## Abstract

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.

## 1. Introduction

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) [1]. 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 $N(0,σ2)$, 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 [1]. Three standard input SOPs $(1, 1, 0, 0)T$, $(1, −0.5, 0.866, 0)T$ and $(1, −0.5, −0.866, 0)T$has been suggested as they obviously have a larger probability to result in a smaller M3E than other input SOPs [1].

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 [2]. 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 $Fout$($Fout$has been defined in Ref [1].) 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 criterion

In this section, the measurement errors of both input and output SOPs will be considered. From Eq. (10) of Ref [1], we have

$ΔM˜=Fout−1(ΔFin−ΔFoutM˜−ΔFoutΔM˜)$
All variables in Eq. (1) have been defined in Ref [1]. When $‖Fout−1‖⋅‖ΔFout‖<1$, M3E, which is depicted by the matrix norm $‖ΔM˜‖$, is bounded by
$‖ΔM˜‖‖M˜‖≤Cond(Fout)1−Cond(Fout)‖ΔFout‖‖Fout‖(‖ΔFin‖‖Fin‖+‖ΔFout‖‖Fout‖)$
where, $Cond(Fout)=‖Fout‖⋅‖Fout−1‖$ is the CN. It is obvious that this upper bound seriously depends on the value of CN. The smaller CN is, the more possible$‖ΔM˜‖/‖M˜‖$ is to be smaller in a single test, regardless of the actual noise realizations of $‖ΔFin‖/‖Fin‖$ and $‖ΔFout‖/‖Fout‖$. Actually, the CN has been widely used to evaluate the measurement uncertainty of a measurement system [8].

When the Frobenius matrix norm is adopted, the detailed expression of the CN is calculated as

$Cond(Fout)=(1+ρouts2)sout02+(1+ρoutt2)tout02+(1+ρoutu2)uout02+sout02tout02uout02(Bout12+Bout22)|M˜|×{sout02tout02[(1+ρouts2)(1+ρoutt2)−(1+ρoutsρouttcosαout)2]+sout02uout02[(1+ρouts2)(1+ρoutu2)−(1+ρoutsρoutucosβout)2]+tout02uout02[(1+ρoutt2)(1+ρoutu2)−(1+ρouttρoutucosγout)2]}(Bout12+Bout22)−4[sout02tout02aout2+sout02uout02bout2+tout02uout02cout2]Bout12+|M˜|[(1+ρouts2)(1+ρoutt2)(1+ρoutu2)−(1+ρoutu2)(1+ρoutsρouttcosαout)2−(1+ρoutt2)(1+ρoutsρoutucosβout)2−(1+ρouts2)(1+ρouttρoutucosγout)2+2(1+ρoutsρouttcosαout)(1+ρoutsρoutucosβout)(1+ρouttρoutucosγout)]sout0tout0uout0|Bout12−Bout22|$
where
${Bout12=ρouts2ρoutt2ρoutu2(1−cos2αout−cos2βout−cos2γout+2cosαoutcosβoutcosγout)Bout22=(aout+bout+cout)(−aout+bout+cout)(aout−bout+cout)(aout+bout−cout)/4aout=ρouts2+ρoutt2−2ρoutsρouttcosαoutbout=ρouts2+ρoutu2−2ρoutsρoutucosβoutcout=ρoutt2+ρoutu2−2ρouttρoutucosγout$
It can be easily noticed that the CN is completely determined by three 4-dimensional (4D) output Stokes vectors, including their powers$(sout0, tout0, uout0)$, their degrees of polarization (DOP) $(ρouts, ρoutt, ρoutu)$ and the three angles $(αout, βout, γout)$between any two of them in Stokes space. From Eqs. (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 [9], the CN must have a local maximum or minimum when
${sout0=tout0=uout0ρouts=ρoutt=ρoutuαout=βout=γout$
In fact, it is easy to verify that this is a local minimum. In this paper, we do not demonstrate whether this local minimum is the global minimum or not; we are only interested in the relationship among the three 4D input Stokes vectors when this local minimum is achieved.

