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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 15 — Jul. 20, 2009
  • pp: 13017–13030
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Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain

H. Dong, Y. D. Gong, Varghese Paulose, P. Shum, and Malini Olivo  »View Author Affiliations


Optics Express, Vol. 17, Issue 15, pp. 13017-13030 (2009)
http://dx.doi.org/10.1364/OE.17.013017


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Abstract

For the measurement of Mueller matrix in an optical system with birefringence and small polarization-dependent loss or gain (PDL/G), we theoretically derive the statistical relationship between the Mueller matrix measurement error and three input states of polarization (SOP). Based on this theoretical relation and simulation results, it can be concluded that the three input SOPs, that are coplanar with an angle of 120° between any two of them in Stokes space, can be considered as a substitute for the best input SOPs which can statistically lead to the minimum measurement error. This conclusion is valid when the PDL/G of the optical system under test is less than 0.35dB.

© 2009 OSA

1. Introduction

To date, several Mueller matrix measurement methods, such as dual rotating retarders polarimetry [7

7. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992). [CrossRef] [PubMed]

], null ellipsometry [8

8. S.-M. F. Nee, “Depolarization and principal mueller matrix measured by null ellipsometry,” Appl. Opt. 40(28), 4933–4939 (2001). [CrossRef]

] and the Stokes methods [9

9. B. J. Howell, “Measurement of the polarization effect of an instrument using partially polarized light,” Appl. Opt. 18(6), 809–812 (1979). [CrossRef] [PubMed]

], have been proposed. All these methods realize the measurement by setting some input states of polarization (SOP) and measuring the corresponding outputs. A potential technique for the fast infrared polarization modulation was also reported based on the high birefringence and low linear diattenuation of ferroelectric liquid crystals in some spectral bands [10

10. D. B. Chenault, R. A. Chipman, K. M. Johnson, and D. Doroski, “Infrared linear diattenuation and birefringence spectra of ferroelectric liquid crystals,” Opt. Lett. 17(6), 447–449 (1992). [CrossRef] [PubMed]

]. In some applications based on Mueller matrix measurement, the smallest number of input SOPs should be used to take the measurements because a series of Mueller matrices are required to be measured in finite time. Two examples of such applications are PMD vector measurement [4

4. 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]

] and polarization measurement in terahertz time-domain spectroscopy (THz-TDS) [11

11. H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).

]. However, most of the above-mentioned methods are not good candidates in such applications.

Actually, it has been demonstrated that at least three input SOPs must be used to realize the Mueller matrix measurement when the system under test has both birefringence and PDL/G [12

12. R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. 37(2), 110–112 (1947). [CrossRef]

,13

13. 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]

]. In 1947, R. C. Jones proposed a Jones matrix measurement approach using the three SOPs (1,1,0,0)T, (1,1,0,0)T and (1,0,1,0)T(“T” denotes the matrix transpose) [12

12. R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. 37(2), 110–112 (1947). [CrossRef]

]. We had also proposed a Mueller matrix measurement approach using the three input SOPs (1,1,0,0)T, (1,0.5,0.866,0)T and (1,0.5,0.866,0)T [13

13. 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]

]. Theoretically, if the input and output SOPs are set and measured without any errors, the error-free Mueller matrix can be measured using three arbitrary input SOPs [13

13. 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]

]. Unfortunately, errors arising from environmental perturbations, imperfect components and alignments, do exist in practice. Hence, the measured input and output SOPs definitely have errors. As a result, these errors will be transferred to the calculated Mueller matrix. The optimizations of polarimeters and noise influences on the SOP measurement have been reported in many papers [14

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

17

17. Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. 46(30), 7354–7364 (2007). [CrossRef] [PubMed]

]. Analysis of the Mueller matrix measurement error, induced by imperfect components and alignments, has also been presented [18

18. S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20(8), 1651–1657 (2003). [CrossRef]

,19

19. D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7(4), 693–700 (1990). [CrossRef]

]. On the other hand, for a pure birefringent system, we have demonstrated that the error of the Mueller matrix measurement is a function of two input SOPs and the statistically minimum error can be achieved when two input SOPs are orthogonal in Stokes space [20

20. H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008). [CrossRef]

]. However, to the best of our knowledge, for a system with both birefringence and PDL/G, the relationship between the error of the calculated Mueller matrix and three input SOPs has not been presented until now.

