Optical activity in multihelicoidal optical fibers - Physical Review Link ...

PHYSICAL REVIEW A 92, 033809 (2015)

Optical activity in multihelicoidal optical fibers C. N. Alexeyev,1,* B. P. Lapin,1 G. Milione,2 and M. A. Yavorsky1 1

2

V. I. Vernadsky Crimean Federal University, Vernadsky Prospekt, 4, Simferopol 295007, Crimea, Ukraine Institute for Ultrafast Spectroscopy and Lasers, Physics Department, City College of New York of the City University of New York, 160 Convent Avenue, New York, New York 10031, USA (Received 9 April 2015; revised manuscript received 13 August 2015; published 8 September 2015; publisher error corrected 23 September 2015) We have studied optical activity in optical fibers that possess multihelical distribution of the refractive index profile. We have demonstrated through an analytical treatment that the effect of optical activity is caused by asymmetric spin-dependent influence of the spin-orbit interaction on the coupling between the fundamental mode and the higher-order modes. We have provided a simple analytical expression for the coefficient of optical activity. DOI: 10.1103/PhysRevA.92.033809

PACS number(s): 42.25.Bs, 42.81.Qb, 42.81.Bm

I. INTRODUCTION

Discovered by Arago and Pasteur, the phenomenon of optical activity—rotation of the elliptical polarization axis— takes place in materials whose refractive indices n+ and n− for left- and right-circularly-polarized (LCP and RCP) light are different [1]. In this case the ellipse rotation angle γ and propagation distance z are connected by the known relation γ = π (n+ − n− )z/λ, λ being the wavelength. In the limiting case of a linearly polarized (LP) light incident onto an optically active medium, this effect is manifested as rotation of the plane of light’s polarization at a constant rate. Basically, it is associated with chirality of the molecules of which the medium consists. Physically, optical activity means the presence of a circular birefringence. Recent research revealed that in metamaterials and nanoengineered artificial media optical activity can be achieved by incorporating structural chirality into a locally nonchiral medium [2]. Apart from the optical activity in bulk media, the polarization of light can be forced to rotate “artificially,” that is, by special engineering of optical systems. For example, one can induce effective optical activity by coiling a monomode fiber. In this case it is the presence of the topological Berry phase that makes the light rotate its polarization plane [3]. Optical activity is also induced in twisted monomode [4] or low-mode optical fibers [5]. Quite recently, this phenomenon was experimentally demonstrated in twisted solid-core photonic-crystal fibers (PCFs) [6,7], which is currently attracting an increasing amount of attention [8]. Being the representatives of a wider class of multihelicoidal fibers, such fibers possess also the ability to change the topological charge of the incoming beam [9]. Optical activity in twisted fibers is due to the presence of two singled-out directions in the transverse cross section of the fiber, which in a nontwisted fiber correspond to eigenpolarization vectors of two fundamental LP modes [10]. Rotation of such axes in twisted fibers induces circular birefringence, the value of which can be calculated with the use of the Jones calculus [11]. However, there is a fundamental difficulty in explaining the reported optical activity in twisted PCFs through the presence of such singled-out directions. Indeed, the PCFs in question

*

Corresponding author: [email protected]

1050-2947/2015/92(3)/033809(8)

