1. Introduction

Spectroscopic Mueller matrix polarimetry (SMMP) is a well-established technique that can measure all 16 elements of the 4×4 Mueller matrix of a material sample as a function of wavelength, which plays an important role in many scientific and technological areas, such as material characterization, biomedical optics, and nano-structure metrology [1–3]. A typical SMMP system consists of a light source and a polarization state generator (PSG) that generates spectroscopic polarization states to interact with the sample, a polarization state analyzer (PSA) equipped with a spectrometer that analyses the spectroscopic polarization states after the interaction. Up to now, several types of SMMP have been developed [4–12]. Among them, spectrally modulated SMMP (SM-SMMP) is a more attractive approach due to its capacity to acquire the spectra of the 16 Mueller elements of the sample in a single integration period of the detector array without mechanically moving components or electric polarization modulation devices (snapshot mode). As such, SM-SMMP is considered as an appropriate solution for in situ monitoring of complex fast processes.

In conventional SM-SMMP, the PSG and PSA are both composed of two high-order retarders and a linear polarizer, which can produce a modulated spectrum that contains separate channels within the Fourier domain. By extracting the desired channels, adequate equations to reconstruct the 16 spectroscopic Mueller elements can be obtained [10,11]. Narrow channel bandwidth is the key factor that restricts the applications of conventional SM-SMMP. As the modulated spectrum comprises more than 25 channels, the channel bandwidth of the conventional SM-SMMP is less than 1/25 of the total bandwidth. Such a narrow bandwidth is more susceptible to cross-talk between adjacent channels. If cross-talk occurs, the spectroscopic Mueller elements cannot be accurately measured. Thus, conventional SM-SMMP is limited to measure the sample whose Mueller elements are varying slowly with wavelength (information contained in the channel are bandlimited to within the bandwidth of a single channel), it may produce unsatisfactory results when measuring the spectroscopic Mueller elements with sharp spectral lines. Recently, several pioneering works have achieved an increase in channel bandwidth of spectrally modulated Stokes polarimetry [13–18]. More recently, our group proposed a hybrid spectro-temporal modulated SMMP, which can efficiently enhance the channel bandwidth [19]. However, the mechanically movable component in the system makes it unable to operate in snapshot mode. Therefore, much work remains to be done in order to accomplish a snapshot SM-SMMP with large channel bandwidth.

In this paper, we propose a modified snapshot spectrally modulated SMMP (MSM-SMMP), which increases the channel bandwidth by a factor of more than 25/7 compared to the conventional SM-SMMP. The optical configuration and operating principle of MSM-SMMP are described in Section 2. Section 3 is the experimental demonstration, and the conclusion is included in Section 4.

2. Theoretical analysis

2.1 Instrumentation

The proposed MSM-SMMP configuration scheme is illustrated in Fig. 1. It is consists of five parts: 1) a collimating optics with a Tungsten-Halogen lamp as a broadband light source, 2) the polarization state generator (PSG) consists of a linear polarizer P and two high-order retarders R₀₁ and R₀₂, 3) the measured sample, 4) the polarization state analyzer (PSA) consists of two non-polarization beam splitters (NPS1 and NPS2) and three high-order retarders R₁, R₂, R₃ respectively followed by a linear analyzer A₁, A₂, A₃, 5) a triple-spectrum sensing module consists of three parabolic mirrors, three multimode optical fibers with a diameter of 1mm and three array sensor spectrometers. The high-order retarders R₀₁, R₀₂, R₁, R₂, and R₃ are built of quartz crystals with the same thickness of 23mm. The polarizer P, the analyzers A₁∼A₃ are α-BBO Glan-Taylor prisms with an extinction ratio of less than 5×10⁻⁶. The transmission axes of the polarizer P and the analyzers A₁∼A₃ are oriented at ${0^ \circ }$ to x-axis. The fast axis of high-order retarders R₀₁, R₀₂, R₁, R₂, and R₃ are oriented at ${45^ \circ }$, ${0^ \circ }$, ${45^ \circ }$, $- {68.4^ \circ }$, and $- {21.6^ \circ }$ to x-axis, respectively.