A general analysis of this relationship, for an optical system with arbitrary PDL/G, is very complicated. In this paper, we present the theoretical and simulation results of this relationship in a simple case, where the system under test has a small PDL/G and the measurement errors of the output SOPs are far larger than those of the input SOPs. Some optical systems satisfy the first precondition. For example, an optical fiber system, composed of SMF, optical isolators and optical couplers, usually has a small PDL. Some measurement setups may approximately satisfy the second precondition. For example, when the input SOPs are generated by precisely rotating a high-quality polarizer and the output SOPs are measured by a fiber-type polarimeter, the measurement errors of the input SOPs must be less than those of the output SOPs.

This paper is organized as follows: firstly, some useful equations are deduced based on the properties of the Mueller matrix in Section 2; secondly, the equations governing Mueller matrix measurement using three arbitrary input SOPs and error propagations are derived in Section 3; the statistical properties of Mueller matrix measurement error are investigated in Section 4 and Section 5; finally, some simulation results are used to verify the theoretical finding.

2. Some properties of Mueller-Jones matrix

It has been demonstrated that the Mueller matrix of a system having both birefringence and PDL/G satisfies the Lorentz transformation [21

21. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53(3), 317–323 (1963). [CrossRef]

,22

22. R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981). [CrossRef]

]. Thus, we can express such a Mueller matrix in the form of a 4×4 complex matrix M˜
M˜=(m11im12im13im14im21m22m23m24im31m32m33m34im41m42m43m44)
(1)
where i=1 and mjk(j,k=1,2,3,4) are the elements of the real Mueller matrix M. Based on this definition and the property of Lorentz transformation [22

22. R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981). [CrossRef]

], we have
M˜TM˜=|M˜|I
(2)
where |M˜|=|M| denotes the determinant of Mueller matrix and I stands for the identity matrix.

It has been demonstrated that such a Mueller matrix can be decomposed as [23

23. S.-Y. Lu and R. A. Chipman; “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

]
M˜=Tu(10T0mR)(     1iDTiDTmD)
(3)
where Tu=m11 is the polarization-independent loss or gain (PIDL/G); 0=(000)T; D=(D1D2D3)T is the PDL/G vector and 0D=D12+D22+D32<1. PDL/G in dB can be calculated using PDL/G=10log10[(1+D)/(1D)] [23

23. S.-Y. Lu and R. A. Chipman; “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

].

Sub-matrix mR is a 3×3 orthogonal matrix, which can be expressed as [23

23. S.-Y. Lu and R. A. Chipman; “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

]
mR=cosϕI+(1cosϕ)rrsinϕr×
(4)
where ϕ denotes the rotation angle in Stokes space; r is a unit vector, standing for the rotation axis; rr is a dyadic; r× is a cross-product operator.

Sub-matrix mD is a 3×3 symmetric matrix, which can be written as [23

23. S.-Y. Lu and R. A. Chipman; “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

]
mD=1D2I+(11D2)DD
(5)
where D=D/D.

From Eqs. (3), (4) and (5), the determinant |M˜| and the Frobenius matrix norm M˜ of M˜, which is defined as M˜=Tr(M˜HM˜), can be calculated as
|M˜|=Tu4(1D2)2and  M˜=2Tu
(6)
where “Tr” denotes the trace of a square matrix and “H” denotes the conjugate transpose. To be compatible with the complex Mueller matrix M˜, a 4-dimensional complex Stokes vector can be defined as S=(is0,s1,s2,s3)T=s0(i,s)T and the dot product of two vectors is defined as ST=s0t0+s1t1+s2t2+s3t3=s0t0(1st)=s0t0(1cosα). Here, α is the angle between S and T. The input and output 4-dimensional complex Stokes vectors are linked by Sout=M˜Sin.