possess six- and fourfold symmetry of the transverse cross sections. Therefore, no unique pair of orthogonal axes can be chosen in the cross section of such fibers despite the implication of Ref. [10]. This fact was pointed out in [12,13]. The way to overcome this obstacle suggested in [6,7] is to allow for the coupling of the fundamental = 0 mode and the modes with higher values of orbital number . In general, a certain representation of such higher-order modes corresponds to orbital angular momentum (OAM) eigenstates known as optical vortices (OVs) [14]. Since the wave front of an OV with the orbital number has the form of an -branch helicoid it can be sensitive to rotation of the transverse cross section, whose symmetry order is also . This idea was implemented in the cited works in the course of exact numerical simulation of the propagation of the fundamental mode in a twisted PCF with a hexagonal cross section. Despite unambiguous theoretical explanation of the observed effect, those seminal papers left beyond their scope some specific questions concerning the nature of the optical activity in twisted PCFs. Due to immense mathematical difficulties connected with the exact analytical treatment of a realistic multicore twisted PC honeycomb fiber the problem in its final stage was solved using computational methods. In [6,7] analytical and numerical treatments were concerned with the most comprehensive and realistic model of a twisted PCF within the framework of an all-vectorial approach. However, despite the acknowledged fact that the observed optical activity is associated with the interplay between orbital and spin degrees of freedom, it remains unclear which physical mechanism is responsible for the phenomenon. In this connection the aim of the present paper is to elucidate the reasons why optical activity appears in twisted fibers with multifold-symmetric transverse profile of the refractive index distribution. We demonstrate the appearance of this effect with the example of the multihelicoidal fiber (MHF) model described in [15] (see Fig. 1). This model has obvious distinctions from the structure of realistic twisted PCFs: the radial dependence of the refractive index distributions of those photonic-crystal structures, as well as the azimuthal dependence, is quite different from that of the suggested model. Moreover, the very mechanism of mode forming in PCFs may be different from the one present in conventional fibers that a modern research suggests [16]. The latest developments in the field, however, have shown that it proves fruitful to

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©2015 American Physical Society

ALEXEYEV, LAPIN, MILIONE, AND YAVORSKY


treat such systems in terms of a discrete rotational symmetry invoking the formalism of helical Bloch modes [17], which implies using partial refractive index functions of the type utilized in [15]. This leads to an insight into a rather essential similarity between the realistic model of a twisted PCF and the MHF—the same type of discrete rotational symmetry. Since for the induced transitions between OAM states the angular distribution of refractive index plays the decisive role [9], one can suppose that the effect of optical activity in twisted PCFs may be also governed by the symmetry properties of the system. Making this assumption, we show through consistent analytical treatment that the effect of optical activity is indeed present in the simple model of a MHF. We prove that it is caused by asymmetric spin-dependent influence of the spin-orbit interaction (SOI) on the coupling between the fundamental = 0 mode and the higher-order OAM modes. We provide simple analytical expressions for the coefficient of the optical activity. II. THE SPIN-ORBIT INTERACTION HAMILTONIAN OF A MULTIHELICOIDAL FIBER

A MHF is an optical fiber that possesses a multihelical refractive index profile as shown in Fig. 1. The refractive index distribution of a MHF has helical branches and has -fold rotational symmetry with respect to the fiber’s axis. Recent research has demonstrated their ability to change the topological charge of the incoming light in both transmitted [9,15] and reflected fields in a Bragg-fiber mode [18]. It was shown that for a certain wavelength the spiraling lattice imparts its “imprinted” charge, which coincides with the order of symmetry , to the incoming field with a well-defined topological charge, changing it by either ± or ±( ± 2) units depending on the type of mode coupling that enables the corresponding transition. MHFs are a natural generalization of chiral (helical) fiber gratings [19], which still evoke a large amount of interest [20]. In the simplest form the refractive index distribution in a MHF can be modeled by the following function [15,21]: n2 (r,ϕ) ≈ n˜ 2 − n2co (r) cos (ϕ − qz) ≡ n˜ 2 − ν 2 ,

(1)

where n˜ 2 = n2co [1 − 2 f (r)] is the refractive index of the unperturbed ideal fiber,(r) = 2 δrfr , is the height of the

FIG. 1. (Color online) The model of an = 6 MHF and the geometry of the problem. The arrow Ein indicates polarization of the incident Gaussian beam. The effect of optical activity is manifested through rotation of the polarization vector Eout of the outcoming beam by the angle γ .

refractive index profile function f , δ 1 is the dimensionless parameter of the cross section’s deformation, nco is the core’s refractive index, q = 2π/H , and H is the pitch of the lattice. Here cylindrical-polar coordinates (r,ϕ,z) are implied. The electric field in optical fibers satisfies the so-called vector wave equation [22], which in long-period spun fibers can be reduced to the equation in the transverse electric field Et [13]: t (Et · ∇ t ln n2 ), 2 + k 2 n2 )Et = −∇ (∇

