Extraction of polarization properties of the individual components of a layered system by using spectroscopic Mueller matrix analysis

Lianhua Jin; Daichi Kobayashi; Eiichi Kondoh; Hiroyuki Kowa; Bernard Gelloz

doi:10.1364/OE.24.009757

1. Introduction

Some electrical or semiconductor devices which are constructed by coating thin film on polymer resins are evaluated with optical probes of polarimetry. Since the coated films and resins exhibit orientation birefringence and intrinsic birefringence, respectively, the evaluation is carried out by comparing the polarization parameters of the resin layer only with those of the layers involving both resin and film. However, the polarization parameters of the films and resin change during the process. Therefore, simultaneous measurement of these parameters would be preferable. Some polarizers with high extinction ratio are constructed by sandwiching stretched polymer films between two protecting plates which exhibit strain induced birefringence. To evaluate the optics, obtaining simultaneously polarization parameters of each layer is also required.

Mueller matrix elements are considered the most complete expression of the polarization properties of a material. For singular polarizing components except for anisotropic reflection ones, the matrix elements are functions of 1-3 polarization parameters [1]. Therefore, the polarization parameters such as the retardation and azimuth angle of a birefringent material, or transmission angle of a polarizer, can be easily obtained by just measuring 2 or 3 elements of the Mueller matrix. When the system consists of multiple components, the Mueller matrix of the system is obtained from multiplication of the matrices of each component, consequently leading to complicated expressions.

So far, several methodologies of Muller matrix decomposing have been addressed [2–5]. The Mueller matrix polar decomposition method [2, 3] proposed by Lu et al. decomposes any Mueller matrices into three matrix factors (a diattenuator, a retarder and a depolarizer), and has been applied to determine the diattenuation, retardation, and depolarization in biomedical diagnosis [6], and geometry of objects [7]. This method can determine the polarization parameters of individual layers of diattenuator-retarder-depolarizer or depolarizer-retarder-diattenuator systems. The generalized matrix equivalence theorem [4] proposed by Savenkov et al. decomposes the matrices into linear and circular phase and linear and circular amplitude anisotropies. The Mueller matrix differential decomposition method [5] proposed by Ortega-Quijano et al. decomposes macroscopic Mueller matrix by using differential calculus, and obtains accumulated polarization parameters of the layered system. These above-mentioned methods are very powerful tools to determine integrated polarization properties for the whole layered system, however cannot determine all the polarization parameters of individual layers of any multilayered systems like those mentioned in the beginning. Several approaches for determining polarization parameters of individual layers have been reported [2, 8–10]. Determining parameters of individual layers for the nondepolarizing system of polarizer-retarder [2, 8], the system of retarder-horizontal or vertical retarding diattenuator-retarder [9], and the system of retarder-reflector-retarder [10] have been implemented. However polarization parameters of individual components of the layered systems with retarder-retarder and retarder-polarizer-retarder, which were applicable to above mentioned industrial inspection field, have not been implemented yet.

In this paper, we propose methods to simultaneously determine the polarization parameters of individual layers from Mueller matrices of multi-layered polarization systems such as retarder-retarder, retarder-polarizer-retarder. The determination was implemented by using Mueller matrices measured at two wavelengths. The Mueller matrix polarimetry using dual rotating-compensator is a successful method originally proposed by Azzam [11] and elaborated on by Hauge [12], and by Goldstein and Chipman [13], and has been used in spectropolarimetry [14]. Spectroscopic Mueller matrix can be obtained with a single measurement [15]. Consequently, the decompositions of Mueller matrices to our polarization system can be implemented with a single measurement.

In the following section, we describe our proposed methods for nondepolarizing systems of two-layer retarder and retarder-polarizer-retarder which are nearly close to the cases mentioned above in the semiconductor and optics fields. We show both simulation results and decomposed results of real polarization systems composed of retarder-retarder and retarder-polarizer-retarder. In the discussion section, we show the applicability of the methods for a system involving an achromatic or nearly achromatic retarder.

