We present the theoretical and simulation results of the relationship between three input states of polarization (SOP) and the Mueller matrix measurement error in an optical system having birefringence and finite polarization-dependent loss or gain (PDL/G). By using the condition number as the criterion, it can be theoretically demonstrated that the three input SOPs should be equally-spaced on the Poincaré sphere and centered on the reversed PDL/G vector to achieve better measurement accuracy in a single test. Further, an upper bound of the mean of the Mueller matrix measurement error is derived when the measurement errors of output Stokes parameters independently and identically follow the ideal Gaussian distribution. This upper bound also shows that the statistically best Mueller matrix measurement accuracy can be obtained when the three input SOPs have the same relationship mentioned above. Simulation results confirm the validity of the theoretical findings.
© 2009 OSA
For an optical system having birefringence and small polarization-dependent loss or gain (PDL/G), we have demonstrated that, by theoretical analysis and simulations, the Mueller matrix measurement error (M3E) seriously depends on the choice of the three input states of polarization (SOP) . When we use the preconditions that 1) the measurement errors of output SOPs are far larger than those of input SOPs and 2) SOP measurement errors independently and identically follow the Gaussian distribution , it can be demonstrated that the minimum M3E is statistically achieved when the three input SOPs are coplanar with an angle of 120° between any two of them in Stokes space . Three standard input SOPs , and has been suggested as they obviously have a larger probability to result in a smaller M3E than other input SOPs .
However, the above conclusion is valid only when the PDL/G is less than 0.35dB. Some optical systems may have a large PDL/G value from several dB to tens of dB, for example, a long-distance optical fiber communication system . In such a system, PDL/G has adverse effect on both analog and digital optical signals [3, 4]. Combined effect of PDL/G and polarization mode dispersion (PMD) gives rise to anomalous pulse broadening and deteriorates the bit error rate [2, 5]. To monitor the PDL/G and PMD values in such a system, its Mueller matrix is required to be accurately measured [6, 7].
When the PDL/G has a finite value, such as 5 or 10 dB, the optimum input SOPs will definitely rely on the PDL/G of the system under test, including both the modulus and the direction of PDL/G vector. In this paper, we will present the detailed theoretical and simulation results of this problem. Firstly, in Section 2, the condition number (CN) of the matrix (has been defined in Ref .) is used to evaluate M3E. By calculating the minimum of this CN, the relationship among the three optimum input SOPs, which can lead to a smaller M3E in a single test, is clearly presented. Secondly, in Sections 3 and 4, the statistical relationship between M3E and the three input SOPs is investigated under the same two preconditions mentioned above. Finally, some simulation results are used to verify the theoretical findings.
2. Optimization using CN as the criterionEq. (1) have been defined in Ref . When , M3E, which is depicted by the matrix norm , is bounded by8].
When the Frobenius matrix norm is adopted, the detailed expression of the CN is calculated asEqs. (3) and (4), it can be observed that the value of the CN will remain unchanged when any two 4D output Stokes vectors are interchanged. Therefore, the CN in Eq. (3) is a symmetric function of three 4D output Stokes vectors. According to the Purkiss Principle , the CN must have a local maximum or minimum when
Due to , considering the input lights are partially polarized, we can obtain the generalized form of Eq. (7) in Re . Then, an equation group, relating the input and output powers, can be derived as1]. Another equation group, relating the input and output DOPs, has already been obtained as 7], and also considering the input lights are partially polarized, we can also obtain the generalized form of Eq. (8) in Re . Then, the third equation group can be written asEqs. (6), (7) and (8), when the above-mentioned local minimum is achieved, it can be seen that 1) if and are satisfied, we have and ; 2) when the input powers or DOPs of three inputs are different, the relative relationship among the three input SOPs and between the three inputs and the PDL/G vector will become complicated.
Fortunately, most of the light sources used in modern polarization measurement systems, such as the tunable laser source in a polarization-mode dispersion measurement system  and the photoconductive switch emitter in a terahertz time-domain spectroscopy system , are completely polarized. Moreover, the input power can remain unchanged by using some well-designed polarization state generation approaches [7, 13]. Therefore, in the rest of this paper, we consider that and are always satisfied for both theoretical analysis and simulations. Further, since all input and output Stokes vectors can be normalized by the input power, can be adopted without any influence to the results.9], the CN in Eq. (9) has a local maximum or minimum when and . Consequently, θ and α are actually related, no matter they are optimized or not, byEq. (11) into Eq. (9) and doing numerical calculations with different values of PDL/G, we can find that 1) this is a local minimum and it is achieved when the minus sign is chosen in Eq. (11); 2) this local minimum is indeed the global minimum. Under this condition, the relationships between the optimum angles , and the PDL/G (in dB) are calculated and plotted in Fig. 1 .
It is evident that is close to 120° when PDL/G is small. This is consistent with the conclusion we have obtained in Ref . When PDL/G increases, the optimum angle decreases. For a given PDL/G vector, the three optimum input Stokes vectors should be equally-spaced on the Poincaré sphere and centered on the reversed PDL/G vector as shown in Fig. 2 .
Figure 3 shows that, for different values of PDL/G, the minimum of the CN is also different. It is obvious that M3E will dramatically increase when the value of PDL/G is up to tens of dB. Therefore, when the system under test has such a big PDL/G, its Mueller matrix cannot be accurately measured by using only three inputs in a single test.
