1. Introduction

Microbend-induced fiber gratings [1, 2] are mechanically induced long-period gratings (LPG) that resonantly couple symmetric and anti-symmetric modes in fibers. They have generated widespread interest over the years [3, 4, 5], because they are potentially attractive low-cost, tunable replacements for UV-induced LPGs. However, a debilitating drawback is that they are inherently polarization dependent, even in perfectly circular fibers. This is because antisymmetric modes are actually almost degenerate linear combinations of four polarized vector modes with different propagation constants.

Demonstrated means to circumvent this problem rely on averaging the grating response for orthogonal states of polarization (SOP) of light by forcing it to “see” both SOPs [3, 6]. However, such techniques do not address the fundamental problem, and negate the low-cost, low-loss, or tunability features of microbend-induced gratings (MIGs). Alternatively, MIGs in very thin fibers (OD ≃10μm) can be shown [7] to preferentially couple light only to polarization-insensitive HE₂₁ modes, but this approach requires fibers that are impractically thin for device applications.

An example of polarization dependence in a MIG is presented in Fig. 1, where the fiber under test was TWRS™. In this paper, we study the fundamental waveguide properties that give rise to the degeneracy splitting of anti-symmetric modes that causes the polarization dependence. We show that this splitting can be analytically described with first-order perturbation corrections to the scalar propagation constant, β. The resultant expressions are tested against measurements of MIGs induced in a standard transmission fiber. The resulting agreement suggests that polarization dependence in MIGs can be controlled and tuned by appropriate fiber designs. The form of the analytic expressions yields insight into what features in the fiber profile control the polarization dependence. An example of a fiber designed using the methods of this paper will be presented in [8].

 

Fig. 1. Measurements of transmitted power as a function of wavelength, for two orthogonal input SOPs. The fiber under test is TWRS™.

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2. Theory

2.1. Resonance Splitting

The dependence of the resonant wavelength on the input SOP observed in Fig. 1 is due to the splitting of the scalar propagation constants β, by vector and other effects. The polarization dependence of resonant wavelength can be described in terms of the propagation constants and group delays. To do so, we compare coupling from the fundamental modes, which we take to be degenerate with propagation constant β ₀, to two higher order modes, with propagation constants β ₁ and β ₂. The grating period is fixed such that for some λ ₀, Λ = 2π/β ₁₀(λ ₀), where we have defined β_ij = β_i , - β_j . We compute δλ, the wavelength shift that renders the second mode β ₂ resonant at wavelength λ = λ ₀ + δλ, to be

We note that β_i̇, is related to the group delay τ_i , by β_i̇, = -ω ² τ_i /2πc.

Equation (1) informs us that there are two quantities that control the wavelength splitting δλ: the difference β ₁₂ between the propagation constants of the two high order modes, and the difference in group velocities $\dot{\beta }$ ₂₀ between the high order and fundamental modes (the two nearly degenerate high-order modes will generally have very similar group velocities). However, we will see in section 2.3 (Eq. (13)) that the resonance width has the same dependence on group velocity difference as (1). It was shown in [9] how the resonance width can be engineered by controlling the value of $\dot{\beta }$ ₂₀ through fiber design. Here, we emphasize the role of the numerator β ₁₂ of (1). Independent control over both β ₁₂ and $\dot{\beta }$ ₂₀ is necessary for full control over both resonance widths and positions.

2.2. Perturbation Theory

The polarization dependence of microbend gratings is due to the vector nature of the electromagnetic fields that propagate in the fiber. In order to study this phenomenon the full vector solutions of the propagating modes must be considered. The theory is developed in many places, e.g. [10], so we will just summarize the necessary results here.

