Abstract

Rotatable retarder fixed polarizer (RRFP) Stokes polarimeters are most commonly used to measure the state of polarization (SOP) of an electromagnetic (EM) wave. Most of commercialized RRFP Stokes polarimeters realize the SOP measurements by rotating a 90° retarder to N(N5)uniformly spaced angles over 360° and performing a discrete Fourier transform of data. In this paper, we address the noise properties of such uniformly-rotating RRFP Stokes polarimeters employing a retarder with an arbitrary retardance. The covariance matrices on the measurement noises of four Stokes parameters are derived for Gaussian noise and Poisson noise, respectively. Based on these covariance matrices, it can be concluded that 1) the measurement noises of Stokes parameters seriously depend on the retardance of the retarder in use. 2) for Gaussian noise dominated RRFP Stokes polarimeters, the retardance 130.48° leads to the minimum overall measurement noises when the sum of the measurement noises of four Stokes parameters (viz., the trace of the covariance matrix) is used as the criterion. The retardance in the range from 126.06° to 134.72° can have a nearly-minimum measurement noise which is only 1% larger than the minimum. On the other hand, the retardance 126.87° results in the equalized noises of the last three Stokes parameters. 3) for Poisson noise dominated RRFP Stokes polarimeters, the covariance matrix is also a fuction of the SOP of the incident EM wave. Even so, the retardance in the range from 126.06° to 134.72° can also result in nearly-minimum measurement noise for Poisson noise. 4) in the case of Poisson noise, N=5,10,12uniformly spaced angles over 360° have special covariance matrices that depend on the initial angle (the first angle in use). Finally, simulations are performed to verify these theoretical findings.

© 2013 OSA

1. Introduction

It is well known that the state of polarization (SOP) of an electromagnetic (EM) wave is completely depicted by a set of four Stokes parameters [1]. Stokes polarimeters are optoelectronic instruments to measure Stokes parameters of an EM wave under test; among them, complete Stokes polarimeters are those that can measure all of four Stokes parameters. Rotatable retarder fixed polarizer (RRFP) Stokes polarimeters are probably the simplest complete Stokes polarimeters, which are constructed with a rotatable retarder (also termed waveplate), a fixed polarizer and a photodetector. A schematic illustration of a RRFP Stokes polarimeter is shown in Fig. 1. Benefiting from such a simple and reliable structure, RRFP Stokes polarimeters have been widely used in many applications and can be easily integrated in polarization measurement systems,such as polarimetric optical time domain reflectometry (POTDR) systems or Terahertz frequency domain spectroscopy (THz-FDS) systems [2,3].

 

Fig. 1 The schematic structure of a RRFP Stokes polarimeter.

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RRFP Stokes polarimeters determine the four Stokes parameters of an incident EM wave by rotating the retarder to N(N4)angular orientations and measuring the corresponding optical intensities with the photodetector. In practice, two types of angular orientations of the retarder are mostly used in RRFP Stokes polarimeters. The first type of angular orientations is N(N5)angles uniformly spaced over 180° or 360° [4]. Most of commercialized RRFP Stokes polarimeters, Thorlabs PAX5720 for instance, adopt such a scheme by rotating a quarter-wave plate (90° retarder) at a constant angular velocity and performing a discrete Fourier transform of data [5]. The second type of angular orientations includes some sets of non-uniformly spaced angles that are chosen by optimizing various figures of merit, for example (−51.7°, −15.1°, 15.1°, 51.7°), (−45°, 0°, 30°, 60°) or (−59.6°, −36.8°, −10.1°, 10.1°, 36.8°, 59.6°) [6,7].

All photodetectors have intrinsic noises, which induce the random fluctuations in the intensity measurements. In practice, two kinds of noises are dominant in the commonly-used photodetectors, which are the signal-independent Gaussian thermal noise and the signal-dependent Poisson shot noise [8,9]. In RRFP Stokes polarimeters, these photodetector noises are important error sources and lead to the measurement noises of four Stokes parameters [6]. D. S. Sabatke et al (2000) and F. Goudail (2009) theoretically investigated the noise properties of RRFP Stokes polarimeters in the presence of Gaussian noise and Poisson noise, respectively [7,10]. They found, for the RRFP Stokes polarimeters employing the second type of angular orientations (four optimized angles), that 1) a 132° retarder and an angle set (−51.7°, −15.1°, 15.1°, 51.7°) are optimum to achieve the smallest measurement noises of Stokes parameters by using the equally weighted variance (EWV) as the figure of merit [7,10]; 2) for the above-mentioned polarimeter structure, a diagonal noise covariance matrix on Stokes parameters can be obtained and noise equalization on the last three Stokes parameters can be achieved for Gaussian noise [7]. These results are very useful in optimizing the RRFP Stokes polarimeters employing the second type of angular orientations, achieving noise equalization in Stokes parameter images and analyzing the Mueller matrix measurement systems employing such Stokes polarimeters.

It has been demonstrated that the RRFP Stokes polarimeters employing N(N5)uniformly spaced angles are more robust than those employing the second type of angular orientations in reducing the measurement errors induced by the imperfections of the retarder [11]. However, the noise properties of the RRFP Stokes polarimeters employing the first type of angular orientations (N5uniformly spaced angles over 180° or 360°) have not been investigated sufficiently to date. For Gaussian noise, one numerical result presented in [7] shows that uniformly spaced angles lead to a slightly larger EWV than four optimized angles (−51.7°, −15.1°, 15.1°, 51.7°) when 90° or 132° retardance is adopted. But there was no analytical result presented to show the relationship between the covariance matrices and the retardance. Hence, the retardances leading to the minimum EWV and noise equlization cannot be found out. Further, there is no result reported for Poisson noise at all, to the best of our knowledge.

