Spectroscopic polarimetry (SP) is a powerful tool for characterization of thin film, polarization optics, semiconductor, and others. However, mechanical polarization modulation of broadband light hampers its application for dynamic monitoring of a sample. In this article, we demonstrate the dynamic SP with features of polarization-modulation-free polarimetry and spectrometer-free spectroscopy benefiting from dual-comb spectroscopy (DCS) using a pair of optical frequency combs (OFCs). DCS enables the direct determination of polarization without the need for polarization modulation by using mode-resolved OFC spectra of amplitude and phase for two orthogonally linear-polarized lights while securing rapid, high-precision, broadband spectroscopy without the need for spectrometer. Effectiveness of the proposed system is highlighted by visualizing the hysteresis property of dynamic response in a liquid-crystal-on-silicon spatial light modulator at a sampling rate of 105 Hz.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Spectroscopic polarimetry (SP) [1–11] is based on simultaneous measurement of polarization and spectrum before and after a broadband light is incident onto a sample. Since change of polarization and spectrum reflects optical property of the sample sensitively, SP is a powerful tool for material characterization in field of fundamental science, industry, and bioscience. For example, spectroscopic ellipsometry [1–9], which is a representative SP with an oblique-incidence reflection configuration, has been applied for evaluation of refractive index and thickness of thin film [1–3], material property such as optical bandgap [4,5] and biosensing of antibody-antigen reaction [6–8].
SP measures broad spectra of a phase difference Δ and an amplitude ratio ψ between two orthogonally linear-polarized lights (x-polarization component and y-polarization component in transmission configuration or p-polarization component and s-polarization component in reflection configuration) when the broadband light with known polarization is incident onto the sample. To obtain these spectra, a spectrum of optical intensity is measured while modulating the polarization of the incident or the output light, and the resulting spectrum is used for determining the spectra of Δ and ψ. Depending on methods of polarization modulation, SP can be classified into three types: rotating-analyzer SP , rotating-compensator SP , and phase modulation SP . The rotating-analyzer SP and the rotating-compensator SP acquire the spectra of Δ and ψ by a combination of a multi-channel spectrometer with a mechanically rotating polarization optics. Although these SPs have own advantages such as easy implementation and moderate precision, such mechanical polarization modulation limits the mechanical stability and the data acquisition time (typically, several tens of milliseconds) of the SP system. On the other hand, the phase modulation SP benefits from non-mechanical, rapid polarization modulation with a photoelastic modulator (PEM), and hence can reduce the data acquisition time (typically, several tens of microseconds). However, since phase modulation with PEM shows a large dependent on the wavelength and the temperature, the precise compensation of phase modulation is required for SP. Furthermore, the fast modulation speed with PEM (tens of kilohertz) is not so good compatible with the use of a multi-channel spectrometer equipped with a camera (frame rate = 10 ∼ 1000 fps). These features make it difficult to apply PEM for broadband SP. In other words, PEM is often used in the monochromatic polarimetry rather than SP.
Recently, optical frequency comb (OFC) [12–14] has attracted attention as a light source for broadband spectroscopy because it can be used as a precise ruler of optical frequency or wavelength by control of a frequency spacing frep and carrier-envelope-offset frequency fceo in comb-tooth-like ultra-discrete multimode spectrum. Full potential of OFC can be used by acquiring mode-resolved OFC spectra of amplitude and phase with dual-comb spectroscopy (DCS) [15–18]. DCS acquires an interferogram between a pair of OFCs with slightly different repetition frequency without the need for mechanical scanning, and then obtains the mode-resolved OFC spectra of amplitude and phase by Fourier transform of the acquired interferogram. Due to its precise, accurate, and rapid acquisition of the broadband spectrum, DCS has been used for spectroscopic applications; examples include gas spectroscopy [19,20], gas thermometry , solid spectroscopy , hyper-spectral imaging , and coherent Raman imaging . Furthermore, DCS has been combined with several modulation techniques for spectroscopic polarimetry [25,26].
