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OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014
Asymptotic theory of strong spin–orbit coupling in optical fiber Steven Golowich MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA ([email protected]) Received September 23, 2013; revised November 15, 2013; accepted November 22, 2013; posted November 25, 2013 (Doc. ID 198163); published December 20, 2013 The spin–orbit coupling of light propagating in optical fiber can be dramatically enhanced by the presence of a highcontrast interface in the refractive index profile, even for modes that are highly paraxial. The resulting modes have spatial and polarization structures that depart greatly from the weak coupling form, and, in particular, are neither orbital nor spin angular momentum eigenstates. We explain the physical origins of this strong-coupling regime with a vector geometric theory of diffraction expansion. © 2013 Optical Society of America OCIS codes: (080.2720) Mathematical methods (general); (080.4865) Optical vortices; (260.6970) Total internal reflection; (260.1960) Diffraction theory. http://dx.doi.org/10.1364/OL.39.000092
The coupling between the spin and orbital degrees of freedom of light [1] explains a variety of dynamical effects involving the propagation of beams in inhomogeneous media. A geometric interpretation for the evolution of polarization along the beam trajectory is given by the Berry phase [2,3]. Conversely, the polarization state of a beam can influence its trajectory, which was first appreciated in the case of reflections at an interface in the form of the Goos–Hänchen [4] and Imbert–Fedorov [5,6] shifts. Later, a Hamiltonian formalism was introduced for the propagation of near-paraxial beams in inhomogeneous media [7–9], which explains the spin Hall effect of light. The theory of spin–orbit coupling was cast in a more general geometric framework in [10–13]. An important category of inhomogeneous optical media that are influenced by spin–orbit coupling is optical fiber. Indeed, manifestations of Berry’s phase in optics were first predicted [14] and observed [15] in a helically wound optical fiber. Most optical fibers are weakly guiding [16] in the sense that the refractive index contrasts responsible for light guidance are small, typically a few percent or less, and propagation is therefore highly paraxial. The paraxial limit of the Maxwell equations is known to decouple the spin and orbital degrees of freedom [17], but post-paraxial corrections induce spin–orbit coupling. A result is an azimuthal shift of speckle patterns in a weakly guided fiber [7,8,18]. The guided mode spectrum of weakly guiding rotationally and translationally invariant fibers break into degenerate subspaces that can be indexed by their orbital and spin angular momentum (OAM and SAM) content, which is a direct result of spin–orbit coupling [19,20]. The refractive index profile of a rotationally symmetric fiber can be specifically tailored to enhance the spin–orbit coupling [21,22]. Fibers with these design features have been used as devices to create and manipulate [23], and to transmit information over [24], OAM states. All of these previous studies of spin–orbit coupling in optical fiber have treated the effect as a weak perturbation of the scalar Maxwell equations. In this Letter, we study a class of fiber structures and associated guided modes that experience strong spin–orbit coupling, despite being highly paraxial. Fibers with an air core 0146-9592/14/010092-04$15.00/0
surrounded by a raised-index ring (Fig. 1) have been proposed for atom guiding [25,26] and communications [27–29]. It has previously been noticed that such fibers can support modes that depart substantially from the weak guidance approximation, in which polarization is treated perturbatively [30]. Here we emphasize two notable results of this strong coupling; that the linearly polarized (LP) degenerate subspaces of the scalar wave equation split into vector components that are widely separated in effective index [28], and that the guided modes cannot be factorized into spatial and spin terms; in particular, they are neither OAM nor SAM eigenstates. We explain the physical origins of these effects with a vector geometric theory of diffraction (GTD) calculation that explicitly relates the Fresnel reflection equations [31] to the spectrum and structure of the guided modes. Figure 2 compares the polarization structures of the HE1;1 modes of “low-contrast” and “high-contrast” fibers. Both structures are ring fibers with r co 16 μm and Δr ri 4 μm; the low-contrast structure has a silica core and cladding, with Δnri 0.004, while the high contrast structure has an air core, silica cladding, and Δnri 0.035; both are manufacturable with current technology. The free-space wavelength λ is 1.55 μm. The black arrows show the polarization of the odd-parity modes [16], which are by construction linear at each point in space. The red ellipses depict the polarization states of the superposieven tions HEodd m;n iHEm;n , which give rise to spatially uniform circular polarization in the low-contrast case, but spatially varying elliptical polarization in the highcontrast case. The low-contrast examples illustrate that, when spin–orbit coupling is weak, a basis of propagating
Fig. 1. Class of refractive index profiles considered. © 2014 Optical Society of America
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
(a) Fig. 2.