(2)

t = (∂/∂x,∂/∂y). where k is the wave number in vacuum and ∇ The right-hand side of this equation is known to determine the exact composition of fiber modes and the fine structure of their spectrum [14,22]. However, in certain cases it proves sufficient to use a reduced form of this equation known as the scalar wave equation: 2 + k 2 n2 )Et = 0. (∇

(3)

The right-hand side of Eq. (2) comprises, in effect, the SOI in optical fibers [14,22], which describes the influence of polarization properties of an electromagnetic wave on the spatial characteristics of its energy propagation. We will develop our theory for the case of weakly guiding fibers: 1. As has been shown in [15], the vector wave equation Eq. (2) can be brought to the following form: 2 ˜ = 0, (4) t + k 2 n2 − (β − q Jˆz˜ )2 + Hˆ so + Vˆso |ψ ∇ ˜ = col(e˜+ ,e˜− ), and e˜± = e± exp(±iqz). The fields where |ψ ˜ t (˜r ,ϕ,˜ ˜ z) = e˜± are introduced in the standard way as E e˜ t (˜r ,ϕ) ˜ exp(iβ z˜ ), where r˜ = r, z˜ = z, ϕ˜ = ϕ − qz, and β is the propagation constant. Also Jˆz˜ = lˆz˜ + τˆ3 is the total AM operator, lˆz˜ = −i∂/∂ ϕ˜ and τî is the Pauli matrix. The operator Hˆ so describes the SOI, which is present in ideal fibers, and reads as [14] (see also [23,24]) Hˆ so = 2ψ + r˜ ψ r˜ + r˜ ψ∇ r˜ τˆ0 + ψ τˆ3 lˆz˜ 0 exp (−2i ϕ) ˜ aˆ + , (5) + exp (2i ϕ) ˜ aˆ − 0 where aˆ ± = r˜ ψ∇ r˜ + r˜ ψr˜ ± ψ lˆz˜ and ψ = fr˜ /˜r . To avoid possible misunderstandings one has to note here that, although the refractive index n˜ 2 depends only on the radial variable, the operator Hˆ so is (ϕ,z) dependent. This comes due to the fact that the SOI-operator comprises in its structure the gradient term on the right of Eq. (2), which is ϕ dependent (see [14] for details). To restore the translational invariance in z of the waveguide equation Eq. (2) one has to make then a z-dependent transform over the total operator [see Eq. (3) of Ref. [15]], which makes Hˆ so depend on the longitudinal variable through the combination ϕ − qz. However, its form is identical to the SOI interaction in an ideal fiber up to the substitution ϕ˜ → ϕ. Moreover, at q = 0 the operator Eq. (5) turns into the standard SOI operator for ideal fibers, which allows one to associate this term with the SOI in ideal fibers. The operator Vˆso describes the SOI induced by the deformation of the cross section. One can make such classification based on the fact that it proves to be proportional to the deformation parameter δ and arises from the gradient term on the right of Eq. (2). Its exact expression can be found in [15]. However, due to its smallness as a cross

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OPTICAL ACTIVITY IN MULTIHELICOIDAL OPTICAL . . .


influence of deformation on the SOI it is irrelevant for the explanation of the effect of optical activity. III. PERTURBATION THEORY FOR COUPLING OF OAM MODES

