1. Introduction

Mueller matrix polarimeter (MMP) is a well-established optoelectronic instrument that can obtain the Mueller matrix containing comprehensive optical information of the material sample under test [1]. The applications of MMP include the optical characterization of thin films [2], the surface sensing in biochemical reactions [3] and the nanostructure metrology [4,5]. Spectroscopic Mueller matrix polarimeter (SMMP) measures all 16 elements of the 4×4 Mueller matrix of a material as a function of wavelength to obtain the properties of the material, such as the complex dielectric function [6], carrier structure [7], crystalline nature [8], and geometric characteristics [9]. The determination of the changed state of polarization at each wavelength in the measured spectral band is essential in SMMP. Generally, it consists of a light source and a polarization state generator (PSG) which generates spectroscopic polarization states to interact with the sample, a polarization state analyzer (PSA) equipped with a spectrometer which analyses the spectroscopic polarization states after the interaction.

Conventional SMMP based on dual-rotating retarder (DRR-SMMP) can provide reliable and precise measurement of the 16 spectroscopic elements of the Mueller matrix in broadband [10–13]. It performs a complete waveform analysis over the fundamental optical cycle in few seconds to extract the 16 spectroscopic elements of the Mueller matrix by Fourier methods. Since the two retarders are driven by synchronous motors rotating synchronously with a frequency ratio of 5ω:Nω (ω is the fundamental frequency, N is an integer ≠ 5 or ≠10, the fundamental optical cycle is determined by $\pi /\omega$), the fundamental frequency is 1/5 of the rotating frequency of the retarder in PSG (or 1/N of the rotating frequency of the retarder in PSA). This long measurement leads to an increased sensitivity to sample motion artifacts. Therefore, DRR-SMMP has not been regarded as a suitable solution for in-situ monitoring of complex rapid processes, and there always has been an increasing demand on faster spectroscopic Mueller matrix polarimetry solutions.

Up to now, several studies have been engaged in aimed at enhancing the measurement speed of MMP. López-Téllez et al., presented an MMP by using four liquid crystal variable retarders (LCVRs) with continually varying voltage and can measure the Mueller matrix in 2s [14]. Alali et al., constructed an MMP by employing four photoelastic modulators (PEMs) and improved the measurement speed to 0.2s [15]. Zhang et al., proposed a high-speed MMP based on dual-PEMs and division-of-amplitude polarimetry, and can acquire the Mueller matrix in about 11µs [16]. These MMPs can only obtain the Muller matrix at a certain wavelength due to the wavelength dependency of the polarization modulation provided by LCVRs or PEMs. To measure the spectroscopic Mueller matrix in broadband, LCVRs or PEMs are required to work with a monochromator, leading to bulky system [17,18]. Spectral modulation is a more attractive approach due to the advantages of compactness, static nature, robustness, and low-cost configuration. Spectral modulated SMMPs can obtain the 16 spectroscopic elements of the Mueller matrix in snapshot mode [19–21]. In such SMMPs, the modulated spectrum recorded by the spectrometer contains more than 25 separate band-limited channels at certain optical path differences (OPD) within the Fourier domain. Since each channel has a bandwidth that is less than 1/25 of the total bandwidth, the spectral resolution of the reconstructed 16 spectroscopic elements of the Mueller matrix is much lower than the intrinsic resolving power of the spectrometer. In addition, such a small bandwidth of each channel is more likely to cause crosstalk between adjacent channels. Recently, several attempts at increasing the bandwidth of each channel for spectral modulated spectroscopic Stokes polarimeter have been reported [22–27]. However, to the best of our knowledge, this issue has not been addressed for SMMP.