Due to $S→out=M˜S→in$, considering the input lights are partially polarized, we can obtain the generalized form of Eq. (7) in Re [1]. Then, an equation group, relating the input and output powers, can be derived as

${sout0=Tusin0(1+ρinsDcosθs)tout0=Tutin0(1+ρintDcosθt)uout0=Tuuin0(1+ρinuDcosθu)$
where Dis the value of PDL/G ; $θs$, $θt$ and $θu$ are the angles between the PDL/G vector and input SOPs $S→in$, $T→in$ and $U→in$ in Stokes space, respectively [1]. Another equation group, relating the input and output DOPs, has already been obtained as [10]
${ρouts=1−(1−D2)(1−ρins2)/(1+ρinsDcosθs)2ρoutt=1−(1−D2)(1−ρint2)/(1+ρintDcosθt)2ρoutu=1−(1−D2)(1−ρinu2)/(1+ρinuDcosθu)2$
From $S→out⋅T→out=|M˜|S→in⋅T→in$ [7], and also considering the input lights are partially polarized, we can also obtain the generalized form of Eq. (8) in Re [1]. Then, the third equation group can be written as
${ρoutsρouttcosαout=1−(1−D2)(1−ρinsρintcosαin)(1+ρinsDcosθs)(1+ρintDcosθt)ρoutsρoutucosβout=1−(1−D2)(1−ρinsρinucosβin)(1+ρinsDcosθs)(1+ρinuDcosθu)ρouttρoutucosγout=1−(1−D2)(1−ρintρinucosγin)(1+ρintDcosθt)(1+ρinuDcosθu)$
From Eqs. (6), (7) and (8), when the above-mentioned local minimum is achieved, it can be seen that 1) if $sin0=tin0=uin0$ and $ρins=ρint=ρinu$are satisfied, we have$θs=θt=θu$ and $αin=βin=γin$ ; 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 [11] and the photoconductive switch emitter in a terahertz time-domain spectroscopy system [12], 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 $sin0=tin0=uin0$ and $ρins=ρint=ρinu=1$ are always satisfied for both theoretical analysis and simulations. Further, since all input and output Stokes vectors can be normalized by the input power, $sin0=tin0=uin0=1$ can be adopted without any influence to the results.

Under the above conditions, Eqs. (3) and (4) can be simplified as

$Cond(Fout)=2(Ds2+Dt2+Du2)+Ds2Dt2Du2(Bout12+Bout22)/(1−D2)2×{Ds2Dt2[4−(1+cosαout)2]+Ds2Du2[4−(1+cosβout)2]+Dt2Du2[4−(1+cosγout)2]}(Bout12+Bout22)−8[Ds2Dt2(1−cosαout)+Ds2Du2(1−cosβout)+Dt2Du2(1−cosγout)]Bout12+2(1−D2)2[4+(1+cosαout)(1+cosβout)(1+cosγout)−(1+cosαout)2−(1+cosβout)2−(1+cosγout)2]DsDtDu|Bout12−Bout22|$
and
${Bout12=1−cos2αout−cos2βout−cos2γout+2cosαoutcosβoutcosγoutBout22=4(1−cosβout)(1−cosγout)−(1+cosαout−cosβout−cosγout)2$
where $Ds=1+Dcosθs$, $Dt=1+Dcosθt$and $Du=1+Dcosθu$. According to the Purkiss Principle [9], the CN in Eq. (9) has a local maximum or minimum when $θs=θt=θu=θ$ and $αin=βin=γin=α$. Consequently, θ and α are actually related, no matter they are optimized or not, by
$cosθ=±1+2cosα3$
By substituting Eq. (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 $αopt$, $θopt$ and the PDL/G (in dB) are calculated and plotted in Fig. 1 .

Fig. 1 The relationships between the optimum angles of inputs and the values of PDL/G when the CN takes the global minimum.

It is evident that $αopt$is close to 120° when PDL/G is small. This is consistent with the conclusion we have obtained in Ref [1]. When PDL/G increases, the optimum angle $αopt$ 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 .

Fig. 2 The relative relationship of three input Stokes vectors and the PDL/G vector on the Poincaré sphere to achieve the minimum of the CN.

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.

Fig. 3 The relationship between the minimum of the CN and the values of PDL/G.

Since the CN in Eq. (9) is a function of six angles $θs, θt, θu$,$αin, βin, γin$, it is impossible to illustrate the whole function. In Fig. 4 , we only present a curve to partially show this function, where $D=0.5195$ (5 dB), $θs=θt=θu=θ$and$αin=βin=γin=α$. Please note, in Fig. 4(a), α is related to θ by Eq. (11) with the minus sign.

Fig. 4 The relationships between the CN and (a) the angle αand (b) the angle θ when the value of PDL/G is 5 dB. The insets show the “zoom in” views of the same data.

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 $N(0,σ2)$, which has been used in Ref [1]. This means that we assume that an ideal polarimeter, which has such statistical properties, is used in the measurement system.

