Polarization properties of chiral fiber gratings

Transcription

1 IOP PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS J. Opt. A: Pure Appl. Opt. 11 ( (10pp doi: / /11/7/ Polarization properties of chiral fiber gratings Gennady Shvets 1, Simeon Trendafilov 1,VictorIKopp, Daniel Neugroschl and Azriel Z Genack 1 Department of Physics, University of Texas at Austin, Austin, TX 7871, USA Chiral Photonics, Inc., Clifton, NJ 0701, USA gena@physics.utexas.edu Received 8 October 008, accepted for publication 6 January 009 Published 6 May 009 Online at stacks.iop.org/jopta/11/ Abstract Recent experiments (Kopp et al 007 J. Opt. Soc. Am. B 4 A48 have demonstrated that the polarization sensitivity of chiral fiber gratings depends strongly on the grating symmetry: double-helix fibers are polarization sensitive while single-helix fibers are not. A coupled-mode perturbation theory is developed and used to explain the polarization properties of chiral fiber gratings. Features of the transmission spectrum such as multiple dips in the spectrum and circular dichroism are also derived and attributed to chiral Bragg scattering of the core modes into the cladding modes of the fiber. Keywords: chirality, optical fiber, mode coupling (Some figures in this article are in colour only in the electronic version 1. Introduction Photonic crystals (PhCs have been called semiconductors of light because of their ability to control light propagation, shape the density of electromagnetic states, and influence the light emission properties. By definition, the refractive index of a PhC is a periodic function of space. Great effort has been expended in fabricating high quality PhCs in two and three spatial dimensions, but even one-dimensional PhCs are very interesting from both fundamental and practical standpoints. Such structures have found use as narrowband filters, edgeemitting lasers, and vertical-cavity surface-emitting lasers [1]. One natural platform for implementing one-dimensional PhCs is an optical fiber with the refractive index periodically varying along the propagation direction. Commonly known as fiber Bragg gratings, these structures serve as narrowband filters in a variety of applications, and are typically manufactured by exposing a photosensitive fiber to periodic ultraviolet illumination []. Recently, a novel approach for imparting periodicity into a glass fiber has been introduced [3]: twisting a fiber with a cross-section that is not circularly symmetric to create a chiral fiber grating [4] (CFG. Depending on the details of the manufacturing process [4], the fiber can be either righthanded or left-handed while having either single-helical or double-helical symmetry. In addition to the relative ease of manufacturing and promising applications as environment sensors, chiral fiber gratings have been shown to exhibit a high degree of circular dichroism (CD and circular birefringence (CB. Therefore, they have potential applications as polarizers and filters. It has been experimentally demonstrated [4, 5] that only the double-helical fibers exhibit CD. While doublehelix fibers are polarization sensitive, single-helix fibers are polarization-insensitive. This appears to contradict recent theoretical work [6] demonstrating the strong polarization sensitivity of a metallic waveguide with helical grooves arranged as a single helix. One goal of this paper is to investigate the polarization properties of single- and double-helical chiral fiber gratings and to explain why the former is polarization insensitive while the latter is not. Another goal is to develop a comprehensive semi-analytic perturbation theory of light propagation in CFGs and to demonstrate theoretically that the propagation of tightly confined core modes is strongly affected by their interaction with the loosely confined cladding modes. In what follows we assume that the CFG consists of a narrow (single-mode core with a high refractive index n co and a wide (multi-mode cladding with a lower refractive index n cl < n co. The cladding cross-section is assumed to be round with radius R cl λ, where λ πc/ω and ω are the vacuum laser wavelength /09/ $ IOP Publishing Ltd Printed in the UK

2 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al and frequency, respectively. The core is assumed to be twisted around the axis of the fiber with the pitch u π/k u to form a single or a double helix. Double-helix CFGs are produced [3] by twisting a glass fiber with a non-circular core cross-section with, for example, elliptical or rectangular shape, at a high rate while passing through a miniature oven. Single-helix CFGs are produced by twisting a glass fiber with a non-concentric core [5]. Long-period gratings with the pitch u λ can resonantly couple co-propagating core and cladding modes and are assumed throughout. The rest of the paper is organized as follows. The general coupled-mode perturbation theory specific to the problem of light propagation in a chiral fiber grating is developed in section. The eigenmodes of a straight optical fiber with a round core are reviewed and classified according to their symmetry in section 3. Specific applications of this theory to single- and double-helix CFGs are described in section 4. The results are summarized in section 5.. Coupled-mode theory of chiral fiber gratings In this section we develop a general coupled-mode theory of light propagation in chiral fiber gratings (CFGs. The general classification of the electromagnetic eigenmodes of a straight cylindrical core step index fiber has been well established for decades, and can be found in a number of classic textbooks [7, 8]. When the waveguide is perturbed by, for example, periodic deformation of its core, different