2. Principle

2.1 Two-layer retarder

Figure 1 shows a two-layer structure consisting of two birefringent layers. The Muller matrix of a nondepolarizing retarder is given by:

L = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 - (1 - \cos δ_{λ}) \sin^{2} 2 θ & (1 - \cos δ_{λ}) \sin 2 θ \cos 2 θ & - \sin δ_{λ} \sin 2 θ \\ 0 & (1 - \cos δ_{λ}) \sin 2 θ \cos 2 θ & 1 - (1 - \cos δ_{λ}) \cos^{2} 2 θ & \sin δ_{λ} \cos 2 θ \\ 0 & \sin δ_{λ} \sin 2 θ & - \sin δ_{λ} \cos 2 θ & \cos δ_{λ} \end{matrix}),

where δ and θ are the retardation and azimuth angle, respectively. The retardation δ is often dependent on the wavelength λ. The matrix elements m₂₂-m₄₄ are related through parameters δ and θ. The Muller matrix of two-layer retarder becomes:

M = L_{2} L_{1} = (\begin{matrix} 1 & 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}) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 \\ 0 & M L_{3 \times 3} \\ 0 \end{matrix}),

where subscripts 1 and 2 represent the order of each layer. Taking cos2θ and sin2θ as Ct and St, and, 1-cos2δ and sin2δ as Cd and Sd, respectively, each element of matrix ML_{3 × 3} at wavelength λ is expressed by:

\begin{array}{l} m_{22, λ} = (1 - C d_{2, λ} S t_{2}^{2}) (1 - C d_{1, λ} S t_{1}^{2}) + C d_{2, λ} S t_{2} C t_{2} C d_{1, λ} S t_{1} C t_{1} - S d_{2, λ} S t_{2} S d_{1, λ} S t_{1} \\ m_{23, λ} = (1 - C d_{2, λ} S t_{2}^{2}) C d_{1, λ} S t_{1} C t_{1} + C d_{2, λ} S t_{2} C t_{2} (1 - C d_{1, λ} C t_{1}^{2}) + S d_{2, λ} S t_{2} S d_{1, λ} C t_{1} \\ m_{24, λ} = - (1 - C d_{2, λ} S t_{2}^{2}) S d_{1, λ} S t_{1} + C d_{2, λ} S t_{2} C t_{2} S d_{1, λ} C t_{1} - S d_{2, λ} S t_{2} (1 - C d_{1, λ}) \\ m_{32, λ} = C d_{2, λ} S t_{2} C t_{2} (1 - C d_{1, λ} S t_{1}^{2}) + (1 - C d_{2, λ} C t_{2}^{2}) C d_{1, λ} S t_{1} C t_{1} + S d_{2, λ} C t_{2} S d_{1, λ} S t_{1} \\ m_{33, λ} = C d_{2, λ} S t_{2} C t_{2} C d_{1, λ} S t_{1} C t_{1} + (1 - C d_{2, λ} C t_{2}^{2}) (1 - C d_{1, λ} C t_{1}^{2}) - S d_{2, λ} C t_{2} S d_{1, λ} C t_{1} \\ m_{34, λ} = - C d_{2, λ} S t_{2} C t_{2} S d_{1, λ} S t_{1} + (1 - C d_{2, λ} C t_{2}^{2}) S d_{1, λ} C t_{1} + S d_{2, λ} C t_{2} (1 - C d_{1, λ}) \\ m_{42, λ} = S d_{2, λ} S t_{2} (1 - C d_{1, λ} S t_{1}^{2}) - S d_{2, λ} C t_{2} C d_{1, λ} S t_{1} C t_{1} + (1 - C d_{2, λ}) S d_{1, λ} S t_{1} \\ m_{43, λ} = S d_{2, λ} S t_{2} C d_{1, λ} S t_{1} C t_{1} - S d_{2, λ} C t_{2} (1 - C d_{1, λ} C t_{1}^{2}) - (1 - C d_{2, λ}) S d_{1, λ} C t_{1} \\ m_{44, λ} = - S d_{2, λ} S t_{2} S d_{1, λ} S t_{1} - S d_{2, λ} C t_{2} S d_{1, λ} C t_{1} + (1 - C d_{2, λ}) (1 - C d_{1, λ}) . \end{array}

