## Abstract

The simple structure of a tunable polarization mode dispersion (PMD) compensator based on a cantilever beam and a high-birefringence linearly chirped fiber Bragg grating is proposed. A cantilever structure is used to introduce a linear strain gradient on the grating, and we can tune the compensated differential group delay (DGD) at a fixed signal wavelength just by changing the displacement at the free end of the beam. Based on numerical simulations, the performance of the cantilever structure as a PMD compensator is assessed for 10-Gbits/s nonreturn-to-zero transmission systems with a large DGD. With this compensator, a significant improvement of system performance can be achieved in the eye pattern of a received signal.

©2003 Optical Society of America

## 1. Introduction

With the increase in data rates, polarization mode dispersion (PMD) as a major limiting factor for long-distance communication has drawn much attention in recent years. PMD in optical fiber arises from the intrinsic mechanisms caused by geometric circular core asymmetries when the fiber is manufactured and extrinsic mechanisms caused by certain external stress. The external stress could be radially compressive forces when fibers lie against one another, compressive and tensile forces when the fibers are bent, and shear force when the fibers are twisted. It is known that PMD is characterized as a random stochastic process along with different environmental factors, such as temperature, stress, and wavelength [1,2]. Therefore, it is necessary to compensate PMD dynamically, which is more challenging than static attenuation and group velocity dispersion (GVD) compensation. Numerous optical and electronic PMD compensation techniques have been proposed in recent years [3]. Typically, an optical PMD compensator consists of three main parts: (1) feedback or feed-forward signals, (2) a control electronic circuit, and (3) a compensating part composed of polarization controller (PC) and a variable time delay line. In general researchers use time delay elements of free-space optics [4], high-birefringence (Hi-Bi) polarization-maintaining fiber, LiNbO_{3} waveguides [5], and fiber Bragg gratings (FBGs) [6].

Among these, a tunable Hi-Bi nonlinearly chirped FBG is promising because it is continuously adjustable, all-fiber based, and compact. The Hi-Bi fiber provides a different time delay for different states of polarization (SOP) and the nonlinear chirp of the grating provides the varying amounts of differential polarization time delay that can be continuously adjusted by mounting the grating on a voltage-controlled piezoelectric element. The Hi-Bi nonlinearly chirped FBG can be written into a Hi-Bi photosensitive fiber by use of a specially designed nonlinearly chirped phase mask [7] or by changing the exposure time along the length of the grating nonlinearly with a linearly chirped phase mask. However, these methods need either a costly nonlinearly chirped phase mask or an accurately controlled exposure time. Moreover, different PMD compensators need different nonlinearly chirped phase masks. Here we introduce a cantilever structure to apply a linear strain gradient on a grating for tunable differential group delay (DGD) of a Hi-Bi linearly chirped FBG. The grating is fabricated on a H2-loaded photosensitive Hi-Bi fiber by use of an UV beam scanning over a linearly chirped phase mask. The performance of this device for 10-Gbits/s nonreturn-to-zero fiber-optic transmission systems is proposed and investigated as a tunable PMD compensator. This proposed technique as an all-fiber solution is inexpensive and is flexible for adjustment of a compensated DGD.

## 2. Theory

Under conditions of constant temperature, it is known that, when a linear chirped FBG is stretched by axial strain *ε*(*z*) at grating position *z*, the period of this point that is due to the physical elastic properties of fiber is

where Λ_{0} is the grating period at position *z*=0 without strain. Constant *c*
_{0} denotes the initial linear chirp of the grating and is represented as
${c}_{0}=\frac{d\Lambda}{\mathrm{dz}}{|}_{\epsilon =0}$
. The induced change in fiber index *dn*_{eff}
(*z*) that is due to photoelastic effects is expressed as [8]

where we have subsumed the photoelastic contributions into *ρ*_{e}
, which is defined by
${\rho}_{e}=\frac{{n}_{\mathit{eff}}^{2}}{2}\left[{P}_{12}-\mu \left({P}_{11}+{P}_{12}\right)\right]$
, in terms of Pockels coefficients *P*_{ij}
and Poisson ratio *µ*. Factor *ρ*_{e}
has a typical value of ≈0.22. Thus Bragg wavelength *λ*(*z*) at grating position *z* becomes

It is straightforward to observe that the reflection wavelength shift of the grating is proportional to the applied strain. If a strain gradient is added to a FBG, the Bragg wavelength varies nonlinearly along the fiber length because of the change in the effective grating period. The nonlinear relationship between the reflection wavelength and its reflection position determines the group delay characteristics of the grating. Therefore, the expected group delay characteristics of the grating can be obtained by an appropriate design of the strain distribution.