As is known, the propagation of light in ideal fibers is governed in the scalar approximation by the Hamiltonian t2 + k 2 n˜ 2 [14,22]. The eigenvalue problem Hˆ 0 |ψ˜ 0 = Hˆ 0 = ∇ 2 ˜ ˜ β |ψ0 determines the eigenfunctions |ψ˜ 0 , known as scalarapproximation ideal fiber modes, and scalar propagation ˜ It is convenient to choose the eigenfunctions |ψ˜ 0 constants β. in the form of OVs |σ,m [14], whose explicit expressions in the basis of linear polarizations |e = col(ex ,ey ) are 1 |σ,m = (6) exp(imϕ)Fm (r). iσ Here σ = ±1 determines the sense of circular polarization, m specifies the topological charge, and the radial function Fm satisfies the standard equation [22]. For simplicity we omit the radial number n, which corresponds to an additional degree of freedom. The basis Eq. (6) is equivalent in the scalar approximation to the vortex basis [25]. These scalarapproximation eigenmodes are also referred to as the OAM modes. As a matter of fact, the fields Eq. (6) nestle in the x and y components of OVs of topological charge m, whereas the z component has the charge of σ + m, which fact makes it impossible to classify the solution Eq. (6) as the OV in a rigorous sense. However, since in the weakly guiding approximation, to which limit we restrict our considerations, the z component is negligible one can conventionally classify them in terms of OVs. This classification causes no ambiguity since there is no other OV in the ideal fiber that is also an eigenmode of the fiber. It should be stressed that here we use the term “OV” in a narrow sense, in connection with the basis of the scalar-approximation ideal fiber modes. In ideal fibers the role of the SOI, whose Hamiltonian can be easily obtained from Hˆ so Eq. (5) by removing tildes over variables, is in forming the known structure of modes and polarization corrections to the scalar propagation constants. In general, the SOI operator couples the states |σ,m with the same index of the total angular momentum J = σ + m, which leads to the known complicated structure of ideal fiber modes [22]. For example, for |J | = 1 there are two degenerate fundamental modes with the structures |1,0 + ε1 |−1,2 and |−1,0 + ε2 |1, − 2, where εi are certain coefficients. For weakly guiding fibers these coefficients prove to be small. This is also the case for other values of |J |, so that in this approximation one can reliably use the classification of fiber modes implied in Eq. (6). In other words, the effect of the SOI on the coupling of modes with different moduli of orbital numbers is negligible. However, this interaction is most effectively manifested for degenerate scalar-approximation states |σ,m, which belong to the same eigenvalue β˜m2 . For such modes the SOI leads to lifting of fourfold degeneracy (at m = 0 there are four |±1, ± |m| eigenvectors that belong to the same eigenvalue) and forming the known set of TE, TM, HE, and EH modes in weakly guiding fibers. It should be emphasized that for the weak-guidance approximation the modes consist only of scalar-approximation modes |σ,m with the same value

of |m|. One should also note that, generally, OVs |σ,m are not the eigenfunctions of the operator comprising the SOI: for example, the |1,−1 OV gets converted in the ideal fiber into the |−1,1 OV. However, all the higher-order OVs with |m| > 1 are again the eigenmodes of the SOI Hamiltonian. Quite different is the role of the term ν 2 in Eq. (1), which describes a perturbation of the fiber’s form in the transverse cross section. Consider first a nontwisted fiber described by the ν 2 (q = 0) ≡ ν02 perturbation term. If this term is relatively small, the eigenvectors of the corresponding ¯ = β¯ 2 |ψ ¯ remain almost eigenvalue problem (Hˆ 0 − ν02 )|ψ ¯ ≈ |ψ˜ 0 . The the same (for the nondegenerate case) and |ψ propagation constants β˜ are renormalized in compliance with the results of perturbation theory [26]: β¯n2 − βñ2 ≈ Vnn +

|Vnm |2 , βñ2 − β˜m2 m

(7)

where the matrix element for an operator Vˆ is Vij ≡ i|Vˆ |j (|i ≡ |σi ,mi ) and the inner product should be defined as + ∗ ∗ | = dS, (8) (+ − ) − S S being the total transverse cross section of the fiber. In this way, the effect of perturbation is manifested mainly in renormalization of the propagation constant of the mode in question. It is the idea of [6,7] that the form of perturbation present in the cross section of PCFs leads to coupling between the fundamental m = 0 mode and the OAM mode with m = . As a result of this interaction the propagation constants for RCP and LCP fundamental modes may acquire different corrections, which would manifest itself in induced circular birefringence and optical activity. However, even a superficial analysis shows that if one chooses Vˆ in Eq. (7) as ν02 , the corrections to the propagation constants of RCP and LCP fundamental modes prove to be the same. That is, in the absence of a twist, chirality, the coupling in question is symmetric and no optical activity could be observed. To determine whether the presence of a twist alone is sufficient to get the asymmetry of the coupling consider first the scalar waveguide equation Eq. (3) at q = 0, which can be represented as 2 ˜ =0 t + k 2 n2 − (β − q Jˆz˜ )2 |ψ ∇ (9) by discarding the operators Hˆ so and Vˆso in Eq. (4). Equation (9) represents a special type of eigenvalue problem, in which the perturbation depends on the eigenvalue β. In this situation to establish the corrections to the propagation constants of zero-approximation eigenvectors one has to build the matrix G of the operator in the left-hand side of Eq. (9) over the basis of those eigenvectors of Hˆ 0 which can be coupled by the perturbation operator. Since the operator Jˆz˜ does not couple different states the set of such eigenvectors is determined solely by the perturbation term ν 2 . This term is scalar (that is, proportional to the unity matrix) and unable to couple states with different polarization. Instead, since it is proportional to exp(iϕ) ˜ + exp(−iϕ) ˜ it couples the state vectors whose orbital numbers m and m differ by : m − m = ±. As we are interested in renormalization of the propagation constants