eigenmodes of the straight fiber become coupled. Coupledmode perturbation theory describing this coupling has been developed [7 9] and recently refined [10, 11] to include highcontrast waveguides with n co n cl n cl. Unperturbed modes of the straight fiber [8] are used as the expansion basis for the perturbed modes and the core deformation is treated as the perturbation responsible for the mode coupling. The end result of the perturbation theory is a set of ordinary differential equations in the form z ã j ( z i β j ã j ( z = i ã k ( z W jk ( z, (1 k where z = ωz/c is the normalized propagation distance along the fiber, ã j and β j = β j /k 0 cβ j /ω are the amplitude and the propagation constants of the jth unperturbed mode, respectively, and W jk (z is the mode-coupling coefficient between the jth and kth modes brought about by the z- dependent core deformation. The mode-coupling coefficients W jk (z are typically computed from two-dimensional spatial profiles E j (x, y and E k (x, y of the unperturbed modes of a round-core fiber and from the shape of the core deformation. Typically, a small number of modes can be kept. Therefore, equation (1 is a convenient simplification of an otherwise fully three-dimensional set of Maxwell s equations for light propagating in a deformed fiber. The specific case of a helical fiber is considered below..1. Starting point: unperturbed and perturbed optical fibers First, consider a chiral fiber obtained by twisting the fiber core with respect to the axis of the fiber. (The fiber core can be displaced from the fiber axis. The dielectric permittivity inside the fiber is thus given by ɛ = ɛ cl + (ɛ co ɛ cl (r r co (θ, z, ( where ɛ co n co and ɛ cl n cl are the core and cladding dielectric constants, respectively, r co is the z-dependent distance of the core cladding boundary from the center of the fiber, and (x is the index function such that (x = 1if x < 0 (inside the core and (x = 0ifx > 0 (outside the core. The core boundary profile r co (θ, z for the chiral fiber can be expanded in harmonic functions: r co (θ, z = r 0 + n r n cos(nθ nk u z. (3 Keeping only the lowest-order expansion coefficient r m (where m = 1 for single-helix fibers and m = for double-helix fibers and assuming that r m r 0, we obtain ɛ = ɛ cl + (ɛ co ɛ cl (r r 0 + (ɛ co ɛ cl δ(r r 0 r m cos(mθ mk u z = ɛ u (r + (ɛ co ɛ cl δ(r r 0 r m cos(mθ mk u z, (4 where ɛ u (r ɛ cl + (ɛ co ɛ cl (r r 0 is the dielectric permittivity profile of an unperturbed fiber with core radius r 0. The implication of equation (4 is that the dielectric permittivity of a chiral fiber is expressed as the sum of the dielectric permittivities of the unperturbed circular crosssection fiber and the helical perturbation: ɛ(r ɛ u (r + δɛ(r,θ,z,whereδɛ = (ɛ co ɛ cl δ(r r 0 r m cos(mθ mk u z. The perturbation theory developed below uses the analytically known electromagnetic eigenmodes of the unperturbed fiber described by the dielectric permittivity profile ɛ u (r as the expansion basis. Simple and widely known symmetry properties of these eigenmodes enable us to make quantitative predictions about the polarization sensitivity of perturbed (chiral fibers. Note that the core radius of the unperturbed fiber r 0 is the effective radius: it is obtained numerically by expanding the core boundary according to equation (3. As an example of a double-helical fiber, consider a chiral fiber produced by twisting an elliptic fiber core with the major and minor axes a = μm andb = 1.6 μm, respectively. This core s boundary is represented by equation (3 with the following coefficients: r 0 = 0.89 μm, r = 0.1 μm, r 4 = μm. Only r m s with even m s are non-vanishing. In what follows only the coefficients r 0 and r are kept for doublehelical fibers. As an example of a single-helix fiber, we consider a chiral fiber obtained by twisting a round core with radius r c = 4.15 μm displaced by the distance = 1 μmfrom the fiber axis. The core s boundary is similarly represented by equation (3 with r 0 = 4.09 μm, r 1 = 1.0 μm, r = 0.1 μm, etc. In what follows only the coefficients r 0 and r 1 are kept for single-helix fibers... General formalism of the coupled-mode theory of light propagation in perturbed fibers In order to treat the problem of the polarization properties of single- and double-helical CFGs we use an established tool for

3 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al studying periodic structures coupled-mode theory. Coupledmode theory in its general form is presented in standard optical waveguide textbooks [7 9]. Additional work expanded the standard treatments by improving their accuracy for the case of high refractive index contrast [10, 11]. However, these approaches do not include the case of chiral fibers addressed below. We treat the twisted core as a perturbation of a round cylindrically symmetric step index fiber and derive analytic formulae that are sufficiently general for treating helices of any multiplicity, including single- and double-helical CFGs. Perturbation theories similar to those used in quantum mechanics have been developed [11, 1] for Maxwell s equations because of the inherent similarities between the Schroedinger equation and Maxwell s equations. The standard approach [1] is to introduce the full six-dimensional state ket ( E i H. Note that truncated (four-dimensional state kets containing only the transverse components of E and H have also been used. Assuming that the single-frequency time dependence of the electromagnetic fields is proportional to e iωt, the source free Maxwell s equations H = i ω c ˆɛ E E = i ω c ˆμ H can be expressed in terms of the full state kets as where ˆL t = ( 0 t t 0 (5 ˆL t + ˆL z ω Ŵ = 0, (6 c, ˆL z = ( 0 z e z z e