A schematic diagram of the proposed cantilever structure to induce a linear strain gradient of the grating is shown in Fig. 1(a). The main components of this device consist of a FBG and a uniform rectangular beam with length *L* and thickness *h*. One end of the beam is free, and the other is anchored. A FBG with original length *l* is bonded to the surface of this beam. Deflecting the free end of the beam with displacement *Y* can induce a strain distribution across the surface of the beam, and it can be described by the following equation [9]:

Therefore, when a uniform FBG is mounted on the surface of the beam, it can be transformed into a linearly chirped FBG [10] through a physical change of the grating pitch with the elastic and photoelastic effects that are due to the developed linear strain gradient. For a linearly chirped FBG it can be seen from Eqs. (3) and (4) that it will have the nonlinear chirped characteristics controlled by the adjustment of only one parameter, namely, displacement *Y*. Furthermore, we can predict that one Hi-Bi linearly chirped FBG bounded to the surface of the beam will be transformed into a Hi-Bi nonlinearly chirped FBG because of this strain gradient.

For a chirped FBG, the phase-sensitive spectral characteristics were simulated based on the piecewise-uniform approach (also called transfer matrix formalism) [11,12]. First, we divide the grating into *M* uniform sections, identify each section by one 2×2 matrix, and then multiply all these together to obtain a single 2×2 matrix that describes the whole grating. ${E}_{j}^{+}$ and ${E}_{j}^{-}$ are introduced, respectively, as the complex field amplitudes of the forward and backward propagating waves after they traverse section *j*. The propagation through section *j* is described by matrix *T*
^{(j)} defined such that

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{T}_{1}^{\left(j\right)}=\mathit{cosh}\left(\gamma \Delta z\right)-i\frac{\hat{\sigma}}{\gamma}\mathit{sinh}\left(\gamma \Delta z\right),$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{T}_{2}^{\left(j\right)}=-i\frac{\kappa}{\gamma}\mathit{sinh}\left(\gamma \Delta z\right),$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\gamma =\sqrt{{\kappa}^{2}-{\hat{\sigma}}^{2}},$$

where ${T}_{1}^{\left(j\right)*}$ and ${T}_{2}^{\left(j\right)*}$ are the complex conjugates of ${T}_{1}^{\left(j\right)}$ and ${T}_{2}^{\left(j\right)}$, respectively; Δ*z* is the length of the *j*th uniform section; and *$\widehat{\sigma}$* and *κ* are, respectively, dc (period averaged) and ac coupling coefficients expressed as

where *δ*=2*πn*_{eff}
(*λ*
^{-1}-${\lambda}_{0}^{-1}$) is the detuning from Bragg wavelength *λ*
_{0} related to period Λ_{0} at position *z*=0. Here
${\overline{\mathit{\delta n}}}_{\mathit{eff}}$
is the dc index spatially averaged over a grating period, *ν* is the fringe visibility of the index change, *g*(*z*) denotes the apodization envelope function, and *φ*(*z*) describes the chirp of the grating. The chirp of grating can be expressed with the grating period function along length *z* by the following equation:

Once all the matrices for the individual sections are known, the output amplitudes can be obtained from

Substituting Eqs. (1)–(3) into Eq. (8) and imposing the boundary condition on ${E}_{0}^{+}$=0, ${E}_{M}^{-}$
=0, we can analyze in detail the reflection spectrum and group delay characteristics under strain *ε*(*z*).

Figure 1(b) is a schematic diagram of a Hi-Bi nonlinearly chirped FBG. It is written into a Hi-Bi fiber with a large refractive-index difference Δ*n* between fast and slow polarization axes, which results in a shift in Bragg wavelength Δ*λ*
_{0}=${\lambda}_{0}^{s}$-${\lambda}_{0}^{f}$=2Δ*n*Λ_{0} at the same location for two polarizations. In other words, for a fixed signal wavelength the Bragg reflection occurs at different locations for different polarizations. Therefore, the Hi-Bi linearly chirped FBG can be seen as two different chirped FBGs because of the birefringence. The position difference of the reflection produces a DGD:

between these two orthogonal SOPs. For a Hi-Bi linearly chirped FBG, it is known that the DGD is constant and equals the product of the chromatic dispersion of grating |*C*| and Bragg wavelength shift Δ*λ*
_{0} such that *DGD*=|*C*|Δ*λ*. A key feature of our device is that its DGD is tunable for a given signal wavelength. For that by the displacement of a cantilever beam the chromatic dispersion and wavelength shift of the grating are functions of strain *ε*(*z*).

