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Abstract
High-speed optical polarization characterization is highly desirable for a wide range of applications, including remote sensing, telecommunication, and medical diagnosis. The utilization of the Mueller matrix provides a superior systematic and comprehensive approach to represent polarization attributes when matter interacts with optical beams. However, the current measurement speed of Mueller matrix is limited to only seconds or milliseconds. In this study, we present an ultrafast Mueller matrix polarimetry (MMP) technique based on optical time-stretch and spectral encoding that enables us to achieve an impressive temporal resolution of 4.83 nanoseconds for accurate Mueller matrix measurements. The unique feature of optical time-stretch technology enables continuous, ultrafast single-shot spectroscopy, resulting in a remarkable speed of up to 207 MHz for spectral encoding Mueller matrix measurement. We have employed an effective Mueller linear reconstruction algorithm based on the measured modulation matrix, accounting for all potential non-ideal effects of polarization components like retardance error and azimuth error. To ensure high precision, prior to the actual measurement, high-order dispersion induced by time-stretch requires adjustment through proper modulation matrix design. Upon such correction, both the results of static and rapid dynamic samples measurements exhibit exceptional accuracy with root-mean-square error (RMSE) approximately equal to 0.04 and 0.07 respectively. This presented ultrafast MMP provides a significant advance over preceding endeavors, enabling superior accuracy and increased speed concurrently.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The characterization of light polarization plays a crucial role in numerous applications, including remote sensing, telecommunication, material science, and medical diagnosis [1–3]. The utilization of Mueller matrix offers a superior systematic and comprehensive approach to represent polarization attributes when matter interacts with optical beams. This is particularly advantageous for describing scattering and depolarizing interaction with light [4,5]. The Mueller matrix has been effectively utilized to asses surface characteristics, interface states, thin films properties, and sub-wavelength nanostructures within the realm of material science [6–8]. Furthermore, as a label-free, non-invasive and quantitative tool, the Mueller matrix based imaging has found wide-ranging applications in biomedical diagnosis and human health monitoring [9,10], demonstrating its potential as an invaluable technique. MMP enables the acquisition of all 16 elements of the $4 \times 4$ Mueller matrix in each measurement, thereby offering valuable sample information such as structural parameters and anisotropy. However, conventional MMP methods typically possess a temporal resolution limited to a few seconds or milliseconds, which restricts their application in rapidly evolving samples or dynamic processes like molecular self-assembly occurring within milliseconds or even microseconds [11,12]. Consequently, it is crucial to develop high-temporal-resolution MMP techniques for real-time and in-situ monitoring of complex rapid processes.
To obtain a complete Mueller matrix measurement, it is necessary to generate and analyze a minimum of 16 polarization states, which can be modulated on time, space, or wavelength. Time-modulated polarimetry involves the utilization of rotating birefringent crystals, photo-elastic modulators (PEM), or liquid crystals as retarders with sequentially varying elements over time [4]. These techniques are well-suited for capturing high spatial resolution images of samples that remain temporally static because they are always time-consuming [13]. In contrast, wavelength or spectrally encoding polarimetry utilizes the retardance dispersion of thick birefringent crystals in front of a polarizer to vary the analyzer vector with wavelength to amplitude modulate the Stokes or Mueller parameters onto a spectral domain, resulting in a sinusoidal intensity distribution commonly referred to as channeled polarimetry [14]. With the advancement in spectral acquisition speed, this technique holds great potential for enhancing the measurement speed of Mueller matrix [15,16].
It is noteworthy that over the past few years, there has been a growing focus on improving the temporal resolution within the field of MMP. For instance, Huang et al. developed a full Mueller matrix microscope for rapid polarization imaging, employing rotating quarter-wave plates and dual division of focal plane (DoFP) CCDs, resulting in an improved acquisition time of 9 s [17]. Dubreuil et al. firstly presented snapshot Mueller polarimeter (channeled polarimetry) and a maximum absolute error of 0.08 on measured Mueller coefficients was acquired in 1 ms [16]. Alali et al. proposed and optimized a rapid Mueller matrix imaging using 4 PEMs and reached to 100 $\mathrm{\mu}$s with a mean error of $5{\% }$ [18]. Zhang et al. presented a high-speed Mueller polarimetry measurement technique utilizing dual-PEMs polarization modulation and division of amplitude demodulation principles, achieving a temporal resolution of approximately 11 $\mathrm{\mu}$s for acquiring a full matrix with standard deviations less than 0.005 [11]. Gratiet et al. presented spectral coding polarimetry based on swept laser source and photodiode detection, enabling single acquisitions at speeds as fast as 10 $\mathrm{\mu}$s (100 kHz) with maximum absolute error of 0.04 on Mueller components [19]. Despite various optical methods being employed to enhance the temporal resolution of MMP, only few systems have achieved temporal resolutions above the microsecond level for complete Mueller matrix measurements.