From Eq. (3), we can easily obtain
sout0=Tusin0Ds
(7)
where Ds=1+Dcosθs and θs is the angle between the PDL/G vector and input SOP Sin in Stokes space.

Another useful equation between input and output SOPs is SoutTout=|M˜|SinTin [13

13. 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]

]. Based on this equation, we have

cosαout=11D2DsDt(1cosαin)
(8)

3. Measurement approach and measurement error

In an experimental setup [13

13. 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]

], three input SOPs Sin, Tin, Uin and three output SOPs Sout, Tout, Uout can be measured. Then, the complex Mueller matrix M˜ can be calculated using the following two equations [13

13. 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]

]
|M˜|=SoutTout+SoutUout+ToutUoutSinTin+SinUin+TinUin
(9)
and
FoutM˜=Fin
(10)
where Fout=1|M˜|(isout0   sout1   sout2   sout3itout0     tout1   tout2   tout3iuout0   uout1   uout2uout3Aout0|M˜|Aout1|M˜|   Aout2|M˜|   Aout3|M˜|) and Fin=(isin0sin1sin2sin3itin0   tin1tin2tin3iuin0uin1uin2uin3Ain0Ain1   Ain2   Ain3).Elements Aj(j=0,1,2,3) in Fin and Fout are defined as A0=|s1s2s3t1t2t3u1u2u3|, A1=i|s0s2s3t0t2t3u0u2u3|, A2=i|s0s1s3t0t1t3u0u1u3| and A3=i|s0s1s2t0t1t2u0u1u2|.Firstly, |M˜| is calculated using Eq. (9). Secondly, M˜ can be calculated by substituting |M˜| in Eq. (10).

When the measured output SOPs have errors ΔSout, ΔTout and ΔUout, the exact form of Eq. (9) becomes
|M˜|+Δ|M˜|=[(Sout+ΔSout)(Tout+ΔTout)+(Sout+ΔSout)(Uout+ΔUout)+(Tout+ΔTout)(Uout+ΔUout)]SinTin+SinUin+TinUin
(11)
The measurement error Δ|M˜| can be calculated as
Δ|M˜|=(Tout+Uout)ΔSout+(Sout+Uout)ΔTout+(Sout+Tout)ΔUoutSinTin+SinUin+TinUin
(12)
Please note that we ignore all high-order error terms in this paper.

Similarly, when errors exist, the exact form of Eq. (10) should be
(Fout+ΔFout)(M˜+ΔM˜)=Fin
(13)
and the Frobenius matrix norm of the Mueller matrix error ΔM˜ is
ΔM˜=Fout1ΔFoutM˜=M˜Fin1ΔFoutM˜
(14)
where F1denotes the inverse matrix of F. Equations (12) and (14) are the starting point of the theoretical analysis.

For three input SOPs, α, β and γ are the angles between Sin and Tin, Sin and Uin, Tin and Uin in Stokes space, respectively. Please note that we have ignored the subscript “in” for the three angles. These angles are bounded by 0α,β,γ180 and α+β+γ360. In fact, these angles are not independent. They should satisfy |αβ|γα+β. Then we have the following relations
{αβγα+β          when  αβ  and  α+β180βαγα+β          when  α<β  and  α+β180αβγ360αβ  when  αβ  and  α+β>180βαγ360αβ  when  α<β  and  α+β>180
(15)
Then, (α,β,γ) can only take values on the surface and the inner space of a tetrahedron enclosed by four planesαβγ=0, α+βγ=0,αβ+γ=0, α+β+γ=360 as shown in Fig. 1
Fig. 1 (α,β,γ) can only take values on the surface and the inner space of the red tetrahedron.
.