The reflection spectrum and group delay characteristics of Hi-Bi linearly chirped FBGs under strain *ε*(*z*), which was adjusted by the proposed scheme, are shown in Figs. 2(a) and 2(b), respectively. Here, the grating has a zero dc index change (
${\overline{\mathit{\delta n}}}_{\mathit{eff}}\to 0$
), maximum ac index change
$\nu {\overline{\mathit{\delta n}}}_{\mathit{eff}}=2\times {10}^{-4}$
, and an initial linear chirp of *c*
_{0}=1×10^{-8}. For our H_{2}-loaded photosensitive Hi-Bi fiber, the normalized birefringence is Δ*n*=4×10^{-4}. A super-Gaussian apodization profile for the modulation amplitude of the form
$g\left(z\right)=\mathrm{exp}[-\frac{2}{5}\xb7{\left(\frac{z-\frac{l}{2}}{\frac{l}{3}}\right)}^{6}]$
(a grating length of *l*=80*mm*) is introduced to suppress the sidelobes in the reflection spectrum and group delay ripples. The cantilever beam is 100 mm in length and is 10 mm thick.

The grating period increases when the displacement of the cantilever beam increases, which results in an increase of the reflection wavelength. The period difference at two points of the grating also increases with the increase of displacement *Y*, which results in both a broadening of the reflection bandwidth and a decrease of the delay curve slope of the grating. As shown in Fig. 2, the Bragg wavelength shift between two polarization components that is due to fiber birefringence is approximately 0.42 nm at 1550 nm. After deflecting a displacement of 1 mm at the free end, the 3-dB bandwidth of the grating increased to 3.05 nm from the initial value of 1.84 nm and was 4.17 nm when the displacement was set to 2 mm. The slope of the delay curve decreased from the initial 305.4 to 185.5 ps/nm, and then decreased to the low of 128.3 ps/nm at a given wavelength of 1550 nm, which leads to the decrease of the DGD of the Hi-Bi linearly chirped FBG. According to the Eq. (9), the relation between the DGD of the Hi-Bi linearly chirped FBG and the displacement of the cantilever beam is illuminated in Fig. 2(c). Note that the maximum tolerated strain of the FBG is at least 3800 *µε* [13], which corresponds to a Bragg wavelength shift of approximately 4.6 nm at 1550 nm. The displacement at the free end of the cantilever structure is 2.2 mm, and the maximum value of the strain is 3300 *µε*. Therefore, it can be predicted that a displacement of 2.2 mm at the free end of the cantilever beam is feasible.

It can be seen that this proposed PMD compensator has a large adjustable range that varies from 141.7 to 48.2 ps. By an appropriate choice of grating and birefringence parameters for the photosensitive fiber, we can also obtain any desirable tunable DGD ranges of grating using this simple method. However, it is worthwhile to note that grating reflectivity decreases with an increase of free end displacement of the beam shown in Fig. 2(a). The peak reflectivity was reduced to 92.26% when *Y*=1*mm* from the original value of 98.98%, and then to 83.01% at *Y*=2*mm*, which would introduce fluctuating insertion loss of the PMD compensator and could result in an unfavorable effect on the fiber-optic transmission systems. Fortunately, the power instability from this device during PMD adaptive compensation can be easily overcome by the dynamic gain control of the amplifier in the fiber links.

The spectra of FBGs are sensitive to both strain and temperature. There are two ways to control the spectral behavior of a grating: application of a strain gradient and a temperature gradient. The Bragg wavelength shift of a conventional FBG normally has a temperature coefficient of 0.01 nm/°C [8]. It is therefore necessary to compensate the undesirable temperature sensitivity of a FBG in some applications, such as fiber grating bandpass filters and a chirped FBG as a chromatic dispersion compensator. For use as a PMD compensator, we determined that temperature compensation is unnecessary. Because the PMD compensation is a dynamic process, the temperature changes could have a minor effect on the range of variable DGD, but its effect on PMD compensation could be negligible.