To overcome these limitations, we propose an SMMP based on spectro-temporal modulation (STM-SMMP) with a compact, low-cost and birefringent crystal-based configuration in this work. It uses the principle of spectral polarization modulation and temporal polarization demodulation, and can acquire the 16 spectroscopic elements of the Mueller matrix in broadband with a measurement speed of about 80ms. STM-SMMP has a faster measurement speed than that of DRR-SMMP due to the temporal polarization demodulation provided by the only rotating wave-plate in its PSA. In addition, the spectral polarization modulation based on two high-order retarders with equal thickness in PSG of STM-SMMP can produce five separate channels in the Fourier domain, which leads to a larger bandwidth of each channel than that of the spectral modulated SMMP. We describe the optical configuration and operation principle of STM-SMMP in Section 2. Section 3 is the experimental demonstration of STM-SMMP, and the conclusion is contained in Section 4.

2. Theoretical analysis

2.1 Instrumentation

The schematic of the proposed STM-SMMP setup is portrayed in Fig. 1. It features four simple parts: a collimating optics with a Tungsten-Halogen lamp as a broadband light source, the PSG consists of a polarizer P (α-BBO Glan-Taylor prism with an extinction ratio of less than $5 \times {10^{ - 6}}$, PGT6312, Union Optic, Inc., China) and two high-order retarders R₁ and R₂ with equal thickness (made by quartz crystal with thickness of 23mm, Union Optic, Inc., China), a measured object part, the PSA consists of a rotating super achromatic quarter wave-plate (SAQWP, SAQWP05M-700 with a retardance accuracy of $\lambda /100$ in the spectral range of 325–1100 nm, Thorlabs, Inc., USA) followed by a fixed analyzer A (α-BBO Glan-Taylor prism with an extinction ratio of less than $5 \times {10^{ - 6}}$, PGT6312, Union Optic, Inc., China), and a spectrometer (Ocean FX, Ocean Optics, Inc., USA). The broadband light source is connected to the optical fiber and passes through the collimating optics and the PSG to generate a polarized input beam to interact with the sample under test. The transmitted/ reflected light from the sample enters the PSA. The SAQWP in PSA is rotating by a hollow shaft motor (DRTM 40-D25-HiDS, Owis, Inc., Germany). The spectrometer records the spectra of the light from PSA in the wavenumber range from 10000cm⁻¹ to 16667cm⁻¹ (600nm ∼1000nm). The fast axis of high-order retarders R₁ and R₂ are oriented at 45° and 0° to x axis, respectively. The initial orientation of the SAQWP’s fast axis, the transmission axes of polarizer P and the analyzer A are oriented at 0° to x-axis. The sample is placed at an incident angle of 65° for reflection configuration and normal incidence for transmission configuration, respectively.

 

Fig. 1. Schematic of the proposed STM-SMMP setup (a) transmission configuration (b) reflection configuration.

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2.2 Measurement method

In SMMP, the 16 spectroscopic elements of Mueller matrix are measured to extract wealthy information about the sample under investigation. Herein, we use the Stokes vector and Mueller matrix approach to analyze the measurement method. The 4×4 Mueller matrix of the sample that contains 16 spectroscopic elements ${m_{xy}}(\sigma )$ ($x,y \in [0,3]$, $\sigma$ is wavenumber that defined as the reciprocal of wavelength) is denoted by ${\textbf{M}_\textrm{s}}$ as described in Eq. (1).

According to Fig. 1, the intensity of the radiation at the spectrometer is expressed as

Equations (3) and (4) respectively describe the four Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ contained in $\vec{S} = {[\begin{array}{{cccc}} {{S_0}(\sigma )}&{{S_1}(\sigma )}&{{S_2}(\sigma )}&{{S_3}(\sigma )} \end{array}]^T}$ and the recorded radiation $I(\theta ,\sigma )$, which are determined by multiplying the matrices in Eq. (2).

To obtain the 16 spectroscopic Mueller matrix elements ${m_{xy}}(\sigma )$ ($x,y \in [0,3]$), the Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ are determined firstly by the PSA, which is a rotating wave-plate Stokes polarimeter. Equation (4) can be rewritten as a truncated Fourier series with a fundamental frequency of ${\omega _0}$ as shown in Eq. (5).

