## Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach |

Optics Express, Vol. 20, Issue 18, pp. 20466-20481 (2012)

http://dx.doi.org/10.1364/OE.20.020466

Acrobat PDF (1547 KB)

### Abstract

Dual photoelastic modulator polarimeter systems are widely used for the measurement of light beam polarization, most often described by Stokes vectors, that carry information about an interrogated sample. The sample polarization properties can be described more thoroughly through its Mueller matrix, which can be derived from judiciously chosen input polarization Stokes vectors and correspondingly measured output Stokes vectors. However, several sources of error complicate the construction of a Mueller matrix from the measured Stokes vectors. Here we present a general formalism to examine these sources of error and their effects on the derived Mueller matrix, and identify the optimal input polarization states to minimize their effects in a dual photoelastic modulator polarimeter configuration. The input Stokes vector states leading to the most robust Mueller matrix determination are shown to form Platonic solids in the Poincaré sphere space; we also identify the optimal 3D orientation of these solids for error minimization.

© 2012 OSA

## 1. Introduction

*I*,

*Q*,

*U*, and

*V*, where

*I*encodes the intensity,

*Q*and

*U*give the degree and orientation of linear polarization, and

*V*gives the degree and direction of circular polarization [1, 2

2. E. Collett, *Field Guide to Polarization* (SPIE Press, 2005). [CrossRef]

*S*= [

*I*,

*Q*,

*U*,

*V*]

^{⊤}, which is said to be ‘normalized’ when a factor of

*I*is divided out (thus making the first element equal to 1). While Stokes vectors are non-Euclidian (i.e. they have no magnitude or direction in a physical sense) it is still possible to interpret them geometrically: when normalized, the parameters

*q*=

*Q/I*,

*u*=

*U/I*, and

*v*=

*V/I*(referred to as ‘polarization parameters’) all fall between −1 and 1. In a Stokes vector representing fully polarized light, the normalized polarization parameters satisfy

*q*

^{2}+

*u*

^{2}+

*v*

^{2}= 1, and so if we consider a 3-dimensional plot where

*x*=

*q*,

*y*=

*u*and

*z*=

*v*, a vector

*ξ⃗*= [

*q*,

*u*,

*v*]

^{⊤}representing full polarization lies on a sphere of unit radius centred about the origin, which is known as the ‘Poincaré sphere’ [2

2. E. Collett, *Field Guide to Polarization* (SPIE Press, 2005). [CrossRef]

3. H. Poincaré, *Théorie mathématique de la lumière* (Gauthiers-Villars, 1892). [PubMed]

*ξ⃗*represents partial polarization, it will lie inside the Poincaré sphere, and if it represents random polarization, it will be equal to 0⃗.

*S*, interacts with a material, its polarization state usually changes, and the modified beam will have a corresponding Stokes vector

^{in}*S*. The change in polarization can be represented by the matrix/vector equation where

^{out}*M*is called the ‘Mueller matrix’, and it fully describes the material’s effect on polarized light [2

2. E. Collett, *Field Guide to Polarization* (SPIE Press, 2005). [CrossRef]

5. N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton. **2**, 145–156 (2009). [CrossRef]

6. D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. **9**, 213–220 (2004). [CrossRef] [PubMed]

7. X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt. **11**, 041105 (2006). [CrossRef] [PubMed]

*κ*, which quantifies the numerical robustness of the system [9

9. A. Ambirajan and D. C. Look Jr., “Optimum angles for a polarimeter: part 1,” Opt. Eng. **34**, 1651–1655 (1995). [CrossRef]

21. J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. **8**, 807–814 (2006). [CrossRef]

*A*, and then calculates the corresponding condition number where ||·|| is generally an induced norm (as shown in Eq. (3)). The condition number quantifies how ‘invertible’

*A*is, where a small value of

*κ*means very invertible, and a large value means nearly singular. In practice, the determination of a sample’s Mueller matrix usually requires one to numerically invert a matrix which represents the system or the measured data; a popular strategy is to orient/adjust the system elements such that

*κ*(

*A*) is minimized, so as to make this inversion as robust as possible, which in turn minimizes the resulting sample Mueller matrix’s sensitivity to measurement errors.