Equation (3) contains in total 8 unknown parameters: Ct_i, St_i, Cd_i,λ, and, Sd_i,λ (i = 1, 2). To determine unknown parameters of nonlinearly coupled system like Eq. (3), the Broyden-Fletcher-Goldfarb-Shanno (BSGS) quasi-Newton method [16–20] is available. To solve for the unknown variables, this method undergoes iterative differential computation until convergence is achieved. During calculation, the quasi-Newton method is often combined with a least-square optimization step. When this method is applied to solve Eq. (3), several solutions are obtained, showing that Eq. (3) is an underdetermined system. To decrease the freedom of parameters in Eq. (3), we used the wavelength dependence of the retardation, and constructed 18 coupled systems for 12 unknown parameters: Ct_i, St_i, Cd_i,λ₁, Cd_i,λ₂, Sd_i,λ₁, and Sd_i,λ₁. Due to the dispersion, Cd_i and Sd_i exhibit wavelength dependence while Ct_i and St_i do not. Here, the sum of squares to be minimized is:

e = \sum_{k = 1}^{2} \sum_{i = 2}^{4} \sum_{j = 2}^{4} {(m_{i j, λ k, m e a s u r e d} - m_{i j, λ k})}^{2},

where m_{ij,λk,measured} and m_ij,λk are values measured at wavelength λ_k, and values determined by Eq. (3), respectively. The 12 unknown parameters can be simultaneously determined from Eq. (4) and the 18 coupled systems. The birefringence parameters of each layer are deduced from:

Fig. 1 Two birefringent layer structure

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\begin{array}{l} δ_{λ 1} = arc \tan \frac{S d_{λ 1}}{1 - C d_{λ 1}}, (- 180^{\circ} \leq δ_{λ 1} \leq 180^{\circ}), \\ δ_{λ 2} = arc \tan \frac{S d_{λ 2}}{1 - C d_{λ 2}}, (- 180^{\circ} \leq δ_{λ 2} \leq 180^{\circ}), \\ θ = \frac{1}{2} arc \tan \frac{S t}{C t}, (- 90^{\circ} \leq θ \leq 90^{\circ}) . \end{array}

2.2 Retarder –polarizer-retarder

The Muller matrix of a linear polarizer is given by:

P = \frac{1}{2} (\begin{matrix} 1 & \cos 2 φ & \sin 2 φ & 0 \\ \cos 2 φ & \cos^{2} 2 φ & \cos 2 φ \sin 2 φ & 0 \\ \sin 2 φ & \cos 2 φ \sin 2 φ & \sin^{2} 2 φ & 0 \\ 0 & 0 & 0 & 0 \end{matrix}),

where ϕ is the angle of the transmission axis. Then, the Muller matrix of the retarder -polarizer-retarder system shown in Fig. 2 is:

M_{L 2 P L 1} = L_{2} P L_{1} = \frac{1}{2} (\begin{matrix} 1 & m_{12} & m_{13} & m_{14} \\ m_{21} \\ m_{31} & M L P L_{3 \times 3} \\ m_{41} \end{matrix}),

Where elements m₂₁, m₃₁, m₄₁, m₁₂, m₁₃, and m₁₄ at wavelength λ are, respectively,

\begin{array}{l} m_{12, λ} = \cos 2 φ (1 - C d_{1, λ} S t_{1}^{2}) + \sin 2 φ (C d_{1}_{, λ} S t_{1} C t_{1}) \\ m_{13, λ} = \cos 2 φ (C d_{1, λ} S t_{1} C t_{1}) + \sin 2 φ (1 - C d_{1, λ} C t_{1}^{2}) \\ m_{14, λ} = - \cos 2 φ (S d_{1, λ} S t_{1}) + \sin 2 φ (S d_{1, λ} C t_{1}), \end{array}

and,

\begin{array}{l} m_{21, λ} = \cos 2 φ (1 - C d_{2, λ} S t_{3}^{2}) + \sin 2 φ (C d_{2}_{, λ} S t_{2} C t_{2}) \\ m_{31, λ} = \cos 2 φ (C d_{2, λ} S t_{2} C t_{2}) + \sin 2 φ (1 - C d_{2, λ} C t_{2}^{2}) \\ m_{41, λ} = \cos 2 φ (S d_{2, λ} S t_{2}) - \sin 2 φ (S d_{2, λ} C t_{2}), \end{array}

