## Abstract

Polarization dependence in microbend gratings is an inherent problem. We formulate simple analytical expressions to describe it, and demonstrate their effectiveness via a comparison with experimental results on a standard transmission fiber. The ability to control polarization dependence with fiber design potentially enables replacing UV-LPGs within low-cost, tunable microbend gratings.

©2005 Optical Society of America

## 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 ≃10*μ*m) can be shown [7] to preferentially couple light only to polarization-insensitive HE_{21} 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].

## 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 = 2*π*/*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*_{i}*̇*, is related to the group delay *τ*_{i}
, by *β*_{i}*̇*, = -*ω*
^{2}
*τ*_{i}
/2*πc*.

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

$${n}_{0}^{2}\left(r\right)={n}_{\mathrm{co}}^{2}\left(1-2\Delta f\left(r\right)\right)$$

$${n}_{\mathrm{ell}}^{2}\left(r,\theta \right)={c}_{\mathrm{ell}}r\phantom{\rule{1.1em}{0ex}}\mathrm{cos}\left(2\theta \right)f\prime \left(r\right)$$

$${c}_{\mathrm{ell}}=\frac{1}{2}{e}^{2}{\Delta n}_{\mathrm{co}}^{2}\phantom{\rule{1.6em}{0ex}}$$

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

$${\phantom{\rule{.2em}{0ex}}I}_{2}\left(\ell \right)=\ell \mathit{\int}\mathit{rdrd\theta}\frac{1}{r}F{\left(r\right)}^{2}f\prime \left(r\right)\phantom{\rule{1.2em}{0ex}}$$

$$\phantom{\rule{.2em}{0ex}}{I}_{3}=\int \mathit{rdrd\theta rF}{\left(r\right)}^{2}f\prime \left(r\right).\phantom{\rule{.5em}{0ex}}$$

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*:

$$\phantom{\rule{.2em}{0ex}}\u3008\ell =0,m,\xb7,\xb7\mid {V}^{\left(\mathrm{vect}\right)}\mid \ell =0,m,\xb7,\xb7\u3009=\phantom{\rule{6.0em}{0ex}}$$

$$\phantom{\rule{2.5em}{0ex}}-2\Delta \frac{\pi}{4}\left(\begin{array}{ccc}{3I}_{1}+{I}_{2}\left(1\right)& & {I}_{1}+{3I}_{2}\left(1\right)\\ & {I}_{1}-{I}_{2}\left(1\right)\phantom{\rule{.9em}{0ex}}{I}_{1}-{I}_{2}\left(1\right)& \\ & {I}_{1}-{I}_{2}\left(1\right)\phantom{\rule{.9em}{0ex}}{I}_{1}-{I}_{2}\left(1\right)& \\ {I}_{1}+{3I}_{2}\left(1\right)& & 3{I}_{1}+{I}_{1}\left(1\right)\end{array}\right)\phantom{\rule{.2em}{0ex}}$$

$$\u3008\ell >1,m,\xb7,\xb7\mid {V}^{\left(\mathrm{vect}\right)}\mid \ell >1,m,\xb7,\xb7\u3009=-2\Delta \frac{\pi}{4}\left(\begin{array}{cccc}{I}_{1}& & & {I}_{2}\left(\ell \right)\\ & {I}_{1}& -{I}_{2}\left(\ell \right)& \\ & -{I}_{2}\left(\ell \right)& {I}_{1}& \\ {I}_{2}\left(\ell \right)& & & {I}_{1}\end{array}\right).\phantom{\rule{.2em}{0ex}}$$

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 TE_{01}, TM_{01}, and two HE_{21} modes:

$$\delta {\beta}_{\mathrm{TM}}^{2}=2\mathrm{\pi \Delta}\left({I}_{1}+{I}_{2}\left(1\right)\right)\phantom{\rule{1.0em}{0ex}}{\mathbf{e}}_{t}^{\left(\mathrm{TM}\right)}=\mid 1,m,\mathrm{cos},\hat{\mathbf{x}}\u3009+\mid 1,m,\mathrm{sin},\hat{\mathbf{y}}\u3009$$

$$\delta {\beta}_{\mathrm{HE}}^{2}=\mathrm{\pi \Delta}\left({I}_{1}-{I}_{2}\left(1\right)\right)\phantom{\rule{0.5em}{0ex}}{\mathbf{e}}_{t}^{\left(\mathrm{HE}\right)}=\mid 1,m,\mathrm{cos},\hat{\mathbf{x}}\u3009-\mid 1,m,\mathrm{sin},\hat{\mathbf{y}}\u3009,$$

$$\phantom{\rule{12.5em}{0ex}}\mid 1,m,\mathrm{sin},\hat{\mathbf{x}}\u3009+\mid 1,m,\mathrm{cos},\hat{\mathbf{y}}\u3009.$$

#### 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

$$\frac{{\mathit{da}}_{1}}{\mathit{dz}}={\mathit{iK}}_{\mathrm{1,0}}{e}^{-\mathit{i\alpha z}}{a}_{0},$$

with *α* = *β*
_{01} - *β*
_{Λ}, *β*
_{01} = *β*
_{0} - *β*
_{1}, *β*
_{Λ} = 2*π*/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*/2*L*. Off resonance, with initial conditions of *a*
_{0} = 1,*a*
_{1} = 0,

$$x=\frac{{L}^{2}}{{\pi}^{2}}{\left({\beta}_{01}-{\beta}_{\Lambda}\right)}^{2}.$$

If we take λ_{0} 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 LP_{15}.

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 LP_{01} 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.

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 1.55*μ*m. 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.

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 _{1,2-5}modes. 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 *β*s (i.e. the difference between the largest and smallest vector*β*) agreed to better than a percent. The agreement was best for the LP_{11} 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.

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