*p*-norms are used to induce a matrix norm on

*A*(as in Eq. (3)), although this approach lacks a clear physical interpretation: suppose the matrix

*A*acts on a Stokes vector

*S*(as is often the case), then it is difficult to assign any physical meaning to the

*p*-norm ||

*S*||

*= (*

_{p}*I*+

^{p}*Q*+

^{p}*U*+

^{p}*V*)

^{p}^{1/}

*, and thus even more difficult to interpret the induced matrix norm*

^{p}20. I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express **16**, 2091–2108 (2008). [CrossRef] [PubMed]

*κ*with different norms resulted in significantly different system configurations. The state of affairs was well summarized by Twietmeyer and Chipman [17

17. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express **16**, 11589–11603 (2008). [CrossRef] [PubMed]

23. W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. **103**, 043104 (2008). [CrossRef]

## 2. Analysis

23. W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. **103**, 043104 (2008). [CrossRef]

*n*different states, which should first be passed directly into the analyzer (the dual PEM system) so as to measure the Stokes vector of each, before repeating the process with the sample in the beam path. The first set of Stokes vectors (those coming directly from the PSG) will be referred to as ‘input states’, and the second set (the ones measured with the sample in place) as ‘output states’.

*Since the experimenter is free to choose the input Stokes vectors, it is desirable to find the set(s) of such vectors which will produce the most error-resistant Mueller matrix results.*Note that while the signal intensity

*I*can be useful in polarimetry, we chose to work with normalized Stokes vectors, partly for mathematical simplicity and partly to account for its uncontrollable fluctuations, and so all input states considered in this paper are assumed to be normalized (i.e.,

*I*= 1).

### 2.1. Determining Mueller matrices from Stokes vectors

*n*≤ 4 this a true (rather than approximate) equality, although for 1 ≤

*n*≤ 3 it is not possible to isolate

*M*. When

*n*= 4, the Mueller matrix can be determined as

*M*= 𝕊

*(𝕊*

^{out}*)*

^{in}^{−1}, provided that 𝕊

*is invertible. Taking*

^{in}*n*> 4 can be useful, as such an approach can potentially reduce noise and generally improve accuracy [11

11. P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express **18**, 23095–23103 (2010). [CrossRef] [PubMed]

16. M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. **41**, 2488–2493 (2002). [CrossRef] [PubMed]

20. I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express **16**, 2091–2108 (2008). [CrossRef] [PubMed]

*M*which satisfies the equality, we must instead find

*M*which provides the best fit, or rather, which minimizes the RMS size of the discrepancy 𝕊

*−*

^{out}*M*𝕊

*, where the RMS of a matrix*

^{in}*A*will be denoted 〈

*A*〉. Note that for the rest of this paper the notation || · || will be understood to represent the Frobenius norm, and 〈·,·〉 the matrix inner product from which it follows.

_{4×4}→ ℝ (where 𝕄

_{4×4}denotes the set of all 4 × 4 matrices) as f(

*X*) = 〈𝕊

*−*

^{out}*X*𝕊

*〉, we have that f(*

^{in}*M*) is a minimum. In analogy to variational calculus, we then perturb this minimum with an arbitrary matrix

*η*multiplied by a scalar

*ε*, and set the derivative

*∂*f(

_{ε}*M*+

*εη*)

^{2}(we consider the RMS squared for convenience) to zero: for any

*η*. It follows immediately from the properties of inner products ([8] Section 6.1) that 𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}

*M*

^{⊤}− 𝕊

*(𝕊*

^{in}*)*

^{out}^{⊤}= 0 if the equality is to hold for arbitrary

*η*∈ 𝕄

_{4×4}. Isolating for

*M*, we find that the Mueller matrix providing the best RMS fit is where (𝕊

*)*

^{in}^{+}= (𝕊

*)*

^{in}^{⊤}[𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}]

^{−1}is the well-known Moore-Penrose pseudoinverse of 𝕊

*[11*

^{in}11. P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express **18**, 23095–23103 (2010). [CrossRef] [PubMed]

14. 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**, 802–804 (2000). [CrossRef]