We will initially assume a longitudinally invariant fiber, in which case the transverse electric field E _t can be decomposed into its Fourier components

where x _t = x x̂ +y ŷ are the transverse coordinates and the transverse time- and z-harmonic fields e _t (x _t ) satisfy

with

We will assume the fiber profile n(x _t ) to be nearly circular, with elliptical deviations away from circularity

where 1 - e ² is the squared ratio of minor and major axes. We assume without loss of generality that the former is along the y-axis. With the definition V ^(ell) = k ² $n_{\text{ell}}^2$, the vector Maxwell equations with elliptical profile perturbations becomes

Though numerical solution of the full vector equation (2) is feasible, considerably more intuition can be built through a perturbative approach, which also has sufficient accuracy for the designs and modes that we will consider. The unperturbed guided modes obey

We choose a coordinate representation of the eigenmodes of (3) that is purely real. The eigen-modes |ℓ,m,v,p̂⟩ are indexed by an angular index ℓ = 0,1,2,…, radial index m = 1,2,3,…, angular dependence v(∙) = cos(∙) or sin(∙), and polarization p̂ = x̂ or ŷ. The wavefunctions take the form

where the radial wavefunction F_ℓm obeys the radial wave equation

The solutions to the scalar wave equation divide into the degenerate subspaces LP _ℓ,m , of dimension two when ℓ = 0 and four when ℓ > 0. To compute the splittings in wavenumber β that are induced by vector perturbations at first order in perturbation theory, we require the perturbation matrices in the basis of (4), which we order as |ℓ,m,cos, x̂), |ℓ,m,cos,ŷ)⟩, |ℓ,m,sin,x̂)⟩, and |ℓ, m, sin, ŷ⟩. The results are expressed in terms of the integrals

The elliptical perturbation gives rise to nonzero matrix elements only in the case ℓ = 1:

The vector perturbation matrices take different forms depending on the value of ℓ:

The total perturbation matrix, the sum of (6) and (7), is block diagonal, which simplifies the diagonalization. When ellipticity is absent the eigenvalues and vectors take a particularly simple form, as we recover the TE₀₁, TM₀₁, and two HE₂₁ modes:

2.3. Resonance Widths

The previous section dealt with the splittings among the LP _ℓ,m propagation constants that determine the wavelengths at which these modes will be resonantly coupled to the fundamental modes, at fixed grating period. In this section we describe the widths, in wavelength, of this resonant coupling. We assume that the microbend grating gives rise to an index perturbation of the form

where r, θ are transverse spatial coordinates, z is the distance along the fiber axis, and Λ is the grating period.

We will work with the approximation that only two modes are resonantly coupled, which covers the situations of interest. Because the two coupled modes will both be forward traveling with these long-period gratings, with the further assumptions of forward initial conditions and weak coupling, the coupled amplitude equations are

with α = β ₀₁ - β _Λ, β ₀₁ = β ₀ - β ₁, β _Λ = 2π/Λ, and the coupling coefficients given by

In (11), the coupling strength κ is proportional to the amplitude of the microbends, which is determined by the pressure applied.

Equations (10) can be solved exactly. We cast the solution into a useful form by first setting the coupling strength such that on resonance (β ₀₁ = β _Λ), complete transfer of power from one mode to the other occurs over a grating of length L; this choice is κ = π/2L. Off resonance, with initial conditions of a ₀ = 1,a ₁ = 0,

If we take λ₀ to be the resonant coupling wavelength, then from (12), we find that the half-width δλ (ε) at which |a ₁|² has dropped to a value 1 - ε is

where

3. Experimental Verification

MIGs were induced in a standard transmission fiber (TWRS™) with a special coating with index n _coat = 1.3765, which is lower than the silica glass cladding, to ensure that well-guided cladding modes exist in the fiber. A variety of grating periods were chosen to induce LPG-resonances in the 1500nm range, for cladding modes ranging from LP ₁₂ through LP₁₅.

The resonant wavelength of a MIG is given by λ _res = Λ(n ₀ -n ₁), where Λ is the grating period and n ₀₁ are the effective indexes of the two coupled modes; here, the couplingis from the fundamental LP₀₁ to higher-order LP_1m modes. Figure 2(a)shows the calculated phase matching curves, alongside the measurements, for these modes. Only the phase matching to the scalar LP modes is presented, as the vector corrections are too small to see on this scale.

 

Fig. 2. (a) Phase matching curves. The lines denote the predictions from theory; the symbols are the experimentally measured values. (b)Polarization splitting: δλ_ℓ,m is the maximal difference in resonance wavelength between the fundamental and LP _ℓm modes obtained by varying the input polarization.