## 3. Statistical properties of $Δ|M˜|$

In the following, we still consider that the measurement errors of the input SOPs can be neglected as we have done in Ref [1]. Under the conditions $Δsoutj, Δtoutj,Δuoutj (j=1,2,3,4)∼N(0,σ2)$, we know that $〈Δ|M˜|〉=0$. Hence, the variance of $Δ|M˜|$ is needed to evaluate the measurement uncertainty. It can be derived as

$Var(Δ|M˜|)=2Tu2σ2KΔ|M|$
where

$KΔ|M|=(Ds+Dt)2+(Ds+Du)2+(Dt+Du)2−(1−D2)(3−cosαin−cosβin−cosγin)(3−cosαin−cosβin−cosγin)2$

Obviously, $KΔ|M|$ 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 $θs=θt=θu=θ$, $αin=βin=γin=α$. When $KΔ|M|$ 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.

Fig. 5 The relationships between the optimum angles and the PDL/G. Solid lines are based on the minimum of $KΔ|M|$ and dash lines are based on the minimum of the CN.

## 4. Upper bound of $〈‖ΔM˜‖〉$

When D has a finite value, from Eq. (14) of Ref [1], we have

$‖ΔM˜‖=‖M˜Fin−1ΔFoutM˜‖$
Due to the existence of the complex Mueller matrix $M˜$, it is difficult to calculate $〈‖ΔM˜‖〉$ starting from Eq. (14). To overcome the mathematical difficulty, we can start from the following equation
$FinM˜−1=Fout$
Based on the relations that $M˜−1=M˜T/|M˜|$ [1] and $(M˜+ΔM˜)T=M˜T+ΔM˜T$, it can be derived that
$ΔM˜T=Fin−1ΔF˜out$
where
$F˜out=( isout0 sout1 sout2 sout3 itout0 tout1 tout2 tout3 iuout0 uout1 uout2 uout3Aout0|M˜| Aout1|M˜| Aout2|M˜| Aout3|M˜|)$
$ΔF˜out=ΔF˜out1−ΔF˜out2$
In Eq.(18),$ΔF˜out1=(iΔsout0 Δsout1 Δsout2 Δsout3iΔtout0 Δtout1 Δtout2 Δtout3iΔuout0 Δuout1 Δuout2 Δuout3ΔAout0|M˜| ΔAout1|M˜| ΔAout2|M˜| ΔAout3|M˜|)$and$ΔF˜out2=Δ|M˜||M˜|( 0 0 0 0 0 0 0 0 0 0 0 0Aout0 Aout1 Aout2 Aout3)$.

Based on the definition of the Frobenius matrix norm, it is easy to know that $‖ΔM˜‖=‖ΔM˜T‖$. Then, by substituting Eq. (18) into Eq. (16), we have

$‖ΔM˜‖=‖ΔM˜T‖=‖Fin−1ΔF˜out1−Fin−1ΔF˜out2‖=Tr(ΔF˜out1HFΔF˜out1+ΔF˜out2HFΔF˜out2−ΔF˜out1HFΔF˜out2−ΔF˜out2HFΔF˜out1)$
where$F=(Fin−1)HFin−1$.

Unfortunately, we cannot directly calculate $〈‖ΔM˜‖〉$ because of the difficulties in mathematics. As an alternative, we can calculate an upper bound as