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

20. I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express **16**, 2091–2108 (2008). [CrossRef] [PubMed]

*n*= 4,

*M*

^{+}reduces to

*M*

^{−1}, and so Eq. (6) becomes

*M*= 𝕊

*(𝕊*

^{out}*)*

^{in}^{−1}when 𝕊

*is square. Thus, the more general pseudoinverse encompasses the traditional matrix inverse.*

^{in}### 2.2. Quantifying Mueller matrix errors

*S*, but that our measurements are subject to some error, and rather than obtaining the ‘true’ vector

*S*, we measure

*S*+

*δS*, where

*δS*is composed of the random errors associated with each Stokes parameter of

*S*. While it is straightforward to measure and quantify the random errors associated with measured Stokes vectors, it is somewhat more complicated to do so with the Mueller matrix calculated from these vectors.

*δM*is the deviation from the ‘true’ Mueller matrix of the sample that results from error-prone Stokes vector measurements. We will define the error matrices

*δ*𝕊

*and*

^{in}*δ*𝕊

*) and Mueller error (*

^{out}*δM*) are unknown; the experimenter will determine the Mueller matrix of the sample in question as

*M*+

*δM*, by applying Eq. (6) to get

*δM*on 𝕊

*from this equation, and so it is necessary to invoke the matrix analogue of a Taylor series, which will be applied to the pseudoinverse in Eq. (9). For a reasonably well-behaved function (more precisely, a*

^{in}*C*

^{1}function) g : ℝ → ℝ, the Taylor series truncated to first order provides a very good approximation for g on a small interval. (In more exact terms, g(

*x*

_{0}+d

*x*) ≈ g(

*x*

_{0}) + dg = g(

*x*

_{0}) + g′(

*x*

_{0})d

*x*is a good approximation for small d

*x*.) In order to linearize the pseudoinverse, we will use the analogous relation where ||

*B*|| is small. The differential of the pseudoinverse, as found by Golub and Pereyra [24

24. G. H. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Numer. Anal. **10**, 413–432 (1973). [CrossRef]

*I*is the

_{n}*n*×

*n*identity matrix. Substituting Eq. (11) into Eq. (10) and evaluating for

*A*= 𝕊

*and*

^{in}*∂A*=

*δ*𝕊

*we can linearize Eq. (9). When expanding said equation, it is possible to make some simplifications. By carrying out the multiplication, we find that we can use Eq. (6) to get additive*

^{in}*M*terms on both sides of the equation, which can be cancelled. Also, given the physical nature of the problem, we can assume that 𝕊

*≃*

^{out}*M*𝕊

*is a very good approximation, and thus we can treat it as an equality when performing simplifications. Finally, we find that there are cross terms involving both*

^{in}*δ*𝕊

*and*

^{in}*δ*𝕊

*. Terms formed with both of these matrices should be exceedingly small, and since we have already truncated our approximation to first order by linearizing, such terms can be neglected. Applying these simplifications leaves*

^{out}*, the matrix of input Stokes vectors, over which the experimenter has full control. This result indicates that the propagation of random errors into the Mueller matrix depends strongly on choice of input polarization states.*

^{in}### 2.3. Maximizing the robustness of determined Mueller matrices

#### 2.3.1. Root mean square of Mueller errors

*δM*〉, be minimized. This global error reduction is essential since Mueller matrices are frequently used to measure physical properties of a sample, often by considering individual matrix elements [6

6. D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. **9**, 213–220 (2004). [CrossRef] [PubMed]

5. N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton. **2**, 145–156 (2009). [CrossRef]

*AB*|| ≤ ||

*A*|| ||

*B*|| (which follows quickly from the Cauchy-Schwarz inequality) and the triangle inequality ||

*A*+

*B*|| ≤ ||

*A*|| + ||

*B*||, which gives us the upper bound on the RMS Mueller error

*)*

^{in}^{+}, which is dependent on the polarization states chosen as inputs. The goal then becomes to find a set of input Stokes vectors such that ||(𝕊

*)*

^{in}^{+}|| is minimized, so as to minimize the effect of the Stokes errors on 〈

*δM*〉.

*)*

^{in}^{+}|| decreases as

*n*grows, which can be interpreted physically as an ‘averaging out’ of errors. On the other hand, 〈

*δM*〉 grows proportionally to

*n*

^{1/2}, which can be interpreted as resulting from accumulation of additional noisy measurements. To analyze the balance between these two effects, we will first consider ||(𝕊

*)*

^{in}^{+}||, before combining the two factors.