93
(b)
Comparison between HE1;1 fields of low-contrast (a) and high-contrast (b) fibers.
states can be chosen such that the elements factor into a product of spatial and spin parts [16]; in particular, these modes are both OAM and SAM eigenstates that carry integral values of OAM and SAM in units of ℏ∕photon. The strongly spin–orbit coupled fields, by contrast, are not. A further contrast is the magnitude of HEodd around the ring, which is constant in (a), 1;1 but not in (b) [30,32]. To gain a better understanding of the physical origins of the strong spin–orbit coupling regime in this class of fibers, we apply the Keller–Rubinow (KR) method [33] for constructing an asymptotic expansion for the guided mode eigenvalue problem. This method consists of defining a closed congruence of rays, consisting of a number of normal congruences that are connected to one another by reflection or passing through a caustic surface. In the case of a ring fiber, there are two normal congruences, one consisting of outgoing and the other of incoming rays. Each forms a simple covering of the space extending inward from the outer boundary to either a caustic or the inner boundary. The guided modes are determined by a quantization condition, described below. We note that a similar construction explains the spin–orbit coupling of nonparaxial light in free space, with the coupling arising from the Berry phase [34]. The waves corresponding to each normal congruence are obtained by expressing the electric field as a power series in ω−1 . Because the index is constant inside the ring, a conventional geometric optics expansion will suffice [12,35,36]. Since we are interested in the guided modes, which are harmonic in z, we may write ∞ X
e−iωt−τr Er; t eiβz−ωt em;n r⊥ es ; m;n r⊥ −iωs s0
and obey Snell’s law of reflection at the interfaces. We index them by η, the angle between a ray and the z axis, and a0 , the minimum radius the ray would attain were the inner reflecting boundary not present (i.e., that of the caustic surface). Both η and a0 are conserved along the ray paths, and, by allowing these parameters to vary, we find a two-parameter family of pairs of outgoing and incoming congruences. The polarization of the field e0 m;n along the ray remains constant between reflections, while its magnitude can be shown to vary as r 2 − a20 −1∕4 . The propagation constant is related to the ray invariants by β nri k0 cos η, with k0 2π∕λ. The eigenvalue equations that determine the allowed values of η and a0 are derived from the condition that the fields be single valued, which may be written as I ω
Γ
∇τ · dl 2 πn − γ;
(2)
where n ∈ Z, Γ is a closed curve on the two-sheeted covering space, and γ is a phase factor resulting from the two reflections. The covering space is topologically a torus, so we must consider two independent generators of the fundamental group as the contours Γ in Eq. (2). The generators that we choose are shown in Fig. 3. The first (blue) is a ring surrounding, but just outside, the fiber core, so it is wholly contained in one of the two sheets. Application of Eq. (2) to this contour yields nri k0 a0 sin η m;
(3)
(1)
where r⊥ x; y are the transverse coordinates, ω the optical frequency, and β the propagation constant. The function em;n represents the spatial dependence of a mode, with azimuthal and radial mode indexes m and n. The functions τ and es m;n satisfy, respectively, the eikonal and transport equations, which are found order by order in ω−1 by substitution into the Maxwell equations. We will restrict attention to the leading order term e0 m;n . The rays are straight lines between the two interfaces
Fig. 3. Paths corresponding to the two GTD eigenvalue equations for the ring fiber.