In this paper, we address the noise properties of the RRFP Stokes polarimeters employing N(N5)uniformly spaced angles by theoretical analysis and simulations. The noise covariance matrices on Stokes parameters are derived analytically for both Gaussian noise and Poisson noise. These noise covariance matrices show that the measurement noises of Stokes parameters depend seriously on the retardance of the retarder. For Gaussian noise dominated systems, it can be concluded that 1) the retardance 130.48° can lead to the smallest EWV; 2) the retardance in the range from 126.06° to 134.72° can result in a nearly-minimum EWV by only 1% larger than the minimum; 3) the retardance 126.87° can result in the equalized measurement noises of the last three Stokes parameters. For Poisson noise dominated systems, it can be found that 1) the measurement noises of Stokes parameters depend on the SOP under test; 2) the retardance in the range from 126.06° to 134.72° can still result in nearly-minimum measurement noise for Poisson noise; 3) N=5,10,12uniformly spaced angles over 360° have slightly different covariance matrices that depend on the initial angle. To verify these theoretical findings, simulations are performed to show the good agreements between the simulation results and theoretical results.

2. Covariance matrix of the measurement noises of Stokes parameters

Four Stokes parameters are usually written in a form of Stokes vector S=(s0,s1,s2,s3)T, where the superscript “T” denotes the transposition of a vector or a matrix. RRFP Stokes polarimeters determine a Stokes vector S by performing N(N4) intensity measurements I=(I0,I1,,IN)Tcorresponding to N angular orientations of the retarder. In the absence of any errors, two vectors are related by [10]

I=WS
where W is the measurement matrix. When the retarder, with a retardance δ, rotates to N angles θi,i=1,2,,N, the measurement matrix is [7]
W=12(1cos22θ1+cosδsin22θ1sin2(δ/2)sin4θ1sinδsin2θ11cos22θ2+cosδsin22θ2sin2(δ/2)sin4θ2sinδsin2θ21cos22θN+cosδsin22θNsin2(δ/2)sin4θNsinδsin2θN)
When the photodetector noises are dominant, from of Eq. (1), the measurement error of the Stokes vector is governed by [7]
ΔS=W+ΔI
where ΔI is a vector representing the intensity measurement errors induced by the photodetector noises. The matrix W+ is the pseudo inverse matrix, which is defined by [10]
W+=(WTW)1WT
where the superscript “-1” denotes the inverse of a square matrix. Because the measurement error vector ΔS consists of four random variables, their variances are ususlly used to assess the measurement noises of Stokes parameters [7,10]. Based on Eq. (3), the covariance matrix on Stokes parameters can be derived as
Γ=(Δs02Δs0Δs1Δs0Δs2Δs0Δs3Δs0Δs1Δs12Δs1Δs2Δs1Δs3Δs0Δs2Δs1Δs2Δs22Δs2Δs3Δs0Δs3Δs1Δs3Δs2Δs3Δs32)=W+(ΔI12ΔI1ΔI2ΔI1ΔINΔI1ΔI2ΔI22ΔI2ΔINΔI1ΔINΔI2ΔINΔIN2)(W+)T
where the sign “” denotes the mean of a random variable. It is evident that the measurement noises of Stokes parameters Δsi2,i=0,1,2,3 depend on the measurement matrix W and the statistical properties of the photodetector noises.

In a photodetector, the intensity measurement error ΔI follows a stationary random process. For a well-calibrated photodetector, ΔI should have a zero mean ΔI=0. Hence, the measurement noise of a photodetector is depicted by the variance σ2=ΔI2. In a photodetector, Gaussian thermal noise and Poisson shot noise are two fundamental noise sources. Gaussian thermal noise is generated by the load resistor, which is described by Gaussian statistics. The variance of Gaussian noise σG2=ΔIG2 is independent on the intensity I [8,9]. Poisson shot noise is a manifestation of the fact that an electric current consists of a stream of electrons that are generated at random times. Mathematically, it is a stationary random process with Poisson statistics. Hence, the variance of Poisson shot noise σP2=ΔIP2=kI(kis constant for a given photodetector and explained in [8] and [9]) is proportional to the intensity I [8,9]. From the properties of Gaussian noise and Poisson noise, the fluctuations ΔIi,i=1,2,,N are statistically independent from one intensity measurement to the other [10]. Hence, it has ΔIiΔIj=0,ij. Therefore, from Eq. (5), it has

Γ=W+(ΔI12000ΔI22000ΔIN2)(W+)T
The noise properties of RRFP Stokes polarimeters can be completely derived from Eq. (6). Such works have been carried out using numercial methods for the RRFP Stokes polarimeters employing four optimized angles [7]. In the following, we will derive the covariance matrices for RRFP Stokes polarimeters employing N(N5)uniformly spaced angles for both Gaussian noise and Poisson noise.

3. Gaussian noise

When Gaussian noise dominates in a RRFP Stokes polarimeter, the variances of the intensity fluctuations are independent on the intensities. Then, it has [8,9]

ΔI12=ΔI22==ΔIN2=σG2
where σG2 denotes the variance of Gaussian noise. Hence, from Eq. (4), Eq. (6) and Eq. (7), it can be derived that
ΓG=σG2W+I(W+)T=σG2[(WTW)1WT]I[W(WTW)1]=σG2(WTW)1(WTW)(WTW)1=σG2(WTW)1
where I denotes a unit matrix. For the RRFP Stokes polarimeters employing Nuniformly spaced angles θ1,θ1+360/N,,θ1+(N1)360/N, it can be directly calculated that
(WTW)1=4NA
where the matrix Ais
A=(3+2cosδ+3cos2δ(1cosδ)24(1+cosδ)(1cosδ)2004(1+cosδ)(1cosδ)28(1cosδ)200008(1cosδ)200002sin2δ)
To derive Eq. (9) and Eq. (10), the following formulas have been used
{i=1Nsin2θi=i=1Nsin4θi=i=1Nsin32θi=i=1Nsin4θicos22θi=i=1Nsin4θisin22θi=i=1Nsin2θicos22θi=i=1Nsin2θisin4θi=0i=1Ncos22θi=i=1Nsin22θi=i=1Nsin24θi=N/2i=1Ncos42θi=i=1Nsin42θi=3N/8i=1Nsin22θicos22θi=N/8
The formulas in Eq. (11) are valid when N=5,7,9,10,and the initial angle θ1 is arbitrary. When N=6or8, some formulas in Eq. (11) are still valid. However, N=6or8 cannot be taken because they will lead to a singular matrix W [7]. Some formulas in Eq. (11) can be found in [12]; others can be easily verified using a simple computer program.