z 0 and Ŵ = ( ˆɛ 0 0 ˆμ are six-by-six matrices, and ˆɛ and ˆμ are permittivity and permeability tensors assumed to be diagonal for the rest of this paper. Specifically, ˆɛ ij = δ ij [ɛ u (r+δɛ(r,θ,z] and ˆμ ij = δ ij. For an unperturbed fiber, the eigenvalue solutions corresponding to the propagation wavenumber β k are k = e iβ k z ψ k (x, y, where the coordinate representation of the ψ k ket is the transverse eigenfunction ψ k (x, y, and ψ k ket satisfies the eigenvalue equation for propagation wavenumber β k : ( ˆL t + ω c Ŵ0 ψ k =β k ˆƔ z ψ k, (7 where ˆƔ z = i ( 0 e z e z 0 and Ŵ0 = ( ɛ u are Hermitian operators for real (lossless ɛs under the inner product of two states ψ and ψ given by ψ ψ = da { E E + ( i H ( i H }, (8 where da dx dy. Equation (7 is a generalized eigenvalue problem for the propagation constant β k. Below, its eigenstates ψ k are used as the expansion basis for the perturbed (CFG fiber. Classification and the analytic expressions for ψ k (r,θ are well known [8, 13]. The orthonormality condition for the state kets is given by [1] ψ j ˆƔ z ψ k =N k δ jk. (9 The physical meaning of N k becomes apparent when equation (9 is expressed in terms of E and H: da e z { E j H k + E k H j } = N k δ jk, (10 revealing the proportionality between Nk and the total optical power carried by the kth eigenmode. Below, all fiber eigenmodes are assumed to be normalized to N k so that the orthonormality condition simplifies to ψ j ˆƔ z ψ k =δ jk. Different mutually orthogonal eigenstates of the unperturbed fiber are coupled by the helical perturbation Ŵ δ Ŵ Ŵ 0 of the dielectric permittivity δɛ = (ɛ co ɛ cl δ(r r 0 r m cos(mθ mk u z. The resulting superposition state evolves according to an equation well known from the timedependent perturbation theory in quantum mechanics: ( ˆL z = ˆL t + ω c Ŵ0 + ω c Ŵδ. (11 Using the standard approach of the time-dependent perturbation theory [14], the solution of the perturbed equation (11 is sought as the sum of the unperturbed eigenstates with timedependent coefficients: = a k (z e iβkz ψ k. (1 k Using the orthonormality condition and switching to the normalized distance z and propagation wavenumbers β j yields the familiar expression for a j : z a j( z = i a k ( z e i( β k β j z ψ j Ŵ δ ψ k, (13 k where the matrix element of the helical perturbation in the rhs of equation (13 isgivenby ψ j Ŵ δ ψ k =(ɛ co ɛ cl r 0 r m π 0 dθ cos(mθ mk u z E j E k (θ,r 0. (14 Note that the slowly varying in z expansion coefficients a k ( z from equation (1 are related to the rapidly varying expansion coefficients ã k ( z from equation (1 asã k ( z = exp (i β k za k ( z. The above set of coupled equations is exact for an infinite number of eigenmodes included in the expansion. Below, we restrict the number of modes to a small subset of strongly coupled eigenmodes. After straightforward manipulation, equation (13 can be expressed as z a j ( z = i a k ( z ( e i( β k β j m k u z W + j,k k + e i( β k β j +m k u z W j,k, (15 where the matrix elements W + jk and W jk of the perturbation operator Ŵ δ are given by π W + jk = 1 (ɛ co ɛ cl r 0 r m dθ E j E k (θ, r 0 e imθ, (16 0 π W jk = 1 (ɛ co ɛ cl r 0 r m dθ E j E k (θ, r 0 e imθ, (17 0 3

4 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al with m = 1 corresponding to the single-helix CFG and m = to the double-helix CFG, respectively. From equations (16 and (17, it follows that W + jk = W kj. It is important to note that the perturbation operator Ŵ δ is singular: it is proportional to δ(r r 0. Therefore, its matrix elements W + j,k and W j,k are evaluated on the surface of the core cladding interface. One well-known implication of this singularity [8] is that the integrals in equations (16 and (17 are ill defined because the radial components of the electric field E k and E j are discontinuous. In other words, there is an ambiguity in which value of the radial electric field E r should be used: the one inside or outside the core. Well-established prescriptions [8] for choosing the proper value of E r (inner or outer exist, and other alternative techniques employing curvilinear coordinates have been developed recently [11]. The ratio between the two values is E r (in /E r (out = ɛ cl /ɛ co. Therefore, when the index contrast between the core and cladding is small, as assumed below, the ambiguity in evaluating the integrals in equations (16and(17 disappears. 3. Eigenmodes of a straight optical fiber with a circular core To evaluate the matrix elements W ± j,k, one needs to know the eigenmodes of an unperturbed (straight fiber with core radius r 0. Classifying these eigenmodes according to the field dependence on the azimuthal angle θ is crucial for understanding the selection rules of the perturbation operator Ŵ δ described below in section 4. Selection rules determine which eigenmodes are coupled by the single- or double-helix perturbation of the straight optical fiber due to the helical twist. Moreover, the knowledge of the propagation constants β k will enable us to predict the values of the helical pitch k u corresponding to the phase-matched (resonant mode coupling. It follows from equation (15 that, depending on which of the perturbation operators (Ŵδ or Ŵ δ + satisfies the selection rules for the unperturbed eigenmodes j and k, the following conditions must be satisfied for resonant mode coupling: β k β j ± m k u = 0, respectively. The electromagnetic eigenmodes of a straight optical fiber with a circular core are well known [8, 13]. Below, we briefly review their classification and derive their propagation properties (dispersion relations ω k versus β k in the limit of a wide-cladding radius R cl λ for the cladding modes and small index contrast n co n cl n cl for