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OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014
and it can be shown that m ∈ Z may be identified with the azimuthal mode index. The second contour (red) in Fig. 3 follows a ray as it reflects off both ring boundaries (ABCD), then returns to the starting point, crossing both sheets. We find nri k0 sin η a22 − a20 1∕2 − a21 − a20 1∕2 a a γ πn − ; − a0 arccos 0 − arccos 0 a2 a1 2
(4)
where n ∈ Z is the azimuthal mode index, and γ is the phase factor resulting from the two reflections. This latter factor requires further discussion, as it depends on the polarization of the field incident on the interfaces, as well as the incidence angles of the rays on the interfaces. The fields throughout a closed congruence are determined by their values on the rays just leaving the inner boundary, and these values must be chosen such that the single-valued condition holds up to the integral multiple of 2π in Eq. (2). Recalling the z harmonic condition in Eq. (1), we must specify these fields in a plane z z0 . In general, this initial data is provided by a complex vector function of the azimuthal variable e0 e0 ϕ. Guided by the exact solution of the Maxwell equations, we choose the value of the field e00 at a single point ϕ 0 and assign the values at other values of ϕ by e0 ϕ Rϕe00 , where Rϕ is the rotation by ϕ around the z axis. The vector e00 is a Jones vector, as it must be orthogonal to the ray passing through the point ϕ 0 at the inner ring boundary, but further information about its value must be obtained as part of the solution of Eq. (4). Given values of a0 and η, we may compute the angle ϕED ϕE − ϕD , referring to Fig. 3. In order for the two congruences of the outgoing and incoming rays to form a closed congruence, the vector field amplitude e00 must map into its rotation RϕED e00 , up to an overall phase, upon the two reflections. This condition may be written as an eigenvalue equation in Stokes space,
Field
ˆ ED † Tˆ AD eˆ 0 eˆ 0 ; Rϕ 0 0
(5)
ˆ refers to the Stokes vector (matrix) where the notation A; associated with a Jones vector (matrix) A [37]. Here, again referring to Fig. 3, T AD is the Jones matrix that maps the Jones vector at point A into that at D, as may be computed from the Fresnel reflection equations [31]. It is the corresponding Stokes matrix Tˆ AD that encodes the strength of the spin–orbit coupling, which is determined by the index contrasts and incidence angles at the two interfaces. The Stokes vector eˆ 00 that determines the polarization state of the fields throughout ˆ ED † Tˆ AD the fiber is identified with the eigenvector of Rϕ with unity eigenvalue. In the weak coupling limit of small contrasts or grazing incidence [e.g., Fig. 2(a)], Tˆ AD tends to the identity, forcing eˆ 00 to be circularly ˆ ED is a rotation around the zˆ axis polarized since Rϕ on the Poincaré sphere. In the general case, eˆ 0 is elliptically polarized [Fig. 2(b)]. Finally, the phase factor γ that we require in Eq. (4) can be read off the corresponding Jones vectors. The mode fields may also be computed by the KR method. It can be shown that the radial and azimuthal components of the electric field associated with each closed congruence have angular dependence of expimϕ and relative phase of exactly π∕2. In Fig. 4, we compare the real-valued field components er r and −ieϕ r of KR and exact solutions, for both the low- and high-contrast profiles. The effective indexes neff β∕k0 of the first 12 solutions of Eqs. (3)–(5) are plotted in Fig. 5 for a profile with Δr ri 2 μm, Δnri 0.035, as a function of r co . For this profile, only the lowest radial mode n 1 is supported for each value of the azimuthal index m. The m 0 modes correspond to meridional rays and result in the transverse fields TE0;1 and TM0;1 . Higher-order modes with m > 0 are classified according to the two possible orientations of the Stokes eigenvector from Eq. (5) and are twofold degenerate under the mapping η → −η. Modes for which the mean spin and orbital angular momenta are aligned correspond to HEm;1 modes [16], while those for which the two are anti-aligned are the EHm;1 modes. Despite the comparable sizes of the ring width and the wavelength, we observe close agreement between the GTD and exact solutions. The agreement improves as r co increases, which is expected as, in the limit, the GTD solutions tend to plane waves, which are the exact solutions to planar step-index
1
1
1
0.8
0.8
0.5
0.6
0.6
0.4
0.4
0.2
0.2
0
0 16
17
18 19 Radius (µm)
(a)
20
0 16
−0.5
17
18
19
Radius ( µm)
(b)
20
−1 16
17
18
19
20
Radius ( µm)
(c)
Fig. 4. (a) Transverse fields of low-contrast HE1;1 , (b) high-contrast HE1;1 , and (c) high-contrast EH3;2 . Exact (solid) and GTD (dashed) er (red) and eϕ (blue) are shown.