To obtain an overall evaluation of the measurement noises, the sum of four measurement noises of Stokes parameters i=03Δsi2 is used, which is equal to the trace of the covariance matrix ΓG. From Eq. (8) and Eq. (9), it is calculated as

i=03Δsi2=Tr[ΓG]=σG2Tr[(WTW)1]=EWVσG2
where “Tr” denotes the trace of a square matrix. The coeffieient EWV, defined in [7], is

EWV=4N3cos3δ+5cos2δ+19cosδ+21cos3δcos2δcosδ+1

The relationship between the EWV and the retardance δof the retarder, governed by Eq. (13), is illustrated in Fig. 2. It is obvious that the EWV reaches its minimum value 10.43 × (4/N) when the retardance is 130.48°. As a comparison, the EWV has a value of 21 × (4/N) when a commonly-used quarter-wave plate (the retardance is 90°) is used, which is 2 times larger than the minimum. Further, it can be easily observed from the inset of Fig. 2 that the EWV only increases by 1% from the minimum in the retardance range from 126.06° to 134.72°. It means the retarder in this retardance range can result in nearly minimum overall measurement noises by using the EWV as the criterion.

 

Fig. 2 The relationship between the EWV and the retardance when Gaussian noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

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On the other hand, the noise equalizationΔs12=Δs22=Δs32 is expected in some applications, such as Stokes parameter imaging [10,13,14]. From Eq. (9) and Eq. (10), it can be easily observed that Δs12=Δs22 is always valid for any retardance. Further, to find where Δs12=Δs32 is valid, two curves in Fig. 3 are plotted to show the relationships between Δs12, Δs32 and the retardance δ, respectively.

 

Fig. 3 The relationships between the meansurement noises of Stokes parameters Δsi2,i=0,1,3 and the retardance. The inset shows the “zoom in” views of the same data.

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It is obvious that when the retardance is 126.87°, the noise equalization Δs12=Δs22=Δs32 can be achieved. Actually, the retardance of 126.87° can lead to a special covariance matrix as

ΓG=σG22N(9500525000025000025)
Based on Eq. (14), a 126.87° retarder and N(N=5,7,9,10,)uniformly spaced angles over 360° can bring in equalized measurement noises of Stokes parameters; and simultaneously, such a measurement scheme can also have a nearly-minimum EWV. By the way, since (WTW)1has been obtained in Eq. (9), the Stokes vector can be calculated from S=W+I=(WTW)1(WTI). It can be easily demonstrated that the above calculation will result in the same Eqs as Eq. (8) in [4] while δ=90, which was derived using the discrete Fourier transform [4]. It means that the matrix-based algorithm used in this paper is equivalent to the discrete Fourier transform-based algorithm when N uniformly spaced angles over 180° or 360° are employed in RRFP Stokes polarimeters.

4. Poisson noise

When Poisson noise dominates, the variances of the intensity fluctuations are proportional to the intensities [810]. Then, it has

ΔIi2=kIi,i=1,2,,N
where the coefficient kis a constant for a given photodetector [8,9]. Further, from Eq. (1) and Eq. (2), we have
ΔIi2=k2{s0+(cos22θ2+cosδsin22θ2)s1+[sin2(δ/2)sin4θ2]s2sinδsin2θ2s3},i=1,2,,N
In the case of Poisson noise, the covariance matrix is
ΓP=W+ΓI(W+)T=(WTW)1(WTΓIW)(WTW)1
where the matrix ΓI=diag(ΔI12,ΔI22,,ΔI32). When N uniformly spaced angles θ1,θ1+360/N,,θ1+(N1)360/N are employed, WTΓIW can be derived as
WTΓIW=Nk8(Bs0+C)
where the matrix Band the matrix C are respectively given by:
B=(11+cosδ2001+cosδ23+2cosδ+3cos2δ80000sin4(δ/2)20000sin2δ2)
C=(1+cosδ2s13+2cosδ+3cos2δ8s1sin4(δ/2)2s2sin2δ2s33+2cosδ+3cos2δ8s15+3cosδ+3cos2δ+5cos3δ16s1sin4(δ/2)(1+cosδ)4s2sin2δ(1+3cosδ)8s3sin4(δ/2)2s2sin4(δ/2)(1+cosδ)4s2sin4(δ/2)(1+cosδ)4s10sin2δ2s3sin2δ(1+3cosδ)8s30sin2δ(1+3cosδ)8s1)
To derive Eq. (18), Eq. (19.1) and Eq. (19.2), the formulas in Eq. (11) and the following formulas have been used
{i=1Ncos62θi=i=1Nsin62θi=5N/16(20.1)i=1Ncos42θisin22θi=i=1Ncos22θisin42θi=N/16(20.2)i=1Ncos22θisin24θi=i=1Nsin22θisin24θi=N/4(20.3)i=1Ncos42θisin4θi=i=1Nsin42θisin4θi=i=1Ncos22θisin22θisin4θi=i=1Nsin34θi=0(20.4)i=1Ncos42θisin2θi=i=1Ncos22θisin32θi=i=1Nsin24θisin2θi=i=1Nsin52θi=i=1Ncos22θisin4θisin2θi=i=1Nsin22θisin4θisin2θi=0(20.5)
The formulas in Eq. (20) are valid for Nuniformly spaced angles θ1,θ1+360/N,,θ1+(N1)360/N. Here, N=7,9,11,13,14, and the initial angle θ1 can be arbitrary. However, when N=5or10, the formulas in Eq. (20.5) are invalid and the results are dependent on the initial angleθ1. When N=12, the formulas in Eq. (20.1) ~Eq. (20.4) are invalid and the results are dependent on the initial angle θ1. For the cases of N=5,10or12, the analytical results are presented in Appendix I and Appendix II, respectively.