$〈‖ΔM˜‖〉≤〈‖ΔM˜‖2〉=〈Tr(ΔF˜out1HFΔF˜out1)〉+〈Tr(ΔF˜out2HFΔF˜out2)〉−〈Tr(ΔF˜out1HFΔF˜out2)〉−〈Tr(ΔF˜out2HFΔF˜out1)〉$
The four terms in Eq. (20) are calculated as
$〈Tr(ΔF˜out1HFΔF˜out1)〉=2σ2{2∑j=13fjj+f44{4[Ds3Dt3(1−cosαin)+Ds3Du3(1−cosβin)+Dt3Du3(1−cosγin)]/(1−D2)3−[Ds4Dt4(1−cosαin)2+Ds4Du4(1−cosβin)2+Dt4Du4(1−cosγin)2]/(1−D2)4}}$
$〈Tr(ΔF˜out2HFΔF˜out2)〉=f44Ds2Dt2Du2(Bout12+Bout22)(1−D2)3Var(Δ|M˜|)|M˜|$
$〈Tr(ΔF˜out1HFΔF˜out2)〉=〈Tr(ΔF˜out2HFΔF˜out1)〉=0$
In Eqs. (21) and (22), $fjj$ are the diagonal elements of the matrix F, which are
${f11={[4−(1+cosγin)2](Bin12+Bin22)−8(1−cosγin)Bin12}/(Bin12−Bin22)2f22={[4−(1+cosβin)2](Bin12+Bin22)−8(1−cosβin)Bin12}/(Bin12−Bin22)2f33={[4−(1+cosαin)2](Bin12+Bin22)−8(1−cosαin)Bin12}/(Bin12−Bin22)2f44=2[4+(1+cosαin)(1+cosβin)(1+cosγin)−(1+cosαin)2−(1+cosβin)2−(1+cosγin)2]/(Bin12−Bin22)2$
where
${Bin12=1−cos2αin−cos2βin−cos2γin+2cosαincosβincosγinBin22=4(1−cosβin)(1−cosγin)−(1+cosαin−cosβin−cosγin)2$
Finally, we have
$〈‖ΔM˜‖〉≤〈‖ΔM˜‖2〉=K‖ΔM‖σ$
where
$K‖ΔM‖=22∑j=13fjj+f44{4[Ds3Dt3(1−cosαin)+Ds3Du3(1−cosβin)+Dt3Du3(1−cosγin)](1−D2)3−[Ds4Dt4(1−cosαin)2+Ds4Du4(1−cosβin)2+Dt4Du4(1−cosγin)2−Ds2Dt2Du2(Bout12+Bout22)KΔ|M|](1−D2)4}$
Apparently, this upper bound is completely determined by $αin, βin, γin$ and $θs, θt, θu$. And it is easy to know that $K‖ΔM‖$ is also a symmetric function of $(αin, βin, γin)$ and $(θs, θt, θu)$. Also from the Purkiss Principle and numerical calculation, $K‖ΔM‖$ takes its global minimum when $αin=βin=γin=α$,$θs= θt= θu=θ$. From Eq. (27), the relationship between the optimum angles and the value of PDL/G, when $K‖ΔM‖$ takes the global minimum, can be shown in Fig. 6 using solid lines. As a comparison, the optimum angles, corresponding to the CN and$KΔ|M|$, 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.

Fig. 6 The relationships between the optimum angles and the PDL/G. Solid lines are based on the minimum of $K‖ΔM‖$ and dash lines are based on the minimum of the CN and$KΔ|M|$.

For different values of PDL/G, the minimum of $K‖ΔM‖$ 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.

Fig. 7 The relationship between the minimum of $K‖ΔM‖$ and the values of PDL/G.

When the value of PDL/G is 5 dB, the relationship between $K‖ΔM‖$ and the angle α is shown in Fig. 8 . The curve in Fig. 8 gives us a partial information of the function of $K‖ΔM‖$.

Fig. 8 The relationship between $K‖ΔM‖$and the angle αwhen the value of the PDL/G is 5 dB. The inset shows the “zoom in” view of the same data.

## 5. Simulation results

To verify the theoretical finding in Section 4, simulations are performed. The parameters of the system under simulation are 1) Birefringence:$ϕ=5π/3$, $r=(0.66, 0.74, −0.1296)$; 2) PDL/G: $0, $D⌢=(0, 0, 1)T$and 3) PIDL/G: $Tu=1$. In Section 4, the theoretical result shows that the upper bound does not depend on the PIDL/G $Tu$. Then, we take $Tu=1$ in the following simulations. In this paper, we only show the simulation results when $αin=βin=γin=α$and $θs= θt= θu=θ$. Then, the three input SOPs can be written as

${S→in=(i, sinθ, 0, cosθ)TT→in=(i, −sinθ/2, 3sinθ/2, cosθ)TU→in=(i, −sinθ/2, −3sinθ/2, cosθ)T$
In the simulations, $〈‖ΔM˜‖〉$ is calculated using 10000 independent noise realizations with $σ=0.03$. For three 4D output Stokes vectors, this means that 120000 random values have been generated. In Fig. 9 , the theoretical upper bound (in red) and the simulation results (in blue) of $〈‖ΔM˜‖〉$ 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.

Fig. 9 The relationships between the theoretical upper bound (in red), simulation results (in blue) of $〈‖ΔM˜‖〉$ and (a) the angle αand (b) the angle θ when the values of PDL/G are 5 dB, 10 dB and 13 dB, respectively.

## 6. Conclusion

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 [14], 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 [1];
• 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.

## Acknowledgements

This work is supported by Singapore A-star, Singapore Bioimaging Consortium, SBIC Grant Ref: SBIC RP C-014/2007.