#### 2.3.2. Optimal selections of *n* input Stokes vectors

*contains the Stokes vectors representing the input beams used to determine the sample’s Mueller matrix, and so the experimenter has full control over it. However, it is not immediately clear how the choice of input Stokes vectors relates to ||(𝕊*

^{in}*)*

^{in}^{+}||, the norm of the pseudoinverse, and thus the resultant uncertainty in

*M*via Eq. (13).

*(𝕊*

^{in}*)*

^{in}^{⊤}is symmetric, which implies that there exist matrices Ω and

*D*such that 𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}= Ω

*D*Ω

^{−1}where

*D*is a 4 × 4 diagonal matrix containing the eigenvalues of 𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}:

*λ*

_{1},...,

*λ*

_{4}([8] Section 6.5). Taking the inverse of both sides, we find that [𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}]

^{−1}= Ω

*D*

^{−1}Ω

^{−1}, where

*D*

^{−1}= diag(1/

*λ*

_{1},..., 1/

*λ*

_{4}). Since the trace is invariant under eigendecomposition ([8] Section 6.5), the norm of the pseudoinverse becomes

*(𝕊*

^{in}*)*

^{in}^{⊤}

*v⃗*=

*λv⃗*. Multiplying both sides to the left by

*v⃗*

^{⊤}, this becomes ||(𝕊

*)*

^{in}^{⊤}

*v⃗*||

^{2}=

*λ*||

*v⃗*||

^{2}, which implies that the eigenvalues cannot be negative (since ||

*x⃗*|| ≥ 0 ∀

*x⃗*). Thus, to minimize the norm of ||(𝕊

*)*

^{in}^{+}|| we must maximize the eigenvalues of 𝕊

*(𝕊*

^{in}*)*

^{in}^{−1}.

^{2}and ℝ

^{3}respectively. To extend this idea to 4 dimensions, we would like to find a matrix 𝕊

*such that 𝕊*

^{in}*(𝕊*

^{in}*)*

^{in}^{⊤}is diagonal. (Technically, we only need the columns of 𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}to be orthogonal, although since the determinant is invariant under eigendecomposition, we can consider the diagonal case without loss of generality.)

*n*matrix 𝕊

*whose rows are formed by the*

^{in}*n*-dimensional vectors

*to be orthogonal. To confirm that such a diagonal matrix does in fact produce the largest possible determinant, it is useful to examine the matrix derivative [25*

^{in}25. J. Dattorro, *Convex Optimization & Euclidean Distance Geometry* (Meboo Publishing USA, 2005). [PubMed]

*X*= 𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}= diag(

*λ*

_{1},...,

*λ*

_{4}), we see that the derivative is also a diagonal matrix. This implies that perturbing any of the off-diagonal elements of 𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}will decrease det[𝕊

*(𝕊*

^{in}*)*

^{in}^{⊤}] since their current values (of zero) already represent critical points. The elements lying along the diagonal, however, are non-zero since they are not critical points (i.e. det(

*X*) increases along with any of the diagonal elements), although we shall see that they are constrained by the physical nature of the problem.

*: its columns are formed by the input Stokes vectors (assumed to be normalized), which represent fully polarized light. This physical interpretation limits the magnitude of the rows of 𝕊*

^{in}*, and so for a given number of input states (columns of 𝕊*

^{in}*),*

^{in}*n*, this puts an upper bound on the determinant in question.

*is composed as and so for all 1 ≤*

^{in}*i*≤

*n*we have that

*n*, the number of Stokes vectors used as inputs. (The norm squared of the first row is necessarily equal to

*n*.)

*to be orthogonal in order to minimize the Mueller error. Since the first row is composed entirely of ones, the dot product*

^{in}*r⃗*

_{1}·

*r⃗*= ∑

_{i}*(*

_{j}*r⃗*)

_{i}*must be equal to zero for 2 ≤*

_{j}*i*≤ 4, and so the sum of each particular polarization parameter (e.g.

*r⃗*

_{2}|| =

*n*and the other rows be zero. Of course, such a solution would result in a determinant of zero, in which case (𝕊

*)*

^{in}^{+}would not exist, and there would be no way to find the Mueller matrix. Clearly it is necessary to examine the rows individually, and to do so we will make use of Lagrange multipliers ([26] Section 14.8).