Then, from Eq. (6), Eq. (17) and Eq. (18), the covariance matrix can be derived a

ΓP=2kN(As0+D)
where the matrix Dis
D=(1+7cosδ+7cos2δ+cos3δ2(1cosδ)2s11+6cosδ+cos2δ(1cosδ)2s1s22s31cosδ1+6cosδ+cos2δ(1cosδ)2s14(1+cosδ)(1cosδ)2s102s31cosδs204(1+cosδ)(1cosδ)2s102s31cosδ2s31cosδ01+3cosδ2sin2δs1)
Equations (21) and (22) mean that the covariance matrix for Poisson noise can be decomposed into two parts: the first part is proportional to the incident optical power s0, but independent on the incident SOP (s1,s2,s3)T; furthermore, the matrix A is the same as that shown in Eq. (10). The second part is the function of the incident SOP (s1,s2,s3)T, which make the measurement noises of Stokes parameters SOP-dependent.

Similarly, to evaluate the measurement noises using an overall factor, the sum of the four measurment noises, i=03Δsi2, can be calculated from Eq. (21) as

i=03Δsi2=Tr(ΓP)=2kN(C0s0+C1s1)
where the two coefficients are
C0=3cos3δ+5cos2δ+19cosδ+21cos3δcos2δcosδ+1,C1=cos4δ+8cos3δ+27cos2δ+42cosδ+182(cos3δcos2δcosδ+1)
Obviously, the coefficient C0 is equal to (N/4)EWV, which has been shown in Fig. 2. For comparison, both C0and C1are plotted in Fig. 4. It can be observed that when the retardance is 130.97°, the coefficient C1will be zero. This means the i=03Δsi2 will be free from the incident SOP when a 130.97° retarder is used. However, the coefficient C0reaches its minimum at a different retardance of 130.48° as shown in Fig. 2. Anyway, in the retardance range from 126.06° to 134.72°, C0 is far larger than |C1|, which can be clearly observed in the inset of Fig. 4. Further, because s0|s1| is always valid, i=03Δsi2 only slightly depends on the incident SOP in the retardance range from 126.06° to 134.72°. Therefore, for Poisson noise, the retardance in the above range can still lead to a nearly-minimum overall measurement noise.When the retardacne is 126.87°, it can be calculated that
ΓP=k4N(9s01.4s15s0+7s1s210s35s0+7s125s0+5s1010s3s2025s0+5s1010s310s3025s05s1)
Hence, for Poisson noise, Δs12,Δs22and Δs32 depend on the incident SOP. It is still valid that Δs12=Δs22, but Δs32 cannot be equal to Δs12,Δs22 unless s1=0. It means that the noise equalization can hardly be reached in the Poisson noise dominated measurements. Such a conclusion can also be drawn for the second type of angular orientations: a 132° retarder and four angles (−51.7°, −15.1°, 15.1°, 51.7°) can lead to a covarivance matrix as
ΓP=k2(s0s1s2s3s13s0s22s32s2s3s2s22s33s0+s12s1s32s2s32s13s0s1)
Hence, Δs12Δs22Δs32unless s1=0.

 

Fig. 4 The relationship between C0, C1 defined in Eq. (23) and the retardance when Poisson noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

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Last but not least, in the derivations of the Eqs in Section 3 and Section 4, Nuniformly spaced angles over 360°, viz. θ1,θ1+360/N,,θ1+(N1)360/N, have been used. From the measurement matrix shown in Eq. (2), it is easy to understand that the same results can be achieved for Nuniformly spaced angles over 180°, viz. θ1,θ1+180/N,,θ1+(N1)180/N [11]. In the case of 180°, N can be any integer larger than 4. In this case, for Poisson noise, when N=5, the covariacne matrix is described by Eq. (30) in Appendix I; when N=6, the covariance matrix is described by Eq. (36) in Appendix II; however, when N=10 or N=12, the covariacne matrix is still depicted by Eq. (21).

5. Simulation results

Simulations are performed to verify the theoretical findings in Section 3, Section 4, Appendix I and Appendix II. Simulations are carried out for 100 different incident SOPs, which uniformly span the whole Poincaré Sphere. Due to the page limit, only the results of the SOP (1,1/3,1/3,1/3)T are shown in this paper. In the first step, Gaussian noise is considered and the variance is taken as σG2=0.01. N=5uniformly-spaced angles over 180°, viz. (−90°, −54°, −18°, 18°, 54°) are used in the simulations. The simulations are performed in the retardance range from 40° to 170° with 1° step with the following procedures: 1) for a given retardance δ, calculate the measurement matrix W; 2) calculate the error-free intensity vector using I=WS; 3) generate 360000 Gaussian noise error vectors ΔIwith the above-given variance; 4) calculate the mesurement errors with ΔS=W1(I+ΔI)S for each noise realization; 5) calculate the variance of ΔS. Both the simulation results and the theoretical results calculated using Eq. (8) ~Eq. (10) are plotted in Fig. 5 for comparison. It is obvious that, for the measurement noises of all of four Stokes parameters, good agreements between the simulation results and the theoretical results are achieved.

 

Fig. 5 Simulation and theoretical results of Δsi2,i=0,1,2,3 for Gaussian noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

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In the second step, Poisson noise is taken into simulation and the coefficient kis taken as k=0.01. The simulation procedures are the same as the above except Poisson noises are applied here. Firstly, N=5uniformly-spaced angles over 180°, viz. (−90°, −54°, −18°, 18°, 54°) are still used in the simulation. Because N=5, the covariance matrix should be calculated from Eq. (30) that is derived in Appendix I. However, since θ1=90,then e1=sin(10θ1)=0,thus the theoretical results of the Δsi2,i=0,1,2,3 can still be calculated from Eq. (21). Both the simulation results and the theoretical results are shown in Fig. 6. Once again, good consistences between the simulation results and the theoretical results can be observed from all of four figures in Fig. 6.

 

Fig. 6 Simulation and theoretical results of Δsi2,i=0,1,2,3 for Poisson noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

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Secondly, N=5uniformly-spaced angles over 180° are chosen as (−81°, −45°, −9°, 27°, 63°). Because θ1=81, then e1=sin(10θ1)=1, hence the theoretical results of the Δsi2,i=0,1,2,3 cannot be obtained from Eq. (21), but Eq. (30) that is derived in Appendix I. The simulation results, together with two sets of theoretical results calculated from Eq. (21) and Eq. (30) respectively, are shown in Fig. 7. From the results in Fig. 7, we can conclude that 1) for N = 5 uniformly-spaced angles, the covariance matrix is determined by Eq. (30) when Possion noise dominates; 2) the difference between Eq. (30) and Eq. (21) is fairly small, especially when the retardance is getting larger.