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10. H. Dong, J. Q. Zhou, M. Yan, P. Shum, L. Ma, Y. D. Gong, and C. Q. Wu, “Quasi-monochromatic fiber depolarizer and its application to polarization-dependent loss measurement,” Opt. Lett. 31(7), 876–878 ( 2006). [CrossRef]   [PubMed]

11. H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 ( 2007). [CrossRef]

12. E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q ( 2005). [CrossRef]

13. H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 ( 2009). [CrossRef]

14. M. Reimer and D. Yevick, “Least-squares analysis of the Mueller matrix,” Opt. Lett. 31(16), 2399–2401 ( 2006). [CrossRef]   [PubMed]

### References

• View by:
• |
• |
• |

1. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017 .
[CrossRef] [PubMed]
2. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]
3. K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994).
[CrossRef]
4. E. Lichtmann, “Performance degradation due to polarization dependent gain and loss in lightwave systems with optical amplifiers,” IEEE Photon. Technol. Lett. 5, 1969–1970 (1993).
5. B. Huttner and N. Gisin, “Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,” Opt. Lett. 22(8), 504–506 (1997).
[CrossRef] [PubMed]
6. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]
7. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]
8. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995).
[CrossRef]
9. W. C. Waterhouse, “Do symmetric problems have symmetric solutions?” Am. Math. Mon. 90(6), 378–387 (1983).
[CrossRef]
10. H. Dong, J. Q. Zhou, M. Yan, P. Shum, L. Ma, Y. D. Gong, and C. Q. Wu, “Quasi-monochromatic fiber depolarizer and its application to polarization-dependent loss measurement,” Opt. Lett. 31(7), 876–878 (2006).
[CrossRef] [PubMed]
11. H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]
12. E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]
13. H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009).
[CrossRef]
14. M. Reimer and D. Yevick, “Least-squares analysis of the Mueller matrix,” Opt. Lett. 31(16), 2399–2401 (2006).
[CrossRef] [PubMed]

#### 2009 (2)

H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009).
[CrossRef]

#### 2007 (2)

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

#### 2005 (1)

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

#### 1997 (2)

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

#### 1995 (1)

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

#### 1994 (1)

K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994).
[CrossRef]

#### 1993 (1)

E. Lichtmann, “Performance degradation due to polarization dependent gain and loss in lightwave systems with optical amplifiers,” IEEE Photon. Technol. Lett. 5, 1969–1970 (1993).

#### 1983 (1)

W. C. Waterhouse, “Do symmetric problems have symmetric solutions?” Am. Math. Mon. 90(6), 378–387 (1983).
[CrossRef]

#### Ambirajan, A.

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

#### Castro-Camus, E.

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

#### Dong, H.

H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009).
[CrossRef]

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

#### Fraser, M. D.

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

#### Gisin, N.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

#### Gong, Y. D.

H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009).
[CrossRef]

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

#### Hong, M. H.

H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009).
[CrossRef]

#### Huttner, B.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

#### Johnston, M. B.

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

#### Kikushima, K.

K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994).
[CrossRef]

#### Lichtmann, E.

E. Lichtmann, “Performance degradation due to polarization dependent gain and loss in lightwave systems with optical amplifiers,” IEEE Photon. Technol. Lett. 5, 1969–1970 (1993).

#### Lloyd-Hughes, J.

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

#### Look, D. C.

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

#### Ning, G. X.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

#### Paulose, V.

H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009).
[CrossRef]

#### Shum, P.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

#### Suto, K.

K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994).
[CrossRef]

#### Tan, H. H.

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

#### Waterhouse, W. C.

W. C. Waterhouse, “Do symmetric problems have symmetric solutions?” Am. Math. Mon. 90(6), 378–387 (1983).
[CrossRef]

#### Wu, C. Q.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

#### Yan, M.

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

#### Yoneda, E.

K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994).
[CrossRef]

#### Yoshinaga, H.

K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994).
[CrossRef]

#### Zhou, J. Q.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

#### Am. Math. Mon. (1)

W. C. Waterhouse, “Do symmetric problems have symmetric solutions?” Am. Math. Mon. 90(6), 378–387 (1983).
[CrossRef]

#### IEEE J. Lightwave Technol. (1)

K. Kikushima, K. Suto, H. Yoshinaga, and E. Yoneda, “Polarization dependent distortion in AM-SCM video transmission systems,” IEEE J. Lightwave Technol. 12(4), 650–657 (1994).
[CrossRef]

#### IEEE Photon. Technol. Lett. (2)

E. Lichtmann, “Performance degradation due to polarization dependent gain and loss in lightwave systems with optical amplifiers,” IEEE Photon. Technol. Lett. 5, 1969–1970 (1993).

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, and C. Q. Wu, “Virtual generalized Mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers,” IEEE Photon. Technol. Lett. 19(1), 27–29 (2007).
[CrossRef]

#### Opt. Commun. (3)

H. Dong, Y. D. Gong, V. Paulose, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz-time domain spectroscopy,” Opt. Commun. 282(18), 3671–3675 (2009).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent loss in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

#### Opt. Eng. (1)

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

#### Proc. SPIE (1)

E. Castro-Camus, J. Lloyd-Hughes, M. D. Fraser, H. H. Tan, C. Jagadish, and M. B. Johnston, “Detecting the full polarization state of terahertz transients,” Proc. SPIE 6120, 61200Q (2005).
[CrossRef]

### Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (9)

Fig. 1

The relationships between the optimum angles of inputs and the values of PDL/G when the CN takes the global minimum.