*)*

^{in}^{+}, which was shown to be equivalent to minimizing

*λ*are the eigenvalues of 𝕊

_{i}*(𝕊*

^{in}*)*

^{in}^{⊤}, for 1 ≤

*i*≤ 4. Having imposed the condition of orthogonality, we know that

*λ*

_{1}=

*n*, and

*λ*= ||

_{j}*r⃗*||

_{j}^{2}, for 2 ≤

*j*≤ 4. Thus, we wish to minimize

*r⃗*||

_{j}^{2}=

*n*/3 for all 2 ≤

*j*≤ 4.

*n*= 4: the tetrahedron is one of five ‘Platonic Solids’ [29

29. M. Atiyah and P. Sutcliffe, “Polyhedra in physics, chemistry and geometry,” Milan J. Math. **71**, 33–58 (2003). [CrossRef]

*n*= 4 this corresponds to a tetrahedron, for

*n*= 6 it is an octahedron, for

*n*= 8 it is a cube, for

*n*= 12 an icosahedron, and finally for

*n*= 20, a dodecahedron. These solutions are represented graphically in Fig. 2.

*)*

^{in}^{+}|| for a given

*n*≥ 4 is

*n*here (e.g. the ones forming the vertices of Platonic solids on the Poincaré sphere), there are almost surely other solutions to Eqs. (21)–(23) for values of

*n*which we have not considered. However, Eq. (24) allows us to examine the effects of varying

*n*analytically, without having to find other solutions to the derived conditions, thus allowing us to compare the merits of all optimal configurations without having to know the exact (geometrical) form of each.

#### 2.3.3. Optimal number of measurements

*n*≥ 4 one is able to find an optimal configuration of Stokes vectors (such as the ones presented in Fig. 2), the effect of

*n*, the number of input/output Stokes vectors, on the Mueller error has yet to be determined.

*n*. And so, the question of

*which*configurations (in Fig. 2 or otherwise) to use remains unaddressed using this method. As discussed in Section 1, the answer must come from considering the unique types of errors which can occur in dual PEM Stokes polarimeters.

#### 2.3.4. Errors in relative modulation phase

*Q*,

*U*, and

*V*(i.e. the polarization parameters) are measured by multiplying the magnitude of the frequency-specific oscillations in the signal by the sign of the relative phase between the signal oscillations and the PEM reference frequency. Here we consider a system like the one described by Guan et al. [23

23. W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. **103**, 043104 (2008). [CrossRef]

*α*= 45° and

*β*= 22.5°, as shown in Fig. 1(b)), and so to measure

*U*, for example, we use

*U*= |

*U*|sgn(

*θ*), where the absolute value of

*U*is the amplitude measured by the lock-in amplifier and

*θ*is the relative phase. Thus far we have been considering noise in the final Stokes parameters rather than in their constituent quantities: the measured amplitudes and relative phases. While this approach was useful and remains valid, there is another type of error specific to this kind of system. If the relative phase

*θ*is close to 0, then fluctuations (noise in

*θ*) which cause it to jump above and below 0 can result in enormous errors if the amplitude is nonzero.

*Q*| = |

*U*| = |

*V*| for all measured Stokes vectors. Of course, the experimenter has no control over the output vectors without

*a priori*knowledge of the Mueller matrix being examined, and so in order to develop a general framework, we must settle for a reduction of input vector phase errors.

*Q*| = |

*U*| = |

*V*| (or rather, where |

*q*| = |

*u*| = |

*v*|, although the two conditions are equivalent for any given Stokes vector since the dividing

*I*term is the same for all of the parameters), and that these form the vertices of a cube—one of the optimal configurations found in Section 2.3.2 (shown in Fig. 2(c) Media 3). However, before concluding that said cubic configuration provides a global optimum, it is important to note that there are infinitely many ways to rotate a cube inside the Poincaré sphere (this holds true for all solutions in Fig. 2), and it is not immediately clear what effect such a rotation has.