We recently presented an ultrafast MMP with a remarkable time resolution of 10 ns (100 MHz) [20], achieved by combining spectral encoding polarimetry and optical time-stretch technology. By compensating for the thickness errors of the retarders in the polarimetry module, we have successfully measured Mueller matrices of specific static samples and rapidly changing phase retardation, yielding average RMSE of approximately 0.05 and 0.08, respectively. In experimental setups, various systematic errors are intertwined, such as thickness errors, misalignment errors, stochastic noise fluctuations, and other nonideal effects associated with polarization elements [21]. Consequently, it becomes quite challenging to precisely calibrate and compensate for each type of error. Notably, fluctuations in light sources and detectors can also be interpreted as systematic error in fast MMP systems, but this issue can be significantly mitigated in previous methods due to their time-consuming accumulation.
In this paper, to the best of our knowledge, we present a novel high-speed spectral encoding MMP with a groundbreaking measurement speed of up to 207 MHz. Notably, we achieve significantly improved accuracy with lower RMSE of approximately to 0.04 and 0.07 for static and dynamic samples respectively compared to previous studies [20]. This remarkable improvement is primarily attributed to the utilization of an efficient calibration procedure and linear reconstruction algorithm based on the measured modulation matrix. The presented procedure involves measuring the Mueller matrix of both the polarization state generator (PSG) and polarization state analyzer (PSA) within the effective spectral range through a precise calibration process. Subsequently, by employing matrix pseudoinverse operation, we can reconstruct the Mueller matrix of the sample under investigation. And all non-ideal polarization elements’ effects are encompassed in the calibrated matrix while ensuring a least-squares fit between data and effect signal [22,23]. Consequently, our approach enables faster measurements without compromising on accuracy as evidenced by achieving lower RMSE errors compared to prior research efforts.
The remaining sections of this paper are structured as follows. In Section 2, we thoroughly analyze and derive the theoretical model for the presented ultrafast MMP. Section 3 provides a detailed description of the experimental setup and calibration method, along with an analysis of the measured results obtained from both static and dynamic samples. Finally, in Section 4, we present our concluding remarks for this study.
2. Theoretical model
The presented time-stretch based MMP measurement system comprises a broadband pulsed light source, a time-stretch module, a MMP spectral encoding module and a high-speed detection module, as shown in Fig. 1. The femtosecond mode-locked laser (MLL) serves as the light source, which is firstly spectral encoded for polarization state using the retardance dispersion of thick birefringent crystals. Subsequently, the optical pulses are time-stretched through group velocity dispersive (GVD) of a dispersive medium [24,25]. This process transforms the wavelength spectrum of each optical pulse into a temporal waveform, that is, a wavelength to time mapping is realized, which is also known as dispersive Fourier transformation [26]. Finally, the stretched pulses are detected and sampled by a high-speed photodetector and a real-time oscilloscope to obtain a serial temporal data-stream. It should be noted that when high-order dispersion exists in the dispersive fiber(mainly third-order dispersion), the wavelength-to-time mapping becomes nonlinear [27], their relationship can be expressed as
where ${D}$ is the dispersion coefficient and ${S}$ is the dispersion slope.A typical spectral encoding MMP mainly consists of PSG and PSA. And the measurement sample is placed between them in a transmission spatial light path as shown in Fig. 1. Then the Mueller matrix of the sample is modulated on its spectrum, whose intensity is serially detected in time and can be expressed as
The advantages of this procedure are as follows. Firstly, all the non-ideal effects of polarization elements in PSG and PSA are treated as a whole and represented by the calibrated modulation matrix [29]. In contrast, if these non-ideal effects need to be calibrated separately, they can only be calibrated in the Fourier transform based method (Channeled splitting method). For instance, its primary error influencing factors, the analysis window position misalignment and the retardance error of the four retarders, could be measured and calibrated separately [15,20]. However, measuring and calibrating the azimuth errors of retarders and polarizers is complex in MMP [21]. Secondly, the pseudoinverse procedure readily handles overdetermined data while providing a least-squares fit of the data to signals, and it effectively eliminates noise as much as possible [4]. Thirdly, compensating for high-order dispersion of the time-stretch module can be omitted compared to Fourier methods in [20]. Since the final digital sampling has equal time intervals but unequal wavenumber intervals according to Eq. (1), its nonlinear relationship should be considered to reduce spectral leakage during the Fourier transform. However, the matrix pseudoinverse-based method can be immune to the influence caused by unequal wavelength intervals. Finally, this matrix reconstruction method is easy to implement and comprehend.
