## Abstract

In this paper, we completely study the wavelength dependency of differential group delay (DGD) in uniform fiber Bragg gratings (FBG) exhibiting birefringence. An analytical expression of DGD is established. We analyze the impact of grating parameters (physical length, index modulation and apodization profile) on the wavelength dependency of DGD. Experimental results complete the paper. A very good agreement between theory and experience is reported.

©2005 Optical Society of America

## 1. Introduction

Fiber Bragg grating technology is widely used to realize both optical communication components and optical sensors devices [1]. Fiber Bragg gratings are generally fabricated by irradiation of a UV laser beam onto a single side of an optical fiber. It is commonly admitted that this side-written fabrication process induces a small quantity of birefringence that combines with the intrinsic fiber birefringence to cause polarization dependence of the grating [2,3]. While this small amount of birefringence can be hardly detected in the grating amplitude response, it leads to differential group delay (DGD) and polarization dependent loss (PDL).

For both telecommunication and sensor applications, it becomes of high importance to study these grating polarization properties. Indeed, due to the increasing bit-rate, high-speed wavelength division multiplexing (WDM) transmission systems are less tolerant to polarization effects [4]. Polarization dependence of gratings used as chromatic dispersion compensator or filters leads to degradations of system performances. In this frame, using Bragg grating components implies to know their polarization properties. In the same way, polarization properties of Bragg gratings take importance in optical sensing. Indeed, a new technique was recently presented in order to discriminate temperature and strain effects by monitoring the PDL of a uniform grating [5].

Polarization properties of gratings have been already investigated in the past: wavelength dependency of PDL for different gratings as well as DGD of chirped gratings have been reported in [5–6] and in [7–10], respectively. However, although uniform Bragg gratings are widely used for communications and have great interest in sensors applications, their differential group delay has never been investigated.

In this paper, we propose to study the wavelength dependency of DGD in the case of uniform FBG written into single mode fiber. The Jones formalism and the solution of the coupled mode theory are used in order to derive an analytical expression of DGD. The impact of the grating parameters and the birefringence value obtained after the writing process are analyzed. We show that the DGD evolutions with wavelength strongly depend on the birefringence, the grating physical length, the modulation index and the apodization profile. To complete the study, we performed DGD measurements on Bragg gratings written into a single mode fiber. A very good agreement is obtained with simulated results, allowing to confirm the validity of our analysis. Moreover, the fitting process between experimental and simulated results allows to get the birefringence value after the writing process. The knowledge of this value is useful for some application as mentioned in [5,11].

## 2. Background theory

Birefringence in optical fibers is defined as the difference in refractive index between a particular pair of orthogonal polarization modes (called the eigenmodes or modes *x* and *y*). In gratings obtained by a single side written process, the index change is not uniform through the fiber cross-section and it causes photo-induced birefringence [2,3]. Photo-induced birefringence combines with the intrinsic fiber birefringence to give a global birefringence value. The order of magnitude of the global birefringence obtained after the writing process typically ranges between 10^{-6} and 10^{-5} [2,3,5].

The refractive index for both the *x* and *y* modes is then defined by Eq. (1) where *n*_{eff}
is the fiber effective refractive index of an FBG without birefringence and *Δn* is the birefringence obtained after the writing process.

In general, *n*_{eff}
and *Δn* are wavelength dependent. However, since the investigated wavelength range (the grating bandwidth) is relatively small (order of magnitude of 1 nm), the wavelength dependency of *n*_{eff}
and *Δn* is neglected in Eq. (1).

The birefringence causes the two orthogonal polarization modes to experience different couplings through the grating. Consequently, the transmitted signal is the combination of the transmitted signals corresponding to both the *x* and *y* polarization modes. The input signal can be represented by a bi-dimensional complex Jones vector (Eq. (2)) which defines its state of polarization (SOP). In Eq. (2), *M*_{x(y)}
and *ξ*_{x(y)}
represent the amplitude and the phase angle of the *x*(*y*) component of the input electric field, respectively.

Using a Cartesian coordinates system such that the reference axes correspond to the grating eigenmodes, the Jones vector of the transmitted signals is defined by Eq. (3) where the FBG transmission properties are described by the diagonal Jones Matrix *J*. In Eq. (3), *t*_{x(y)}
denotes the transmission coefficient of the uniform Bragg grating corresponding to the *x*(*y*) mode.

The wavelength dependency of the complex transmission coefficient *t* of a uniform FBG can be derived from the coupled mode theory [12]. Taking into account the birefringence, Eq. (4) gives the two transmission coefficients *t*_{x}
and *t*_{y}
corresponding to the eigenmodes induced by *Δn*.