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of the fundamental = 0 modes, the set of eigenvectors should comprise either the states |1,0, |1,,|1,−

(10)

or the states with opposite polarizations, |−1,0, |−1,,|−1,−.

(11)

The sets Eqs. (10) and (11) form the basis over which the matrix G should be built. Naturally, one can expand them by adding the states with orbital numbers multiples of . However, the influence of such states on the renormalization of the value of β˜0 would be negligibly small. The eigenvalue equation Gx = 0, where the vector x is assumed to be decomposed over the basis Eq. (10) or Eq. (11), in the standard manner gives the equation in β: det G = 0. The desired equation has the form 2 β˜ − (β − q)2 Q Q 0 2 2 0 Q β˜ − (β − q − q) 2 2 ˜ Q 0 β − (β − q + q) = 0.

(12)

Here Q = σ,0|k ν |σ, is the corresponding coupling constant: 2 2

k 2 n2 δ Q = − √ co , N0 N

(13)

∞ where the normalization coefficient is Nm = 0 xFm2 dx. Equation (12) corresponds to the problem of a three-level system with two degenerate levels. It is easily verified that at q = 0 an analogous equation for a two-level system yields in a limiting case corrections to the propagation constants in agreement with Eq. (7). As is known, the eigenvalue problems for coupling matrices of the type implied by Eq. (12) can describe different physical effects. One of such effects takes place for large enough values of q and lies in mutual conversion of fundamental modes and OVs in MHFs [9,15,18]. Assuming that the twisting is relatively weak we will be interested only in the small-q region of twists. We also restrict our considerations to small perturbations: |Q| |β˜2 − β˜02 |. These approximations simplify the process of solving Eq. (12). Indeed, at q = 0, Q = 0 there are four zeroapproximation spectrum branches: β1,2 = ±β˜0 and β3,4 = ±β˜ . To find the correction to the spectrum of a forwardpropagating fundamental mode β1 = β˜0 it is convenient to

introduce a small detuning ε+ : ε+ = β − β˜0 − q.

(14)

Also one can allow for the fact that |β˜0 − β˜ | β˜0 [22], which leads to the following approximation: β˜02 − β˜2 ≈ 2β˜0 ξ , where ξ = β˜0 − β˜ . Then, linearizing Eq. (12), in ε+ one readily obtains the desired correction to the propagation constant of the forward-propagating RCP fundamental mode: ε+ =

4β˜02 (ξ 2

Q2 ξ . − 2 q 2 ) − 2Q2

(15)

It should be noted that this correction turns out to be quadratic in the perturbation coefficient Q. This fact complies with Eq. (7) since for such perturbation Vnn = 0 and the correction should be quadratic in perturbation. To obtain the analogous correction to the forward-propagating LCP fundamental mode one has to use the basis Eq. (15) for constructing the corresponding secular equation, which in this case reads as 2 β˜ − (β + q)2 Q Q 0 2 2 ˜ 0 Q β − (β + q − q) 2 2 ˜ Q 0 β − (β + q + q) = 0.