In the practical application, the waveform given in Eq. (5) can be obtained by using the spectrometer performing L integrals of the irradiance over the fundamental optical cycle of ${T_0} = \pi /{\omega _0}$, which leads to raw flux data {g_l, l = 1, 2, . . ., L} of the form in Eq. (7).

In Eq. (5), the frequency $4{\omega _0}$ carries the highest-order nonzero Fourier coefficient. Therefore, at least $L \ge 4$ integrations over the fundamental optical cycle ${T_0} = \pi /{\omega _0}$ are required to extract all the Fourier coefficients described by Eqs. (6). Then, one can generate at least $L \ge 4$ equations in 4 unknowns to determine the Stokes parameters ${S_x}(\sigma ),x \in [0,3]$.

Based on Eq. (3), the determined Stokes parameters ${S_x}(\sigma )$ ($x \in [0,3]$) can be rewritten as

Each of ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ include five different carrier frequency components 0, ${\pm} \delta (\sigma )$ and ${\pm} 2\delta (\sigma )$, which carry the information about ${I_0}(\sigma )[{{m_{x0}}(\sigma ) + (1/2){m_{x2}}(\sigma )} ]$, ${I_0}(\sigma ){m_{x1}}(\sigma )/2$, ${I_0}(\sigma )[{ - {m_{x2}}(\sigma ) - i{m_{x3}}(\sigma )} ]/4$ and ${I_0}(\sigma )[{ - {m_{x2}}(\sigma ) + i{m_{x3}}(\sigma )} ]/4$, respectively. This modulated spectrum with different carrier frequencies is similar to the channeled spectrum in Stokes channeled polarimetry [29]. Therefore, the demodulation method in Stokes channeled polarimetry can be used to determine the 16 spectroscopic Mueller matrix elements ${m_{xy}}(\sigma )$ ($x,y \in [0,3]$) from ${S_x}(\sigma )$ ($x \in [0,3]$). The autocorrelation functions of ${S_x}(\sigma )$ ($x \in [0,3]$) described in Eq. (9) are calculated firstly by inverse Fourier transformation.

With the proper choice of the thicknesses of the high-order retarders R₁ and R₂, the five components contained in ${C_x}(h)$ that centered at 0, ${\pm} D$ and ${\pm} 2D$ can be separated from one another with equidistance over the OPD domain. Extracting the desire channels by frequency windowing technique and performing Fourier transformation $\Im \{{\bullet} \}$, the following equations can be obtained

In Eq. (10)–Eq. (12), the phase terms ${e^{i\delta (\sigma )}}$, ${e^{i2\delta (\sigma )}}$ and the light intensity ${I_0}(\sigma )$ are independent of the sample that is being measured. Therefore, they can be calibrated by measuring a reference sample with known spectroscopic Mueller matrix elements. Here, we chose a polarizer with its polarization direction relative to x axis aligned at 22.5° as the reference sample, whose Mueller matrix is represented in Eq. (13).

Based on Eq. (8), Eq. (9) and Eq. (13), the autocorrelation function of the first Stokes parameter $S_0^{refer}(\sigma )$ of the Stokes vector of the light from the reference sample can be obtained, which is represented by Eq. (14).

Finally, the 16 spectroscopic Mueller matrix elements ${m_{xy}}(\sigma )$ ($x,y \in [0,3]$) can be determined from Eq. (10)–Eq. (12) and Eq. (15)–Eq. (17) as

