Separating azimuthal orientation dependence in polarization measurements of anisotropic media

Pengcheng Li; Donghong Lv; Honghui He; Hui Ma

doi:10.1364/OE.26.003791

1. Introduction

Polarized light measurements can provide abundant information on the microstructure and optical properties of the sample, and have been widely applied to material and biomedical studies [1,2]. For instance, degree of polarization (DOP) imaging can enhance the contrast between cancerous and healthy region in human skin [3,4]. Further studies show that Mueller matrix imaging can serve as a noninvasive tool for the diagnosis of colon [5], cervix [6], and thyroid [7] cancerous tissues. However, the value of many polarization parameters, such as the DOP, and the majority of Mueller matrix elements, are often sensitive to the orientation of anisotropic samples [6,8], or the reference orientation of the polarization state generator and analyzer. Therefore, separating the orientation dependence in these parameters becomes an important task for quantitatively characterizing anisotropic samples.

Previous studies provide many methods that can separate characteristic properties of Mueller matrix from the orientation effects. Lu and Chipman [9] proposed Mueller matrix polar decomposition (MMPD) method to obtain the value of diattenuation, retardance, depolarization and the orientation of extraordinary axis of linear retardance. Ossikovski [10] proposed differential formalism to separate elementary polarization properties and their uncertainties. Jiang et al. [11,12] proposed the rotating linear polarization imaging (RLPI) technique which reduces the orientation dependence in the upper left 3 × 3 Mueller matrix elements. He et al. [13] extended this strategy to full 4 × 4 Mueller matrix and named it as Mueller matrix transformation (MMT). He proposed rotation invariant parameter b to sense the scattering by sub-wavelength particles, and t₁ or A to describe the degree of anisotropy. The MMT method also allows us to extract the orientations of anisotropy [14], but could be affected when the illumination or detection directions are not normal to the sample [15]. Arteaga et al. [16] studied three types of anisotropic Mueller matrices that maintain the 0, 90° linear, ±45° linear or circular polarized states, and proposed corresponding anisotropy coefficients such as circular anisotropy (m₁₄ + m₄₁) and (m₂₃ − m₃₂). Ushenko et al. [17] studied the effects of linear birefringence and dichroism, proposed azimuth invariant parameters m₁₁, m₄₄, (m₂₂ + m₃₃), (m₂₃ − m₃₂) for the characterization of biological tissues, and m₁₄ or m₄₁ ≠ 0 for determine the presence of linear or circular dichroism in blood plasma films [18].

Recently, Gil [19] developed a general theory of invariant quantities of Mueller matrix under rotation and retarder transformation, showed that there are 15 independent invariants under dual-rotation transformation, given the condition that rotating both incident and output beam synchronously. It is proposed that the dual-retarder transformation can be physically realized by sandwiching the medium by two retarders. In this paper, we will show that in the particular case of half wave plate (HWP), it can also be realized by mirror transformation of the sample.

Following Gil’s work, we study the rotation and mirror transformation of Mueller matrix, and formally derive a set of rotation invariant parameters and corresponding orientation parameters. We examine and verify their physical meanings with MC simulations based on the sphere-cylinder scatterers embedded in birefringence media model [20]. Further experiments on transparent sticky tapes show their potential applications in identifying contributions by multiple anisotropic components.

2. Rotation invariant parameters of Mueller matrix

Consider a polarized light with Stokes vector S_in traveling toward z direction as in Fig. 1 (a), rotate the polarization ellipse for an angle α turns the Stokes vectors into S_out = R(α)S_in, where [15]

R (α) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & cos (2 α) & - sin (2 α) & 0 \\ 0 & sin (2 α) & cos (2 α) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

which is identical with the Mueller matrix of an optical rotator [21].

Fig. 1 (a) Azimuth rotation transformation of the Stokes vector for angle α, right-hand as the positive direction. (b) Mirror transformation of the Stokes vector with mirror plane (blue) pass axis z and lies at direction α.