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS TE0,1 HE1,1 HE2,1 HE3,1 HE4,1 HE5,1 TM0,1 EH1,1 EH2,1 EH3,1 EH4,1 EH5,1
1.456
n
eff
1.454 1.452 1.45 1.448 8
10
12 14 Air Core Radius (µm)
16
Fig. 5. Effective index as function of r co for air-core fiber, with Δr ri 2 μm, Δnri 0.04. Exact (solid) and GTD (dashed) solutions are shown.
waveguides [16]. In the low-contrast limit, the HEl1;m and EHl−1;m modes become degenerate and form the LPl;m groups of the scalar wave equation [16]. A consequence of enhanced spin–orbit coupling is the large intraLP group splitting seen in Fig. 5, which is two orders of magnitude larger than that of typical weakly guiding fibers [28]. Finally, we comment on some topics for future work. The fields near a caustic are not accurately expressed in GTD, so the methods of this Letter are limited to modes for which the caustic surface would be located far inside the inner ring boundary. Also, the degeneracy of the LP modes in the small-contrast limit is the result of delicate cancellations that are not captured by the leading order GTD that we consider. We believe that a more refined expansion can remedy both of these problems. In summary, we have demonstrated that the presence of a high-contrast interface in the refractive index profile can dramatically enhance the spin–orbit coupling of light propagating in an optical fiber, even one that supports only highly paraxial modes. Two consequences of this strong coupling are that the propagating modes cannot be decomposed into basis elements that are OAM and SAM eigenstates, and the scalar degeneracies in effective index are strongly lifted. Instead, the polarization structures of these modes are seen to be elliptical and spatially varying. We have explained the physical origins of this strong coupling regime through a vector GTD expansion. The close agreement with the exact solutions demonstrates quantitatively how these effects are explained by the polarization dependence of the Fresnel equations of reflection at an interface. In particular, spatial dispersion in the Fresnel equations [13] plays no role in our calculations. The enhancement of spin–orbit coupling offers a new degree of freedom for shaping the polarization structure of optical fiber modes, and offers avenues for structuring light in more general optical systems. This work is sponsored by the Defense Advanced Research Projects Agency under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and not necessarily endorsed by the United States government.
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References 1. S. M. Rytov, Dokl. Akad. Nauk SSSR 18, 263 (1938). 2. V. V. Vladimirsky, Dokl. Akad. Nauk SSSR 31, 222 (1941). 3. M. V. Berry, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 392, 45 (1984). 4. F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947). 5. F. I. Fedorov, Dokl. Akad. Nauk SSSR 105, 465 (1955). 6. C. Imbert, Phys. Rev. D 5, 787 (1972). 7. V. S. Liberman and B. Y. Zeldovich, Phys. Rev. A 46, 5199 (1992). 8. A. Y. Savchencko and B. Y. Zel’dovich, J. Opt. Soc. Am. B 13, 273 (1996). 9. M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 83901 (2004). 10. K. Y. Bliokh and Y. P. Bliokh, Phys. Lett. A 333, 181 (2004). 11. K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, Nat. Photonics 2, 748 (2008). 12. K. Y. Bliokh, J. Opt. A 11, 094009 (2009). 13. K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013). 14. R. Y. Chiao and Y.-S. Wu, Phys. Rev. Lett. 57, 933 (1986). 15. A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986). 16. A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983), Vol. 190. 17. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975). 18. A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Y. Zeldovich, Phys. Rev. A 45, 8204 (1992). 19. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, 1972). 20. A. V. Volyar, V. Z. Zhilaitis, V. G. Shvedov, M. S. Soskin, and T. A. Fadeeva, Atmos. Ocean. Opt. 11, 981 (1998). 21. S. Golowich and S. Ramachandran, Opt. Express 13, 6870 (2005). 22. S. Ramachandran, S. Golowich, M. F. Yan, E. Monberg, F. V. Dimarcello, J. Fleming, S. Ghalmi, and P. Wisk, Opt. Lett. 30, 2864 (2005). 23. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, Opt. Lett. 37, 2451 (2012). 24. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013). 25. H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, Ultramicroscopy 61, 91 (1995). 26. M. Hautakorpi, A. Shevchenko, and M. Kaivola, Opt. Commun. 237, 103 (2004). 27. K. Oh, S. Choi, Y. Jung, and J. Lee, J. Lightwave Technol. 23, 524 (2005). 28. S. Golowich, P. Kristensen, N. Bozinovic, P. Gregg, and S. Ramachandran, in Frontiers in Optics (Optical Society of America, 2012), paper FW2D.2. 29. P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, and S. Ramachandran, in Conference on Lasers and Electro-Optics (CLEO 2013) (Optical Society of America, 2013), paper CTu2K.2. 30. M. Hautakorpi and M. Kaivola, J. Opt. Soc. Am. A 22, 1163 (2005). 31. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999). 32. S. Lee, J. Park, Y. Jeong, H. Jung, and K. Oh, J. Lightwave Technol. 27, 4919 (2009). 33. J. B. Keller and S. I. Rubinow, Ann. Phys. 9, 24 (1960). 34. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, Phys. Rev. A 82, 063825 (2010). 35. V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, 1991). 36. K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 70, 026605 (2004). 37. J. P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014