Equations (8) and (9) have 6 unknown parameters: Ct_i, St_i, Cd_i,λ, Sd_i,λ (i = 1, 2), cos2ϕ and sin2ϕ for 3 coupled system, respectively. To increase the number of coupled system, here the Mueller matrices at two wavelengths are needed, and only the case of

S d_{i, λ} = \sqrt{1 - C d_{i, λ}^{2}}, (0^{\circ} \leq δ_{i, λ} \leq 180^{\circ})

is considered. Here, we also use the BSGS quasi-Newton method to simultaneously obtain the unknown parameters Ct_i, St_i, Cd_i_,λ₁,_, Cd_i_,λ₂, cos2ϕ and sin2ϕ. The sum of squares to be minimized is

e = \sum_{k = 1}^{2} \sum_{i = 2}^{4} {(m_{1 i, λ k, m e a s u r e d} - m_{1 i, λ k})}^{2},

for determining Ct₁, St₁, Cd₁_,λ₁,_, Cd₁_,λ₂, cos2ϕ and sin2ϕ, and

e = \sum_{k = 1}^{2} \sum_{i = 2}^{4} {(m_{i 1, λ k, m e a s u r e d} - m_{i 1, λ k})}^{2},

for Ct₂, St₂, Cd₂_,λ₁,_, Cd₂_,λ₂, cos2ϕ and sin2ϕ.

Fig. 2 Multi-layered system of retarder –polarizer-retarder

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After the 10 unknown parameters are simultaneously determined, the polarization parameters of each layer are deduced from

\begin{array}{l} δ_{i, λ 1} = arc \cos (1 - C d_{i, λ 1}), (0^{\circ} \leq δ_{λ 1} \leq 180^{\circ}), \\ δ_{i, λ 2} = arc \cos (1 - C d_{i, λ 2}), (0^{\circ} \leq δ_{λ 2} \leq 180^{\circ}), \\ θ_{i} = \frac{1}{2} arc \tan \frac{S t_{i}}{C t_{i}}, (- 90^{\circ} \leq θ \leq 90^{\circ}), \\ φ = \frac{1}{2} arc \tan \frac{\sin 2 φ}{\cos 2 φ}, (- 90^{\circ} \leq θ \leq 90^{\circ}) . \end{array}

3. Simulation and experimental results

To verify the validity of the above mentioned extraction methods, the Muller matrices of different multilayers were simulated, and the extracted polarization parameters from these matrices were listed in Table 1. The initial values of each unknown variables were assumed to be between −1 and 1. The iteration number was set to 400. The convergence of e in Eqs. (4), (10), and (11) was about 10⁻¹¹ for all extraction.

Table 1. Polarization parameters used for simulation and extracted parameters by using proposed methods.

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To investigate the practicality of the methods proposed here, the Mueller matrices of train of optics were measured with commercial transmission Mueller matrix spectroscopic ellipsometer (RC2, J.A. Woollam Co., Inc.). (The spectroscopic sensor of this instrument has defects at the wavelengths near 655 nm, which will affect the measurement results through all experiments.) The measurement principle of this instrument originates from the two-rotating-compensator polarimeter [11]. The polarization elements used were commercial retarder films (Edmund Optics) and polarizer film (Edmund Optics). The extraction was carried out for the wavelength range of 600-800 nm. During extraction, the Mueller matrix at wavelength 600 nm and matrices at wavelengths from 601 to 800 nm were used sequentially.

Figure 3 shows the polarization parameters of two retarders extracted from the Mueller matrix of the train of these retarders (see Appendix). The extracted results were compared with parameters measured individually for each retarder. The results shown in Fig. 3 show an excellent agreement between the extracted and measured polarization parameters of retarders for the wavelength range of 630-800 nm. However, in the range of 601-630 nm, the extracted and measured values are quite different. In this case, the poor extraction can be attributed to the very small dispersion of the retardation of retarders, both of which were less than 5°.

Fig. 3 (Upper) Extracted results of individual retardation and azimuth angle of two retarders train, and measurement results of singular retarders. (Lower) Differences of Δδ and Δθ between the extracted and measured results.

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Next the Mueller matrices of the train of retarder - polarizer - retarder were measured (see Appendix) and the individual polarization parameters were extracted. Figure 4 shows the extracted values and comparison with the individually measured ones. Excellent agreements for polarization parameters were again obtained in the range of 630-800 nm.