4. Statistical properties of Δ|M˜|

From Eq. (12), the smallest |Δ|M˜|| may be achieved corresponding to different input SOPs in different tests because the measurement errors of the output SOPs vary. Therefore, the relationship between the statistical parameter of Δ|M˜| and input SOPs should be considered. Since a good polarimeter should have a completely random measurement error with zero mean, then ΔSout=ΔTout=ΔUout=0. stands for the mean of a random variable. Obviously, Δ|M˜|=0. Thus, we need to use the variance Var(Δ|M˜|) to evaluate the uncertainty. A smaller variance Var(Δ|M˜|) means the larger possibility of|M˜| of having a smaller measurement error. In this paper, we assume all SOP measurement errors follow the Gaussian distributions, that is, Δsoutj,Δtoutj,Δuoutj(j=1,2,3)N(0,σ2). Then, Var(Δ|M˜|) can be calculated as
Var(Δ|M˜|)=(Δ|M˜|)2=2Tu2σ2[(Ds+Dt)2+(Ds+Du)2+(Dt+Du)2(3cosαcosβcosγ)](3cosαcosβcosγ)2
(16)
Obviously, this variance depends not only on the relative relationship among three input SOPs, namely α, β and γ, but also on the relative relationship between three input SOPs and the PDL/G vector, namely θs, θt and θu. It also depends on the magnitudes of the PDL/G and the PIDL/G. When the PDL/G is small, that isD<<1, 1D21 and Ds=Dt=Du1 are tenable. Then
Var(Δ|M˜|)D<<12Tu2σ2KΔ|M|
(17)
where KΔ|M|=9+cosα+cosβ+cosγ(3cosαcosβcosγ)2.

Equation (17) means thatVar(Δ|M˜|)D<<1 is completely determined by α, β and γ if Tu is not taken into consideration. From Eq. (17), it is easy to know that Var(Δ|M˜|)D<<1 is the smallest when KΔ|M| takes its minimum value. Actually, all (α,β,γ), with the same KΔ|M|, form a curved surface as shown in Fig. 2
Fig. 2 Curved surfaces formed by (α,β,γ) with the same values ofKΔ|M|.
. From Fig. 2, KΔ|M| has the minimum value when α=β=γ=120, which means three input SOPs are coplanar and have angles of 120° between any two of them in Stokes space.

To illustrate more clearly, two special cases are plotted in Fig. 3
Fig. 3 KΔ|M| of Eq. (17) as a function of angles between three inputs (a) three angles are identical and (b) three inputs are coplanar and two of them are identical. The insets show the “zoom in” views of the same data.
: (a) α=β=γ and (b) α=β  and  γ=min(2α,2π2α). In Fig. 3(a), “orthogonal inputs” is equivalent toα=β=γ=90. In Fig. 3(b), “Jones inputs” is equivalent to α=β=90  and  γ=180.

5. Upper limit of ΔM˜

When D<<1, the matrix mD in Eq. (5) is almost equal to I. Then, the complex Mueller matrix M˜ is close to Tu(10T0mR). Since mRH=mRT=mR1 [23