Asymptotic theory of strong spin–orbit coupling in optical fiber Steven Golowich MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA ([email protected]) Received September 23, 2013; revised November 15, 2013; accepted November 22, 2013; posted November 25, 2013 (Doc. ID 198163); published December 20, 2013 The spin–orbit coupling of light propagating in optical fiber can be dramatically enhanced by the presence of a highcontrast interface in the refractive index profile, even for modes that are highly paraxial. The resulting modes have spatial and polarization structures that depart greatly from the weak coupling form, and, in particular, are neither orbital nor spin angular momentum eigenstates. We explain the physical origins of this strong-coupling regime with a vector geometric theory of diffraction expansion. © 2013 Optical Society of America OCIS codes: (080.2720) Mathematical methods (general); (080.4865) Optical vortices; (260.6970) Total internal reflection; (260.1960) Diffraction theory. http://dx.doi.org/10.1364/OL.39.000092
The coupling between the spin and orbital degrees of freedom of light [1] explains a variety of dynamical effects involving the propagation of beams in inhomogeneous media. A geometric interpretation for the evolution of polarization along the beam trajectory is given by the Berry phase [2,3]. Conversely, the polarization state of a beam can influence its trajectory, which was first appreciated in the case of reflections at an interface in the form of the Goos–Hänchen [4] and Imbert–Fedorov [5,6] shifts. Later, a Hamiltonian formalism was introduced for the propagation of near-paraxial beams in inhomogeneous media [7–9], which explains the spin Hall effect of light. The theory of spin–orbit coupling was cast in a more general geometric framework in [10–13]. An important category of inhomogeneous optical media that are influenced by spin–orbit coupling is optical fiber. Indeed, manifestations of Berry’s phase in optics were first predicted [14] and observed [15] in a helically wound optical fiber. Most optical fibers are weakly guiding [16] in the sense that the refractive index contrasts responsible for light guidance are small, typically a few percent or less, and propagation is therefore highly paraxial. The paraxial limit of the Maxwell equations is known to decouple the spin and orbital degrees of freedom [17], but post-paraxial corrections induce spin–orbit coupling. A result is an azimuthal shift of speckle patterns in a weakly guided fiber [7,8,18]. The guided mode spectrum of weakly guiding rotationally and translationally invariant fibers break into degenerate subspaces that can be indexed by their orbital and spin angular momentum (OAM and SAM) content, which is a direct result of spin–orbit coupling [19,20]. The refractive index profile of a rotationally symmetric fiber can be specifically tailored to enhance the spin–orbit coupling [21,22]. Fibers with these design features have been used as devices to create and manipulate [23], and to transmit information over [24], OAM states. All of these previous studies of spin–orbit coupling in optical fiber have treated the effect as a weak perturbation of the scalar Maxwell equations. In this Letter, we study a class of fiber structures and associated guided modes that experience strong spin–orbit coupling, despite being highly paraxial. Fibers with an air core 0146-9592/14/010092-04$15.00/0
surrounded by a raised-index ring (Fig. 1) have been proposed for atom guiding [25,26] and communications [27–29]. It has previously been noticed that such fibers can support modes that depart substantially from the weak guidance approximation, in which polarization is treated perturbatively [30]. Here we emphasize two notable results of this strong coupling; that the linearly polarized (LP) degenerate subspaces of the scalar wave equation split into vector components that are widely separated in effective index [28], and that the guided modes cannot be factorized into spatial and spin terms; in particular, they are neither OAM nor SAM eigenstates. We explain the physical origins of these effects with a vector geometric theory of diffraction (GTD) calculation that explicitly relates the