Fig. 2

The relative relationship of three input Stokes vectors and the PDL/G vector on the Poincaré sphere to achieve the minimum of the CN.

Fig. 3

The relationship between the minimum of the CN and the values of PDL/G.

Fig. 4

The relationships between the CN and (a) the angle αand (b) the angle θ when the value of PDL/G is 5 dB. The insets show the “zoom in” views of the same data.

Fig. 5

The relationships between the optimum angles and the PDL/G. Solid lines are based on the minimum of $K Δ | M |$ and dash lines are based on the minimum of the CN.

Fig. 6

The relationships between the optimum angles and the PDL/G. Solid lines are based on the minimum of $K ‖ Δ M ‖$ and dash lines are based on the minimum of the CN and $K Δ | M |$ .

Fig. 7

The relationship between the minimum of $K ‖ Δ M ‖$ and the values of PDL/G.

Fig. 8

The relationship between $K ‖ Δ M ‖$ and the angle αwhen the value of the PDL/G is 5 dB. The inset shows the “zoom in” view of the same data.

Fig. 9

The relationships between the theoretical upper bound (in red), simulation results (in blue) of $〈 ‖ Δ M ˜ ‖ 〉$ and (a) the angle αand (b) the angle θ when the values of PDL/G are 5 dB, 10 dB and 13 dB, respectively.