As can be seen from the above measurement method, the 16 spectroscopic elements of the Mueller matrix can be measured in two steps. Firstly, using PSA to measure the Stokes parameters of the light from the sample, then, reconstruct the Mueller elements by Fourier methods. Similar to DRRMMP, the 16 spectroscopic elements of the Mueller matrix are obtained by a complete waveform analysis over the fundamental cycle ${T_0} = \pi /{\omega _0}$, which is similar to the DRR-SMMP. In Eq. (7), the fundamental frequency ${\omega _0}$ is the rotating frequency of SAQWP. As mentioned in Section1, the two retarders in DRR-SMMP are rotating at the frequency ratio of 5ω:Nω(ω is the fundamental frequency of DRR-SMMP, N is an integer ≠ 5 or ≠10) for generating adequate harmonics, which are used to reconstruct the 16 spectroscopic elements of the Mueller matrix. Since SAQWP is the only rotating component in STM-SMMP, the fundamental frequency ${\omega _0}$ can be set as either 5ω or Nω. In this case, the measurement speed of STM-SMMP is 5 or N times faster than that of DRR-SMMP. Besides, each of the acquired Stokes parameters described by Eq. (8) contains 5 separate channels. By proper choice of thicknesses of the high-order retarders R₁ and R₂, these channels can be equally spaced over the OPD domain with maximized bandwidth of each channel. When STM-SMMP and spectral modulated SMMP have equal total bandwidth, for a separable channel structure, cancellation of channels makes the bandwidth of each channel in STM-SMMP is much larger than that of the existing spectral modulated SMMP. The increase in the bandwidth of each channel is particularly important, because larger bandwidth of each channel can effectively avoid crosstalk between adjacent channels and improve the spectral resolution of the reconstructed 16 spectroscopic elements of the Mueller matrix [30–32].

3. Experimental results

To prove the feasibility of the proposed STM-SMMP in the measurement of complete spectroscopic Mueller matrix, we performed a series of measurements on an achromatic quarter-wave plate (AQWP) oriented at different azimuths and SiO₂ thin films deposited on silicon wafers with different thicknesses. In the experiment, the SAQWP is mounted on the hollow shaft motor with rotating frequency of ${\omega _0} = 12.5\pi \textrm{ }rad/s$. Then, the fundamental optical cycle of STM-SMMP is ${T_0} = \pi /{\omega _0} = 80ms$. The integration time of the spectrometer is set as 5ms so that 16 integrals of the irradiance over the fundamental optical cycle (ie., $L = 16$) are obtained. This over determined condition in PSA ensures the accuracy of the measurement of the Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ of the transmitted/reflected light from the sample [33]. The measurement accuracy of the proposed STM-SMMP is estimated through comparisons with the results obtained by a commercial SMMP (VASE ELLIPSOMETER, J. A. Woollam, Inc., USA) that performed the same experiment. It should be noted that the azimuths and the retardances of the high order retarders R₁ and R₂ in PSG, and the retardance of the rotating SAQWP in PSA have been calibrated separately before the measurement [34,35].

3.1 Measurement of an achromatic quarter-wave plate

The measurements on an AQWP (SAQWP05M-700 with a retardance accuracy of $\lambda /100$ in the spectral range of 325–1100 nm, Thorlabs, Inc., USA) rotated at three different azimuths of ${12^ \circ }$, ${57^ \circ }$ and ${82^ \circ }$ are performed firstly. Figure 2 shows the four spectroscopic Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ of the transmitted light from the AQWP under test, which are determined by the PSA of STM-SMMP. It is evident from the figure that ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ are modulated with the wavenumber dependences of the phase retardances of the high-order retarders. The magnitudes of the autocorrelation functions of ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ obtained by inverse Fourier transformation are shown in Fig. 3. As can be seen, the magnitudes of the channels except $|{{A_{00}}(h)} |$ contained in $|{{C_0}(h)} |$ are almost zero, in agreement with the theoretical value of the first row of the Mueller matrix of the AQWP ($\left[ {\begin{array}{{cccc}} {{m_{00}}(\sigma )}&0&0&0 \end{array}} \right]$). The desired five channels are separated from one another in $|{{C_x}(h)} |(x \in [1,3])$. The total bandwidth is about 1.042mm and the channels ${A^\ast }_{x2}$, ${A^\ast }_{x1}$, ${A_{x0}}$, ${A_{x1}}$, and ${A_{x2}}$ in $|{{C_x}(h)} |(x \in [1,3])$ are centered at −0.4195mm, −0.208mm, 0mm, 0.208mm, and 0.4195mm, respectively. Based on Eq. (8), these channels should be equally spaced over the OPD axis in theory. The non-equally spaced distribution of the channels in Fig. 3 are caused by the slight difference in thickness between the two high-order retarders R₁ and R₂ in the experimental setup.