Fresnel reflection equations [31] to the spectrum and structure of the guided modes. Figure 2 compares the polarization structures of the HE1;1 modes of “low-contrast” and “high-contrast” fibers. Both structures are ring fibers with r co 16 μm and Δr ri 4 μm; the low-contrast structure has a silica core and cladding, with Δnri 0.004, while the high contrast structure has an air core, silica cladding, and Δnri 0.035; both are manufacturable with current technology. The free-space wavelength λ is 1.55 μm. The black arrows show the polarization of the odd-parity modes [16], which are by construction linear at each point in space. The red ellipses depict the polarization states of the superposieven tions HEodd m;n iHEm;n , which give rise to spatially uniform circular polarization in the low-contrast case, but spatially varying elliptical polarization in the highcontrast case. The low-contrast examples illustrate that, when spin–orbit coupling is weak, a basis of propagating
Fig. 1. Class of refractive index profiles considered. © 2014 Optical Society of America
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS
(a) Fig. 2.
93
(b)
Comparison between HE1;1 fields of low-contrast (a) and high-contrast (b) fibers.
states can be chosen such that the elements factor into a product of spatial and spin parts [16]; in particular, these modes are both OAM and SAM eigenstates that carry integral values of OAM and SAM in units of ℏ∕photon. The strongly spin–orbit coupled fields, by contrast, are not. A further contrast is the magnitude of HEodd around the ring, which is constant in (a), 1;1 but not in (b) [30,32]. To gain a better understanding of the physical origins of the strong spin–orbit coupling regime in this class of fibers, we apply the Keller–Rubinow (KR) method [33] for constructing an asymptotic expansion for the guided mode eigenvalue problem. This method consists of defining a closed congruence of rays, consisting of a number of normal congruences that are connected to one another by reflection or passing through a caustic surface. In the case of a ring fiber, there are two normal congruences, one consisting of outgoing and the other of incoming rays. Each forms a simple covering of the space extending inward from the outer boundary to either a caustic or the inner boundary. The guided modes are determined by a quantization condition, described below. We note that a similar construction explains the spin–orbit coupling of nonparaxial light in free space, with the coupling arising from the Berry phase [34]. The waves corresponding to each normal congruence are obtained by expressing the electric field as a power series in ω−1 . Because the index is constant inside the ring, a conventional geometric optics expansion will suffice [12,35,36]. Since we are interested in the guided modes, which are harmonic in z, we may write ∞ X
e−iωt−τr Er; t eiβz−ωt em;n r⊥ es ; m;n r⊥ −iωs s0
and obey Snell’s law of reflection at the interfaces. We index them by η, the angle between a ray and the z axis, and a0 , the minimum radius the ray would attain were the inner reflecting boundary not present (i.e., that of the caustic surface). Both η and a0 are conserved along the ray paths, and, by allowing these parameters to vary, we find a two-parameter family of pairs of outgoing and incoming congruences. The polarization of the field e0 m;n along the ray remains constant between reflections, while its magnitude can be shown to vary as r 2 − a20 −1∕4 . The propagation constant is related to the ray invariants by β nri k0 cos η, with k0 2π∕λ. The eigenvalue equations that determine the allowed values of η and a0 are derived from the condition that the fields be single valued, which may be written as I ω
Γ
∇τ · dl 2 πn − γ;
(2)
where n ∈ Z, Γ is a closed curve on the two-sheeted covering space, and γ is a phase factor resulting from the two reflections. The covering space is topologically a torus, so we must consider two independent generators of the fundamental group as the contours Γ in Eq. (2). The generators that we choose are shown in Fig. 3. The first (blue) is a ring surrounding, but just outside, the fiber core, so it is wholly contained in one of the two sheets. Application of Eq. (2) to this contour yields nri k0 a0 sin η m;
(3)
(1)
where r⊥ x; y are the transverse coordinates, ω the optical frequency, and β the propagation constant. The function em;n represents the spatial dependence of a mode, with azimuthal and radial mode indexes m and n. The functions τ and es m;n satisfy, respectively, the eikonal and transport equations, which are found order by order in ω−1 by substitution into the Maxwell equations. We will restrict attention to the leading order term e0 m;n . The rays are straight lines between the two interfaces
Fig. 3. Paths corresponding to the two GTD eigenvalue equations for the ring fiber.