Fig. 4 (Upper) Extracted results of linear birefringence parameters and transmission angle from the train of retarder-polarizer-retarder, and measurement results of single optical elements. (Lower) Differences (Δδ, Δθ, Δϕ) between the extracted and measured results.

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4. Discussion

As described in previous sections, the methods proposed here require the spectroscopic Mueller matrices, which are available in a single measurement, and the retarders should be chromatic. When both retarders are simultaneously achromatic, the methods are not valid, while when one retarder is chromatic, and the other one is achromatic or nearly achromatic, those are still valid. Table 2 shows the simulation results for the situations with an achromatic or nearly achromatic retarder. The polarization parameters of chromatic retarders used are same as those in Table 1. For the system of retarder-polarizer-retarder, when the polarization parameters of the polarizer and chromatic retarder are decided from spectroscopic Mueller matrix by using Eq. (10) (or Eq. (11)), the polarization parameters of the other retarder can be easily determined by substituting the polarization parameter ϕ of the polarizer into the left Eq. (11) (or Eq. (10)). Since two unknown parameters (cos2ϕ and sin2ϕ) for Eq. (11) (or Eq. (10)) are eliminated, the polarization parameters of the other retarder can be easily extracted from single wavelength Mueller matrix. Namely, for both multi-layered system, when one retarder is chromatic, the other retarder can be achromatic.

Table 2. Polarization parameters used for simulation and extracted parameters for the systems including an achromatic or nearly achromatic retarder and a chromatic retarder.

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5. Conclusions

Methods of extraction of the polarization parameters of individual layers for retarder-retarder and retarder-polarizer-retarder systems were proposed. This method requires Mueller matrices measured at two wavelengths and the chromatic retardation property of a layer is needed to determine all the polarization parameters simultaneously. The polarization parameters of each layer were extracted experimentally and compared with individually measured values. Excellent agreements were obtained.

Since spectroscopic Mueller matrix is available in a single measurement, the methods will be effective analysis tools for inspection and process-control applications in semiconductor and optics fields.

Appendix

Mueller matrices measured for trains of two retarders and retarder-polarizer-retarders were illustrated in Fig. 5 and Fig. 6, respectively.

Fig. 5 Muller matrix elements m₁₂-m₄₄ measured for the train of two retarders

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Fig. 6 Muller matrix elements m₁₂-m₄₄ measured for the train of retarder-polarizer-retarder

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References and links

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Multi-layered system	Polarization parameters (deg)			Extracted parameters (deg)
Retarder-retarder	δ₁_,λ₁ = 70	δ₁_,λ₂ = 50	θ₁ = 40	δ₁_,λ₁ = 70.0003	δ₁_,λ₂ = 49.9997	θ₁ = 39.9999
Retarder-retarder	_,λ₁ = 60	δ₂_,λ₂ = 54	θ₂ = 30	δ₂_,λ₁ = 60.0001	δ₂_,λ₂ = 54.0004	θ₂ = 30.0000
Retarder-polarizer-retarder	δ₁_,λ₁ = 50	δ₁_,λ₂ = 40	θ₁ = 30	δ₁_,λ₁ = 50.0005	δ₁_,λ₂ = 40.0004	θ₁ = 30.0002
	δ₂_,λ₁ = 60	δ₂_,λ₂ = 44	θ₂ = 10	δ₂_,λ₁ = 60.0004	δ₂_,λ₂ = 44.0003	θ₂ = 10.0012
	ϕ = 50			ϕ = 50.0001