 

Fig. 2. Spectroscopic Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ of the light transmitted from the AQWP sample with azimuth at (a) 12°(b) 57° (c) 82°.

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Fig. 3. Magnitude of the autocorrelation functions of the spectroscopic Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ of the light transmitted from the AQWP sample with azimuth at (a) 12°(b) 57° (c) 82°.

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By filtering the desired channels with Hamming window and performing the reconstruction procedure described in Section2, the 16 spectroscopic elements of the Mueller matrix of the AQWP oriented at the three azimuths are determined, which are shown in Fig. 4. As can be seen, experimental results and the Mueller elements determined by the commercial SMMP are highly consistent. The zero theoretical values of the elements ${m_{33}}(\sigma )$, ${m_{0a}}(\sigma )$ and ${m_{a0}}(\sigma )(a \in [1,3])$ can be observed as well as the symmetry in ${m_{12}}(\sigma ) = {m_{21}}(\sigma )$ and ${m_{13}}(\sigma ) ={-} {m_{31}}(\sigma )$. Since the measured AQWP has a retardance accuracy of $\lambda /100$ in the spectral range of 325nm∼1100 nm, the theoretical values of the non-zero elements except ${m_{00}}(\sigma )$ are mainly dependent on the azimuths, which can be also observed in Fig. 4. The residual error for each spectroscopic Mueller element shown in the inset of Fig. 4 is less than 0.02 over most of the spectral range, which demonstrate the feasibility of STM-SMMP in the measurement of the 16 spectroscopic elements of the Mueller matrix.

 

Fig. 4. The 16 spectroscopic elements of the Mueller matrix of the AQWP sample with azimuth at (a) 12°(b) 57° (c) 82° . Dash lines indicate experimental data; solid lines indicate data measured by commercial equipment. Residuals are shown in the insets of the figures.

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3.2 Measurements of SiO₂ thin films deposited on Si substrate

To further investigate the feasibility of the proposed STM-SMMP, SiO₂ thin films deposited on Si substrate with the nominal thicknesses of 25nm, 500nm and 1000nm are measured by using the reflection configuration of the experiment setup. SiO₂ thin film is an isotropic sample and its Mueller matrix is diagonal with the non-zero elements of ${m_{00}} = {m_{11}} = 1$; ${m_{01}} = {m_{10}} ={-} \cos 2\psi$, ${m_{22}} = {m_{33}} = \sin 2\psi \cos \Delta $ and ${m_{23}} ={-} {m_{32}} = \sin 2\psi \sin \Delta $, where $\psi$ and $\Delta $ are defined by the ratio of complex Fresnel reflection coefficients ${r_p}$ and ${r_s}$ as ${r_p}/{r_s} = \tan \psi {e^{i\Delta }}$. The Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ of the reflected light from the sample surface that determined by the PSA are shown in Fig. 5. Figure 6 shows the magnitudes of the autocorrelation functions of these spectroscopic Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$. It can be seen that the magnitudes of $|{{A_{k2}}(h)} |$ and $|{A{\ast_{k2}}(h)} |$ in $|{{C_k}(h)} |$ (k=0,1) are almost zero as well as $|{{A_{p1}}(h)} |$, $|{A{\ast_{p1}}(h)} |$ in $|{{C_p}(h)} |$ (p=2,3) due to the zero theoretical values of the eight elements ${m_{02}}(\sigma )$, ${m_{03}}(\sigma )$, ${m_{12}}(\sigma )$, ${m_{13}}(\sigma )$, ${m_{20}}(\sigma )$, ${m_{21}}(\sigma )$, ${m_{30}}(\sigma )$, and ${m_{31}}(\sigma )$ in the Mueller matrix of the sample. In addition, the difference in thickness of the two retarders in the experimental setup results in non-negligible amplitude distortion in channels ${A_{x0}}(h),x \in [1,3]$. This distortion is caused by two channels corresponding to the OPD introduced by the residual of the retarder’s thickness, which can be removed by taking another measurement by rotating the polarizer in PSG to orthogonal direction.