In Eq. (4), *α*_{x(y)}
and *σ*_{x(y)}
correspond to *α* and *σ* parameters defined in [12] where *n*_{eff}
is replaced by *n*_{eff,x(y)}
so that *α*_{x(y)}
= [*κ*
^{2}-${{\sigma}_{x\mathit{\left(}y\mathit{\right)}}}^{\mathit{2}}$
]^{1/2} , *κ* = *πδn*/*λ*, *σ*_{x(y)}
= *2π* (*n*_{eff,x(y)}
+*δn*)/*λ*- *π*/*Λ* and *λ* is the wavelength. All these parameters depend on the grating physical parameters: the index modulation *δn*, the grating period *Λ* and the grating length *L*. Resonant wavelength *λ*_{max,x(y)}
, corresponding to the wavelength at which *t*_{x(y)}
presents a minimum, is given by *λ*_{max,x(y)}
= 2(*n*_{eff,x(y)}
+ *δn*)*Λ*.

## 3. Wavelength dependency of differential group delay: analytical result

The differential group delay, denoted by *Δτ*, is defined as the difference in propagation time between the two eigenmodes: *Δτ* = |*τ*_{x}
- *τ*_{y}
| where *τ*_{x(y)}
is defined as the derivative, versus frequency ω, of the phase of *E*_{t,x(y)}
. Considering Eq. (3) and Eq. (4), it is possible to derive a theoretical expression of DGD for any arbitrary input SOP. For the sake of simplicity, we have calculated it when the input state of polarization is the π/4 linear polarization state so that *M*_{x}
= *M*_{y}
= 1/√2 and *ξ*_{x}
= *ξ*_{y}
= 0. In this case, the group delay *τ*_{x(y)}
is the derivative, versus frequency ω, of the phase *θ*_{x(y)}
of the complex transmission coefficient *t*_{x(y)}
. Since all coefficients in Eq. (4) are wavelength dependent, the derivative of the phase gives a quite extended result. Considering that *δn* is very small compared with *n*_{eff}
, some terms in the development can be neglected. *τ*_{x(y)}
is finally expressed by Eq. (5). Expressions of *τ*_{x}
and *τ*_{y}
can then be used to obtain the complete expression of DGD from |*τ*_{x}
- *τ*_{y}
|.

## 4. Simulation results

In this section, we analyze the impact of the grating parameters and the birefringence on the wavelength evolution of DGD. Let us first discuss the effects of birefringence on the transmission spectrum. Fig. 1 presents transmission coefficients obtained for typical grating parameters values and for different classical birefringence values *Δn* obtained after the writing process. In this case, Fig. 1 shows that the three curves are superimposed so that the influence of these amounts of birefringence is not perceived in the transmitted spectrum. Consequently, contrary to gratings written into polarization maintaining fiber, birefringence does not lead to spectral separation between the two maxima corresponding to each polarization states. However, it leads to DGD spectrum.

In the following, in order to make easier the comparison between different simulation curves, DGD evolutions are plotted versus the normalized wavelength *λ*/*λ*_{max}
where *λ*_{max}
is defined by (*λ*_{max,x}
+ *λ*_{max,y}
)/*2*. It also corresponds to the resonant wavelength of the grating without birefringence.

In order to obtain the DGD for uniform gratings, Eq. (5) is used. All curves were simulated with *n*_{eff}
= 1.4514 and *Λ* = 530 nm. All other parameters values used for simulations are written into the figures. The wavelength dependency of DGD for uniform gratings as a function of *Δn*, *L* and *δn* are depicted in Fig. 1, Fig. 2(a) and Fig. 2(b).

As a general comment, all the simulated evolutions are symmetric. As expected, the increase of the *Δn* value leads to a general increase of the DGD amplitudes. However, the value of *Δn* has no impact on the relative shape of the DGD evolution.

The variations of *L* or *δn* have the same effects on the DGD evolution. Indeed, strong gratings are obtained by increasing either *L* or *δn*. In this case, the transmission spectrum is saturated and presents a rapid variation at the edges of the main rejection band. The same
Fig. 2(a) and Fig. 2(b)): for high values of *L* and *δn*, DGD presents a greater variation at the edges of the main transmission band. Moreover, very high values of DGD (few tens of ps) can be reached when *L* and *δn* increase, the value of *Δn* remaining constant. This analysis shows that the DGD evolutions with wavelength strongly depend on both the birefringence and the grating physical parameters.

One other important parameter is the apodization profile of the grating. In order to simulate the effect of the apodization on the wavelength dependency of DGD, we have implemented in the transfer matrix method [13] the DGD analysis presented in sections 2 and 3. We used two Gaussian apodization profiles with two different Gaussian window parameters *G* (as defined in [14]). The weak and the strong apodization profiles correspond to *G* = 1 and *G* = 3, respectively. Amplitude and DGD responses of the two apodized gratings are plotted in Fig. 3(a) and Fig. 3(b), respectively.