More recently, DCS was applied for spectroscopic ellipsometry without the need for polarization modulation, namely dual-comb spectroscopic ellipsometry or DCSE . In DCSE, ultrahigh-resolution spectra of Δ and ψ is directly obtained from mode-resolved OFC spectra of amplitude and phase for the p-polarization component and s-polarization component of the output light. Such polarization-modulation-free approach benefits from the immunity to mechanical vibrational noise, thermal instability, and polarization-wavelength dependency, which were often problems in existing mechanical-modulation-based or non-mechanical-modulation-based SP. Effectiveness of DCSE was successfully demonstrated in the determination of thickness in SiO2 film on Si substrate. On the other hand, there is a considerable need for dynamic monitoring of SiO2 film process in semiconductor production and antibody-antigen reaction in biosensing. DCSE is one potential candidate for such dynamic monitoring because it has a potential to boost the data acquisition rate up to kilohertz due to its modulation free capability and rapid spectroscopy.
In this article, for proof of concept in dynamic DCSE, we constructed dual-comb SP with a normal-incidence reflection configuration, we here call it DCSP, and apply it for investigation of dynamic property of a liquid-crystal-on-silicon spatial light modulator.
2. Experimental setup
Figure 1 shows an experimental setup of DCSP. A pair of mode-locked erbium-doped fiber OFCs (Neoark Co., Japan, OCLS-HSC-D100-TKSM, center wavelength = 1562 nm, spectral bandwidth = 50 nm; signal OFC, fceo1 = 10.5 MHz, frep1 = 100 MHz ; local OFC, fceo2 = 10.5 MHz, frep2 = 99.99895 MHz; Δfrep = frep2 – frep1 = 1050 Hz) were used for light sources in DCSP. While fceo1, frep1, and fceo2 are phase-locked to a rubidium frequency standard (Rb-FS, Stanford Research Systems, Inc., FS725, frequency = 10 MHz, accuracy = 5×10−11, instability = 2×10−11 at 1 s), the local OFC was tightly and coherently locked to the signal OFC with a constant Δfrep using a narrow-linewidth continuous-wave (CW) laser (CWL, Redfern Integrated Optics, Inc., Santa Clara, California, USA, PLANEX, center wavelength = 1550 nm, FWHM < 2.0 kHz) for an intermediate laser via control of frep2 [20,22,23]. This enables coherent averaging of the interferogram between the signal and local OFCs.
We here define x- and y-polarization components to be horizontally- and vertically-polarized components, respectively. Linear polarization of two OFCs was directed at an angle of +45° to x-polarization component using a polarization controller (PC, extinction ratio = 100,000). After passing through a non-polarizing beamsplitter cube (BS, Thorlabs Inc., Newton, NJ, USA, BS018, split ratio = 50:50), the signal OFC light was incident to a sample perpendicular to its surface. We here used a reflective spatial light modulator based on liquid crystal on silicon (LCOS) technology (SLM, Santec, Komaki, Aichi, Japan, SLM-100, phase stability < 0.002π rad, phase resolution = 10 bit) for a dynamic sample. Principal polarization axis of SLM was set to be parallel to x-polarization component, which tilts by -45° from the polarization angle of OFC1 beam. We confirmed the normal incidence of the signal OFC to the SLM surface by checking the spatial overlapping between the incident light and the reflected light. For more precise analysis, the parameter fitting of the model function for SLM will be useful. After being reflected by the sample, the signal OFC light was spatially overlapped with the local OFC light with +45° linear polarization by BS. This results in generation of the interferogram between the signal and local OFCs. An optical band-pass filter (BPF, Thorlabs Inc., Newton, NJ, USA, FB1560-12, pass band = 1560 ± 12 nm) was used to suppress the aliasing effect in the acquisition of the interferogram. The x-polarization component and y-polarization component of the generated interferogram was separated by a polarization beam splitter (PBS, Thorlabs Inc., Newton, NJ, USA, PBS104, extinction ratio > 1,000), and then was detected by a pair of photodetectors (PDs, Thorlabs Inc., Newton, NJ, USA, PDA10CF-EC, wavelength = 800-1700 nm, RF bandwidth < 150 MHz). Temporal waveform of the detected electric signal was acquired by a digitizer (National Instruments Corp., Austin, Texas, USA, NI PCI-5122, sampling rate = frep2 = 99,998,950 samples/s, number of sampling points = 95237, resolution = 14 bit). The acquisition rate of interferogram were equal to Δfrep. We acquired consecutive interferograms and integrated them as necessary to obtain appropriate signa-to-noise ratio. Fourier transform of acquired temporal waveforms gives the mode-resolved OFC spectra of amplitude and phase in the x-polarization component and y-polarization component of the interferogram. The resulting spectra were used for calculation of the Δ and ψ spectra of the sample. We beforehand measured an instrumental function of DCSP by using a gold flat mirror as a standard material and calibrated the experimental data of a sample by comparing between them.