3. Experimental setup
The presented ultrafast MMP measurement system is illustrated in Fig. 2. To ensure a sufficiently high optical power entering the free space, an optical time-stretch technique is implemented prior to spectral encoding, which differs from Fig. 1 but achieves the same effect [30]. A custom-made MLL generates an optical pulse train with a repetition period of approximately 9.66 ns ($\sim$103.5 MHz) and a pulse width of around 100 fs. The pulses are separated into two arms equally. And pulses in one arm are temporally delayed half the period and then recombined with another arm, resulting in a temporally interlaced pulse train with a doubled repetition rate of approximately 207 MHz. The pulse train is then bandpass filtered to approximately 7.6 nm bandwidth centered at 1550 nm wavelength region. After passing through dispersion compensating fiber with GVD value of −580 ps/nm and dispersion slope value of −2.076 $\rm {ps/nm^{2}}$, the femtosecond laser pulses are stretched to approximately 4.5 ns wide. An optical amplifier is used to amplify the average power and then the amplified light is converted into a spatial beam with a spot diameter of 2 mm using a fiber collimator.
Fig. 2. Configuration diagram of the presented ultrafast MMP measurement system. (a) The broadband pulsed source and the time-stretch module that can connect to (b) or (c). (b) The DRR-MMP system for measuring PSG and PSA in (c). (c) The present MMP measurement system. OC: Optical Coupler. PD: PhotoDetector. DCF: Dispersion Compensation Fiber.
The next spectral encoding module consists of a PSG and PSA. The PSG is composed of a linear polarizer $(P_{1})$ and retarders $(R_{1}, R_{2})$, and the PSA is symmetrical to PSG. Here the retarders $R_{1}$, $R_{2}$, $R_{3}$ and $R_{4}$ are made of ${\rm YVO}_{4}$ birefringent crystal with a thickness of $e: e: 5e: 5e \left ( e = 1.6 {\rm mm} \right )$, respectively. Their optical axes are set parallel to the retarders and maximal birefringence $\Delta n$ is 0.2039 at 1550 nm. The fast axis of $R_{2}$ and $R_{3}$ and the transmission axis of $P_{1}$ and $P_{2}$ are set parallel, and the fast axis of $R_{1}$ and $R_{4}$ are oriented at $45^{\circ }$ relative to $R_{2}$, $R_{3}$. The working wavelength range should exceed the complete free spectral range (FSR) of the spectrum, which can be expressed as $\Delta \lambda = {\lambda _c^{2}} / { \left ( e \Delta n \right ) }$, where $\lambda _c$ is the central wavelength, and then the FSR can be calculated as 7.364 nm.
To accurately measure the analyzing vectors ${\mathbf {A}}_n$ of PSA and the generating vectors ${\mathbf {G}}_n$ of PSG, a calibration procedure needs to be performed using a common dual rotating retarder Mueller matrix polarimetry (DRR-MMP). As a time-sequenced polarimetry technique, DRR-MMP has longer measurement time but has reached a mature state of development and simple configuration playing an important role in modern polarimetric research [9,31]. DRR-MMP also consists primarily of two parts, which we refer to as DPSG and DPSA respectively. Here the DPSG is inserted before PSG, comprising a fixed horizontal linear polarizer $P_{G}$ and a rotatable quarter-wave plate $Q_{G}$ driven by a rotation stage (Thorlabs DDR100/M). Similarly, the DPSA is symmetrical to the DPSG and inserted after PSA.
The final spatial light that carries Mueller matrix information is then collected into a fiber by a collimator and detected by an InGaAs PIN photodetector (Lightsensing LSIHPD-A12) with a 0.9 A/W responsivity at 1550 nm and a bandwidth of 12 GHz. The electric signal output from the photodetector is then acquired and digitized using a real-time oscilloscope (OSC, LeCroy LabMaster 10Zi) with a bandwidth of 36 GHz and sampling rate of 80 GS/s. Finally, the digital data is processed using MATLAB on a computer. A fiber OC with a power splitting ratio of 90:10 is inserted before the photodetector to simultaneously acquire the spectrum using an optical spectrum analyzer (OSA).