### Equations (28)

$Δ M ˜ = F out − 1 ( Δ F in − Δ F out M ˜ − Δ F out Δ M ˜ )$
$‖ Δ M ˜ ‖ ‖ M ˜ ‖ ≤ Cond ( F out ) 1 − Cond ( F out ) ‖ Δ F out ‖ ‖ F out ‖ ( ‖ Δ F in ‖ ‖ F in ‖ + ‖ Δ F out ‖ ‖ F out ‖ )$
$C o n d ( F o u t ) = ( 1 + ρ o u t s 2 ) s o u t 0 2 + ( 1 + ρ o u t t 2 ) t o u t 0 2 + ( 1 + ρ o u t u 2 ) u o u t 0 2 + s o u t 0 2 t o u t 0 2 u o u t 0 2 ( B o u t 1 2 + B o u t 2 2 ) | M ˜ | × { s o u t 0 2 t o u t 0 2 [ ( 1 + ρ o u t s 2 ) ( 1 + ρ o u t t 2 ) − ( 1 + ρ o u t s ρ o u t t cos α o u t ) 2 ] + s o u t 0 2 u o u t 0 2 [ ( 1 + ρ o u t s 2 ) ( 1 + ρ o u t u 2 ) − ( 1 + ρ o u t s ρ o u t u cos β o u t ) 2 ] + t o u t 0 2 u o u t 0 2 [ ( 1 + ρ o u t t 2 ) ( 1 + ρ o u t u 2 ) − ( 1 + ρ o u t t ρ o u t u cos γ o u t ) 2 ] } ( B o u t 1 2 + B o u t 2 2 ) − 4 [ s o u t 0 2 t o u t 0 2 a o u t 2 + s o u t 0 2 u o u t 0 2 b o u t 2 + t o u t 0 2 u o u t 0 2 c o u t 2 ] B o u t 1 2 + | M ˜ | [ ( 1 + ρ o u t s 2 ) ( 1 + ρ o u t t 2 ) ( 1 + ρ o u t u 2 ) − ( 1 + ρ o u t u 2 ) ( 1 + ρ o u t s ρ o u t t cos α o u t ) 2 − ( 1 + ρ o u t t 2 ) ( 1 + ρ o u t s ρ o u t u cos β o u t ) 2 − ( 1 + ρ o u t s 2 ) ( 1 + ρ o u t t ρ o u t u cos γ o u t ) 2 + 2 ( 1 + ρ o u t s ρ o u t t cos α o u t ) ( 1 + ρ o u t s ρ o u t u cos β o u t ) ( 1 + ρ o u t t ρ o u t u cos γ o u t ) ] s o u t 0 t o u t 0 u o u t 0 | B o u t 1 2 − B o u t 2 2 |$
${ B o u t 1 2 = ρ o u t s 2 ρ o u t t 2 ρ o u t u 2 ( 1 − cos 2 α o u t − cos 2 β o u t − cos 2 γ o u t + 2 cos α o u t cos β o u t cos γ o u t ) B o u t 2 2 = ( a o u t + b o u t + c o u t ) ( − a o u t + b o u t + c o u t ) ( a o u t − b o u t + c o u t ) ( a o u t + b o u t − c o u t ) / 4 a o u t = ρ o u t s 2 + ρ o u t t 2 − 2 ρ o u t s ρ o u t t cos α o u t b o u t = ρ o u t s 2 + ρ o u t u 2 − 2 ρ o u t s ρ o u t u cos β o u t c o u t = ρ o u t t 2 + ρ o u t u 2 − 2 ρ o u t t ρ o u t u cos γ o u t$
${ s out 0 = t out 0 = u out 0 ρ out s = ρ out t = ρ out u α out = β out = γ out$
${ s out 0 = T u s in 0 ( 1 + ρ in s D cos θ s ) t out 0 = T u t in 0 ( 1 + ρ in t D cos θ t ) u out 0 = T u u in 0 ( 1 + ρ in u D cos θ u )$
${ ρ out s = 1 − ( 1 − D 2 ) ( 1 − ρ in s 2 ) / ( 1 + ρ in s D cos θ s ) 2 ρ out t = 1 − ( 1 − D 2 ) ( 1 − ρ in t 2 ) / ( 1 + ρ in t D cos θ t ) 2 ρ out u = 1 − ( 1 − D 2 ) ( 1 − ρ in u 2 ) / ( 1 + ρ in u D cos θ u ) 2$
${ ρ out s ρ out t cos α out = 1 − ( 1 − D 2 ) ( 1 − ρ in s ρ in t cos α in ) ( 1 + ρ in s D cos θ s ) ( 1 + ρ in t D cos θ t ) ρ out s ρ out u cos β out = 1 − ( 1 − D 2 ) ( 1 − ρ in s ρ in u cos β in ) ( 1 + ρ in s D cos θ s ) ( 1 + ρ in u D cos θ u ) ρ out t ρ out u cos γ out = 1 − ( 1 − D 2 ) ( 1 − ρ in t ρ in u cos γ in ) ( 1 + ρ in t D cos θ t ) ( 1 + ρ in u D cos θ u )$
$Cond ( F out ) = 2 ( D s 2 + D t 2 + D u 2 ) + D s 2 D t 2 D u 2 ( B out 1 2 + B out2 2 ) / ( 1 − D 2 ) 2 × { D s 2 D t 2 [ 4 − ( 1 + cos α out ) 2 ] + D s 2 D u 2 [ 4 − ( 1 + cos β out ) 2 ] + D t 2 D u 2 [ 4 − ( 1 + cos γ out ) 2 ] } ( B out 1 2 + B out2 2 ) − 8 [ D s 2 D t 2 ( 1 − cos α out ) + D s 2 D u 2 ( 1 − cos β out ) + D t 2 D u 2 ( 1 − cos γ out ) ] B out 1 2 + 2 ( 1 − D 2 ) 2 [ 4 + ( 1 + cos α out ) ( 1 + cos β out ) ( 1 + cos γ out ) − ( 1 + cos α out ) 2 − ( 1 + cos β out ) 2 − ( 1 + cos γ out ) 2 ] D s D t D u | B out 1 2 − B out2 2 |$
