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]
  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]
  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. A84(4), 043815 (2011). [CrossRef]
  16. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt.4(2), S7–S16 (2002). [CrossRef]
  17. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A27(3), 381–389 (2010). [CrossRef]
  18. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A18(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. A20(11), 2163–2171 (2003). [CrossRef] [PubMed]
  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. A19(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]
  22. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-gauss and bessel-gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A19(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]
  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]
  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. SPIE4403, 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. A45(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,” Nature444(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]

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