the core modes Core modes of a single-mode fiber: the LP 01 band First, consider the core modes. The size of the core and the dielectric permittivity contrast between the core and cladding determine the number of modes confined by the core. Quantitatively, this number is determined by the so-called fiber parameter V co = k 0 r 0 ɛcore ɛ cl. For V co <.4, the fiber is single mode, supporting a single doubly degenerate propagating electromagnetic mode. Double degeneracy arises because the mode can be either x- ory-polarized. These two polarizations can be combined to produce two degenerate circularly polarized modes: right-hand circularly polarized (RCP and left-hand circularly polarized (LCP modes. The naming convention for these modes is HE 11 or LP 01. The physical meaning of this notation is as follows. The HE classification refers to the symmetry and nodal properties of the axial fields H z and E z. The name itself (HE 11 implies that the modes possess non-vanishing H z and E z, and that the azimuthal dependence of the fields is e ±imθ for RCP and LCP modes, respectively, where m = 1 (first index of the HE 11 name. The second (radial index labels the mode number for a given value of m. For example, both E z and H z of the HE 11 have a single node (at r = 0 inside the fiber. The LP classification refers to the symmetry and nodal properties of the transverse electric field, e.g., E x. For example, the transverse electric field of the LP 01 possesses the m = 0 harmonic and has a single node inside the fiber. While the general expressions for the electromagnetic fields and for the dispersion relation are rather lengthy [13], considerable simplification can be obtained in the limit of small index contrast, (n co n cl n cl. The dispersion relation of the HE 11 core mode is implicitly given by J 0 (u co u co J 1 (u co = K 0(w co w co K 1 (w co, (18 where u co = r 0 k0 n co β co and w co = r 0 βco k 0 n cl. Numerically solving for u co and w co yields the propagation constant of the core doublet β co. Equation (18 is derived under the approximation of infinitely extended cladding. In the small contrast limit, the (unnormalized electric field components of the circularly polarized LP 01 core eigenmode are given by the following expressions: inside the core, for 0 r r 0 : E r = i β ( cor 0 uco J 0 r e ±iθ uco r 0 E θ =± β ( cor 0 uco J 0 r uco r 0 E z = 1 ( uco J 1 r e ±iθ, r 0 e ±iθ and inside the cladding, for r > r 0 : E r = i β ( cor 0 J 0 (u co uco K 0 (w co K wco 0 r e ±iθ r 0 E θ =± β cor 0 uco J 0 (u co K 0 (w co K 0 E z = J 1(u co K1 (w co K 1 r r 0 ( wco ( wco r 0 r e ±iθ e ±iθ (19 (0 where u co, w co, and β co of the core modes are obtained by numerically solving equation (18; the + ( sign of the harmonic angular dependence corresponds to LCP (RCP waves, respectively. Note that, as one of the by-products of the small index contrast approximation, both the tangential and radial components of the electric field are continuous across the core/cladding interface. 4

5 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al Figure 1. Longitudinal electric field E z for the two orthogonally polarized HE 1 cladding modes. Arrows represent the transverse electric field. These two linearly polarized modes can be combined to form the RCP and LCP modes. 3.. Cladding modes of a large-cladding fiber: the LP 0n doublet bands The influence of a very small core r 0 R cl on the fields structure of the cladding modes is negligible for the lowestorder (few nodes inside the fiber cladding modes. Defining the cladding fiber parameter as V cl = k 0 R cl ɛcl 1, considerable simplifications can be obtained in limit of V cl 1. The exact expressions for the electric and magnetic fields of an arbitrary cladding mode, as well as their exact dispersion relations, can be found in [8, 13]. Here, in the relevant limit of V cl 1, we only concentrate on several modes that are coupled to the HE 11 core modes through the selection rules that follow from equations (16 and(17. The relevant cladding modes belong to two distinct types: (a the doubly degenerate band LP 0n consisting of HE 1n cladding modes (where n, and (b an almost-degenerate quartet band LP 1n consisting of TE n, TM n, and doubly degenerate HE n modes. As will be shown below, the LP 0n cladding band is coupled to the LP 01 core band when the CFG has double-helix symmetry. Likewise, the LP 1n cladding quartet is coupled to the LP 01 core band when the CFG has single-helix symmetry. The two mutually orthogonal cladding modes corresponding to the LP 0 (or HE 1 degenerate doublet are shown in figure 1. These modes were computed using the commercial COMSOL software without making either the wide-cladding or the small-core approximation. The two mutually orthogonal linear polarizations can be combined to obtain the RCP and LCP modes. The dispersion relation of the LP 0n cladding band modes satisfies the same dispersion relation given by equation (18 as the core band modes, except that in the definition of u and w one should substitute r 0 by R cl, n co by n cl,and n cl by 1. In the limit of w 1 (corresponding to V cl 1 we find that the dispersion relation becomes J 0 (u = 0, where u = R cl k0 n cl β. Therefore, the propagation constant can be estimated as β LP0n = k0 n cl j 0n Rcl k 0 n cl j 0n k 0 n cl Rcl, (1 where j 0n is the nth root satisfying J 0 ( j 0n = 0. The electric field components of the circularly polarized LP 0n cladding band modes inside the cladding can be found by making the above-mentioned substitution into equation (19: E r = iβ LP0n R cl J 0 ( j 0n R cl r e ±iθ, E θ = ±β LP0n R cl J 0 ( j 0n R cl r e ±iθ, and E z = j 0n J 1 ( j 0n R cl r e ±iθ,where± corresponds to the LCP (RCP modes, respectively. These