The workflow of DRR-MMP can be summarized as follows: $Q_{A}$ is rotated every $5^{\circ }$ from $5^{\circ }$ to $180^{\circ }$, $Q_{G}$ is rotated five times faster than $Q_{A}$. After each incremental rotation, both stages are stopped completely and a spectral intensity reading is recorded by OSA as shown in Fig. 2. For the calibration procedure, we choose an OSA to acquire the spectrum due to its the lower ground noise compared to high-speed photodetector and OSC. The resulting data set is then processed according to the algorithms in [13]. Keeping only DPSG and DPSA in the optical path shown in Fig. 2, we take air as standard reference sample to calculate and compensate the errors in orientational alignment and errors caused by nonideal retardation elements of DRR-MMP. After calibration of DRR-MMP, the Mueller matrix spectrum of air is measured with an average RMSE value of 0.0037, which matches well with the theoretical matrix. Then PSG and PSA are respectively placed into the optical path showed in Fig. 2, and their Mueller matrices in the working wavelength range are measured by the well-calibrated DRR-MMP. The generating vectors ${\mathbf {G}}_n$ is the first column of the Mueller spectrum of PSG, while the analyzing vectors ${\mathbf {A}}_n$ is the first row of the Mueller spectrum of PSA. It is important to note that the wavelengths of ${\mathbf {G}}_n$ and ${\mathbf {A}}_n$ should strictly match the wavelength of the serial temporal sampling data-stream according to Eq. (2). Subsequently, we obtain the modulation matrix $\mathbf {W}$ using Eq. (3) and Eq. (4), and its pseudoinverse $\mathbf {W}^{-1}$ is partially shown in Fig. 4(a), which is a real matrix with 325 rows and 16 columns.
4. Simulation verification
To validate the efficacy of the proposed system, a series of simulations were conducted. The wavelength range selected spans from 1542.6 to 1549.7 nm, encompassing almost an entire Free Spectral Range (FSR). The birefringence of ${\rm YVO}_{4}$ was calculated using the SellMeier equation provided in [32]. The sampling number N was set at 325, consistent with the experimental, and their wavenumber intervals were kept equal. The primary sources of systematic errors comes from both retarder thickness and assembly alignment inaccuracies. Table 1 illustrates how these error values affect the RMSE of Mueller matrix for air. It is observed that RMSE induced by phase errors in retarders can be effectively mitigated through FFT-based correction methods employed in our previous studies [20], whereas RMSE caused by assembly alignment errors cannot be rectified as easily. However, all these error-induced RMSE can be significantly reduced to nearly zero through linear reconstruction methods, aligning with the analysis presented in section 2.
Table 1. Simulation results of RMSE for air’s Mueller matrix, obtained through the previous FFT method and the linear reconstruction method.
The simulation can incorporate the wavelength-to-time nonlinear mapping, resulting from the high-order dispersion in the time-stretch module. It is worth noting that the RMSE of FFT methods is 0.1911 when considering the presence of the nonlinear factors, which has been compensated in [20]. In contrast, the present method yield an RMSE of approximately ${10}^{-15}$ by simply ensuring consistency among all simulated wavelengths.
So far, the current linear reconstruction method appears to be flawless. However, in reality, the entire system is highly susceptible to noise interference. Firstly, a random noise distribution is introduced into the spectrum of the air sample. As depicted in Fig. 3(a), the RMSE of Mueller matrix for air gradually diminishes SNR increases from 10 dB to 35 dB. It can be observed that an SNR greater than 29 dB yields an RMSE below 0.01, which aligns closely with Siu’s previous studies [23]. Another source of noise arises from measurement errors in modulation matrix ${\mathbf {W}}$ caused by fluctuations in light intensity and other non-ideal factors. Random noise can directly affect the elements of the matrix ${\mathbf {G}}_n$ and ${\mathbf {A}}_n$. The amplitudes range from square root of 0.5 $\times$ ${10}^{-3}$ to 5 $\times$ ${10}^{-3}$, which is consistent with the measured RMSE value of air using DRR-MMP at 0.0037. The RMSE obtained from the linear reconstruction method against the noise power is depicted in Fig. 3(b). Subsequently, both types of noise can be simultaneously considered, and their corresponding results are illustrated in Fig. 3(c). It can be observed that the result RMSE ranges approximately between 0.02 to 0.06.
Fig. 3. Simulated RMSE against the noise on light intensity noise (a), the noise on the elements of the modulation matrix (b), and their combined effect (c).