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The Mueller matrix of sample M after rotated around z axis for α is [15]

M^{'} = R (α) MR (- α)

In Gil’s study, this formula is valid when rotating the input and output beam in their own reference system synchronously [19]. However in RLPI [11,12] and MMT [13,15] studies, it is more common to fix the input and output reference system and rotate the sample itself, in which case the valid condition for invariants is the illumination and detection directions are both collinear with the rotating axis of the sample. Equation (2) is valid for forward detection, as well as backward detection if choose certain reference system for the detector, as will be discussed in Section 5.1.

Expanded formula of M′ is [19]:

R (α) MR (- α) = [\begin{matrix} m_{11} & m_{12} c_{2} - m_{13} s_{2} & m_{12} s_{2} + m_{13} c_{2} & m_{14} \\ m_{21} c_{2} - m_{31} s_{2} & b + (\tilde{b} c_{4} - \tilde{β} s_{4}) & β + (\tilde{b} s_{4} + \tilde{β} c_{4}) & m_{24} c_{2} - m_{34} s_{2} \\ m_{21} s_{2} + m_{31} c_{2} & - β + (\tilde{b} s_{4} + \tilde{β} c_{4}) & b - (\tilde{b} c_{4} - \tilde{β} s_{4}) & m_{24} s_{2} + m_{34} c_{2} \\ m_{41} & m_{42} c_{2} - m_{43} s_{2} & m_{42} s_{2} + m_{43} c_{2} & m_{44} \end{matrix}]

where s_n = sin(nα), c_n = cos(nα),

b = \frac{1}{2} (m_{22} + m_{33})

\tilde{b} = \frac{1}{2} (m_{22} - m_{33})

β = \frac{1}{2} (m_{23} - m_{32})

\tilde{β} = \frac{1}{2} (m_{23} + m_{32})

Mueller matrix in this article are not normalized by default. However, since m₁₁ is invariant under any rotation and retarder transformation [19], the conclusions can also be applied to Mueller matrices normalized by m₁₁.

Based on Eq. (3), Gil [19] proposed an abundant set of rotation invariants, including the four “corners” of Mueller matrix Eqs. (8)–(11) and “edges” Eqs. (12)–(15) and their combinations such as diattenuation D = |D| and polarizance P = |P|, as well as Eq. (18), Eq. (6), tr M, det M, indices of polarimetric purity (IPP), eigenvalues of the N-matrix N = GM^T GM (G ≡ diag(1, −1, −1, −1)), singular values of m, and P^T D, P^T mD, P^T m^T D from the block form of Mueller matrix. We compared these parameters with earlier studies [11–13,17,18,22], and propose the following set of rotation invariant parameters that have shown potential applications for the study of anisotropic samples:

m_{11}

k_{C} = m_{44} \in [- 1, 1]

D_{C} = m_{14} \in [- 1, 1]

P_{C} = m_{41} \in [- 1, 1]

P_{L} = \sqrt{m_{21}^{2} + m_{31}^{2}} \in [0, 1]

D_{L} = \sqrt{m_{12}^{2} + m_{13}^{2}} \in [0, 1]

q_{L} = \sqrt{m_{42}^{2} + m_{43}^{2}} \in [0, 1]

r_{L} = \sqrt{m_{24}^{2} + m_{34}^{2}} \in [0, 1]

P_L, P_C are named as linear and circular polarizance, D_L, D_C are linear and circular diattenuation [19]. The combination (P_C + D_C) represents the circular dichroism anisotropy [16, 23]. So far q_L and r_L have no names. We will show that they represent the capability of transforming between linear and circular polarizations.

Denote the central 2×2 block of the Mueller matrix as B, we have its trace, determinant, Frobenius norm as rotation invariants:

tr B = m_{22} + m_{33} = 2 b

| B | = (m_{22} m_{33} - m_{23} m_{32})

{‖ B ‖}_{F} = \sqrt{m_{22}^{2} + m_{33}^{2} + m_{23}^{2} + m_{32}^{2}}

Equation (3) shows that all elements of B have the same amplitude:

t_{1} = \sqrt{{\tilde{b}}^{2} + {\tilde{β}}^{2}} = \frac{1}{2} \sqrt{{(m_{22} - m_{33})}^{2} + {(m_{23} + m_{32})}^{2}}

which has been proposed [13] as the degree of anisotropy. This is not a new independent parameter since it can also be written as

t_{1} = \frac{1}{2} \sqrt{{‖ B ‖}_{F}^{2} - 2 | B |}

[12], and parameter

β = \frac{1}{2} (m_{23} - m_{32})

which represents the circular birefringence (optical rotation) anisotropy [16] is also not new since

4 (b^{2} + β^{2}) = {‖ B ‖}_{F}^{2} + 2 | B |

. So far the set {b, t₁, β} draws more attention of experimentalists [7,17,18,22] than the set {tr B, |B|, ‖B‖_F}.