 

Fig. 7 Simulation and theoretical results of Δsi2,i=0,1,2,3 for Poisson noise and the five uniformly spaced angles (−81°, −45°, −9°, 27°, 63°). The inset shows the “zoom in” views of the same data.

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Finally, N=12uniformly-spaced angles over 360° are chosen as (0°, 30°, 60°, …, 300°, 330°), the simulation results for Poisson noise, together with two sets of theoretical results calculated from Eq. (21) and Eq. (36) that is derived in Appendix II, respectively, are shown in Fig. 8. It is obvious that 1) for N=12uniformly-spaced angles, the covariance matrix is determined by Eq. (36) when Possion noise dominates; 2) the difference between Eq. (36) and Eq. (21) is fairly small, especially when the retardance is getting larger.

 

Fig. 8 Simulation and theoretical results of Δsi2,i=0,1,2,3 for Poisson noise and the twelve uniformly spaced angles (0°, 30°, 60°, …, 300°, 330°). The inset shows the “zoom in” views of the same data.

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7. Conclusion

In this paper, we investigated, by theoretical analysis and simulation validation, the measurement noises of Stokes parameters induced by photodetector noises in RRFP Stokes polarimeters employing N(N5)uniformly spaced angles over 180° or 360°. The noise covariance matrices on Stokes parameters are derived analytically for both Gaussian noise and Poisson noise dominated situations. These noise covariance matrices show that the measurement noises of Stokes parameters depend heavily on the retardance of the retarder. For Gaussian noise dominated systems, the retardance 130.48° can lead to the smallest EWV and the retardance 126.87° can result in the equalized measurement noises of Stokes parameters. For Poisson noise, the measurement noises of Stokes parameters are dependent on the incident SOP under test. If using the sum of the measurement noises of four Stokes parameters as the figure of merit, the retardance in the range from 126.06° to 134.72° can result in nearly-minimum measurement noise for both Gaussian and Poisson noise. In the case of Poisson noise, N=5,10,12uniformly spaced angles over 360° have slightly different covariance matrices that depend on the initial angle.

Appendix I: Covariance matrix when Poisson noise dominates and N=5or10

When Poisson noise dominates in the photodetector and N=5or10uniformly spaced angles θ1,θ1+360/N,,θ1+(N1)360/N are used, Eq. (20.5) becomes

{i=1Ncos42θisin2θi=i=1Nsin52θi=Nπsin(10θ1)/50i=1Ncos22θisin32θi=Nπsin(10θ1)/50i=1Ncos22θisin4θisin2θi=Nπcos(10θ1)/25i=1Nsin22θisin4θisin2θi=Nπcos(10θ1)/25i=1Nsin24θisin2θi=2Nπsin(10θ1)/25
These formulas can be verified using a simple computer program. Due to the existence of these non-zero terms, there will one more matrix being added to the matrix shown in Eq. (18), which is
WTΓIW=Nk8(Bs0+C+E)
where the matrix E is
E=π50sinδ(1cosδ)2(00000e1s3e2s3e2s2e1s10e2s3e1s3e2s1+e1s20e2s2e1s1e2s1+e1s20)
where e1=sin(10θ1)and e2=cos(10θ1). Then, Eq. (21) becomes
ΓP=2kN(As0+D+F)
where the matrix F is
F=16πsinδ25((1+cosδ)2e12(1cosδ)2s3(1+cosδ)e1(1cosδ)2s3(1+cosδ)e2(1cosδ)2s3e1s1e2s24(1cosδ)(1+cosδ)e1(1cosδ)2s32e1(1cosδ)2s32e2(1cosδ)2s3e2s2e1s12sin2δ(1+cosδ)e2(1cosδ)2s32e2(1cosδ)2s32e1(1cosδ)2s3e2s1+e1s22sin2δe1s1e2s24(1cosδ)e2s2e1s12sin2δe2s1+e1s22sin2δ0)
Therefore, when N=5or10, the measurement noises of the first three Stokes parameters Δs02,Δs12and Δs22depend on the initial angle θ1and s3; but Δs32 is not affected. When θ1=n×18,n=0,±1,±2,, sin(10θ1)=0; then Δs02,Δs12,Δs22and Δs32will be free from the matrix Feven if N=5or10. For example, θ1=0or±90can be used to avoid the influence of the matrix F.

Appendix II: Covariance matrix when Poisson noise dominates and N=12

When Poisson noise dominates in the photodetector and N=12uniformly spaced angles θ1,θ1+360/N,,θ1+(N1)360/N are used, Eq. (20.1) ~ Eq. (20.4) becomes

{i=1Ncos62θi=5N/16+3cos(12θ1)/8i=1Nsin62θi=5N/163cos(12θ1)/8i=1Ncos42θisin22θi=N/163cos(12θ1)/8i=1Ncos22θisin42θi=N/16+3cos(12θ1)/8i=1Ncos22θisin24θi=N/43cos(12θ1)/2i=1Nsin22θisin24θi=N/4+3cos(12θ1)/2i=1Ncos42θisin4θi=i=1Nsin42θisin4θi=3sin(12θ1)/4i=1Ncos22θisin22θisin4θi=3sin(12θ1)/4i=1Nsin34θi=3sin(12θ1)
Correspondingly, Eq. (18) is changed to
WTΓIW=Nk8(Bs0+C+G)
where the matrix Gis
G=(1cosδ)232(00000g1g200g2g100000)
where two factors in Eq. (AII.3) are
g1=cos(12θ1)s1+sin(12θ1)s2,g2=sin(12θ1)s1cos(12θ1)s2
In this case, the covariance matrix becomes
ΓP=2kN(As0+D+H)
where the matrix His
H=((1+cosδ)2g12(1cosδ)(1+cosδ)g11cosδ(1+cosδ)g21cosδ0(1+cosδ)g11cosδ2g11cosδ2g21cosδ0(1+cosδ)g21cosδ2g21cosδ2g11cosδ00000)
Therefore, when N=12, the measurement noises of the first three Stokes parameters Δs02,Δs12and Δs22 depend on the initial angle θ1and s2; but Δs32 is not affected. When tan(12θ1)=s1/s2is valid, g1=0; then Δs02,Δs12,Δs22and Δs32will be free from the matrix Heven when N=12 is taken. Different from the case of N=5or10, the initial angle θ1 cannot be determined without the knowledge of the incident SOP to avoid the influence of the matrix H.