 

Fig. 5. Spectroscopic Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ of the light reflected from the SiO₂ thin film samples with thicknesses of (a) 25 nm (b) 500 nm (c) 1000nm.

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Fig. 6. Magnitude of the autocorrelation functions of the spectroscopic Stokes parameters ${S_x}(\sigma )\textrm{ }(x \in [0,3])$ of the light reflected from the SiO₂ thin film samples with thicknesses of (a) 25 nm (b) 500 nm (c) 1000nm.

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The 16 spectroscopic elements of the Mueller matrix of the SiO₂ thin films sample determined by STM-SMMP and the commercial SMMP are shown in Fig. 7. The symmetry in the eight non-zero elements ${m_{00}}(\sigma ) = {m_{11}}(\sigma )$, ${m_{01}}(\sigma ) = {m_{10}}(\sigma )$, ${m_{22}}(\sigma ) = {m_{33}}(\sigma )$ and ${m_{23}}(\sigma ) ={-} {m_{32}}(\sigma )$ are apparent. These non-zero elements except ${m_{00}}(\sigma )$ and ${m_{11}}(\sigma )$ are varied with ellipsometric parameters that depend sensitively on the thicknesses of the SiO₂ thin films. The agreement between the experimental results and the Mueller elements determined by the commercial SMMP is good, with the residuals less than 0.02 over the spectral range of the measurement. The residuals in Fig. 4 and Fig. 7 come mainly due to the calibration errors of system parameters, including the azimuth of the elements, the retardances of the high order retarders R₁ and R₂ and the retardance of the SAQWP, and a slight error can be generated by the signal processing performed in the spectral Fourier domain. The residual error can be reduced by performing multiple repeated calibrations of the system parameters before the measurements.

 

Fig. 7. The 16 spectroscopic elements of the Mueller matrix of the SiO₂ thin film samples with thicknesses of (a) 25 nm (b) 500 nm (c) 1000 nm. Dash lines indicate experimental data; solid lines indicate data measured by commercial equipment. Residuals are shown in the insets of the figures.

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It is worth mentioning that the fundamental frequency ${\omega _0} = 12.5\pi \textrm{ }rad/s$ of STM-SMMP in the experiment equals to the rotating frequency of the retarder in PSG of the DRR-SMMP in Ref. [12]. Therefore, the measurement speed of STM-SMMP is much faster than that of the DRR-SMMP in Ref. [12], which takes 0.4s to complete the measurement. An even faster measurement speed can be achieved by setting the rotating frequency of the hollow shaft motor to higher value. While, in the case that spectral modulated SMMP with 25 channels has a total bandwidth of 1.042mm with equally spaced channel structure, the bandwidth of each channel in STM-SMMP is 5 times larger than that of the spectral modulated SMMP [19]. It can be expected that the proposed STM-SMMP will be a suitable solution for the case that the complete 16 spectroscopic elements of the Mueller matrix cannot be measured accurately by the existing SMMPs.

4. Conclusion

In this work, a spectroscopic Mueller matrix polarimeter that uses the principle of spectral polarization modulation and temporal polarization demodulation with compact, low-cost and birefringent crystal-based configuration has been developed. Compared with conventional SMMP based on dual-rotating retarder, the proposed SMMP has a faster measurement speed due to the temporal polarization demodulation provided by the only rotating wave-plate in the system. It can acquire the 16 spectroscopic elements of the Mueller matrix with a measurement speed of 80ms in the experiment. An even faster measurement speed can be achieved by setting the rotating frequency of the hollow shaft motor to higher value. Since the two high-order retarders with equal thickness in system can produce five separated channels in Fourier domain, the proposed SMMP has a larger bandwidth of each channel than that of spectral modulated SMMP. The reliable measurement with high speed makes the proposed SMMP ideal for fast measurement on the complete spectroscopic Mueller matrix in various industrial applications.