 

Fig. 1. Schematic of the proposed modified snapshot spectral modulated SMMP

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2.2 Measurement method

Using Mueller calculations, the triple modulated spectra ${I_n}(\sigma ),n = 1,2,3$ recorded by the spectrometers can be determined as

Table 1. The detail expressions of the ${D_{zn}}$ coefficients

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Eq. (1) shows that each of the modulated spectra ${I_n}(\sigma )$ includes seven different carrier frequency components 0, ${\pm} \delta (\sigma )$, ${\pm} 2\delta (\sigma )$ and ${\pm} 3\delta (\sigma )$, which respectively carries the information about $1/32{K_n}{S_0}(\sigma ){D_{0n}}$, $1/32{K_n}{S_0}(\sigma ){D_{ {\pm} 1n}}$, $1/32{K_n}{S_0}(\sigma ){D_{ {\pm} 2n}}$, and $1/32{K_n}{S_0}(\sigma ){D_{ {\pm} 3n}}$. Therefore, inverse Fourier transform of every single ${I_n}(\sigma )$ produces seven separated channels equally spaced in the optical path difference (OPD):

Like the conventional SM-SMMP, to reconstruct the 16 Mueller elements, the ${D_{zn}}$ coefficients must be obtained firstly [11]. By filtering the channels ${A_{n0}}(h)$, ${A_{n1}}(h - L)$, ${A_{n2}}(h - 2L)$, ${A_{n3}}(h - 3L)$ and performing Fourier transform, we can obtain the following equations:

Since the phase terms ${e^{i\delta }}$, ${e^{2i\delta }}$, ${e^{3i\delta }}$ and ${K_n}{S_0}(\sigma )$ in Eq. (3a)∼Eq. (3d) are independent of the measured sample, they can be easily calibrated by measuring a linear polarizer with its polarization direction aligned at 45°relative to x axis [21–24]. Then, ${D_{0n}}$, ${D_{1n}}$, ${D_{2n}}$, and ${D_{3n}}$ can be determined. To reconstruct the 16 Mueller elements ${m_{kl}}(\sigma )(k,l \in [0,3])$, a 21-dimension vector F is defined as

From the magnitudes of ${D_{1n}}$, ${D_{2n}}$, ${D_{3n}}$, and ${D_{0n}}$ in Table 1, the $21 \times 16$-dimension transformation Matrix $\textrm{W}$ can be found, and we have

Then, the vector ${\mathbf{V}_\mathbf{m}}$ can be obtained by

In the above measurement method, the specific ${\alpha _n}$ (${45^ \circ }$, $- {68.4^ \circ }$, and $- {21.6^ \circ }$) are obtained by shuffled complex evolution for minimizing the condition number (CN) of $\mathbf{W}$ that ensures the insensitivity of the reconstructed Mueller elements to fluctuations in vector $\mathbf{F}$[13,25]. The matrix $\mathbf{W}$ has a minimum CN of 43.43, which is similar to the CN of measurement matrix of the dual-rotating retarder SMMP in Ref. [26]. In addition, each of ${C_n}(h)$ has the same total bandwidth, which is defined as the twice summation of the OPDs produced by the high-order retarders R₀₁, R₀₂, and R_n, and the seven separate channels in ${C_n}(h)$ are equally spaced in OPD domain. Thus, the channel bandwidth of the filtered channels for the reconstruction is 1/7 of the total bandwidth of every single ${C_n}(h)$.

3. Experimental results

To demonstrate the feasibility of the proposed MSM-SMMP, we firstly performed a measurement on an achromatic quarter wave plate (AQWP) with an azimuth of ${12^ \circ }$. Figure 2 shows the triple modulated spectra ${I_n}(\sigma )$ recorded by the spectrometers. As can be seen, ${I_n}(\sigma )$ are modulated with the wavenumber dependences of the phase retardances of the high-order retarders. The inverse Fourier transforms of ${I_n}(\sigma )$ that produce the separate channels in OPD are shown in Fig. 3. The channels in Fig. 3(a), Fig. 3(b), and Fig. 3(c) have almost the same total bandwidth of 1.24mm. By performing the reconstruction described in Section 2, the 16 spectroscopic Mueller elements of the AQWP are reconstructed, which are shown in Fig. 4. The zero theoretical values of ${m_{01}}(\sigma )$, ${m_{02}}(\sigma )$, ${m_{03}}(\sigma )$, ${m_{10}}(\sigma )$, ${m_{20}}(\sigma )$, ${m_{30}}(\sigma )$, and ${m_{33}}(\sigma )$ can be observed as well as the symmetry in ${m_{12}}(\sigma )\textrm{ = }{m_{21}}(\sigma )$, ${m_{13}}(\sigma )\textrm{ = } - {m_{31}}(\sigma )$, and ${m_{23}}(\sigma )\textrm{ = } - {m_{32}}(\sigma )$[20].