For a sample that is α-rotation symmetric:

R (α) M_{α sym} R (- α) - M_{α sym} = 0

When α ≠ nπ (n ∈ ℤ), Eq. (20) leads to P_L, D_L, q_L, r_L = 0, when $α \neq n \frac{π}{2}$ , it also leads to t₁ = 0. Therefore, the non-zero of these parameters can indicate the existence of anisotropy.

The above deduction is based purely on the coordinate transformation theory, therefore can apply to any optic sample (within the fundamental linear interaction assumption) [23], but requires both illumination and detection systems configured collinear with the rotating axis of the sample. For point illumination (which is common in MC simulation), the theory only applies to the center pixel, or the average result of a circular area around the center. For uniform surface illumination (which is common in experiments), the theory applies to the full detected image. Breaking of the invariance of these parameters can indicate the inclination in the experimental setup [13,15].

3. Azimuth orientation parameters

Many biological tissues contain aligned fibrous microstructures [14]. Previous MMT studies proposed

α = \frac{1}{4} atan 2 (m_{23} + m_{32}, m_{22} - m_{33})

α = \frac{1}{2} atan 2 (m_{13}, m_{12})

for mapping the local orientations α (previously denoted as x [14,22]) of silk fiber phantom and human papillary thyroid carcinoma tissues [14] (here atan2 is the arctan function that takes in two parameters (dy, dx) thus able to return the angle in full 2π range), and

α_{3} = \frac{1}{2} atan 2 (m_{42}, m_{34})

for mapping the local orientation of fibrous human liver cancer tissues [22]. It was shown that Eq. (21) has higher degeneracy for predicting the orientations but better immunity to noise than Eqs. (22) and (23).

In this paper, we attempt to explain these formulas by studying the mirror symmetry property of the sample. Consider a longitudinal section plane that pass through axis z and lies at direction α, as shown in Fig. 1 (b). This mirror transforms 0° linearly polarized light into 2α linearly polarized light. It also transforms right circular polarization into left circular polarization. After considering the transformation of four base Stokes vectors, we can deduce the intermediate transformation matrix:

H (α) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & cos (4 α) & sin (4 α) & 0 \\ 0 & sin (4 α) & - cos (4 α) & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}]

which is identical with the Mueller matrix of a HWP oriented at α, also known as pseudo-rotator [21].

Since the inverse operation of H(α) is itself, after similar discussion we get that the Mueller matrix after such mirror transformation is M′ = H(α)MH(α). For α = 0:

H (0) MH (0) = [\begin{matrix} m_{11} & m_{12} & - m_{13} & - m_{14} \\ m_{21} & m_{22} & - m_{23} & - m_{24} \\ - m_{31} & - m_{32} & m_{33} & m_{34} \\ - m_{41} & - m_{42} & m_{43} & m_{44} \end{matrix}]

Therefore, for a sample that is symmetric through xOz plane, its Mueller matrix obeys this form:

M_{mirsym} = [\begin{matrix} m_{11} & m_{12} & 0 & 0 \\ m_{21} & m_{22} & 0 & 0 \\ 0 & 0 & m_{33} & m_{34} \\ 0 & 0 & m_{43} & m_{44} \end{matrix}]

Thus, if we want to extract α from the rotated M′_mirsym:

M_{mirsym}^{'} = R (α) M_{mirsym} R (- α) = [\begin{matrix} m_{11} & m_{12} c_{2} & m_{12} s_{2} & 0 \\ m_{21} c_{2} & b + \tilde{b} c_{4} & \tilde{b} s_{4} & - m_{34} s_{2} \\ m_{21} s_{2} & \tilde{b} s_{4} & b - \tilde{b} c_{4} & m_{34} c_{2} \\ 0 & - m_{43} s_{2} & m_{43} c_{2} & m_{44} \end{matrix}]