Acknowledgments

This work is supported by A-STAR of Singapore and JST of Japan joint project with A-STAR/SERC/SICP grant No. 1021630069 and is also supported by the National Natural Science Foundation of China (Grant No. 61107087).

References and Links

1. D. Goldstein, Polarized Light (Marcel Dekker, Inc., 2003), Chap. 27.

2. H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys. 93, 7321–7324 (2007).

3. A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas. 53(1), 86–94 (2004). [CrossRef]  

4. L. Giudicotti and M. Brombin, “Data analysis for a rotating quarter-wave, far-infrared Stokes polarimeter,” Appl. Opt. 46(14), 2638–2648 (2007). [CrossRef]   [PubMed]  

5. Manual of Thorlabs PAX Series polarimeters, http://www.thorlabs.com/NewGroupPage9.cfm?ObjectGroup_ID=1564

6. A. Ambirajan and D. C. Look Jr., “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995). [CrossRef]  

7. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25(11), 802–804 (2000). [CrossRef]   [PubMed]  

8. G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, Inc., 2002), Chap. 4.

9. R. Q. Hui and M. O’Sullivan, Fiber Optics Measurement Techniques (Elsevier Academic, 2009), Chap. 1.

10. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. 34(5), 647–649 (2009). [CrossRef]   [PubMed]  

11. H. Dong, M. Tang, and Y. D. Gong, “Measurement errors induced by deformation of optical axes of achromatic waveplate retarders in RRFP Stokes polarimeters,” Opt. Express 20(24), 26649–26666 (2012). [CrossRef]   [PubMed]  

12. F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. 49(4), 683–693 (2010). [CrossRef]   [PubMed]  

13. H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express 17(15), 13017–13030 (2009). [CrossRef]   [PubMed]  

14. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25(16), 1198–1200 (2000). [CrossRef]   [PubMed]  

References

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  1. D. Goldstein, Polarized Light (Marcel Dekker, Inc., 2003), Chap. 27.
  2. H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).
  3. A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
    [CrossRef]
  4. L. Giudicotti and M. Brombin, “Data analysis for a rotating quarter-wave, far-infrared Stokes polarimeter,” Appl. Opt.46(14), 2638–2648 (2007).
    [CrossRef] [PubMed]
  5. Manual of Thorlabs PAX Series polarimeters, http://www.thorlabs.com/NewGroupPage9.cfm?ObjectGroup_ID=1564
  6. A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
    [CrossRef]
  7. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett.25(11), 802–804 (2000).
    [CrossRef] [PubMed]
  8. G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, Inc., 2002), Chap. 4.
  9. R. Q. Hui and M. O’Sullivan, Fiber Optics Measurement Techniques (Elsevier Academic, 2009), Chap. 1.
  10. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett.34(5), 647–649 (2009).
    [CrossRef] [PubMed]
  11. H. Dong, M. Tang, and Y. D. Gong, “Measurement errors induced by deformation of optical axes of achromatic waveplate retarders in RRFP Stokes polarimeters,” Opt. Express20(24), 26649–26666 (2012).
    [CrossRef] [PubMed]
  12. F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt.49(4), 683–693 (2010).
    [CrossRef] [PubMed]
  13. H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express17(15), 13017–13030 (2009).
    [CrossRef] [PubMed]
  14. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett.25(16), 1198–1200 (2000).
    [CrossRef] [PubMed]

2012 (1)

2010 (1)

2009 (2)

2007 (2)

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

L. Giudicotti and M. Brombin, “Data analysis for a rotating quarter-wave, far-infrared Stokes polarimeter,” Appl. Opt.46(14), 2638–2648 (2007).
[CrossRef] [PubMed]

2004 (1)

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

2000 (2)

1995 (1)

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Bénière, A.

Brombin, M.

Dereniak, E. L.

Descour, M. R.

Dong, H.

Galtarossa, A.

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

Giudicotti, L.

Gong, Y. D.

Goudail, F.

Ito, H.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Kemme, S. A.

Look, D. C.

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Nawahara, A.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Olivo, M.

Palmieri, L.

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

Paulose, V.

Phipps, G. S.

Sabatke, D. S.

Sato, T.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Shum, P.

Suizu, K.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Sweatt, W. C.

Tang, M.

Tyo, J. S.

Yamashita, T.

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Appl. Opt. (2)

IEEE Trans. Instrum. Meas. (1)

A. Galtarossa and L. Palmieri, “Reflectometric measurements of polarization properties in optical-fiber links,” IEEE Trans. Instrum. Meas.53(1), 86–94 (2004).
[CrossRef]

J. Appl. Phys. (1)

H. Ito, K. Suizu, T. Yamashita, A. Nawahara, and T. Sato, “Radom frequency accessible broad tunable terahertz-wave source using phase-matched 4-dimethylamino-N-methyl-4-stilbazolium Tosylate crystal,” J. Appl. Phys.93, 7321–7324 (2007).

Opt. Eng. (1)

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Other (4)

D. Goldstein, Polarized Light (Marcel Dekker, Inc., 2003), Chap. 27.

G. P. Agrawal, Fiber-Optic Communication Systems (John Wiley & Sons, Inc., 2002), Chap. 4.

R. Q. Hui and M. O’Sullivan, Fiber Optics Measurement Techniques (Elsevier Academic, 2009), Chap. 1.

Manual of Thorlabs PAX Series polarimeters, http://www.thorlabs.com/NewGroupPage9.cfm?ObjectGroup_ID=1564

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Figures (8)

Fig. 1
Fig. 1

The schematic structure of a RRFP Stokes polarimeter.

Fig. 2
Fig. 2

The relationship between the EWV and the retardance when Gaussian noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

Fig. 3
Fig. 3

The relationships between the meansurement noises of Stokes parameters Δ s i 2 ,i=0,1,3 and the retardance. The inset shows the “zoom in” views of the same data.