*R*(

*ϕ*,

*θ*,

*ψ*) which rotates vectors in ℝ

^{3}by the Euler angles ([30] Section 13.8). The analogous matrix for Stokes vectors which rotates the polarization components around the Poincaré sphere by the same angles but leaves the intensity unchanged is given by where 0⃗ = [0, 0, 0]

^{⊤}. Thus, for a matrix 𝕊

*of input Stokes vectors,*

^{in}*R*′𝕊

*describes all possible rotations of the polarization components inside the Poincaré sphere. Since*

^{in}*R*is a rotation matrix, it is orthogonal ([8] Section 6.10), and it is trivial to show that

*R*′ must be orthogonal as well (i.e. that (

*R*′)

^{⊤}= (

*R*′)

^{−1}). Consider a matrix of input states 𝕊

*, whose column Stokes vectors are rotated arbitrarily in polarization space (the Poincaré sphere). The pseudoinverse of the resulting matrix is*

^{in}*R*′𝕊

*)*

^{in}^{+}|| = ||(𝕊

*)*

^{in}^{+}||, and so it is invariant under an arbitrary rotation. Thus, any set of input Stokes vectors, including the optimal configurations found in Section 2.3.2 can be freely rotated in polarization space.

*Consequently, not only does a cubic configuration of Stokes vectors (*Fig. 2(c)) yield maximum Mueller matrix robustness to noise, but it also minimizes the risk of phase errors, since it is the only solid that can be naturally oriented such that every vertex falls precisely in the areas which are least prone to phase errors.

## 3. Simulated results

*n*≥ 4 Stokes vectors at random, all of which were normalized and represented fully polarized states. (The results presented here are based on the experimentally-derived matrix considered by Ghosh et al. [5

5. N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton. **2**, 145–156 (2009). [CrossRef]

*Q*,

*U*, and

*V*) with a probability shown in Fig. 4, such that as each corresponding normalized parameter approached ±1, the probability of a phase error approached 0, and as each approached 0 the probability of a phase error approached 1/2.

*M*+

*δM*, as in Eq. (9), from which the original Mueller matrix was subtracted to isolate

*δM*. This procedure was repeated for ∼ 10

^{4}sets of randomly chosen Stokes vectors, uniformly distributed over 4 ≤

*n*≤ 20. The results of each set are shown as blue dots in Fig. 5 ( Media 7). To test the validity of the results from Section 2.3, we also performed the same procedure 50 times using the derived cubic configuration, and plotted the maximum observed Mueller error, 〈

*δM*〉

*in red on Fig. 5 ( Media 7) for comparison.*

_{max}*δM*〉 and ||(𝕊

*)*

^{in}^{+}|| is very large for 4 randomly chosen input Stokes vectors, but as

*n*increases, the points become more tightly clustered and the Mueller error decreases. Fig. 6 shows the various 2-dimensional projections of this plot.

*)*

^{in}^{+}|| decrease as

*n*grows. The theoretically predicted minimum in Eq. (25) has been overlaid on the plot and provides an excellent fit to the lower bound, which suggests that for every

*n*, some of the randomly chosen sets of input vectors correspond approximately to optimal configurations, like the ones shown in Fig. 2. These optimal sets are the ones for which 〈

*δM*〉 is minimized, for a given

*n*. Examining Fig. 6(b) ( Media 9) we see that the lower bound of the Mueller error is approximately constant for all

*n*. Since the states with the lowest Mueller error should correspond to the optimal configuration of Stokes vectors, the observed result that 〈

*δM*〉

*is largely independent of*

_{min}*n*is consistent with Eq. (25). Finally, examining Fig. 6(c) ( Media 10) we see that the cubic configuration (the red dot) does in fact provide the lowest Mueller error of all of the simulated configurations, and therefore it allows for the most robust possible Mueller matrix measurements with a dual PEM Stokes polarimeter.