We can use the following formulas:

α_{1} = \frac{1}{4} atan 2 (m_{23}^{'} + m_{32}^{'}, m_{22}^{'} - m_{33}^{'}) if t_{1} \neq 0

α_{P} = \frac{1}{2} atan 2 (m_{31}^{'}, m_{21}^{'}) if P_{L} \neq 0

α_{D} = \frac{1}{2} atan 2 (m_{13}^{'}, m_{12}^{'}) if D_{L} \neq 0

α_{q} = \frac{1}{2} atan 2 (m_{42}^{'}, - m_{43}^{'}) if q_{L} \neq 0

α_{r} = \frac{1}{2} atan 2 (- m_{24}^{'}, m_{34}^{'}) if r_{L} \neq 0

The sign convention in Eqs. (28)–(32) is based on the assumption that b̃, m₁₂, m₂₁, m₃₄ > 0 and m₄₃ < 0 when α = 0. If the truth is the opposite, then both input parameters (dy, dx) should flip their sign (or add π to the result of atan2).

4. Physical interpretations of invariant parameters and orientation parameters

We begin by looking at two fundamental types of anisotropy: dichroism and birefringence [23]. The Mueller matrix for a linear diattenuator with amplitude transmission coefficient p_x, p_y [21] and rotated for α is:

R (α) M_{D} (p_{x}, p_{y}) R (- α) = [\begin{matrix} m_{11} & D c_{2} & D s_{2} & 0 \\ D c_{2} & m_{11} c_{2}^{2} + p_{x} p_{y} s_{2}^{2} & \frac{1}{4} {(p_{x} - p_{y})}^{2} s_{4} & 0 \\ D s_{2} & \frac{1}{4} {(p_{x} - p_{y})}^{2} s_{4} & m_{11} s_{2}^{2} + p_{x} p_{y} c_{2}^{2} & 0 \\ 0 & 0 & 0 & p_{x} p_{y} \end{matrix}]

where

m_{11} = \frac{1}{2} (p_{x}^{2} + p_{y}^{2})

,

D = \frac{1}{2} (p_{x}^{2} - p_{y}^{2}) = D_{L} = P_{L}

. We can calculate that

t_{1} = \frac{1}{4} {(p_{x} - p_{y})}^{2} \in [0, \frac{1}{4}]

(range [0,

\frac{1}{2}

] for normalized Mueller matrices), which can be regard as the degree of anisotropy. In this case parameters α₁, α_D, α_P all give the orientation of maximum transmission.

The Mueller matrix for a linear retarder of δ [21] and rotated for α is:

R (α) M_{R} (δ) R (- α) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & c_{2}^{2} + s_{2}^{2} cos δ & s_{2} c_{2} (1 - cos δ) & - s_{2} sin δ \\ 0 & s_{2} c_{2} (1 - cos δ) & c_{2}^{2} cos δ + s_{2}^{2} & c_{2} sin δ \\ 0 & s_{2} sin δ & - c_{2} sin δ & cos δ \end{matrix}]

We can see that q_L = r_L = sin δ, whose physical meaning is the capability of transforming between linear and circular polarizations. This explains q_L = r_L = 0 for HWP, due to the fact that HWP acts as a pseudo-rotator hence does not change the ellipticity. The sign convention in Eqs. (31) and (32) gives the orientation of fast axis, if δ is limited to [0, π] (for δ ∈ (π, 2π] this is equivalent to retard (2π − δ) and exchange the fast and slow axes). We can calculate that $t_{1} = \frac{1}{2} (1 - cos δ) = {sin}^{2} (δ / 2) \in [0, 1]$ . Therefore t₁ = 1 for HWP and drop to 0 when δ = 0 or 2π, which meets our expectation as the degree of anisotropy.