3.1 Static characterization of SLM
The SLM used here generates arbitrary 2D phase patterns on a LCOS pixel-by-pixel basis by changing a control parameter for SLM, namely SLM grayscale value. We first investigated the change of Δ and ψ in SLM by DCSP when the SLM grayscale value in the solid image is changed from 0 to 960 step by step at interval of 120. We acquired 1000 temporal waveforms of interferogram (total acquisition time = 952 ms), integrated them, and performed Fourier transform of it. Red pots of Figs. 2(a) and 2(b) show 3D plots of Δ and ψ with respect to optical frequency ( = 191.5 ∼ 192.5 THz, sampling interval = frep1) and SLM grayscale value ( = 0, 120, 240, 360, 480, 600, 720, 840, and 960). For comparison, specification values of SLM for Δ and ψ are indicated as mesh surface. Little dependence of Δ and ψ on optical frequency was confirmed due to flat optical-frequency response of SLM and/or the limited spectral bandwidth of OFC1. On the other hand, the Δ plot shows a linear dependence on the SLM grayscale value while the ψ plot maintains the constant value with respect to the SLM grayscale value (≈ 0.8 rad). Since SLM changes only the optical phase of the x-polarization component to the fixed optical phase of the y-polarization component in the reflected light at this configuration, the behavior of Δ well reflects the polarization modulation by SLM: +45° linearly polarized light for Δ = 0 rad, right-handed circularly polarized light for Δ =π/2 rad, +135° linearly polarized light for Δ =π rad, left-handed circularly polarized light for Δ = 3π/2 rad, and +225° linearly polarized light for Δ = 2π rad, respectively. The +45° linearly polarized light for Δ = 0 rad is equivalent to the +225° linearly polarized light for Δ = 2π rad although there is a large difference of SLM grayscale value between them. In this way, the SLM could cover the optical phase change over 2π.
To evaluate Δ and ψ in SLM more precisely, we extracted the data of Δ and ψ at 192.000 THz from Figs. 2(a) and 2(b) as shown by red plots in Figs. 2(c) and 2(d). For comparison, we indicate specification values of SLM as blue lines in Figs. 2(c) and 2(d). The measure values of Δ and ψ were good agreement with the specification value of them. When the measurement accuracy of Δ and ψ (ΔRMSE and ψRMSE) is defined as a root mean square error (RMSE) between the measured value and the specification value in Δ and ψ, ΔRMSE and ψRMSE were 1.6×10−1 rad and 1.4×10−2 rad, respectively. The reason why ΔRMSE is an order of magnitude more than ψRMSE might be due to polarization dependence of complexed reflectance or transmittance in detection optics such as BS and PBS.
We next investigated the temporal stability of Δ and ψ in SLM when its SLM grayscale value was set to be a fixed value of 500. We here define the standard deviation of Δ and ψ (ΔSTD and ψSTD) as their temporal stability. Red plots in Figs. 3(a) and 3(b) show ΔSTD and ψSTD at 192.000 THz with respect to the accumulation time, respectively. For comparison, we also measured ΔSTD and ψSTD at 192.000 THz as indicated by blue plots in Figs. 3(a) and 3(b) when a gold flat mirror was set as a sample. ΔSTD and ψSTD at 192.000 THz was decreased in them when the accumulation time was increased. Also, ΔSTD and ψSTD in SLM was higher than those in the mirror, implying a little instability of Δ and ψ in SLM. ΔSTD at an accumulation time of 1 s for SLM was reasonably coincides with the specification value of phase stability in SLM (< 0.002π rad).
3.2 Dynamic characterization of SLM
We next investigated characteristics of temporal response in SLM. To this end, we measured the transient change of Δ and ψ in SLM by DCSP when the SLM grayscale level in the solid image was changed as a step function from 0 to 190 which corresponds to change of polarization from a + 45° linearly polarized light to a right-handed circularly polarized light. We acquired 10 temporal waveforms of interferogram (total acquisition time = 9.52 ms), integrated them, and performed Fourier transform of it. Δ and ψ were obtained at a sampling interval of 9.52 ms within a time window size of 3.5 s by repeating the procedure above. Figure 4(a) and 4(b) shows 3D plots of Δ and ψ with respect to optical frequency and elapsed time. Δ shows similar behavior reflecting a step-function-like change at all optical frequencies while ψ maintained the same level before and after the change of polarization.