5. Experimental results
In order to evaluate our reconstruction and calibration methods, air is measured and the acquired time-domain data has 325 points within 4.06 ns, as shown in Fig. 4(b). By multiplying the matrix $\mathbf {W}^{-1}$ with the normalized data-stream according to Eq. (5), we can get the Mueller matrix of air as shown in Fig. 4(c) and its RMSE value is 0.0434. This value aligns closely with the simulation results shown in Fig. 3(c) and provides preliminary evidence supporting the feasibility of our system. Additionally, utilizing multiple periods of the time-domain signal allowed us to calculate a SNR of 23.9 dB, represented by a red pentagram in Fig. 3(a).
Fig. 4. Reconstruction of Mueller matrix (c) of air from the modulation matrix (a) and the acquired light intensity (b).
To further demonstrate the stability and accuracy of the presented MMP system, we conducted a series of static measurements on a linear polarizer, a quarter-wave plate and a half-wave plate with the azimuth varying between $0^{\circ }$ and $90 ^{\circ }$ every $10^{\circ }$. The RMSE results for every measured Mueller matrix are shown by the red curve in Fig. 5. The average RMSE values for the three samples are 0.0415, 0.0419 and 0.0498 respectively, indicating excellent agreement with theoretical matrices. For the sake of contrast, the RMSEs of the Mueller matrix constructed using the previous Fourier transform based method are shown by the blue curve in Fig. 5, whose average values are 0.0594, 0.0509 and 0.0655 respectively. This clearly demonstrates that our newly presented MMP method significantly reduces RMSE while improving speed compared to the previous method.
Fig. 5. RMSE of measured Mueller matrices of linear polarizer (a), quarter-wave plate (b) and half-wave plate (c) with the azimuth varying between $0^{\circ }$ and $90^{\circ }$.
Now that the accuracy of the presented MMP system has been verified through a series of static experiments, its performance for Mueller matrix measurement during rapid dynamic processes will be demonstrated. We choose a free space EOM (Thorlabs EO-AM-NR-C3) as a rapidly changing dynamic sample, which acts as a voltage-controlled phase retarder and its $V_{\pi }$ is about 550 V at 1550 nm and maximum response frequency is about 100 MHz. Limited by the output voltage and the bandwidth of the high-voltage driver amplifier (Thorlabs HVA200), we apply a sinusoidal voltage with a frequency of 500 kHz and an amplitude of $\pm$ 100 V to the EOM, resulting in a sinusoidal phase retardation correspondingly. The applied sinusoidal voltage with two full periods (4000 ns) and the sampled data-stream from the OSC are shown in Fig. 6(a) and (b), where a single full sampling period with 387 sampling points (4.83 ns) is shown in Fig. 6(c). By resolving each sampling period to Mueller matrix, we make a comparison between all measured 826 matrices and their corresponding theoretical matrices, as shown in Fig. 6(e). The deviation between them, that is the RMSE shown by the red line in Fig. 6(d), has an average value of 0.0759. It can be observed that the two matrices are basically consistent, thereby demonstrating the acceptable accuracy of our presented MMP system in the Mueller matrix measurement of rapid dynamic processes. Notably, it should be emphasized that the average RMSE of rapid changing sample in previous method is 0.0887.
Fig. 6. (a) Sinusoidal voltage applied on EOM. (b) The detected time-domain signal from an OSC with a time length of 4000 ns. (c) One sample period of the detected signal in 4.83 ns. (d) RMSE of measured Mueller matrices of EOM under modulation. (e) The theory (red) and the measured (black) Mueller matrix of the EOM under modulation.
6. Conclusion
In this paper, based on optical time-stretch and spectral encoding Mueller matrix polarimetry, we have developed a high-speed MMP with a temporal resolution of 4.83 ns in a single measurement of the complete Mueller matrix. By utilizing an efficient linear reconstruction method, our presented MMP system achieves simultaneous improvements in both lower RMSE and faster speed compared to our previous research. The reconstruction Mueller matrix of static and rapid dynamic samples show high accuracy with RMSE values approximately equal to 0.04 and 0.07 respectively. The residual errors are mainly from the spectrum sampling noise and the measurement error in the modulation matrix, as confirmed through simulation verification. Therefore, we anticipate that our presented MMP system holds great potential for investigating the mechanism of various rapid changing process, such as in molecular self-assembly process, the particle migration process, and the phase transition process in material under high temperature and pressure.
Funding
National Natural Science Foundation of China (62175092); Basic and Applied Basic Research Foundation of Guangdong Province (2023A1515011155); Guangzhou Basic and Applied Basic Research Foundation (202201010088).
Disclosures
The authors declare no conflicts of interest
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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