Since the CN in Eq. (9) is a function of six angles ,, it is impossible to illustrate the whole function. In Fig. 4 , we only present a curve to partially show this function, where (5 dB), and. Please note, in Fig. 4(a), α is related to θ by Eq. (11) with the minus sign.
In this section, we use the CN as the criterion to find out the appropriate input SOPs. The results show that the minimum CN is achieved when the three input SOPs are equally-spaced on the Poincaré sphere and centered on the reversed PDL/G vector. The larger the PDL/G, the closer the three input SOPs is to the reversed PDL/G vector.
From Eq. (2), M3E depends on not only the CN, but also on the noise realization. When many tests can be performed, the mean of M3E should be investigated. To carry out such an investigation, we must know the statistical properties of the Stokes parameter measurement errors in advance. In this paper, we assume that all Stokes parameter measurement errors independently and identically follow the Gaussian distribution , which has been used in Ref . This means that we assume that an ideal polarimeter, which has such statistical properties, is used in the measurement system.
3. Statistical properties of
In the following, we still consider that the measurement errors of the input SOPs can be neglected as we have done in Ref . Under the conditions , we know that . Hence, the variance of is needed to evaluate the measurement uncertainty. It can be derived as
Obviously, is also a symmetric function of the three input SOPs. Based on the Purkiss Principle and numerical calculations, its global minimum is also achieved when , . When takes the global minimum, the relationships between the optimum angles and the PDL/G is shown in Fig. 5 using solid lines. As a comparison, the optimum angles, corresponding to the CN, are also plotted in Fig. 5 using dashed lines. It is clear that they are different corresponding to the same value of PDL/G.
4. Upper bound ofEq. (14). To overcome the mathematical difficulty, we can start from the following equation1] and , it can be derived thatEq.(18),and.
Unfortunately, we cannot directly calculate because of the difficulties in mathematics. As an alternative, we can calculate an upper bound asEq. (20) are calculated asEqs. (21) and (22), are the diagonal elements of the matrix F, which areEq. (27), the relationship between the optimum angles and the value of PDL/G, when takes the global minimum, can be shown in Fig. 6 using solid lines. As a comparison, the optimum angles, corresponding to the CN and, are also plotted in Fig. 6 using dashed lines. It is clear that, although the value of PDL/G is the same, different criteria lead to different optimum angles.
For different values of PDL/G, the minimum of is also different. As shown in Fig. 7 , M3E will dramatically increase when the value of PDL/G is up to tens of dB. Therefore, when the system under test has such a big PDL/G, the mean of the Mueller matrix also cannot be accurately measured.
5. Simulation results
To verify the theoretical finding in Section 4, simulations are performed. The parameters of the system under simulation are 1) Birefringence:, ; 2) PDL/G: , and 3) PIDL/G: . In Section 4, the theoretical result shows that the upper bound does not depend on the PIDL/G . Then, we take in the following simulations. In this paper, we only show the simulation results when and . Then, the three input SOPs can be written asFig. 9 , the theoretical upper bound (in red) and the simulation results (in blue) of are plotted with three values of PDL/G: 5, 10 and 13 dB. Please note, in Fig. 9(a), α is related to θ by Eq. (11) with the minus sign. It is evident that the simulation results have the same profiles as the theoretical upper bounds. And the minimum M3Es are achieved with the angles determined by the solid lines in Fig. 6.
We presented the relationship between three input SOPs and M3E in a system having birefringence and a finite PDL/G. Firstly, by using the CN as the criterion, it has been demonstrated that the three optimum input SOPs should be equally-spaced and centred on the reversed PDL/G vector for achieving a smaller M3E in a single test. Secondly, the statistical relationship, which is expressed as an upper bound of the mean of M3E, has been derived when SOP measurement errors follow the same Gaussian distribution. This upper bound also tells us that the minimum M3E will be statistically achieved when the three input SOPs are equally-spaced and centred on the reversed PDL/G vector. Finally, the simulation results confirm the validity of the proposed conclusion.
This conclusion can be used in real measurements. If the PDL/G vector of the system under test can be known before the measurement, the three input SOPs, with the relative relationship shown in Section 2, should be adopted in a single test. If the PDL/G vector is unknown, the polarimeter has the statistical properties we have used in this paper and the measurement can be repeated for many times , the measurement can be performed using the following steps:
- 1) The PDL/G vector can be measured in the first test using three “not-too-bad” input SOPs, for example, the three inputs we suggested in Ref ;
- 2) The second test is carried out using three input SOPs optimized using the relative relationship shown in Section 4 and the PDL/G vector measured in the first test. Then a more accurate PDL/G vector can be obtained;
- 3) The third test is carried out based on the knowledge of the more accurate PDL/G to result in a further more accurate PDL/G vector;
- 4) The measurement is repeated in this iterative way. Then the final averaged measurement result will be statistically the best.
Moreover, if the measurement errors of input SOPs cannot be neglected, the three statistically optimum input SOPs will depend on not only the PDL/G vector, but also on the birefringence. A detailed analysis will be presented in another paper.
On the other hand, if the polarimeter used in the measurements does not have the statistical properties we adopted in this paper, the statistical relationship between M3E and the three input SOPs will not be as same as the one we obtained in this paper. It will be polarimeter-dependent. Further analysis will be presented in other papers.
This work is supported by Singapore A-star, Singapore Bioimaging Consortium, SBIC Grant Ref: SBIC RP C-014/2007.
References and Links
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