 

Fig. 2. The triple modulated spectra recorded by the spectrometers (a) ${I_1}(\sigma )$ (b) ${I_2}(\sigma )$ (c) ${I_3}(\sigma )$

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Fig. 3. Inverse Fourier transform of (a) ${I_1}(\sigma )$ (b) ${I_2}(\sigma )$ (c) ${I_3}(\sigma )$

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Fig. 4. The reconstructed 16 spectroscopic Mueller elements of an AQWP rotated at the azimuth of 12°.

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The integration time of the spectrometers is set to be 8ms in the experiment. The measurement time of 50ms is a suggestion in this paper for averaging and signal processing steps. The measurement accuracy of the proposed MSM-SMMP is estimated through comparisons with the results obtained by a commercial SMMP (VASE ELLIPSOMETER, J.A.Woollam, Inc., USA) that performed the same measurement. The experimental results and the Mueller elements measured by the commercial equipment are highly consistent, and the residual errors shown in the inset of Fig. 4 are less than 0.02 over most of the spectral range. The residual errors come mainly due to the calibration errors of the system parameters, including the azimuths of the system components, the retardances of the high-order retarders, and a slight error can be due to the signal processing performed in the Fourier domain.

To further investigate the feasibility of the proposed MSM-SMMP, the AQWP rotated at the azimuths, denoted by $\theta$, in the range of ${0^ \circ }\sim {90^ \circ }$ with an interval of ${22.5^ \circ }$ are measured. Figure 5 illustrates the reconstructed 16 Mueller matrix elements at the azimuths corresponding to the wavenumber of $16340c{m^{ - 1}}$, $13340c{m^{ - 1}}$ and $10340c{m^{ - 1}}$. As can be seen, the elements ${m_{01}}$, ${m_{02}}$, ${m_{03}}$, ${m_{10}}$, ${m_{20}}$, ${m_{30}}$, and ${m_{33}}$ are independent with the azimuth of the measured AQWP, which is consistent with the fact that these elements have zero theoretical values. For a perfect AQWP, the elements varied with azimuth have the function form of ${m_{11}} = {\cos ^2}2\theta$, ${m_{21}} = {m_{12}} = \cos 2\theta \sin 2\theta$, ${m_{31}} ={-} {m_{13}} ={-} \sin 2\theta$, ${m_{32}} ={-} {m_{23}} ={-} \cos 2\theta$, and ${m_{22}} = {\sin ^2}2\theta$[20]. Since the AQWP has a retardance accuracy of $\lambda /100$ in the spectral range of 325nm∼1100nm, the reconstructed elements ${m_{11}}$, ${m_{12}}$, ${m_{13}}$, ${m_{21}}$, ${m_{22}}$, ${m_{23}}$, ${m_{31}}$, and ${m_{32}}$ are mainly dependent on the azimuths, and have good agreement with the theoretical values, which can be also observed in Fig. 5. These results prove the feasibility of the proposed MSM-SMMP in the snapshot measurement of the complete spectroscopic Mueller matrix, and the measurement speed of 50ms can satisfy most applications for in-situ monitoring of fast processes [27–29]. An even faster measurement speed can be achieved by setting the integration time of the spectrometers to a lower value. Furthermore, it can be expected in the case that the conventional SM-SMMP with 25 channels and has a total bandwidth of 1.24mm with equally spaced channel structure [10,12], the proposed MSM-SMMP can improve the channel bandwidth by a factor of 25/7 than that of the conventional SM-SMMP.

 

Fig. 5. The reconstructed 16 spectroscopic Mueller elements of an AQWP rotated at the azimuth of $\theta$ corresponding to the wavenumber of $16340c{m^{ - 1}}$, $13340c{m^{ - 1}}$ and $10340c{m^{ - 1}}$

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4. Conclusion

In this work, a modified spectroscopic Mueller matrix polarimetry based on spectral modulation has been developed. Compared with conventional SM-SMMP, the proposed MSM-SMMP has larger channel bandwidth due to the cancelation of carrier frequencies in the modulated spectrum. It can acquire the complete spectroscopic Mueller matrix in snapshot mode. The reliable measurement with large channel bandwidth makes the proposed snapshot MSM-SMMP ideal for in-situ monitoring of fast processes in various industrial applications.