In the case of only one single pure anisotropy is presented, we have

P_{L} = D_{L}

q_{L} = r_{L}

α_{P} - α_{D} = 0 (or \pm \frac{π}{2})

α_{q} - α_{r} = 0 (or \pm \frac{π}{2})

β = 0

m_{14} and m_{41} = 0

Equation (26) indicates that Eqs. (37)–(40) are the necessary (not sufficient) conditions for a Mueller matrix at arbitrary azimuth orientation M′ to be longitudinally mirror symmetric (physically speaking: exists a mirror symmetry plane that pass axis z, mathematically speaking: can be block diagonized to Eq. (26) by transformation Eq. (2)). They are true for single pure anisotropies, but not for multiple crossing (not parallel or perpendicular) anisotropy effects, which can be reflected from the breaking of some of these conditions.

When multiple crossing dichroism effects coexist, we multiply R(α_b)M_D(p_bx, p_by)R(−α_b) · · · R(α_a)M_D(p_ax, p_ay)R(−α_a) and find that condition Eqs. (35) and (40) holds but Eqs. (37) and (39) are broken. The physical meaning of α_P, α_D become unclear, but in the special case of the serial combination of linear total polarizers (p_by, · · · , p_ay = 0), parameters α_D and α_P give the transmission orientation of the first (α_a) and last (α_b) polarizer respectively, while α₁ gives their average $\frac{1}{2} (α_{a} + α_{b})$ .

When multiple crossing birefringence effects coexist, we multiply R(α_b)M_R(δ_b)R(−α_b) · · · R(α_a)M_R(δ_a)R(−α_a) and find that condition Eqs. (36) and (40) holds but Eqs. (38) and (39) are broken, providing (α_r − α_q) as an indicator of coexistence of multiple birefringences, as shown in Fig. 2 (c).

Fig. 2 Experimental measurement of transparent sticky tape (a) Mueller matrix in transmission, m₁₁ is plotted in [0, 1] grayscale, other elements are normalized by m₁₁. (b) $β \in (- \frac{1}{2}, \frac{1}{2})$ . (c) The difference in orientation prediction (α_r − α_q) (degree) is shown as hue and r_L is shown as brightness. (d)(e)(f) Orientation predictions by α_q, α_r, α₁, length and color of the small bars represent q_L, r_L, t₁. (f) To solve the degeneracy problem in α₁ [14] we compare it with α_r and add $\frac{π}{2}$ to α₁ if $| α_{1} - α_{r} | > \frac{π}{4}$ .

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Previous study on pathological slides [7] show that the signal of m₁₄, m₄₁ are relatively not significant. We find out that only in the crossing coexistence of dichroism and birefringence can we observe the breaking of Eq. (40). Multiplication of M_R(δ)R(α)M_DR(−α) gives m₁₄ = 0 but m₄₁ ≠ 0, which imply dichroism happens before birefringence (this could be the sample is layered, or there is oblique reflection on the incident surface of the material). If the order is reversed then m₄₁ = 0 but m₁₄ ≠ 0. When both m₁₄, m₄₁ ≠ 0, it indicates these two types of anisotropy are mixing together, which is observed in the MC simulation [20] of cylinder scattering Fig. 3.

Fig. 3 MC simulation of anisotropic Mueller matrix in transmission. (a) Two groups of cylinder scatterers crossing overlapped, one group is fixed at x orientation, radius r_c = 0.3 μm, n = 1.56, scattering coefficient μ_c = 15 cm⁻¹. The other group changes its radius r_c2 = 0.1, 0.3, 0.6 μm (dot, cross, square) and scattering coefficient μ_c2 = 5 or 20 cm⁻¹ (blue thin, orange thick). When r_c2 = r_c and μ_c2 = μ_c the m₁₄, m₄₁ are always 0 (green line). (b) One group of cylinder scatterers crossing overlap with medium birefringence, fast axis fixed at x orientation, medium refractive index n = 1.33, Δn = 0.00005. Horizontal axis α is the intersection angle between two anisotropies. The simulation use λ = 633 nm illumination, thick 1 cm sample, diameter 1 cm detector.

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Figure 3 (a) indicates that cylinder scatterering contain both birefringence and dichroism effects. Both (a) (b) confirms that m₁₄, m₄₁ drop to exactly 0 when mirror symmetry is fulfilled (α = 0 or $\frac{π}{2}$ ).