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The index profile of the fiber under test is presented in Fig. 3(a), along with the calculated radial mode fields for the LP ₁₂ through LP ₁₅ modes. We note that the fiber profile in Fig. 3 was a scaled version of the design profile for the fiber. In order to correct for differences between this ideal profile and the actual profile, the index values in the central portion of the design profile were uniformly scaled by a single constant factor chosen to maximize the agreement observed in Fig. 2(a).

With the effects of vector and geometric profile perturbations included, there is a splitting among the wavelengths at which the various degenerate LP _1m modes are resonantly coupled to the fundamental mode by a grating of fixed period. We denote by δλ_ℓ,m the maximal difference in resonance wavelength between the fundamental and LP _ℓ,m modes obtained by varying the input polarization. In Fig. 2(b), the experimental measurements of the splitting δλ_ℓ,m for several cladding modes is presented, along with the model predictions given by (1), with and without geometric (elliptical) perturbations. In each case, the grating period was chosen so the resonance wavelength was close to 1.55μm. We observe that the circular fiber predictions have the correct trend and agree to better than a factor of two with the experimental results, but much better agreement is obtained with a geometric perturbation of e ² = 0.003 included. We emphasize that we do not consider this agreement as evidence that this fiber is completely described by an elliptical profile with this ellipticity. In fact, it is likely that in addition to geometric perturbations away from circularity, stress-induced birefringence is also present and contributes to the splitting [11]. However, we note that the value of e ² that results in the agreement in Fig. 2(b) is too small to measure directly; the same would be true of birefringence, which is even harder to measure than ellipticity. Therefore, what we can conclude from Fig. 2(b) is that the circular fiber predictions are a useful guide to what will be observed, and that the corrections necessary for exact agreement are small.

 

Fig. 3. (a) Fiber index profile plotted along with radial mode fields for the LP₁₂ to LP₁₅ modes; (b) index profile near the cladding-coating boundary, again with mode fields.

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The integrands in (5) are all proportional to f′(r), the radial derivative of the normalized index profile. This dependence suggests that large jumps in the profile, at points where the field is non-zero, can induce large splittings δβ. In the case of the TWRS™ fiber, by far the largest value of f′ occurs at the cladding-coating boundary. Figure 3(b) shows a close-up view of this boundary, along with the mode fields of the LP _1,2-5modes. For these modes, with e ² = 0, it happens that I ₁ ≫ I ₂(1), and that the contributions to I ₁ from the cladding-coating boundary accounts for over 70% of the full value. This is true despite the suppression of these contributions by the fields, which are rapidly dropping to zero at the boundary.

Some comments on the use of perturbation theory to compute the splittings reported in Fig. 2(b) are in order, in light of the large index contrast at the cladding-coating interface and its dominance on the vector corrections to the propagation constants. Though the index jump is large, the values of the field are small for the modes we consider, which is advantageous for the perturbation expansion. In order to confirm its validity, we considered the step-index fiber obtained by ignoring the core of the TWRS™ fiber with the low-index coating. We compared the perturbative results to the exact values of the splittings among the propagation constants of each of the LP_1,1-5 groups. In each LP _{ℓ, m} mode group, the perturbative and exact calculated range of vector βs (i.e. the difference between the largest and smallest vectorβ) agreed to better than a percent. The agreement was best for the LP₁₁ group, and became progressively worse as ℓ increased. Overall, the worst deviation from an exact splitting was 8.7%, for the value of β _HE - β _TE in the LP ₁₅ group. We may conclude that the calculations reflected in Fig. 2(b) are reliable. However, we emphasize that while the perturbative results (7) are important guides to intuition, there is no reason that an exact vector Maxwell solver cannot be used to perform calculations.

4. Conclusions

Microbend-induced gratings offer the promise of low-cost, tunable devices, but suffer from inherent polarization dependence, even in perfectly circular fibers. This polarization dependence arises from the fundamental physical processes that govern propagation in the fiber. We have shown that simple analytical expressions derived by perturbation theory are sufficient to explain experimental observations of polarization dependence in microbend gratings, as demonstrated with a TWRS™ fiber. The form of the expressions gives valuable insight into which features of the profile and mode fields are responsible for the splitting. In particular, large jumps in the index profile can result in large polarization dependence. This insight can guide the tuning of polarization dependence via fiber design.