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OPTICS LETTERS / Vol. 39, No. 1 / January 1, 2014
and it can be shown that m ∈ Z may be identified with the azimuthal mode index. The second contour (red) in Fig. 3 follows a ray as it reflects off both ring boundaries (ABCD), then returns to the starting point, crossing both sheets. We find nri k0 sin η a22 − a20 1∕2 − a21 − a20 1∕2 a a γ πn − ; − a0 arccos 0 − arccos 0 a2 a1 2
(4)
where n ∈ Z is the azimuthal mode index, and γ is the phase factor resulting from the two reflections. This latter factor requires further discussion, as it depends on the polarization of the field incident on the interfaces, as well as the incidence angles of the rays on the interfaces. The fields throughout a closed congruence are determined by their values on the rays just leaving the inner boundary, and these values must be chosen such that the single-valued condition holds up to the integral multiple of 2π in Eq. (2). Recalling the z harmonic condition in Eq. (1), we must specify these fields in a plane z z0 . In general, this initial data is provided by a complex vector function of the azimuthal variable e0 e0 ϕ. Guided by the exact solution of the Maxwell equations, we choose the value of the field e00 at a single point ϕ 0 and assign the values at other values of ϕ by e0 ϕ Rϕe00 , where Rϕ is the rotation by ϕ around the z axis. The vector e00 is a Jones vector, as it must be orthogonal to the ray passing through the point ϕ 0 at the inner ring boundary, but further information about its value must be obtained as part of the solution of Eq. (4). Given values of a0 and η, we may compute the angle ϕED ϕE − ϕD , referring to Fig. 3. In order for the two congruences of the outgoing and incoming rays to form a closed congruence, the vector field amplitude e00 must map into its rotation RϕED e00 , up to an overall phase, upon the two reflections. This condition may be written as an eigenvalue equation in Stokes space,
Field
ˆ ED † Tˆ AD eˆ 0 eˆ 0 ; Rϕ 0 0
(5)
ˆ refers to the Stokes vector (matrix) where the notation A; associated with a Jones vector (matrix) A [37]. Here, again referring to Fig. 3, T AD is the Jones matrix that maps the Jones vector at point A into that at D, as may be computed from the Fresnel reflection equations [31]. It is the corresponding Stokes matrix Tˆ AD that encodes the strength of the spin–orbit coupling, which is determined by the index contrasts and incidence angles at the two interfaces. The Stokes vector eˆ 00 that determines the polarization state of the fields throughout ˆ ED † Tˆ AD the fiber is identified with the eigenvector of Rϕ with unity eigenvalue. In the weak coupling limit of small contrasts or grazing incidence [e.g., Fig. 2(a)], Tˆ AD tends to the identity, forcing eˆ 00 to be circularly ˆ ED is a rotation around the zˆ axis polarized since Rϕ on the Poincaré sphere. In the general case, eˆ 0 is elliptically polarized [Fig. 2(b)]. Finally, the phase factor γ that we require in Eq. (4) can be read off the corresponding Jones vectors. The mode fields may also be computed by the KR method. It can be shown that the radial and azimuthal components of the electric field associated with each closed congruence have angular dependence of expimϕ and relative phase of exactly π∕2. In Fig. 4, we compare the real-valued field components er r and −ieϕ r of KR and exact solutions, for both the low- and high-contrast profiles. The effective indexes neff β∕k0 of the first 12 solutions of Eqs. (3)–(5) are plotted in Fig. 5 for a profile with Δr ri 2 μm, Δnri 0.035, as a function of r co . For this profile, only the lowest radial mode n 1 is supported for each