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Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain

Abstract: A generalized Mueller matrix method (GMMM) is proposed to measure the polarization mode dispersion (PMD) in an optical fiber system with polarization-dependent loss or gain (PDL/G). This algorithm is based on the polar decomposition of a 4 4× matrix which corresponds to a Lorentz transformation. Compared to the generalized Poincaré sphere method, the GMMM can measure PMD accurately with a relatively larger frequency step, and the obtained PMD data has very low noise level.


Introduction
According to the principle state theory, polarization mode dispersion (PMD) is fully described by the principal states of polarization (PSP), differential group delay (DGD) and differential attenuation slope (DAS) between the two PSPs (or equivalently the complex PMD vector Λ + Ω = i W ) in an optical fiber system with polarization-dependent loss or gain (PDL/G) [1][2][3]. To date, many PMD measurement methods have been reported. Among them, Jones matrix eigenanalysis (JME) method is considered to be competent for measuring the PMD in such a system with PDL/G. However, the existing JME method does not present all information of the complex PMD vector [4]. The complex plane method [5] and the generalized Poincaré sphere method [3] need to calculate the derivative of polarization states with respect to optical frequency, which may cause "inaccuracies in the creation and measurement of small differences between optical frequencies and low experimental signalto-noise ratio in Stokes parameters differences due to polarimeter inaccuracies and other errors" [6]. In an optical fiber system without PDL/G, the Mueller matrix method (MMM) has been demonstrated to be a better technique to attain the low-noise high-resolution PMD data [6]. Very recently, we have derived theoretically the governing equation between the Mueller matrix and the complex PMD vector based on the fact that the Mueller matrix in an optical fiber system with both birefringence and PDL/G satisfies the Lorentz transformation [7]. In this work, we will show that it is possible to find a generalized Mueller matrix method (GMMM) for PMD measurement in an optical fiber system with PDL/G. The paper is organized as follows. In Section 2, we briefly introduce the determination of the Mueller matrix using three input polarization states. In Section 3, we propose a method to extract the complex PMD vector from the measured Mueller matrices based on a polar decomposition technique. Finally in Section 4, experimental results on an optical fiber system with bendinginduced PDL confirm the validity and advantages of our GMMM compared with the generalized Poincaré sphere method.

Determination of Mueller matrix
When both birefringence and PDL/G exist in an optical system, the 4 4 × Mueller matrix M has been demonstrated to be corresponding to a Lorentz transformation [8], i.e.  [9]. There are 9 independent bilinear equations relating the elements of the Mueller matrix [9], and it has been shown that only 7 Mueller matrix elements are independent in this case [10]. Therefore, in principle, three non-normalized input polarization states (including the optical power) are sufficient for the determination of Mueller matrix. The output and input Stokes parameters are related by a Mueller matrix as [9], hence these 4 elements can be calculated according to Eq. (4).

Determination of the complex PMD vector, DGD and DAS
If the input polarization state is fixed, the output polarization states at two consecutive optical frequencies can be related by [6] is a unit vector, φ is the rotation angle around rˆ in Stokes space.
is a vector representing PDL/G in frequency domain. And we know [7] ⎟ ⎟ ( are the real and imaginary parts of the complex PMD vector W . Comparing Eq. (6) and Eq. (9), assuming the higher-order PMD effects between the two frequencies are negligible, we have Here φ and rˆcan be extracted from R m based on same equations as the Eqs. (6), (7) and (8) in Ref [6]. To avoid measurement ambiguities [4], the condition π ω < Δ Ω should be satisfied. Finally, DGD and DAS can be calculated based on the following equation [3] ⎟ ⎠ To verify the proposed theory and measurement method, an experimental setup is established as shown in Fig. 1. The fiber system under test (FUT) comprises three Hi-Bi fiber sections (PANDA type, with 3m-long length and 4ps DGD for each) interleaved by two single-mode fiber sections (1m-long for each). In particular, the Hi-Bi fiber in the middle is spooled on a drum with a 1.5cm diameter, so bending-induced PDL exists in this system [11]. The tunable laser source emits completely polarized light with a stable polarization state and a 700kHz linewidth. The computer-controlled polarization controller (PC) as well as the in-line polarimeter are used to generate and monitor the appropriate input Stokes parameters. The polarimeter in the end is used to measure the output Stokes parameters. All components except FUT are controlled by a computer through GPIB port, USB port and I/O card. . The wavelength range is from 1520nm to 1570nm. Based on the same measured data, we calculate the DGD and DAS using both GMMM and the generalized Poincaré sphere method. The obtained results are compared in Fig. 2. The wavelength step sizes are chosen as 0.05nm, 0.1nm, 0.2nm and 0.5nm in different measurements to verify which PMD measurement algorithm tolerates larger frequency step size for attaining low-noise PMD data. From Fig. 2, we clearly know that GMMM and the generalized Poincaré sphere method give close results of DGD and DAS and exhibit large noise when the wavelength step size is less than 0.1nm. But when the wavelength step size is larger, GMMM can well maintain its measurement accuracy, and with lower noise; the generalized Poincaré sphere method, however, gives incorrect results due to its using differentiation.

Experimental setup and results
We point out that the PDL can also be obtained from measured Mueller matrices based on the following equation [12] The measured relationship between PDL and wavelength in FUT is plotted in Fig. 3.

Conclusion
We presented a generalized Mueller matrix method for PMD measurement in an optical fiber system with PDL/G based on the polar decomposition of a 4 4 × matrix which satisfies the Lorentz transformation. Compared to the generalized Poincaré sphere method, the GMMM allows a larger frequency step size in measurements, and its results appear to be accurate with high signal-to-noise ratio.