(16)

Introducing the detuning ε− = β − β˜0 + q one obtains for the correction ε− to the propagation constant of the LCP fundamental mode the same result Eq. (15): ε− = ε+ . This means that in the scalar approximation the system features no optical activity even in the presence of coupling between the modes with different orbital numbers. It is useful to note that the structure of the coupling matrices Eqs. (12) and (16) implies that the eigenmodes of the system should rather be the superposition of the scalar-approximation eigenmodes Eqs. (10) and (11) than such pure OVs. For example, the exact expression for the RCP fundamental mode should have a composition ∝ |1,0 + η1 |1, + η2 |1, − , where η1,2 1 are coefficients determined from the coupling matrix Eq. (12). However, the small admixture of the higherorder fields |1, ± to the field |1,0 would be experimentally undetectable even for such a relatively simple system. This is why we neglect all such amplitude corrections and focus our attention on phase corrections to propagation constants of the fundamental modes, which can have a cumulative effect and, in this way, be detectable in the experiment.

IV. SPIN-ORBIT-INTERACTION-INDUCED OPTICAL ACTIVITY IN MHFs

To allow for the vectorial nature of optical activity it is necessary to include the SOI term Hˆ so Eq. (5) into the structure of the total Hamiltonian. In the following we will restrict our consideration to the case > 2, thus leaving elliptical spun fibers beyond the scope of the paper. In this case the third term in Hˆ so gives no contribution to the matrix elements. Also its first term only slightly renormalizes scalar propagation constants in the diagonal elements and therefore can be neglected. The most important is the second term ψ τˆ3 lˆz˜ , which lifts the degeneracy in propagation constants for both |1,,|1,− and |−1,,|−1,− OVs [14]. The eigenvalue equation for the matrix built over the basis Eq. (10) will have the form 2 β˜ − (β − q)2 0 Q Q

Q

0 = 0, 2 2 ˜ β − α − (β − q + q) Q

β˜2 + α − (β − q − q)2 0 033809-4

(17)



where the spin-orbit coupling constant α = /r02 N and r0 is the core’s radius. This equation is equivalent to 2 2 β˜0 − (β − q)2 β˜2 − (β − q)2 − q 2 2 − 2 [a + 2q(β − q)]2 − 2Q2 β2 − (β − q)2 − q 2 2 = 0.

(18)

Linearizing it in the detuning Eq. (14), allowing for ξ β0 Q,α, one can obtain for the spectrum correction for the forwardpropagating RCP fundamental mode ε+ ≈

Q2 ξ . 2β˜02 (ξ 2 − 2 q 2 ) − 2qα β˜0 2

(19)

It should be noted that although, formally, this expression is defined at all q such that the denominator is not zero, it should comply with the smallness of ε+ assumed in Eq. (14), at which condition the nondegenerate perturbation theory applies. In this way, this expression is invalid at q ≈ ξ . As is known [15], at such twist rates the fundamental mode strongly hybridizes with the OV, leading to an intense energy exchange between the coupled fields. In this resonance area a special treatment of the problem might be in order. On the contrary, at large q this expression is true and leads to vanishing of the correction ε+ , which complies with a general principle of the decrease of the polarization mode dispersion with increase of the twist rate. At small twist rates (q ξ ) decomposing this result in a power series in q, one obtains for the desired correction ε+ ≈

Q2 Q2 2 α + q. 2ξ β˜02 2ξ 3 β˜03

(20)

It should be emphasized that taking into consideration both |1, and |1,− levels along with the |1,0 level proves to be crucial for the effect. For example, disregarding |1,− level and reducing the problem to a two-level one would result in the presence of a circular birefringence of the system even at a zero twist. To determine the correction for the forward-propagating LCP fundamental mode one has to build the corresponding matrix over the basis Eq. (11). In this case the desired eigenvalue equation for has the form 2 β˜ − (β + q)2 Q Q 0 (21) 0 Q β˜2 − α − (β + q − q)2 = 0. Q 0 β˜ 2 + α − (β + q + q)2

It is easily noticed that in the detuning ε− one obtains the same eigenvalue Eq. (18) upon changing the sign of the SOI constant: α → −α. In this way, the desired correction at small twist rates is ε− ≈

Q2 Q2 2 α − q. 2ξ β˜02 2ξ 3 β˜03 (22)

This leads to one of the main results of this paper—the expression for the optical activity A in -helicoidal fibers: A≡