## 4. Discussion

14. 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**, 802–804 (2000). [CrossRef]

15. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. **41**, 965–972 (2002). [CrossRef]

17. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express **16**, 11589–11603 (2008). [CrossRef] [PubMed]

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

27. A. Ambirajan and D. C. Look Jr., “Optimum angles for a polarimeter: part 2,” Opt. Eng. **34**, 1656–1658 (1995). [CrossRef]

28. R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A **5**, 681–689 (1988). [CrossRef]

## 5. Conclusions

*optimal configurations*of input Stokes vectors, and we found that vectors forming Platonic solids on the Poincaré sphere satisfy these conditions. We found that, to first order, the number of Stokes vectors in an

*optimal set*does not affect the robustness of the Mueller matrix measurements, and simulations confirmed this result. Finally, we deduced that in the case of a dual PEM Stokes polarimeter which is prone to phase errors, input Stokes vectors forming a cube on the Poincaré sphere whose vertices are equidistant from the areas prone to such errors represents a global optimum—a result which was also confirmed by simulation. We plan to report on experimental tests of these results in a future publication. Overall, this work presents a general framework for polarimetric optimization strategy, as well as furnishing practical ‘recipes’ for an optics researcher viz. experimental methodology for robust Mueller matrix determination with a dual-PEM polarimeter.

## References and links

1. | G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Phil. Soc. |

2. | E. Collett, |

3. | H. Poincaré, |

4. | H. Mueller, “Memorandum on the polarization optics of the photoelastic shutter,” Report No. 2 of the OSRD project OEMsr-576, (1943). |

5. | N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton. |

6. | D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. |

7. | X. Guo, M. F. G. Wood, and I. A. Vitkin, “Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter,” J. Biomed. Opt. |

8. | S. H. Friedberg, A. J. Insel, and L. E. Spence, |

9. | A. Ambirajan and D. C. Look Jr., “Optimum angles for a polarimeter: part 1,” Opt. Eng. |

10. | E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films |

11. | P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express |

12. | A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films |

13. | A. D. Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. |

14. | 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. |

15. | S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. |

16. | M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. |

17. | K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express |

18. | J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. |

19. | J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. |

20. | I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express |

21. | J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. |

22. | G. H. Golub and C. F. V. Loan, |

23. | W. Guan, G. A. Jones, Y. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. |

24. | G. H. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Numer. Anal. |

25. | J. Dattorro, |

26. | J. Stewart, |

27. | A. Ambirajan and D. C. Look Jr., “Optimum angles for a polarimeter: part 2,” Opt. Eng. |

28. | R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A |

29. | M. Atiyah and P. Sutcliffe, “Polyhedra in physics, chemistry and geometry,” Milan J. Math. |

30. | A. P. Arya, |

**OCIS Codes**

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(170.1470) Medical optics and biotechnology : Blood or tissue constituent monitoring

(170.4090) Medical optics and biotechnology : Modulation techniques

(230.4110) Optical devices : Modulators

(260.5430) Physical optics : Polarization

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 6, 2012

Revised Manuscript: July 19, 2012

Manuscript Accepted: July 22, 2012

Published: August 21, 2012

**Virtual Issues**

Vol. 7, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

D. Layden, M. F. G. Wood, and I. A. Vitkin, "Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach," Opt. Express **20**, 20466-20481 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20466

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### References

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- E. Collett, Field Guide to Polarization (SPIE Press, 2005). [CrossRef]
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- N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton.2, 145–156 (2009). [CrossRef]
- D. Côté and I. A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt.9, 213–220 (2004). [CrossRef] [PubMed]
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- E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004). [CrossRef]
- P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23095–23103 (2010). [CrossRef] [PubMed]
- A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films455, 112–119 (2004). [CrossRef]
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- K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express16, 11589–11603 (2008). [CrossRef] [PubMed]
- J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt.41, 619–630 (2002). [CrossRef] [PubMed]
- J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett.25, 1198–1200 (2000). [CrossRef]
- I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express16, 2091–2108 (2008). [CrossRef] [PubMed]
- J. Zallat, S. Aïnouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt.8, 807–814 (2006). [CrossRef]
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- J. Stewart, Calculus Early Transcendentals, 6th ed. (Thompson Brooks/Cole, 2008).
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- R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, “General analysis and optimization of the four-detector photopolarimeter,” J. Opt. Soc. Am. A5, 681–689 (1988). [CrossRef]
- M. Atiyah and P. Sutcliffe, “Polyhedra in physics, chemistry and geometry,” Milan J. Math.71, 33–58 (2003). [CrossRef]
- A. P. Arya, Introduction to Classical Mechanics, 2nd ed. (Prentice-Hall, 1998).

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