expressions are valid in the smallcore and large-cladding approximation Cladding modes of a large-cladding fiber: the LP 1n quartets Another class of cladding modes corresponds to the series of almost-degenerate LP 1n quartets, where n is the radial number marking the number of nodes of the transverse electric field. The LP 1n quartets consist of the TE n,tm n, and the doubly degenerate HE n modes. In the limit of the small low-contrast fiber core V co V cl and wide fiber V cl 1 the dispersion relations of all four modes of the nth quartet can be computed from J 1 (u = 0, where u = R cl k0 n cl β. The propagation constant β of the nth quartet modes is given by β LP1n = k0 n cl j 1n Rcl k 0 n cl j 1n k 0 n cl Rcl, ( where j 1n is the nth root satisfying J 1 ( j 1n = 0. As an example, cladding modes of the LP 11 nearly degenerate quartet are shown in figure. This specific quartet consists of the TE 1 and TM 1 modes and a degenerate HE 1 doublet. As figure clearly illustrates, all four modes of the quartet have a vanishing transverse electric field on the axis. This is in contrast to the modes of the LP 0n doublets shown in figure 1 whose transverse field peaks on the axis of the fiber. Note that the TE mode has vanishing E z = E r = H θ = 0field components (see figure, top left while the TM mode has vanishing H z = H r = E θ = 0 field components. For completeness, we present the electric field components of the TE n,tm n,andhe n modes. These formulae are valid in the small-core and large-cladding approximation. For 5

6 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al Figure. Longitudinal electric field E z for the TM 1 (top right and the two orthogonal linear polarizations of the HE 1 mode (bottom left and bottom right; longitudinal magnetic field H z for the TE 1 (top left mode. Arrows represent the transverse components of the electric field. The two linearly polarized HE 1 modes can be combined to form RHP and LHP modes, respectively. the TE n modes, the only non-vanishing component of the electric field is E θ = J 1 ( j 1n r/r cl.forthetm n modes, the nonvanishing components of the electric field are E z = J 0 (ur/r cl and E r = iβ LP1n R cl J 1 ( j 1n r/r cl /j 1n. Finally, for the HE n modes we find that E r = iβ LP1n R cl J 1 ( j 1n r/r cl e ±iθ, E θ = ±β LP1n R cl J 1 ( j 1n r/r cl e ±iθ,ande z = j 1n J ( j 1n r/r cl e ±iθ, where ± corresponds to the LCP (RCP modes, respectively. Finally, we comment on the accuracy of the approximations that were used in determining the field profiles and propagation constants of the core and cladding modes. There are two key assumptions used in analyzing the cladding modes: (a that the cladding is very wide (or V cl 1, and (b that thecoreisverysmall(orv co V cl. Assumption (a is fairly standard, and very (exponentially accurate for the tightly confined core modes as long as R cl r 0. Under assumption (b, the cladding modes are assumed to be the modes of a step index fiber in which the cladding of our fiber plays the role of a core surrounded by air. Our full electromagnetic simulations performed using the COMSOL software indicate that the exact propagation constants differ from the ones obtained using the assumptions (a and (b by no more than one part in Selection rules for single- and double-helix chiral fiber gratings and their polarization sensitivity Selection rules for the chiral perturbation operator Ŵ δ ± with the matrix elements W ± jk given by the equations (16 and(17 can be deduced from the symmetry of the unperturbed eigenstates ψ j described in section 3. The strength of the coupling between modes j and k depend on the value of the W ± jk. This value of W + jk is non-vanishing only when the following selection rule is satisfied: M j + M k + m = 0, where M j and M k are the modes azimuthal numbers: the E z and H z components of the modes have exp (imθ dependence. This relation constitutes the selection rule for the Ŵ δ + operator. Likewise, the selection rule for the Ŵδ operator is M j + M k m = 0. The effectiveness of mode coupling depends not only on the magnitude of the transition matrix operator, but also on the satisfaction of the chiral equivalent of the Bragg scattering conditions. From equation (15, these are β k β j = mk u for the Ŵ δ + operator and β k β j = mk u for the Ŵδ operator. Therefore, the Ŵ + δ operator imparts momentum to the jth mode while the Ŵ δ operator subtracts momentum from the jth mode. When the selection rule and the Bragg condition for 6

7 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al two modes are satisfied, there is a strong resonant interaction between them. As will be shown below, this interaction manifests itself in the periodic energy exchange between the two modes. The Bragg condition β k β j = ±mk u has a clear physical meaning: any m-helical perturbation has the period m = u /m. For example, the period of a doublehelical CFG is half of its pitch: = u /. A summary of the angular and momentum selection rules for the W ± jk matrix elements of the Ŵ δ ± operators is given below: M j + M k +m = 0, β j +β k mk u = 0 forŵ + δ (3 M j +M k m = 0, β j +β k +mk u = 0 forŵδ. (4 Because in practice only the core modes can be launched or detected, it is most important to identify which operators connect the core modes to which cladding modes. Moreover, because the mode-coupling chiral perturbation is localized at the core radius, only the core cladding mode interaction is expected to be strong. Most of the energy of the cladding modes is concentrated far away from the core. Therefore, the cladding cladding mode interaction is expected to be weak. Note that the circularly polarized core modes are not coupled to each other by the single-helical