π (n+ − n− ) π nco 2 δ 2 3 ≈ q. λ λξ 3 r02 N0 N2

is a complicated refractive index function in PCFs. However, there may be certain features in this function that are more critical than others. To examine this matter one should note that the model refractive index distribution used in this paper has the same type of rotational symmetry as in the reported PCFs. However, here we have introduced only the main harmonic cos ϕ˜ into its structure. For a realistic -fold rotationally symmetric distribution n(r,ϕ) in the Fourier-decomposition there will be present all higher harmonics:

n(r,ϕ) = a0 (r) + [an (r) cos nϕ + bn (r) sin nϕ], n

(23)

It is useful to notice that the optical activity induced by the mechanism of asymmetric intermodal coupling is rather sensitive to the “distance” ξ = β˜0 − β˜ between the propagation constants of the modes. However, since for weakly guiding fibers |β˜0 − β˜ | < k nco and ξ depends slowly on , the main influence on the magnitude of the effect on changing is exerted through the form of radial functions represented in the normalization coefficient N . It is quite understandable that the direct comparison of this result with those of numerical simulation for a realistic PCF is impossible due to the drastic differences between the models [27]. Although we do not aim to achieve quantitative coincidence of the results for the twisted PCFs and MHFs, it is worth emphasizing the main distinctions in the models to get a better understanding of how our results may still be useful in the study of realistic twisted PCFs. The first obvious distinction

(24) where the radial distribution of the harmonicsis found through weighted integration with n(r,ϕ). Based on the results obtained above, one can say that there should be contributions to the effect from all higher harmonics in the refractive index decomposition. As we will show later, the main contribution is provided by the first harmonic, so that this circumstance is not crucial. More essential is the influence exertedby the form of

the radial dependence of the lowest harmonic a2 (r) + b2 (r), on which the values of the exchange integrals involved in the final formula, as well as the values of the propagation constants, strongly depend. Since in our model we assumed a step-index radial dependence this is likely to be the main source of quantitative discrepancies. The second distinction between the models is in the high index contrast of the PCFs in question and the approximation

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of weak guidance assumed in our model. It should be noted that a high index contrast can be compatible with a weak guidance at large values of the waveguide parameter V . In other cases the nonparaxiality of propagation exerts its main influence through changing the form of the dispersion equation on the propagation constant. To allow for this circumstance it is not necessary to reconfigure the very scheme of calculations. Instead, one can modify the theory in the following way. In the matrix Eq. (17) one should replace in the diagonal elements (22) and (33) β˜2 ± α instead of β+2 and β−2 , respectively, where β± are the values of the propagation constants for OVs |±1,, calculated with the use of the exact (nonscalarapproximation) dispersion equation [22]. Assuming that the SOI splits the scalar propagation constant β˜ symmetrically, to obtain the SOI-induced splitting it is sufficient to make the following substitution in the corresponding equations: α → 12 (β+2 − β−2 ) ≈ β˜0 (β+ − β− ). For the difference of refractive indices n± of RCP and LCP fundamental modes this gives the following result: n+ − n− = 1 (β + 2

k 3 n4co δ 2 2 q(β+ − β− ) , N0 N β˜02 (β0 − β )3

β− ).

FIG. 2. (Color online) The difference n+ − n− in refractive index units (RIU) vs twist rate q. Fiber parameters are λ = 8×10−7 m: nco = 1.5, = 0.28, r0 = 3×10−5 m, V = 263.

(25)

