NUSOD 2008
Mueller Matrix based Modeling of Nonlinear Polarization Rotation in a Tensile-Strained Bulk SOA Michael J. Connelly and Li-Qiang Guo Optical Communications Research Group, Dept. Electronic and Computer Engineering, University of Limerick, Limerick, Ireland, E-mail:
[email protected]. Abstract- A numerical model of non-linear polarization rotation in a tensile-strained bulk semiconductor optical amplifier is presented. The model uses a wideband model of the SOA to determine the carrier density and the TE and TM mode gains. The carrier density distribution is used to determine the phase difference between the TE and TM components of the amplified probe. The SOA Mueller matrix can be modeled as a diattenuator and phase shifter. The Mueller matrix is used to determine the Stokes vector of the amplified probe if its input Stokes vector is known. Stokes vectors can be easily measured using a polarization analyzer. The model is used to predict the polarization rotation of a probe signal induced by a co-propagating pump. I.
INTRODUCTION
*
There has been considerable progress in the exploitation of optical nonlinearities in semiconductor optical amplifiers (SOAs). Much attention has been paid to the carrier-induced nonlinearities, which lead to nonlinear polarization rotation (NPR) in SOAs. NPR is a mixture of cross-phase and crossgain-modulation effects which leads to a rotation of the probe polarization in the presence of an intense pump light [1]. NPR shows great potential for applications in optical networks for use as wavelength converters and all-optical logic. The tensilestrained bulk SOA has attracted much interest due to its relative ease of fabrication and commercial devices are now available. Such SOAs introduce tensile train in the amplifier bulk active region to achieve different TE and TM material gain coefficients. This difference is used to compensate for the different TE and TM confinement factors in order to equalize the SOA TE and TM gains. In this paper we model the NPR induced on a probe signal due to a co-propagating pump signal in a tensile-strained bulk SOA. We use a wideband model to determine the SOA carrier density distribution and the TE/TM gains. The carrier density distribution is used to calculate the net phase shift between the TE and TM components of the probe. We can then formulate the SOA Mueller matrix which can be used to predict the Stokes vector of the output probe signal. II.
∆φ =
λ probe
∫ [Γ 0
TE
N TE ( z ) − ΓTM N TM ( z )]dz
M = M dia M ret where the diattenuator Mueller matrix is given by
M dia
GTE 1 GTE = 2
+ GTM − GTM 0 0
GTE − GTM GTE + GTM 0 0
0 0 2 GTE GTM 0
0 0 0 2 GTE GTM
GTE and GTM are the single-pass modal gains. The phase shifter Muller matrix is given by
M ret
1 0 = 0 0
0 1
0 0
sin ∆φ cos ∆φ 0 0
0 cos ∆φ 0 − sin ∆φ
The Stokes vector of the output probe is then given by
S = MS i where S i is the Stokes vector of the input probe. The ellipticity angle of the output probe is given by
(
ε = 0.5 tan −1 S 3
*
This research was supported by Science Foundation Ireland Investigator Grant 02/IN1/I42.
978-1-4244-2307-1/08/$25.00 ©2008 IEEE
L
where N TE / TM and ΓTE are spatially dependent TE and TM refractive indices and confinement factors respectively. The refractive index difference between the TE and TM modes depends on the injected carrier density, which in this model includes the effects of band-filling and free-carrier absorption. The polarization rotation properties of the SOA can be represented by a Mueller matrix M composed of a diattenuator (as the TE and TM components of the probe experience different gains) followed by a retarder (phase shifter) [4],
MULLER MATRIX MODEL
The SOA modeled in this paper is a tensile-strained SOA from Kamelian Ltd. [2]. The steady-state characteristics of the SOA, including the carrier density spatial distribution, can be determined using the steady-state model [3]. The phase shift between the TE and TM modes of the output probe is given by
2π
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S12 + S 22
)
NUSOD 2008
III.
NUMERICAL SIMULATIONS
It is of interest to model the effect on the polarization state of the input probe signal as the pump power is changed. This can be done by using the steady-state model, which includes spontaneous emission, to determine the carrier density distribution. The wideband steady-state model can also be used to determine the active region material absorption spectrum, which is necessary to calculate the TE-TM refractive index difference. An experiment was carried out in which a probe signal at a wavelength and power of 1545.6 nm and -8 dBm respectively was injected into the SOA operating at a bias current of 200 mA. The Stokes vector of the input and output probe signal was measured using a polarization analyzer. The Stokes vector of the input probe signal was [1, 0.73, -0.46, 0.13] with a degree of polarization of 87%. This Stokes vector was used to determine the proportion of the input probe signal power coupled to the TE and TM modes. The ellipticity angle of the output probe polarization state was then compared with simulations. A parameter extraction algorithm was used to carry out necessary adjustments to the computed TE-TM refractive index difference to obtain good agreement between experiment and simulations of the probe ellipticity angle as the pump power was varied. This is because the TE-TM phase difference is very sensitive to the TE-TM refractive index difference. Fig. 1. shows comparisons between the experimental and simulated output probe ellipticity angle as the pump power was varied and also for the case where the probe input Stokes vector = [1, 0.75, -0.4, 0.37] (The experimental data in this case is somewhat erratic). The divergence between experiment and simulation at high pump powers may be due to the increased difference between the TE and TM gains at such powers as shown in Fig. 2. The polarization state of the output probe signal can be shown on a Poincare sphere as shown in Fig. 3.
Fig. 2. Simulated probe TE and TM gain versus pump power. The probe power is -8 dBm.
Pump power = -10 dBm
Fig. 3. Simulation of the evolution of the output probe signal Stokes vector as the pump power is varied from -10 to 10 dBm. The input probe Stokes vector is [1, 0.73, -0.46, 0.13]. The pump power increment is 1 dB.
Further results including a more detailed description of the model, numerical algorithms and more extensive simulations will be presented at the conference. REFERENCES [1]
[2] [3] [4]
L.Q. Guo, and M.J. Connelly, "Signal-induced birefringence and dichroism in a tensile-strained bulk semiconductor optical amplifier and its application to wavelength conversion," J. Lightwave Technol., vol. 23, pp. 4037-4045, 2005. C. Michie, A. E. Kelly, J. McGeough, I. Armstrong and I. Andonovic, "Polarization-insensitive SOAs using strained bulk active regions," J. Lightwave Technol., vol. 24, pp. 3920-3927, 2006. M.J. Connelly, "Wide-band steady-state numerical model and parameter extraction of a tensile-strained bulk semiconductor optical amplifier," IEEE J. Quantum Electron., vol. 43, pp. 47-56, 2007. L.Q. Guo and M.J. Connelly, "A Mueller-matrix formalism for modeling polarization azimuth and ellipticity angle in semiconductor optical amplifiers in a pump-probe scheme," J. Lightwave Technol., vol. 25, pp. 410-420, 2007.
Fig. 1. Output probe ellipticity angle versus pump power. The input Stokes vectors are S1 = [1, 0.73, -0.46, 0.13] and S2 = [1, 0.75, -0.4, 0.37].
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