One can observe the well-known impact of the two apodization profiles on the transmission spectrum: the weak apodization profile leads to a small asymmetry for the side-lobes while the strong apodization profile reduces the side-lobes. Concerning the DGD evolution, apodization leads to an asymmetry in comparison with uniform gratings (see Fig. 3(b)). Moreover, apodization reduces the number of side-lobes outside the rejection band as well as the reached maximum values. All these effects are more significant for the strong apodization.

## 5. Experimental results

Experimental results are reported for saturated and non-saturated gratings. Two uniform Bragg gratings were written into boron co-doped photosensitive fiber by means of an Innova FreD argon-ion laser from Coherent. We used the continuous grating writing technique and the interference pattern was created by a 1070.40 nm period phase mask. The grating length and the index modulation were nearly 1 cm and 1 10^{-4} for the weak grating and 0.7 cm and 4 10^{-4} for the strong grating. Moreover, the strong grating was slightly apodized.

For simulation purpose, the physical gratings parameters were reconstructed from the experimental reflected spectrum using a simplex algorithm technique derived from [15]. The reconstructed parameters are the grating period *Λ*, the physical length of the grating *L*, the index modulation *δn* and the apodization profile with the *G* parameter. *Δn* was not taken into account in this reconstruction process since it affects very slightly the shape of the transmitted spectrum. The reconstructed parameters values for the two gratings are shown on the corresponding figures.

The DGD evolution with wavelength was obtained by measuring the Jones matrix of the grating in transmission versus wavelength (Jones matrices eigenanalyzis method) [16]. For each wavelength, DGD value is computed from a pair of Jones matrices separated by a small wavelength step using Eq. (6).

In Eq. (6), *Arg*(*x*) denotes the argument of *x* and *Δω* is the change in optical frequency in rad/s corresponding to the wavelength step. *ρ*_{1}
and *ρ*_{2}
are the eigenvalues of the following matrix product:

where *T*(*ω*) is the measured Jones matrix of the grating at the pulsation *ω*.

In our setup, we used a fully polarized tunable laser source EXFO FLS2600B and a polarimeter Profile PAT9000B. The input light was launched through a polarizer controlled by the polarimeter. The measurements were performed with a 10 pm wavelength step. We checked that connectors and pigtails do not have significant impact on DGD results. During measurements, gratings were free of mechanical constraints and the ambient temperature was maintained constant. Moreover, all the fibers were fixed to avoid polarization instabilities. All these precautions ensured a very good repeatability. The DGD evolutions presented in Fig. 4 result from the average of 5 measurements performed in the same conditions.

To confirm the experimental evolutions, we simulated the DGD with the numerically reconstructed grating parameters. We adjusted the birefringence value to obtain the best fit between the measured and simulated DGD evolutions. We obtained *Δn* equal to 4 10^{-6} and 10 10^{-6} for the weak and the strong grating, respectively. The experimental and simulated DGD evolutions are reported on Fig. 4(a) and Fig. 4(b) for the strong and the weak grating, respectively. A very good agreement is observed between experiment and simulation.

DGD evolution presents relatively weak and constant values in the main rejection band while more important values can be reached at the edges of the main rejection band. For the strong grating, the impact of the slight apodization of the index modulation can be classically detected on the amplitude of the two secondary side-lobes in the transmission spectrum. Moreover, an asymmetry appears in the DGD evolution with wavelength, which can be detected on both simulated and experimental curves. These experimental results are completely confirmed by simulations, proving the validity of this study.

## 6. Conclusion

We reported the evolution with wavelength of differential group delay (DGD) caused by birefringence in uniform FBG. Based on the coupled-mode theory, we derived an expression of DGD for uniform FBG. In order to simulate the effect of the apodization on the wavelength dependency of DGD, we implemented the DGD calculus to the transfer matrix method.

While the DGD in FBG is caused by the birefringence, we showed that the spectral evolution of DGD is directly related to the grating parameters. Considering a constant value for the birefringence, our analysis demonstrated that the grating physical length, the grating index modulation and the apodization profile of the grating have an important impact on the spectrum shape and the maximum values of DGD. Experimental results have completed the study. Very good agreement with simulation result has been presented.

In general, birefringence obtained after the writing process can not be detected in the grating amplitude response since it is relatively weak. In this case, our study could be used to determine this amount of total birefringence since the fitting between the experimental and theoretical evolutions of the DGD requires to adjust the birefringence value. The knowledge of this value can then be useful for some application, as mentioned in section 1.

Furthermore, in the frame of optical sensing, we expect that this analysis will be used to design suitable gratings for the development of new sensor applications based on polarization properties of Bragg gratings.

## Acknowledgments

C. Caucheteur is supported by the Fonds National de la Recherche Scientifique (FNRS). Authors take part to the EU FP6 Network of Excellence e-Photon/ONe (WP5 and WP10). They acknowledge the Belgian Science Policy.

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