perturbation. They can be coupled to each other, albeit non-resonantly, by the double-helical perturbation. Polarization sensitivity (circular birefringence and circular dichroism of a CFG is manifested by the differences in propagation constants and/or transmission spectra of the RCP and LCP core modes. The selection rules given by equations (3 and(4 have important implications for the core cladding mode coupling. For concreteness let us assume that label j corresponds to the core mode and label k to the cladding mode. Then, using the property β j >β k, we conclude from the momentum selection rule that only Ŵδ can resonantly couple the core and cladding modes. That, in turn, implies (using the angular momentum selection rule that M k = M j +m. Recall that if the E z and H z components of the core mode have exp (im j θ dependence, the mode is left-hand (right-hand circularly polarized for M j =+1(M j = 1, respectively. Therefore, depending on the symmetry of the right-hand CFG (single or double helix, circularly polarized core modes will be coupled to different cladding mode propagation bands. Below, we investigate the core cladding mode coupling for double-helix and single-helix CFGs. We will conclude that only double-helix CFGs exhibit strong polarization sensitivity Double-helix CFG: selection rules and numerical examples In the case of a double-helix CFG (m = it follows from the angular momentum selection rule that the Ŵδ operator connects the RCP-polarized jth mode (M j = 1 to the LCPpolarized kth mode (M k =+1 via the W jk matrix element (see equation (15. This cladding mode can belong to any of the LP 0n cladding mode bands described in section 3.. The coupling becomes resonant (i.e. satisfies the momentum selection rule if the helical pitch is chosen to be k u = β co β LP0n. On the other hand, the LCP core mode (M j =+1 is coupled to the M k = +3 cladding mode via the same angular momentum selection rule. Visual inspection of the M k =+3cladding mode indicates that the coupling strength is negligible. Similarly, the Ŵ δ + operator strongly couples the LCPpolarized jth mode (M j =+1 to the RCP-polarized kth mode (M k = 1 owing to the angular momentum selection rule. The same Ŵ δ + operator weakly couples the RCP-polarized (M j = 1 jth mode to the M k = 3cladding mode owing to the same selection rule. However, coupling of the jth (core mode to the kth (cladding mode by the Ŵ + δ operator cannot be resonant because β j >β k. These simple arguments reveal why the right-hand double-helical CFG is a strongly polarization-sensitive optical element: the only strong resonant interactions occur between the co-propagating RCP core mode and the LCP cladding mode which belongs to an LP 0n doublet. The LCP core mode is not expected to be affected by the right-hand double-helical CFG. This simplified two-mode model of the evolution of the a RCP co and a LCP cl modes is described by the following equations: z arcp co = iacl LCP We i( β LP0n β co + k u z z alcp cl = iaco RCP W e i( β LP0n β co + k u z, where the coupling coefficient W W co RCP,LP LCP 0n (5 between the core modes and the cladding modes of the LP 0n doublet is given by ( J 0 (u co J j0n 0 R cl r 0 = (ɛ co ɛ cl r β co β LP0n R cl k0 u co j 0n A co A LP0n. (6 W co RCP,LP LCP 0n The normalization constants A co and A LP0n are introduced for normalizing the power of the core and cladding modes, respectively: [ J 0 (u co + J 1 (u co ] A co = ɛ co u co + ɛ cl J 1 (u co [ K wco K 1 (w co 1 (w co K0 (w co ], A LP0n = ɛ cl [ J j0n 0 ( j 0n + J1 ( j 0n ], (7 where A LP0n is computed in the wide-cladding approximation. For perfect resonant matching corresponding to β co β LP0n = k u it can be shown that the amplitudes of the core and cladding modes evolve according to aco RCP = cos W z and acl LCP =sin W z. Therefore, resonant matching implies that, for certain lengths of the fiber, the amplitude of the core mode vanishes, i.e. the core mode is entirely converted into the cladding mode. Note that β LP0n and β co depend on the frequency ω, so for a fixed helical pitch there will be a set of frequencies ω n for which the resonant matching condition is achieved. Different n values correspond to resonant coupling to different LP 0n cladding mode bands. 7

8 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al Figure 3. Interaction of RCP and LCP core modes with a right double-helix chiral fiber grating. Core: 1.6 μm ellipse, n co = 1.5; cladding: R cl = 10 μm, n cl = 1.444; pitch: u = 197 μm; wavelength: λ = 1.5 μm. Only four modes are included in the simulation: RCP/LCP core modes and CP/LCP cladding modes of the LP 0 band. Exact matching of the propagation constants of the modes is achieved by the choice of u : β co β cl = k u. Figure 4. Interaction of RCP and LCP core modes with a right double-helix chiral fiber grating with a tapered period. Parameters: thesameasinfigure3, except that k u (z changes by ±1.5% between z = 0 and 00 mm while achieving resonant matching at z = 100 mm. For illustrative purposes we have simulated a chiral fiber grating with the following parameters: round cladding with diameter D cl = 0 μm and refractive index n cl = 1.444; axiscentered elliptical core with major and minor axes a = μm and b = 1.6 μm, and refractive index n co = 1.5. The linearly polarized fields for the cladding LP 0 mode of the unperturbed fiber used in obtaining the coupling coefficients are shown in figure 1. Core mode fields are not shown due to their high confinement inside and in the vicinity of the core. The effective refractive indices for the core mode and the cladding mode are β co = and β LP0 = The pitch u = 197 μm of the double helix has been chosen to satisfy the matching condition