${ B out1 2 = 1 − cos 2 α out − cos 2 β out − cos 2 γ out + 2 cos α out cos β out cos γ out B out2 2 = 4 ( 1 − cos β out ) ( 1 − cos γ out ) − ( 1 + cos α out − cos β out − cos γ out ) 2$
$cos θ = ± 1 + 2 cos α 3$
$Var ( Δ | M ˜ | ) = 2 T u 2 σ 2 K Δ | M |$
$K Δ | M | = ( D s + D t ) 2 + ( D s + D u ) 2 + ( D t + D u ) 2 − ( 1 − D 2 ) ( 3 − cos α in − cos β in − cos γ in ) ( 3 − cos α in − cos β in − cos γ in ) 2$
$‖ Δ M ˜ ‖ = ‖ M ˜ F in − 1 Δ F out M ˜ ‖$
$F in M ˜ − 1 = F out$
$Δ M ˜ T = F in − 1 Δ F ˜ out$
$F ˜ out = ( i s out 0 s out 1 s out 2 s out 3 i t out 0 t out 1 t out 2 t out 3 i u out 0 u out 1 u out 2 u out 3 A out 0 | M ˜ | A out 1 | M ˜ | A out 2 | M ˜ | A out3 | M ˜ | )$
$Δ F ˜ out = Δ F ˜ out1 − Δ F ˜ out2$
$‖ Δ M ˜ ‖ = ‖ Δ M ˜ T ‖ = ‖ F in − 1 Δ F ˜ out1 − F in − 1 Δ F ˜ out2 ‖ = Tr ( Δ F ˜ out1 H F Δ F ˜ out1 + Δ F ˜ out2 H F Δ F ˜ out2 − Δ F ˜ out1 H F Δ F ˜ out2 − Δ F ˜ out2 H F Δ F ˜ out1 )$
$〈 ‖ Δ M ˜ ‖ 〉 ≤ 〈 ‖ Δ M ˜ ‖ 2 〉 = 〈 Tr ( Δ F ˜ out1 H F Δ F ˜ out1 ) 〉 + 〈 Tr ( Δ F ˜ out2 H F Δ F ˜ out2 ) 〉 − 〈 Tr ( Δ F ˜ out1 H F Δ F ˜ out2 ) 〉 − 〈 Tr ( Δ F ˜ out2 H F Δ F ˜ out1 ) 〉$
$〈 Tr ( Δ F ˜ out1 H F Δ F ˜ out1 ) 〉 = 2 σ 2 { 2 ∑ j = 1 3 f j j + f 44 { 4 [ D s 3 D t 3 ( 1 − cos α in ) + D s 3 D u 3 ( 1 − cos β in ) + D t 3 D u 3 ( 1 − cos γ in ) ] / ( 1 − D 2 ) 3 − [ D s 4 D t 4 ( 1 − cos α in ) 2 + D s 4 D u 4 ( 1 − cos β in ) 2 + D t 4 D u 4 ( 1 − cos γ in ) 2 ] / ( 1 − D 2 ) 4 } }$
$〈 Tr ( Δ F ˜ out2 H F Δ F ˜ out2 ) 〉 = f 44 D s 2 D t 2 D u 2 ( B out 1 2 + B out2 2 ) ( 1 − D 2 ) 3 Var ( Δ | M ˜ | ) | M ˜ |$
$〈 Tr ( Δ F ˜ out1 H F Δ F ˜ out2 ) 〉 = 〈 Tr ( Δ F ˜ out2 H F Δ F ˜ out1 ) 〉 = 0$
${ f 11 = { [ 4 − ( 1 + cos γ in ) 2 ] ( B in1 2 + B in2 2 ) − 8 ( 1 − cos γ in ) B in1 2 } / ( B in1 2 − B in2 2 ) 2 f 22 = { [ 4 − ( 1 + cos β in ) 2 ] ( B in1 2 + B in2 2 ) − 8 ( 1 − cos β in ) B in1 2 } / ( B in1 2 − B in2 2 ) 2 f 33 = { [ 4 − ( 1 + cos α in ) 2 ] ( B in1 2 + B in2 2 ) − 8 ( 1 − cos α in ) B in1 2 } / ( B in1 2 − B in2 2 ) 2 f 44 = 2 [ 4 + ( 1 + cos α in ) ( 1 + cos β in ) ( 1 + cos γ in ) − ( 1 + cos α in ) 2 − ( 1 + cos β in ) 2 − ( 1 + cos γ in ) 2 ] / ( B in1 2 − B in2 2 ) 2$
${ B in 1 2 = 1 − cos 2 α in − cos 2 β in − cos 2 γ in + 2 cos α in cos β in cos γ in B in 2 2 = 4 ( 1 − cos β in ) ( 1 − cos γ in ) − ( 1 + cos α in − cos β in − cos γ in ) 2$
$〈 ‖ Δ M ˜ ‖ 〉 ≤ 〈 ‖ Δ M ˜ ‖ 2 〉 = K ‖ Δ M ‖ σ$
$K ‖ Δ M ‖ = 2 2 ∑ j = 1 3 f j j + f 44 { 4 [ D s 3 D t 3 ( 1 − cos α in ) + D s 3 D u 3 ( 1 − cos β in ) + D t 3 D u 3 ( 1 − cos γ in ) ] ( 1 − D 2 ) 3 − [ D s 4 D t 4 ( 1 − cos α in ) 2 + D s 4 D u 4 ( 1 − cos β in ) 2 + D t 4 D u 4 ( 1 − cos γ in ) 2 − D s 2 D t 2 D u 2 ( B out 1 2 + B out2 2 ) K Δ | M | ] ( 1 − D 2 ) 4 }$
${ S → in = ( i , sin θ , 0 , cos θ ) T T → in = ( i , − sin θ / 2 , 3 sin θ / 2 , cos θ ) T U → in = ( i , − sin θ / 2 , − 3 sin θ / 2 , cos θ ) T$