23. S.-Y. Lu and R. A. Chipman; “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

], from Eq. (14), we have
ΔM˜Tu2Fin1ΔFout
(18)
where
ΔFout=1|M˜|(ΔFout1ΔFout2)
(19)
In Eq. (19), ΔFout1=(iΔsout0Δsout1   Δsout2Δsout3iΔtout0Δtout1   Δtout2Δtout3iΔuout0     Δuout1Δuout2Δuout3ΔAout0|M˜|ΔAout1|M˜|   ΔAout2|M˜|   ΔAout3|M˜|) and ΔFout2=Δ|M˜||M˜|(isout0   sout1   sout2   sout3itout0     tout1   tout2   tout3iuout0   uout1   uout2uout32Aout0|M˜|2Aout1|M˜|   2Aout2|M˜|   2Aout3|M˜|). By substituting Eq. (19) into Eq. (18), we have
ΔM˜Fin1ΔFout1Fin1ΔFout2=Tr(ΔFout1HFΔFout1+ΔFout2HFΔFout2ΔFout1HFΔFout2ΔFout2HFΔFout1)
(20)
where F=(Fin1)HFin1. Obviously, ΔM˜ depends on the actual noise realization in a single test. Similarly, the statistical parameter should be used to characterize its uncertainty. Unfortunately, we cannot directly calculate the mean of ΔM˜ because of the difficulties in mathematics. As an alternative, we can calculate its upper limit as
ΔM˜ΔM˜2=Tr(ΔFout1HFΔFout1)+Tr(ΔFout2HFΔFout2)Tr(ΔFout1HFΔFout2)Tr(ΔFout2HFΔFout1)
(21)
Under the condition thatD<<1, we can calculate the four terms in Eq. (21) as
Tr(ΔFout1HFΔFout1)=2σ2{2j=13fjj+f44[4(3cosαcosβcosγ)(1cosα)2(1cosβ)2(1cosγ)2]}
(22)
Tr(ΔFout2HFΔFout2)=Var(Δ|M˜|)D<<1|M˜|j=14k=14fjkΩjk
(23)
Tr(ΔFout1HFΔFout2)=Tr(ΔFout2HFΔFout1)=σ23cosαcosβcosγj=13k=13fjkΨjk
(24)
In Eqs. (22), (23) and (24), fjk are the elements of the matrix F, which are
{f11={[4(1+cosγ)2](Bin12+Bin22)8(1cosγ)Bin12}/(Bin12Bin22)2f22={[4(1+cosβ)2](Bin12+Bin22)8(1cosβ)Bin12}/(Bin12Bin22)2f33={[4(1+cosα)2](Bin12+Bin22)8(1cosα)Bin12}/(Bin12Bin22)2f44=2[4+2(1+cosα)(1+cosβ)(1+cosγ)(1+cosα)2(1+cosβ)2(1+cosγ)2]/(Bin12Bin22)2f12=f21={[(1+cosβ)(1+cosγ)2(1+cosα)](Bin12+Bin22)+4[(1+cosα)(1+cosβ)(1+cosγ)+2]}/(Bin12Bin22)2f13=f31={[(1+cosα)(1+cosγ)2(1+cosβ)](Bin12+Bin22)+4[(1+cosβ)(1+cosα)(1+cosγ)+2]}/(Bin12Bin22)2f23=f32={[(1+cosα)(1+cosβ)2(1+cosγ)](Bin12+Bin22)+4[(1+cosγ)(1+cosα)(1+cosβ)+2]}/(Bin12Bin22)2f14=f41=2iBin1[2(1+cosα)+2(1+cosβ)+(1+cosγ)2(1+cosα)(1+cosγ)(1+cosβ)(1+cosγ)4]/(Bin12Bin22)2f24=f42=2iBin1[2(1+cosα)+2(1+cosγ)+(1+cosβ)2(1+cosα)(1+cosβ)(1+cosβ)(1+cosγ)4]/(Bin12Bin22)2f34=f43=2iBin1[2(1+cosβ)+2(1+cosγ)+(1+cosα)2(1+cosα)(1+cosβ)(1+cosα)(1+cosγ)4]/(Bin12Bin22)2
(25)
where
{Bin1=|sin(tin×uin)|=1cos2αcos2βcos2γ+2cosαcosβcosγBin2=|(tinsin)×(uinsin)|=4(1cosβ)(1cosγ)(1+cosαcosβcosγ)2
(26)
In Eq. (23), the terms of Ωjk are given by
{Ω11=Ω22=Ω33=2,Ω44=4(Bin12+Bin22)Ω12=Ω21=1+cosα,Ω13=Ω31=1+cosβ,Ω23=Ω32=1+cosγΩ14=Ω24=Ω34=Ω41=Ω42=Ω43=4iBin1
(27)
In Eq. (24), the terms ofΨjk are given by
{Ψ11=2cosαcosβ,Ψ22=2cosαcosγ,Ψ33=2cosβcosγΨ12=Ψ13=1cosγ,Ψ21=Ψ23=1cosβ,Ψ31=Ψ32=1cosα
(28)
Finally, we have
ΔM˜ΔM˜2=KΔMσ
(29)
where
KΔM=2{2j=13fjj+f44[4(3cosαcosβcosγ)(1cosα)2(1cosβ)2(1cosγ)2]+KΔ|M|j=14k=14fjkΩjkj=13k=13fjkΨjk3cosαcosβcosγ}12
(30)
Apparently, this upper limit is completely determined by α, β and γ. From the calculation, we can find that KΔM has its minimum value also when α=β=γ=120. All (α,β,γ), with the same KΔM, form a curved surface as shown in Fig. 4
Fig. 4 Curved surfaces formed by (α,β,γ) with the same values ofKΔM.
.