Fig. 4
Fig. 4

The relationship between C0, C1 defined in Eq. (23) and the retardance when Poisson noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

Fig. 5
Fig. 5

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Gaussian noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

Fig. 6
Fig. 6

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Poisson noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

Fig. 7
Fig. 7

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Poisson noise and the five uniformly spaced angles (−81°, −45°, −9°, 27°, 63°). The inset shows the “zoom in” views of the same data.

Fig. 8
Fig. 8

Simulation and theoretical results of Δ s i 2 ,i=0,1,2,3 for Poisson noise and the twelve uniformly spaced angles (0°, 30°, 60°, …, 300°, 330°). The inset shows the “zoom in” views of the same data.

Equations (38)

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I =W S
W= 1 2 ( 1 cos 2 2 θ 1 +cosδ sin 2 2 θ 1 sin 2 ( δ/2 )sin4 θ 1 sinδsin2 θ 1 1 cos 2 2 θ 2 +cosδ sin 2 2 θ 2 sin 2 ( δ/2 )sin4 θ 2 sinδsin2 θ 2 1 cos 2 2 θ N +cosδ sin 2 2 θ N sin 2 ( δ/2 )sin4 θ N sinδsin2 θ N )
Δ S = W + Δ I
W + = ( W T W ) 1 W T
Γ=( Δ s 0 2 Δ s 0 Δ s 1 Δ s 0 Δ s 2 Δ s 0 Δ s 3 Δ s 0 Δ s 1 Δ s 1 2 Δ s 1 Δ s 2 Δ s 1 Δ s 3 Δ s 0 Δ s 2 Δ s 1 Δ s 2 Δ s 2 2 Δ s 2 Δ s 3 Δ s 0 Δ s 3 Δ s 1 Δ s 3 Δ s 2 Δ s 3 Δ s 3 2 )= W + ( Δ I 1 2 Δ I 1 Δ I 2 Δ I 1 Δ I N Δ I 1 Δ I 2 Δ I 2 2 Δ I 2 Δ I N Δ I 1 Δ I N Δ I 2 Δ I N Δ I N 2 ) ( W + ) T
Γ= W + ( Δ I 1 2 00 0 Δ I 2 2 0 00 Δ I N 2 ) ( W + ) T
Δ I 1 2 = Δ I 2 2 == Δ I N 2 = σ G 2
Γ G = σ G 2 W + I ( W + ) T = σ G 2 [ ( W T W ) 1 W T ]I[ W ( W T W ) 1 ] = σ G 2 ( W T W ) 1 ( W T W ) ( W T W ) 1 = σ G 2 ( W T W ) 1
( W T W ) 1 = 4 N A
A=( 3+2cosδ+3 cos 2 δ ( 1cosδ ) 2 4( 1+cosδ ) ( 1cosδ ) 2 00 4( 1+cosδ ) ( 1cosδ ) 2 8 ( 1cosδ ) 2 00 00 8 ( 1cosδ ) 2 0 000 2 sin 2 δ )
{ i=1 N sin2 θ i = i=1 N sin4 θ i = i=1 N sin 3 2 θ i = i=1 N sin4 θ i cos 2 2 θ i = i=1 N sin4 θ i sin 2 2 θ i = i=1 N sin2 θ i cos 2 2 θ i = i=1 N sin2 θ i sin4 θ i = 0 i=1 N cos 2 2 θ i = i=1 N sin 2 2 θ i = i=1 N sin 2 4 θ i =N/2 i=1 N cos 4 2 θ i = i=1 N sin 4 2 θ i =3N/8 i=1 N sin 2 2 θ i cos 2 2 θ i =N/8
i=0 3 Δ s i 2 =Tr[ Γ G ]= σ G 2 Tr[ ( W T W ) 1 ]=EWV σ G 2
EWV= 4 N 3 cos 3 δ+5 cos 2 δ+19cosδ+21 cos 3 δ cos 2 δcosδ+1
Γ G = σ G 2 2N ( 9500 52500 00250 00025 )
Δ I i 2 =k I i ,i=1,2,,N
Δ I i 2 = k 2 { s 0 +( cos 2 2 θ 2 +cosδ sin 2 2 θ 2 ) s 1 +[ sin 2 ( δ/2 )sin4 θ 2 ] s 2 sinδsin2 θ 2 s 3 },i=1,2,,N
Γ P = W + Γ I ( W + ) T = ( W T W ) 1 ( W T Γ I W ) ( W T W ) 1
W T Γ I W= Nk 8 ( B s 0 +C )
B=( 1 1+cosδ 2 00 1+cosδ 2 3+2cosδ+3 cos 2 δ 8 00 00 sin 4 ( δ/2 ) 2 0 000 sin 2 δ 2 )
C=( 1+cosδ 2 s 1 3+2cosδ+3 cos 2 δ 8 s 1 sin 4 ( δ/2 ) 2 s 2 sin 2 δ 2 s 3 3+2cosδ+3 cos 2 δ 8 s 1 5+3cosδ+3 cos 2 δ+5 cos 3 δ 16 s 1 sin 4 ( δ/2 )( 1+cosδ ) 4 s 2 sin 2 δ( 1+3cosδ ) 8 s 3 sin 4 ( δ/2 ) 2 s 2 sin 4 ( δ/2 )( 1+cosδ ) 4 s 2 sin 4 ( δ/2 )( 1+cosδ ) 4 s 1 0 sin 2 δ 2 s 3 sin 2 δ( 1+3cosδ ) 8 s 3 0 sin 2 δ( 1+3cosδ ) 8 s 1 )