To evaluate the temporal dynamics of Δ and ψ in SLM more precisely, we extracted the data of Δ and ψ at 192.000 THz from Figs. 4(a) and 4(b) as shown by red and blue plots in Fig. 4(c). Figure 4(d) shows temporally magnified plots of Δ and ψ in the vicinity of a rising edge of Δ in Fig. 4(c). Somewhat blunt response of Δ to the step-function-like input was confirmed by a sufficient number of sampling points in DCSP. From Δ and ψ in Fig. 4(d), we visualized the dynamic change of polarization before and after the rising edge as plots on the Poincare sphere in Fig. 4(e) and Visualization 1. Here, Poincare parameters (S0, S1, S2, and S3) are given as follows1) to (4) are transformed as follows
We also investigated hysteresis property of dynamic response in SLM by comparing temporal response of Δ and ψ between rising edge and falling edge of a step-function-like polarization change. Here, the principle polarization axis of the SLM was slightly tilted from x-polarization component to change both Δ and ψ by the change of SLM grayscale value. We measured Δ and ψ at a sampling interval of 9.52 ms. Figures 5(a) and 5(b) respectively show temporally magnified plots of Δ and ψ when the SLM grayscale level in the solid image was increased from 0 to 30. Δ increases while ψ decreases. We determined the time constant of these slopes to be 84 ms for Δ and 82 ms for ψ by the curve fitting analysis with an exponential decay function. It is reasonable that Δ and ψ shows dynamic behavior similar to each other because SLM controls Δ and ψ by changing the alignment direction of liquid crystal molecules with the external electric field. Figures 5(c) and 5(d) respectively show temporally magnified plots of Δ and ψ when the SLM grayscale level was decreased from 30 to 0. We confirmed the temporal behavior of Δ and ψ opposite to that in Figs. 5(a) and 5(b). The reason for the opposite trend is due to the tilt angle of the principle polarization axis of the SLM. If the principle polarization axis of the SLM is set to be the opposite direction, the trend of Δ and ψ is coincide with each other. Time constants of these slopes were determined to be 32 ms for Δ and 37 ms for ψ, which are in significant agreement with each other again. More importantly, time constants in Figs. 5(a) and 5(b) was significantly different from those in Figs. 5(c) and 5(d). We clearly confirmed the hysteresis property of dynamic response in SLM.
We discuss the reason for hysteresis property of dynamic response in SLM. Under no external electric field, cylindrical molecules of nematic liquid crystal are oriented in parallel to a surface of SLM and perpendicularly to an incident direction of light beam. When the external electric filed is applied in parallel to the incident direction of light beam, a long axis of liquid-crystalline molecules is aligned along the axis of the external electric filed depending on its field strength. SLM controls the polarization of incident light via the birefringence of liquid-crystalline molecules caused by their aligned direction depending on the applied external electric field. When the external electric field is increased as a step function, liquid-crystalline molecules are transiently affected by both the forced driving force and the inertial restoring force, which are opposite to each other. On the other hand, when the external electric field is decreased as a step function, only the inertial restoring force affects liquid-crystalline molecules. Such difference of contributed force between the step-like-increased and step-like-decreased external electric field leads to hysteresis property of dynamic response in SLM.
We next discuss a possibility to further increase the data acquisition rate Δfrep and broaden the spectral bandwidth Δν in DCSP. Δν is related with Δfrep and frep1 as follows
We demonstrated the dynamic characterization of polarization property in LCOS-SLM by DCSP. The proposed DCSP system has a potential to boost the scan rate of Δ and ψ spectra measurement up to 1050 Hz due to the direct polarization determination without the need for polarization modulation benefitting from DCS-based polarimetry. Effectiveness of DCSP was highlighted by visualizing the hysteresis property of dynamic response in SLM at a sampling rate of 105 Hz, indicating the dynamic interaction between the forced driving force and the inertial restoring force in liquid-crystalline molecules of SLM. Although the achieved data acquisition rate and the spectral bandwidth in this article may be still insufficient for some applications of dynamic SP, DCSP has a sufficient room to further boost the data acquisition rate and broaden the spectral bandwidth by use of GHz-spacing OFCs. DCSP will be a powerful tool for dynamic characterization of various samples.
Japan Society for the Promotion of Science (19H00871, 20J23577); Cabinet Office, Government of Japan (Subsidy for Reg. Univ. and Reg. Ind. Creation); Exploratory Research for Advanced Technology (JPMJER1304).
The authors declare no conflicts of interest.
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