To illustrate more clearly, two special cases are plotted in Fig. 5
Fig. 5 KΔM of Eq. (29) as a function of angles between three inputs (a) three angles are identical and (b) three inputs are coplanar and two of them are identical. The insets show the “zoom in” views of the same data.
: (a) α=β=γ and (b) α=β  and  γ=min(2α,2π2α). Strictly speaking, the upper limit shown in Eq. (29) is valid only when there is no PDL/G (D=0). However, when D<<1, it still approximately governs the statistical relationship between three input SOPs and the Mueller matrix measurement error ΔM˜.

6. Simulation results

To verify the theoretical finding in Section 5, 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: 0.02D0.1, Dvaries on the whole Poincaré sphere and 3) PIDL/G: Tu=1. In Section 5, theoretical result shows that the upper limit does not depend on the PIDL/G Tu. This is because we assume D<<1 and the variance σ2is not a function of Tu. Then, we take Tu=1 in the following simulations. Further, we take the first two input SOPs asSin=(i,1,0,0)Tand Tin=(i,cosα,sinα,0)T. In this paper, we only show the simulation results in the same two special cases as those in Section 4 and Section 5: (a) α=β=γand (b) α=β  and  γ=min(2α,2π2α).

In (a),
Uin=(i,cosα,cosαcos2αsinα,sin4α(cosαcos2α)2sinα)T
(31)
and in (b),

Uin=(i,cosα,sinα,0)T
(32)

In the simulations, ΔM˜ is calculated using 1000 independent noise realizations with σ=0.03. For three 4-dimensional Stokes vectors, this means that 12000 random values have been generated. In Fig. 6
Fig. 6 Simulation results of ΔM˜as a function of angles between three inputs corresponding to different values of D (a) three angles are identical and (b) three inputs are coplanar and two of them are identical.
, simulation results, with D=(0,0,1) and different values of D, show that the theoretical upper limit (ULimit) is valid when D0.04 (0.35dB). When D is less than 0.35 dB, α=β=γ=120 can lead to a measurement error that is very close to the minimum. In fact, the real minimum measurement errors, corresponding to different values of D, have been shown in Fig. 6(a). However, to achieve these minimums, the PDL/G vectors must be known before the measurement. If the PDL/G vector is unknown, α=β=γ=120 can be considered as the substitute for the best inputs whenD<<1.

Next, we need to confirm the validity of the above conclusion for all D when D0.04. In Fig. 7
Fig. 7 Simulation results of ΔM˜as a function of angles between three inputs corresponding to different values of D (a) three angles are identical and (b) three inputs are coplanar and two of them are identical.
, D=0.04and D varies on the whole Poincaré sphere. Results show that α=β=γ=120 can be considered as the substitute for the best inputs whatever D is. Based on these results, we can use(1,1,0,0)T, (1,0.5,0.866,0)T and (1,0.5,0.866,0)Tas the standard input SOPs in the measurements. From the results shown in Fig. 6 and Fig. 7, these standard input SOPs can result in obviously better measurement accuracy than Jones inputs and orthogonal inputs.

7. Conclusion

We presented the statistical relationship between three input SOPs and the Mueller matrix measurement error in a system having small PDL/G. This statistical relationship is expressed as an upper limit of the measurement error, which is approximately valid when the PDL/G is small. Based on this upper limit, the minimum Mueller matrix measurement error will be statistically achieved when three input SOPs are coplanar with an angle of 120° between any two of them in Stokes space. From the simulations, these input SOPs are confirmed to lead to a measurement error that is very close to the minimum when the PDL/G of the system under test is less than 0.35 dB. The standard input SOPs (1,1,0,0)T, (1,0.5,0.866,0)T and (1,0.5,0.866,0)Tcan be suggested since they have obviously larger probability to result in better Mueller matrix measurement accuracy than Jones inputs and orthogonal inputs.

Further, if the PDL/G is not very small, three optimum input SOPs will depends on the PDL/G vector. They will be not coplanar any more, but still equally separated. A detailed analysis will be presented in another paper.