for the propagation constants of the interacting modes: β co β LP0 = k u. We have simulated the propagation of an initially linearly polarized core mode through a right-handed double-helix CFG. The results of the numerical simulation are shown in figure 3. The coupling coefficients of the Ŵ δ operator matrix were obtained through the use of numerical solutions for the electric fields of the unperturbed step index fiber with the effective core radius r 0 = 0.89 μm. No further approximations about the mode structure was made, i.e. we did not assume a wide-cladding fiber with V cl 1 or a small guiding core with V co V cl. The infinite set of equation (15 was truncated to four modes: RCP/LCP core modes and RCP/LCP modes of the LP 0 doublet. The pitch u of the core helix was chosen to satisfy the resonance condition for the coupled modes. As figure 3 makes apparent, only the RCP core mode strongly interacts with the LCP cladding mode. At a distance z = 18 mm the RCP component of the initially linearly polarized core mode is entirely converted into the cladding mode. The LCP component of the core mode is unaffected. Thus, at the z = 18 mm distance, this CFG acts as a perfect circular polarizer. It may be desirable to make the CFG fiber a perfect polarizer regardless of its precise length. That can be Figure 5. Interaction of RCP and LCP core modes with a right single-helix chiral fiber grating. Core: round with r 0 = 8.3 μm and refractive index n co = 1.449, which is displaced by = 1 μm. Cladding: R cl = 15 μm, refractive index n cl = and pitch u = 668 μm; wavelength: λ = 1.5 μm. Only six modes are included in the simulation: RCP/LCP core modes and four cladding modes of the LP 11 quartet. accomplished by chirping the CPG s period, i.e. making k u k u (z. If the helical wavenumber is a linear function of z, and the local matching condition β co β cl = k u (z 0 is accomplished at some point z 0 inside the fiber, then total conversion can be achieved regardless of the length of the fiber. We have modeled a chirped-period CFG with the same parameters as in figure 3, except that k u (z changes by ±1.5% between z = 0 and 00 mm while achieving resonant matching at z = 100 mm. The results are shown in figure 4, where a 0 db reduction of the RCP component is accomplished. An even shorter fiber can be used if the coupling matrix coefficients W ij can also be tapered, i.e. varied from negligible to finite, and back to negligible. 8

9 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al 4.. Single-helix CFG: selection rules and numerical examples In the case of a single-helix CFG (m = 1 it follows from the angular momentum selection rule that the Ŵ δ operator connects the RCP-polarized jth mode (M j = 1 to the TE n and TM n modes (M k = 0 via the W jk matrix element (see equation (15. These two cladding modes can belong to any of the LP 1n cladding mode quartets described in section 3.3. Examples of these modes belonging to the LP 11 band are shown in the top panel of figure. The coupling is resonant if the helical pitch is chosen to be k u = β co β LP1n.TheLCP core mode (M j =+1 is coupled to the HE n (M k =+ cladding mode via the same angular momentum selection rule. Note that the HE n cladding mode belongs to the same almostdegenerate quartet LP 1n as TE n and TM n (see the bottom panel of the figure. As shown below, the coupling strength of the LCP core mode to the HE n is the same as the coupling strength of the RCP core mode to the TE n /TM n modes. As before, coupling of the jth (core mode to the kth (cladding mode by the Ŵ δ + operator cannot be resonant because β j >β k. The coupling amplitudes of the RCP core mode to the TE n and TM n modes can be calculated by assuming the approximate unperturbed profiles of the core and cladding modes calculated in the small-core large-cladding approximations in sections 3.1 and 3.3, respectively. We find that these two coupling strengths, W TM and W TE W co RCP TM n, are related to each other through W TM = iw TE, and can be calculated as W co RCP TM n = (ɛ co ɛ cl r 1 β co ɛ cl J 0 (u co J 1 ( j 1n r 0 /R cl R cl β LP1n u co j 1n A co A LP1n, (8 where the normalization constant A LP1n for the LP 1n mode is given by A LP1n = ɛ cl j1n J 0 ( j 1n J ( j 1n, (9 and is analogous to the normalization constant previously introduced in equation (7 forthelp 0n cladding band. Note that A LP1n is always positive because of the interlacing of the Bessel function roots [15]: zeros of the J 1 Bessel function always fall between the zeros of the J 0 and J Bessel functions. It is straightforward to show that an RCP core mode with amplitude a co simultaneously coupled to two resonant modes with the coupling strengths W TM and W TE evolves according to the equation d aco RCP /d z + ( W TM + W TE aco RCP = 0, so the effective interaction constant for the RCP core mode is WW = W co RCP co RCP TM n. Now let us consider the interaction of the LCP core mode with the LP 1n band. As explained earlier, the LCP core mode (M j = +1 is coupled by the Ŵ δ operator to the HE n (M k =+ cladding mode of the LP 1n quartet. The coupling coefficient W HE n is calculated to be W co LCP HE n = (ɛ co ɛ cl r 1 β co β LP1n J 0 (u co J 1 ( j 1n r 0 /R cl R cl k0 u co j 1n A co A LP1n, (30 which can be shown to be exactly equal to the W co RCP TM n in the large-cladding limit of β LP1n = n cl k 0 that follows from equation (. Thus, we predict that the LCP core mode (coupled to the HE n cladding mode oscillates with the same spatial period as the RCP core mode (coupled to the TM n and TM n cladding modes. Therefore, it is predicted that a largeradius single-helix CFG is polarization insensitive, i.e. the circular dichroism/birefringence is very small. To