where β ≡ + Another effect of nonparaxiality is in SOI-induced strong coupling between the scalar-approximation modes |σ,m with the same value of the total angular momentum J . In this way, for example, the scalar-approximation modes |1,0 and |−1,2 get coupled and form two nonparaxial modes |1,0 + c2 |−1,2 and c1 |1,0 + |−1,2. Such phenomena cannot be incorporated into our theory and can be the source of quantitative discrepancies. However, the nonparaxiality seems to be not present in the reported PCFs. Indeed, in Ref. [6] the estimated values of the spin and orbital angular momenta for a RCP fundamental mode were 0.9996 and 0.0022. This indicates that the admixtures of the |−1,2 field due to the SOI (not mentioned explicitly in the paper) and with |1,6m partial fields due to interaction caused by the violation of the form were negligibly small, which is true for the paraxial propagation. Although nonparaxiality of propagation can be to some extent allowed for in the final formulas, the role of paraxiality cannot be reduced to a simple quantitative difference. To understand this point one has to recall that the rigorous solving of Maxwell’s equations for waveguides involves reducing them to equations in the fields’ longitudinal components. This form of equations, however, does not comprise any SOI operator terms and, in this way, does not enable one to single out, on a formal mathematical level, the physical reason for the observed effect. Only in the weakly guiding approximation can one specify such terms in the Hamiltonian-like equation in the transverse components of the electric field [14]. Moreover, beyond the paraxial approximation the very notions of the spin and orbital parts of the angular momentum themselves become less definite, so that it becomes difficult to separate these parts from the total angular momentum [28,29]. Instead, it may prove useful to invoke the notion of an angular momentum flux [30,31]. In this way, paraxial propagation is the only case where our theory is valid. It is interesting to get an estimate of the error in the magnitude of the optical activity introduced by the above-

mentioned distinctions between the models. To this end we consider a twisted hexagonal solid-core fiber with material parameters approximately coinciding with the ones reported in [6,7]. We set r0 ≈ 30 μm, λ = 800 nm, and the rate of rotation q = 10 m−1 . The value of deformation parameter for a hexagon can be assessed as δ ∼ (1 − cos π6 )/2 ≈ 0.07. We also set nco = 1.5 and ncl = 1, where ncl is the refractive index of cladding. Applying Eq. (25) one can plot the dependence of n+ − n− on the twist rate q (Fig. 2). As is seen, for small twist rates we get a satisfactory agreement with the corresponding results for a twisted PCF: a predicted order of n is also 10−9 for such twist rates. This order is preserved for a rather wide range of core radius (see Fig. 3). Note that the value of r0 for a hexagonal core in Refs. [6,7] was approximately 15 μm. The waveguide parameter for these examples is much greater than unity, so that despite the large refractive index contrast the paraxial approximation is still valid and the main

FIG. 3. (Color online) The difference n+ − n− of refractive indices vs core radius r0 for an = 6 MHF at a twist rate q = 10 m−1 . Note that we exclude the area of small r0 since in this range the paraxial approximation is invalid.

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Indeed, to show this let us study a twisted square solid-core fiber with the same material parameters and r0 ≈ 30 μm. The deformation parameter for a square-core fiber is assessed as δ ∼ (1 − cos π4 )/2 ≈ 0.15. For q = 10 m−1 we calculate the contribution due to harmonics with m = 4,8,12. The results for such contributions are given in Fig. 4. As is seen, the main effect is caused by the lowest harmonic. V. CONCLUSION

source of deviation of this result from the previously reported one comes due to the difference in the radial behavior of the refractive index. Neither can this difference be caused by the contributions to the effect of the higher harmonics.

In the present paper we have studied the effect of optical activity in optical fibers that possess multihelical distribution of the refractive index profile. On the basis of an all-analytical approach we have shown that the effect of optical activity is caused by the asymmetric spin-dependent coupling between the fundamental mode and the higher-order modes. The perturbation of the form causes different renormalizations of the propagation constants for right- and left-circularly-polarized fundamental modes. The reason for the asymmetry to exist in such renormalization lies in the presence of the spin-orbit interaction. We have provided a simple analytical expression for the coefficient of optical activity. The given example of the multihelicoidal fibers suggests that the existence of this type of optical activity in photonic crystal fibers is not directly connected either with the specific mechanism of mode formation and guidance or with the intricate pattern of the refractive index. For this phenomenon to appear it is sufficient for the helical-like refractive index distribution to possess discrete rotational symmetry of higher order.

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FIG. 4. (Color online) Contributions to optical activity from three lowest harmonics (in a semilogarithmic scale) in the form of decomposition in a Fourier series for a square-core twisted fiber ( = 4) at a twist rate q = 10 m−1 . Here for the lowest harmonic [n+ − n− ]4 = 6.3×10−8 .

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