## 3. Numerical analysis of system application

#### 3.1 System setup

PMD compensation in a 10-Gbit/s dispersion-managed optical transmission system model as depicted in Fig. 3 is numerically evaluated to demonstrate the effectiveness of our proposed device as shown in Fig. 1. The period dispersion map consists of 85 km of G.652 single-mode fiber (SMF) along with 15 km of dispersion compensation fiber (DCF) and two gain stages. The parameters of the SMF and the DCF are listed in Table. 1, and the average input powers are set to 5 and -2 dBm for SMF and DCF, respectively. Four stages of dispersion map transmission, a total of 400 km, are considered. The erbium-doped fiber amplifier (EDFA) (4.5-dB noise figure) is set to compensate the fiber loss exactly. There is residual dispersion of 22.5 ps/nm under compensation of each map for the optical pulse. In the transmitter, a 10-Gbit/s 2^{6}-1 pseudo- random bit sequence (PRBS) with super-Gaussian (m=2) nonreturn-to-zero (NRZ) pulses is launched into the fiber; see Fig. 4(a). At the input, we used a polarization controller to adjust the signal’s input SOP. The PMD compensator with three degrees of freedom comprised the PC, the device described in Fig. 1(a), and a section of Hi-Bi fiber that was used to adjust the DGD range of the variable delay line. For example, the section of Hi-Bi fiber in our PMD compensator is set to have an opposite 40-ps DGD relative to that of Hi-Bi linearly chirped FBG. Then the variable delay line has a tunable range from 8 to 101 ps. The degree of polarization (DOP) of the output signal is detected as a feedback signal to control the PMD compensator. We used the control algorithm to maximize the DOP by deflecting the cantilever beam and adjusting the PC. It is worthwhile to note that the insertion loss of this PMD compensator is due mainly to the reflectivity of the grating and to the insertion loss of the optical circulator. Here it is assumed lossless in the simulation. In the receiver, the output optical signal from an optical filter with a 40-GHz bandwidth is transferred to an electric signal after square-law detection and is then sent to a fourth-order low-pass electrical Bessel filter with a 7.5-GHz bandwidth. Finally, the eye diagram of the received signal is observed on the oscilloscope. The sampling time is optimized to account for PMD-induced bit pattern shifts. Here we define the parameter of the eye-opening penalty (EOP) that is due to PMD as

where *B*
_{0} is the eye opening with first- and second-order GVD, amplified spontaneous emission (ASE) noise, and Kerr nonlinear effects but without PMD; *B* is the eye opening with all these factors plus PMD.

#### 3.2 Results and discussion

Based on the system described above, one can evaluate its performance by using the coarse-step method [14]. For the calculations the fiber is divided into 400 segments, which provides a good representation of the random mode coupling effect along the link [15]. Figure 4(b) shows the received signal eye diagram after 400-km transmission for a PMD of 0 ps. Figs. 4(c) and 4(d) are eye diagrams of the accumulated DGD of 40 ps without and with PMD compensation. In Figs. 4(e) and 4(f) the link’s DGD is 108 ps. Here, the worst-case launch pattern is assumed, i.e., half of the total power into each principal SOP at the input of fiber. It is straightforward to observe that the eye at the output of fiber links is clearly distorted (0.98-dB EOP with 40-ps DGD in Fig. 4(c) and 15.80-dB EOP with 108-ps DGD in Fig. 4(e)) because of PMD compared with that shown in Fig. 4(b). We achieved a significant improvement in the quality of the received signal eye diagram when the proposed PMD compensator was added as shown in Fig. 4(d) (0.08- dB EOP), and even the link’s DGD is larger than 1-bit period as illustrated in Fig. 4(f) (0.41-dB EOP).

To qualify the complete effectiveness of the proposed PMD compensation techniques, we also take a statistical picture into account by evaluating 10,000 ensembles of fiber for one PMD value based on the Monte Carlo method. Figure 5 shows the results of the received EOP distribution with a link average accumulated DGD of 60 ps before and after PMD compensation. Without compensation, 10.5% of the samples exhibited greater than 5 dB of received EOP. When we use our proposed compensator, there is only 1.14% of the samples whose EOP extends beyond the 5-dB limit. At these points we note that the high EOP could not be handled because of large high-order PMD. Fig. 6 shows a comparison of the worst-case EOP (with a probability of 10^{-3}) as a function of the mean DGD before and after compensation based on the proposed PMD compensation technique. For values of fiber DGD >40 ps, almost a 4-dB reduction of the EOP is obtained by PMD compensation.

## 4. Conclusion

We have proposed a novel tunable PMD compensator by using a Hi-Bi linearly chirped FBG bounded on the surface of a cantilever beam. By deflecting one free end of the beam, we obtained a linear strain gradient on the grating. The various properties of reflectivity, group delay, and DGD of Hi-Bi linearly chirped FBG have been discussed in detail. With an optimal design of the grating and cantilever beam parameters, the compensated DGD at a fixed signal wavelength can be tunable over a much larger range. In addition, the tunable range can also be adjusted by an accessional Hi-Bi fiber. Its performance as PMD compensator was assessed for a 10-Gbit/s NRZ transmission systems with large DGD. We achieved a significant improvement in quality of the eye pattern of the received signal. The proposed PMD compensator can be potentially useful because it is inexpensive, its structure is simple, and it has the flexibility to adjust compensated DGD.

## Acknowledgments

This research was supported by the National Hi-tech Research 863 Project of China (No. 2001AA120204).

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