value of the azimuthal index m. The m 0 modes correspond to meridional rays and result in the transverse fields TE0;1 and TM0;1 . Higher-order modes with m > 0 are classified according to the two possible orientations of the Stokes eigenvector from Eq. (5) and are twofold degenerate under the mapping η → −η. Modes for which the mean spin and orbital angular momenta are aligned correspond to HEm;1 modes [16], while those for which the two are anti-aligned are the EHm;1 modes. Despite the comparable sizes of the ring width and the wavelength, we observe close agreement between the GTD and exact solutions. The agreement improves as r co increases, which is expected as, in the limit, the GTD solutions tend to plane waves, which are the exact solutions to planar step-index
1
1
1
0.8
0.8
0.5
0.6
0.6
0.4
0.4
0.2
0.2
0
0 16
17
18 19 Radius (µm)
(a)
20
0 16
−0.5
17
18
19
Radius ( µm)
(b)
20
−1 16
17
18
19
20
Radius ( µm)
(c)
Fig. 4. (a) Transverse fields of low-contrast HE1;1 , (b) high-contrast HE1;1 , and (c) high-contrast EH3;2 . Exact (solid) and GTD (dashed) er (red) and eϕ (blue) are shown.
January 1, 2014 / Vol. 39, No. 1 / OPTICS LETTERS TE0,1 HE1,1 HE2,1 HE3,1 HE4,1 HE5,1 TM0,1 EH1,1 EH2,1 EH3,1 EH4,1 EH5,1
1.456
n
eff
1.454 1.452 1.45 1.448 8
10
12 14 Air Core Radius (µm)
16
Fig. 5. Effective index as function of r co for air-core fiber, with Δr ri 2 μm, Δnri 0.04. Exact (solid) and GTD (dashed) solutions are shown.
waveguides [16]. In the low-contrast limit, the HEl1;m and EHl−1;m modes become degenerate and form the LPl;m groups of the scalar wave equation [16]. A consequence of enhanced spin–orbit coupling is the large intraLP group splitting seen in Fig. 5, which is two orders of magnitude larger than that of typical weakly guiding fibers [28]. Finally, we comment on some topics for future work. The fields near a caustic are not accurately expressed in GTD, so the methods of this Letter are limited to modes for which the caustic surface would be located far inside the inner ring boundary. Also, the degeneracy of the LP modes in the small-contrast limit is the result of delicate cancellations that are not captured by the leading order GTD that we consider. We believe that a more refined expansion can remedy both of these problems. In summary, we have demonstrated that the presence of a high-contrast interface in the refractive index profile can dramatically enhance the spin–orbit coupling of light propagating in an optical fiber, even one that supports only highly paraxial modes. Two consequences of this strong coupling are that the propagating modes cannot be decomposed into basis elements that are OAM and SAM eigenstates, and the scalar degeneracies in effective index are strongly lifted. Instead, the polarization structures of these modes are seen to be elliptical and spatially varying. We have explained the physical origins of this strong coupling regime through a vector GTD expansion. The close agreement with the exact solutions demonstrates quantitatively how these effects are explained by the polarization dependence of the Fresnel equations of reflection at an interface. In particular, spatial dispersion in the Fresnel equations [13] plays no role in our calculations. The enhancement of spin–orbit coupling offers a new degree of freedom for shaping the polarization structure of optical fiber modes, and offers avenues for structuring light in more general optical systems. This work is sponsored by the Defense Advanced Research Projects Agency under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and not necessarily endorsed by the United States government.
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