check this conclusion, we have simulated the propagation of an initially linearly polarized core mode through a right-handed single-helix CFG. The following parameters of the CFG have been chosen: round cladding with the diameter R cl = 15 μm, refractive index n cl = 1.444; round core with the diameter r 0 = 8.3 μm and refractive index n co = 1.449, which is displaced by = 1 μm. The propagation constants of the core and cladding modes are β co = and β LP11 = , respectively. Only six modes were included in the simulation: RCP/LCP modes of the core and four modes of the LP 11 quartet. The pitch u = 668 μm of the core helix was chosen to satisfy the resonance condition for the coupled modes: β co β LP11 = k u. The results of the numerical simulation are shown in figure 5. As figure 5 makes apparent, both RCP and LCP core modes strongly interact with the cladding modes. The RCP core mode primarily (an to equal degree interacts with the TE 1 /TM 1 modes while the LCP core mode primarily interacts with the M =+memberofthehe 1 doublet. Note that the M = member of the HE 1 doublet is never excited by the righthand single-helix CFG. Note from figure 5 that at distances z = 8.5 mm and 17 mm the core mode is entirely converted into cladding modes Spectral characteristics of the transmission through the CFG Strong interaction between core and cladding modes is enabled by the matching condition for the propagation wavenumbers that corresponds to the chiral version of the Bragg scattering condition: β co (ω β cl (ω = mk u. Here m = 1(m = and β cl = β LP1n (β cl = β LP0n correspond to the single- (double- helical CFG. So far we have assumed a fixed frequency ω corresponding to λ = 1.5 μm. Allowing the frequency of the input signal to vary enables resonant coupling to multiple propagation bands of the cladding modes corresponding to different radial indices n. Transmission spectra for various CFGs have been reported earlier [3, 5]. One notable feature of the observed spectra is the emergence of multiple narrow dips in transmission. Below, we relate these multiple dips to resonant coupling to multiple bands of cladding modes. As an example, we simulate the transmission spectrum through a single-helix CFG with the following parameters: round core with radius r 0 = 3.8 μm and refractive index n co = 1.449, displaced by = 0.9 μm from the fiber axis; round cladding with radius R cl = 50 μm and refractive index n cl = 1.444; helical pitch = 900 μm. The wavelength is scanned through the μm range. Ten modes are included in the simulation: two core modes (LCP and RCP 9

10 J. Opt. A: Pure Appl. Opt. 11 ( GShvetset al circular polarizations. The predictive power of these results makes them useful for tailoring the design of long-period CFGs to achieve particular wavelength and polarization characteristics. Acknowledgments This work was supported by the NSF s STTR grant IIP and IIP and the ARO MURI W911NF Figure 6. Transmission spectrum through the single-helix CFG at the fixed fiber length z = 0.6 mm. Core: round with r 0 = 3.8 μm and refractive index n co = displaced by = 0.9 μm from the axis. Cladding: round with radius R cl = 50 μm and refractive index n cl = Helical pitch: = 900 μm. The two dips correspond to the resonant coupling between the core mode and two different cladding mode quartets: LP 11 and LP 1. and eight cladding modes from the LP 11 and LP 1 quartets. The length of the fiber was chosen to be z = 0.6 mm. The transmission spectrum is shown in figure 6. The 15 db transmission dip corresponds to the resonant coupling of the core modes to the LP 11 band of the cladding modes. The 45 db narrow dip is identified as being caused by the coupling to the LP 1 band. The spectrum is essentially independent of the input polarization because the wide-cladding single-helix CFGs are essentially polarization insensitive. 5. Conclusions In conclusion, we have developed a coupled-mode perturbation theory that can be used to describe the propagation of light through chiral fiber gratings of arbitrary helicity. We have applied the theory to single-helix and double-helix longperiod CFGs, which exhibit distinct polarization properties. Analytical results of the coupled-mode theory explained the polarization insensitivity of single-helix CFGs and the polarization sensitivity of double-helix CFGs, which allows them to sharply discriminate between right- and left-hand References [1] Yokoyama H and Ujihara K 1995 Spontaneous Emission and Laser Oscillation in Microcavities (Boca Raton, FL: CRC Press [] Othonos A and Kalli K 1999 Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Norwood, MA: Artech House Publishers [3] Kopp V, Churikov V, Singer J, Chao N, Neugroschl D and Genack A 004 Science [4] Kopp V and Genack A 003 Opt. Lett [5] Kopp V, Churikov V, Zhang G, Singer J, Draper C, Chao N, Neugroschl D and Genack A 007 J. Opt. Soc. Am. B A48 [6] Shvets G 006 Appl. Phys. Lett [7] Marcuse D 1991 Theory of Dielectric Optical Waveguides (San Diego, CA: Academic [8] Snyder A and Love J 1983 Optical Waveguide Theory (London: Chapman and Hall [9] Yariv A and Yeh P 003 Optical Waves in Crystals: Propagation and Control of Laser Radiation (Hoboken, NJ: Wiley [10] Johnson S, Bienstman P, Skorobogatiy M, Ibanescu M, Lidorikis E and Joannopoulos J 00 Phys. Rev. E [11] Skorobogatiy M, Johnson S, Jacobs S and Fink Y 003 Phys. Rev. E [1] Bresler A D, Joshi G H and Marcuvitz N 1958 J. Appl. Phys [13] Okamoto K 000 Fundamentals of Optical Waveguides (San Diego, CA: Academic [14] Landau L D and Lifshitz E M 1977 Quantum Mechanics (Course of Theoretical Physics vol 3 3rd edn (Oxford: Pergamon [15] Abramowitz M and Stegun I A 1965 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover 10