{ i=1 N cos 6 2 θ i = i=1 N sin 6 2 θ i =5N/16( 20.1 ) i=1 N cos 4 2 θ i sin 2 2 θ i = i=1 N cos 2 2 θ i sin 4 2 θ i =N/16(20.2) i=1 N cos 2 2 θ i sin 2 4 θ i = i=1 N sin 2 2 θ i sin 2 4 θ i =N/4(20.3) i=1 N cos 4 2 θ i sin4 θ i = i=1 N sin 4 2 θ i sin4 θ i = i=1 N cos 2 2 θ i sin 2 2 θ i sin4 θ i = i=1 N sin 3 4 θ i =0(20.4) i=1 N cos 4 2 θ i sin2 θ i = i=1 N cos 2 2 θ i sin 3 2 θ i = i=1 N sin 2 4 θ i sin2 θ i = i=1 N sin 5 2 θ i = i=1 N cos 2 2 θ i sin4 θ i sin2 θ i = i=1 N sin 2 2 θ i sin4 θ i sin2 θ i =0( 20.5 )
Γ P = 2k N ( A s 0 +D )
D=( 1+7cosδ+7 cos 2 δ+ cos 3 δ 2 ( 1cosδ ) 2 s 1 1+6cosδ+ cos 2 δ ( 1cosδ ) 2 s 1 s 2 2 s 3 1cosδ 1+6cosδ+ cos 2 δ ( 1cosδ ) 2 s 1 4( 1+cosδ ) ( 1cosδ ) 2 s 1 0 2 s 3 1cosδ s 2 0 4( 1+cosδ ) ( 1cosδ ) 2 s 1 0 2 s 3 1cosδ 2 s 3 1cosδ 0 1+3cosδ 2 sin 2 δ s 1 )
i=0 3 Δ s i 2 =Tr( Γ P )= 2k N ( C 0 s 0 + C 1 s 1 )
C 0 = 3 cos 3 δ+5 cos 2 δ+19cosδ+21 cos 3 δ cos 2 δcosδ+1 , C 1 = cos 4 δ+8 cos 3 δ+27 cos 2 δ+42cosδ+18 2( cos 3 δ cos 2 δcosδ+1 )
Γ P = k 4N ( 9 s 0 1.4 s 1 5 s 0 +7 s 1 s 2 10 s 3 5 s 0 +7 s 1 25 s 0 +5 s 1 010 s 3 s 2 025 s 0 +5 s 1 0 10 s 3 10 s 3 025 s 0 5 s 1 )
Γ P = k 2 ( s 0 s 1 s 2 s 3 s 1 3 s 0 s 2 2 s 3 2 s 2 s 3 s 2 s 2 2 s 3 3 s 0 + s 1 2 s 1 s 3 2 s 2 s 3 2 s 1 3 s 0 s 1 )
{ i=1 N cos 4 2 θ i sin2 θ i = i=1 N sin 5 2 θ i = Nπsin( 10 θ 1 )/50 i=1 N cos 2 2 θ i sin 3 2 θ i =Nπsin( 10 θ 1 )/50 i=1 N cos 2 2 θ i sin4 θ i sin2 θ i =Nπcos( 10 θ 1 )/25 i=1 N sin 2 2 θ i sin4 θ i sin2 θ i =Nπcos( 10 θ 1 )/25 i=1 N sin 2 4 θ i sin2 θ i =2Nπsin( 10 θ 1 )/25
W T Γ I W= Nk 8 ( B s 0 +C+E )
E= π 50 sinδ ( 1cosδ ) 2 ( 0000 0 e 1 s 3 e 2 s 3 e 2 s 2 e 1 s 1 0 e 2 s 3 e 1 s 3 e 2 s 1 + e 1 s 2 0 e 2 s 2 e 1 s 1 e 2 s 1 + e 1 s 2 0 )
Γ P = 2k N ( A s 0 +D+F )
F= 16πsinδ 25 ( ( 1+cosδ ) 2 e 1 2 ( 1cosδ ) 2 s 3 ( 1+cosδ ) e 1 ( 1cosδ ) 2 s 3 ( 1+cosδ ) e 2 ( 1cosδ ) 2 s 3 e 1 s 1 e 2 s 2 4( 1cosδ ) ( 1+cosδ ) e 1 ( 1cosδ ) 2 s 3 2 e 1 ( 1cosδ ) 2 s 3 2 e 2 ( 1cosδ ) 2 s 3 e 2 s 2 e 1 s 1 2 sin 2 δ ( 1+cosδ ) e 2 ( 1cosδ ) 2 s 3 2 e 2 ( 1cosδ ) 2 s 3 2 e 1 ( 1cosδ ) 2 s 3 e 2 s 1 + e 1 s 2 2 sin 2 δ e 1 s 1 e 2 s 2 4( 1cosδ ) e 2 s 2 e 1 s 1 2 sin 2 δ e 2 s 1 + e 1 s 2 2 sin 2 δ 0 )
{ i=1 N cos 6 2 θ i =5N/16+3cos( 12 θ 1 )/8 i=1 N sin 6 2 θ i =5N/163cos( 12 θ 1 )/8 i=1 N cos 4 2 θ i sin 2 2 θ i =N/163cos( 12 θ 1 )/8 i=1 N cos 2 2 θ i sin 4 2 θ i =N/16+3cos( 12 θ 1 )/8 i=1 N cos 2 2 θ i sin 2 4 θ i =N/43cos( 12 θ 1 )/2 i=1 N sin 2 2 θ i sin 2 4 θ i =N/4+3cos( 12 θ 1 )/2 i=1 N cos 4 2 θ i sin4 θ i = i=1 N sin 4 2 θ i sin4 θ i =3sin( 12 θ 1 )/4 i=1 N cos 2 2 θ i sin 2 2 θ i sin4 θ i =3sin( 12 θ 1 )/4 i=1 N sin 3 4 θ i =3sin( 12 θ 1 )
W T Γ I W= Nk 8 ( B s 0 +C+G )
G= ( 1cosδ ) 2 32 ( 0000 0 g 1 g 2 0 0 g 2 g 1 0 0000 )
g 1 =cos( 12 θ 1 ) s 1 +sin( 12 θ 1 ) s 2 , g 2 =sin( 12 θ 1 ) s 1 cos( 12 θ 1 ) s 2
Γ P = 2k N ( A s 0 +D+H )
H=( ( 1+cosδ ) 2 g 1 2( 1cosδ ) ( 1+cosδ ) g 1 1cosδ ( 1+cosδ ) g 2 1cosδ 0 ( 1+cosδ ) g 1 1cosδ 2 g 1 1cosδ 2 g 2 1cosδ 0 ( 1+cosδ ) g 2 1cosδ 2 g 2 1cosδ 2 g 1 1cosδ 0 0000 )

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