Acknowledgements

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

References and Links

1.

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

2.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004). [CrossRef]

3.

R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998). [CrossRef]

4.

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]

5.

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission though optical devices,” IEEE Photon. Technol. Lett. 4(5), 451–454 (1992). [CrossRef]

6.

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4(9), 1066–1069 (1992). [CrossRef]

7.

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992). [CrossRef] [PubMed]

8.

S.-M. F. Nee, “Depolarization and principal mueller matrix measured by null ellipsometry,” Appl. Opt. 40(28), 4933–4939 (2001). [CrossRef]

9.

B. J. Howell, “Measurement of the polarization effect of an instrument using partially polarized light,” Appl. Opt. 18(6), 809–812 (1979). [CrossRef] [PubMed]

10.

D. B. Chenault, R. A. Chipman, K. M. Johnson, and D. Doroski, “Infrared linear diattenuation and birefringence spectra of ferroelectric liquid crystals,” Opt. Lett. 17(6), 447–449 (1992). [CrossRef] [PubMed]

11.

H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).

12.

R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. 37(2), 110–112 (1947). [CrossRef]

13.

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]

14.

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

15.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34(6), 1656–1658 (1995). [CrossRef]

16.

J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002). [CrossRef] [PubMed]

17.

Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. 46(30), 7354–7364 (2007). [CrossRef] [PubMed]

18.

S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20(8), 1651–1657 (2003). [CrossRef]

19.

D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7(4), 693–700 (1990). [CrossRef]

20.

H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008). [CrossRef]

21.

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53(3), 317–323 (1963). [CrossRef]

22.

R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981). [CrossRef]

23.

S.-Y. Lu and R. A. Chipman; “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

OCIS Codes
(060.2300) Fiber optics and optical communications : Fiber measurements
(060.2310) Fiber optics and optical communications : Fiber optics
(260.3090) Physical optics : Infrared, far
(260.5430) Physical optics : Polarization

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 7, 2009
Revised Manuscript: June 21, 2009
Manuscript Accepted: June 23, 2009
Published: July 15, 2009

Citation
H. Dong, Y. D. Gong, Varghese Paulose, P. Shum, and Malini 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, 13017-13030 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13017


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References

  1. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997). [CrossRef]
  2. J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004). [CrossRef]
  3. R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998). [CrossRef]
  4. 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]
  5. B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission though optical devices,” IEEE Photon. Technol. Lett. 4(5), 451–454 (1992). [CrossRef]
  6. B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4(9), 1066–1069 (1992). [CrossRef]
  7. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992). [CrossRef] [PubMed]
  8. S.-M. F. Nee, “Depolarization and principal mueller matrix measured by null ellipsometry,” Appl. Opt. 40(28), 4933–4939 (2001). [CrossRef]
  9. B. J. Howell, “Measurement of the polarization effect of an instrument using partially polarized light,” Appl. Opt. 18(6), 809–812 (1979). [CrossRef] [PubMed]
  10. D. B. Chenault, R. A. Chipman, K. M. Johnson, and D. Doroski, “Infrared linear diattenuation and birefringence spectra of ferroelectric liquid crystals,” Opt. Lett. 17(6), 447–449 (1992). [CrossRef] [PubMed]
  11. H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).
  12. R. C. Jones, “A new calculus for the treatment of optical systems VI. Experimental determination of the matrix,” J. Opt. Soc. Am. 37(2), 110–112 (1947). [CrossRef]
  13. 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]
  14. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995). [CrossRef]
  15. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34(6), 1656–1658 (1995). [CrossRef]
  16. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002). [CrossRef] [PubMed]
  17. Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. 46(30), 7354–7364 (2007). [CrossRef] [PubMed]
  18. S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20(8), 1651–1657 (2003). [CrossRef]
  19. D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7(4), 693–700 (1990). [CrossRef]
  20. H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008). [CrossRef]
  21. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53(3), 317–323 (1963). [CrossRef]
  22. R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981). [CrossRef]
  23. S.-Y. Lu and R. A. Chipman; “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996). [CrossRef]

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