Multi-layered system	Polarization parameters (deg)			Extracted parameters (deg)
Retarder (chromatic)-retarder (achromatic)	δ₁_,λ₁ = 70	δ₁_,λ₂ = 50	θ₁ = 40	δ₁_,λ₁ = 70.0004	δ₁_,λ₂ = 50.0004	θ₁ = 40.0000
Retarder (chromatic)-retarder (achromatic)	δ₂_,λ₁ = δ₂_,λ₂ = 60		θ₂ = 30	δ₂_,λ₁ = δ₂_,λ₂ = 59.9997		θ₂ = 29.9999
Retarder (chromatic)-retarder (nearly achromatic)	δ₁_,λ₁ = 70	δ₁_,λ₂ = 50	θ₁ = 40	δ₁_,λ₁ = 70.0007	δ₁_,λ₂ = 50.0008	θ₁ = 39.9998
Retarder (chromatic)-retarder (nearly achromatic)	δ₂_,λ₁ = 60	δ₂_,λ₂ = 59	θ₂ = 30	δ₂_,λ₁ = 59.9993	δ₂_,λ₂ = 58.9993	θ₂ = 29.9999
Retarder (chromatic)-polarizer-retarder (achromatic or nearly achromatic)	δ₁_,λ₁ = 50	δ₁_,λ₂ = 40	θ₁ = 30	δ₁_,λ₁ = 50.0003	δ₁_,λ₂ = 40.0003	θ₁ = 30.0002
	δ₂_,λ₁ = δ₂_,λ₂ = 60		θ₂ = 10	δ₂_,λ₁ = δ₂_,λ₂ = 60.0000		θ₂ = 10.0000
	ϕ = 50			ϕ = 50.0001

Multi-layered system	Polarization parameters (deg)			Extracted parameters (deg)
Retarder-retarder	δ₁_,λ₁ = 70	δ₁_,λ₂ = 50	θ₁ = 40	δ₁_,λ₁ = 70.0003	δ₁_,λ₂ = 49.9997	θ₁ = 39.9999
Retarder-retarder	_,λ₁ = 60	δ₂_,λ₂ = 54	θ₂ = 30	δ₂_,λ₁ = 60.0001	δ₂_,λ₂ = 54.0004	θ₂ = 30.0000
Retarder-polarizer-retarder	δ₁_,λ₁ = 50	δ₁_,λ₂ = 40	θ₁ = 30	δ₁_,λ₁ = 50.0005	δ₁_,λ₂ = 40.0004	θ₁ = 30.0002
	δ₂_,λ₁ = 60	δ₂_,λ₂ = 44	θ₂ = 10	δ₂_,λ₁ = 60.0004	δ₂_,λ₂ = 44.0003	θ₂ = 10.0012
	ϕ = 50			ϕ = 50.0001

Multi-layered system	Polarization parameters (deg)			Extracted parameters (deg)
Retarder (chromatic)-retarder (achromatic)	δ₁_,λ₁ = 70	δ₁_,λ₂ = 50	θ₁ = 40	δ₁_,λ₁ = 70.0004	δ₁_,λ₂ = 50.0004	θ₁ = 40.0000
Retarder (chromatic)-retarder (achromatic)	δ₂_,λ₁ = δ₂_,λ₂ = 60		θ₂ = 30	δ₂_,λ₁ = δ₂_,λ₂ = 59.9997		θ₂ = 29.9999
Retarder (chromatic)-retarder (nearly achromatic)	δ₁_,λ₁ = 70	δ₁_,λ₂ = 50	θ₁ = 40	δ₁_,λ₁ = 70.0007	δ₁_,λ₂ = 50.0008	θ₁ = 39.9998
Retarder (chromatic)-retarder (nearly achromatic)	δ₂_,λ₁ = 60	δ₂_,λ₂ = 59	θ₂ = 30	δ₂_,λ₁ = 59.9993	δ₂_,λ₂ = 58.9993	θ₂ = 29.9999
Retarder (chromatic)-polarizer-retarder (achromatic or nearly achromatic)	δ₁_,λ₁ = 50	δ₁_,λ₂ = 40	θ₁ = 30	δ₁_,λ₁ = 50.0003	δ₁_,λ₂ = 40.0003	θ₁ = 30.0002
	δ₂_,λ₁ = δ₂_,λ₂ = 60		θ₂ = 10	δ₂_,λ₁ = δ₂_,λ₂ = 60.0000		θ₂ = 10.0000
	ϕ = 50			ϕ = 50.0001

Extraction of polarization properties of the individual components of a layered system by using spectroscopic Mueller matrix analysis

Abstract

1. Introduction

2. Principle

2.1 Two-layer retarder

2.2 Retarder –polarizer-retarder

3. Simulation and experimental results

4. Discussion

5. Conclusions

Appendix

References and links

Cited By

Figures (6)

Tables (2)

Equations (12)

Optics Express