5. Discussion

5.1. Choice of coordinate system for backward detection

There are two choices for the coordinate system of detector in backward configuration. Figure 4 (a) coincide with the local system of backward scattering light beams. However, in this case the rotation transformation of the reflection Mueller matrix become M′ = R(α)MR(α) and leads to changes of signs in the definition of some of the invariants [24]. Previous MMT studies [13,15] and MC softwares [20, 25] followed [26] used convention Fig. 4 (b) so that the definition of rotation invariants are the same for both forward and backward detection.

Fig. 4 Two coordinate systems for backward detection, (a) coincide with the local system of back scattering light beams, (b) one more mirror reflection before detection. These two schemes give the Mueller matrix of an ideal plane mirror reflection as (a) diag(1, 1, −1, −1), (b) diag(1, 1, 1, 1). (Illumination beam is tilted for drawing input and output separately, to fulfill the invariant condition they should be collinear and normal to the sample.)

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5.2. Improvements to the definitions in previous publications

In earlier MMT studies [13], t₁ is normalized as $A = 2 b t_{1} / (b^{2} + t_{1}^{2})$ to represent the degree of anisotropy, which is no longer recommended since we found A is numerically instable when depolarization is strong. (For example, diag(1, 0, 0, 0) gives A = 0 but diag(1, 0.002, 0.001, 0) gives A = 0.6.)

The definition for the invariants P_L, D_L (previously considered equal thus both named t₂) sometimes have a factor $\frac{1}{2}$ [13,22] and sometimes don’t [14]. In order to be consistent with t₁ (which represents the “amplitude” of the sinusoidal fluctuation in corresponding Mueller matrix elements), the definition without $\frac{1}{2}$ is preferred, as Eqs. (12)–(15).

Due to the similar mistaken belief that m₄₂ = −m₂₄, m₄₃ = −m₃₄, the old definition of $t_{3} = \sqrt{m_{42}^{2} + m_{34}^{2}}$ and Eq. (23) was not transpose symmetric [22], which should be fixed as Eq. (14), Eq. (15) and Eq. (31), Eq. (32).

6. Conclusion

Anisotropies are important information in material and biomedical studies. In this paper, we propose a set of rotation invariant parameters Eqs. (8)–(19), Eqs. (4) and (6) and orientation parameters Eqs. (28)–(32) based on the rotation and mirror transformation theory of Mueller matrix. We analyze their physical meaning by studying the Mueller matrix of linear diattenuator, retarder and their combinations. We show that when one of these two anisotropy effects dominates, P_L, D_L indicate the degree of dichroism and α_P, α_D provide the orientation of maximum transmittance, q_L, r_L indicate the capability of transform between linear and circular polarization and α_q, α_r provide the orientation of fast axis. In both cases t₁ can be regard as the degree of anisotropy, but t₁ can only goes beyond $\frac{1}{2}$ when there exists birefringence effect.

Crossing coexistence of the anisotropy effects will affect the above conclusions, and can be indicated from the breaking of conditions Eqs. (35)–(40), which are necessary conditions for the sample to be longitudinal mirror symmetric. α_P ≠ α_D and α_q ≠ α_r are the symbols for the coexistence of multiple dichroism or birefringence respectively, while m₁₄, m₄₁ ≠ 0 is the symbol of crossing coexistence of both dichroism and birefringence, and can provide clues on the sequence of these two effects.

Parameters from MMT theory are explicit elementary functions of a few Mueller matrix elements, therefore have better computational efficiency compared with MMPD and other matrix decomposition approaches, and suitable for analyzing objects which are difficult to measure the full Mueller matrix.

Funding

National Natural Science Foundation of China (NSFC) (61527826, 11374179, and 61405102).

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Separating azimuthal orientation dependence in polarization measurements of anisotropic media

Abstract

1. Introduction

2. Rotation invariant parameters of Mueller matrix

3. Azimuth orientation parameters

4. Physical interpretations of invariant parameters and orientation parameters

5. Discussion

5.1. Choice of coordinate system for backward detection

5.2. Improvements to the definitions in previous publications

6. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (40)

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