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Dual-comb spectroscopic polarimetryfor dynamic characterization ofpolarization property

Koresawa, Hidenori, Gouryeb, Marc, Shibuya, Kyuki,
Mizuno, Takahiko, Hase, Eiji, et al.

Hidenori Koresawa, Marc Gouryeb, Kyuki Shibuya, Takahiko Mizuno, Eiji
Hase, Yu Tokizane, Ryo Oe, Takeo Minamikawa, Takeshi Yasui, "Dual-comb
spectroscopic polarimetry for dynamic characterization of polarization
property," Proc. SPIE 11671, Real-time Measurements, Rogue Phenomena,
and Single-Shot Applications VI, 116710H (5 March 2021); doi:
10.1117/12.2583107
Event: SPIE LASE, 2021, Online Only
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Dual-comb spectroscopic polarimetry for dynamic characterization ofpolarization property
Hidenori Koresawa,a Marc Gouryeb,a,b Kyuki Shibuya,a,c,d Takahiko Mizuno,c,d,e Eiji Hase,d,e Yu
Tokizane,e Ryo Oe,a Takeo Minamikawa,c,d,e and Takeshi Yasui*c,d,eaGraduate School of Advanced Technology and Science, Tokushima University, 2-1 MinamiJosanjima, Tokushima, Tokushima 770-8506, Japanb
College of Sciences and Technology, University of Bordeaux, 351 cours de la Libération, TalenceCedex 33405, FrancecGraduate School of Technology, Industrial and Social Sciences, Tokushima University, 2-1
Minami-Josanjima, Tokushima, Tokushima 770-8506, JapandJST, ERATO, MINOSHIMA Intelligent Optical Synthesizer Project, 2-1 Minami-Josanjima,
Tokushima, Tokushima 770-8506, Japane
Institute of Post-LED Photonics (pLED), Tokushima University, 2-1 Minami-Josanjima,
Tokushima, Tokushima 770-8506, Japan
ABSTRACTSpectroscopic 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-freepolarimetry and spectrometer-free spectroscopy benefiting from dual-comb spectroscopy (DCS) using a pair of opticalfrequency 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.
Keywords: Spectroscopic polarimetry, Optical frequency comb, Dual-comb spectroscopy

1. INTRODUCTION
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 propertyof 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 ∆ and an amplitude ratio ψ between two orthogonally linearpolarized 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 thepolarization of the incident or the output light, and the resulting spectrum is used for determining the spectra of ∆ and ψ.
Depending on methods of polarization modulation, SP can be classified into three types: rotating-analyzer SP [9],rotating-compensator SP [10], and phase modulation SP [11]. The rotating-analyzer SP and the rotating-compensator SP
acquire the spectra of ∆ 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, suchmechanical polarization modulation limits the mechanical stability and the data acquisition time (typically, several tens
Real-time Measurements, Rogue Phenomena, and Single-Shot Applications VI, edited by
Daniel R. Solli, Georg Herink, Serge Bielawski, Proc. of SPIE Vol. 11671, 116710H© 2021 SPIE · CCC code: 0277-786X/21/$21 · doi: 10.1117/12.2583107
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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 wavelengthand 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 witha 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 opticalfrequency or wavelength by control of a frequency spacing frep and carrier-envelope-offset frequency fceo in comb-toothlike ultra-discrete multimode spectrum. Full potential of OFC can be used by acquiring mode-resolved OFC spectra ofamplitude 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 moderesolved 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 [21], solid spectroscopy [22], hyper-spectral imaging [23], andcoherent Raman imaging [24]. Furthermore, DCS has been combined with several modulation techniques forspectroscopic polarimetry [25,26].
More recently, DCS was applied for spectroscopic ellipsometry without the need for polarization modulation,
namely dual-comb spectroscopic ellipsometry or DCSE [27]. In DCSE, ultrahigh- resolution spectra of ∆ and ψ isdirectly obtained from mode-resolved OFC spectra of amplitude and phase for the p-polarization component and spolarization component of the output light. Such polarization-modulation-free approach benefits from the immunity tomechanical 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 wassuccessfully demonstrated in the determination of thickness in SiO2 film on Si substrate. On the other hand, there is aconsiderable 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-onsilicon 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; OFC1, fceo1 = 10.5 MHz,
frep1 = 100 MHz ; OFC2, fceo2 = 10.5 MHz, frep2 = 99.99895 MHz; ∆frep = frep2 – frep1 = 1050 Hz) were used for lightsources 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 OFC2 wastightly and coherently locked to the OFC1 with a constant ∆frep 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 OFC1 and OFC2.

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Figure 1. Experimental setup. OFC1 and OFC2, optical frequency combs; CWL, narrow-linewidth CW laser; Rb-FS,
rubidium frequency standard; PC, polarization controller; BS, non-polarizing beamsplitter cube; SLM, reflective spatial light
modulator; BPF, 1560 ±12 nm band-pass filter; PBS, polarization beam splitter; PDs, photodetectors.

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 apolarization 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 OFC1 light was incident to a sample perpendicular toits 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 thepolarization angle of OFC1 beam. We confirmed the normal incidence of the OFC1 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 OFC1 light was spatially overlappedwith the OFC2 light with +45° linear polarization by BS. This results in generation of the interferogram between the
OFC1 and OFC2. An optical band-pass filter (BPF, Thorlabs Inc., Newton, NJ, USA, FB1560-12, pass band = 1560 ± 12nm) was used to suppress the aliasing effect in the acquisition of the interferogram. The x-polarization component and ypolarization 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,

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sampling rate = frep2 = 99,998,950 samples/s, number of sampling points = 95237, resolution = 14 bit). The acquisition
rate of interferogram were equal to ∆frep. We acquired consecutive interferograms and integrated them as necessary toobtain 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 ∆ 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 ofa sample by comparing between them.

3. RESULT
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 ∆ 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 Fouriertransform of it. Red pots of Figs. 2(a) and 2(b) show 3D plots of ∆ 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 ∆ and ψ are indicated as mesh surface. Little dependence of ∆ 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 ∆ 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 thex-polarization component to the fixed optical phase of the y-polarization component in the reflected light at thisconfiguration, the behavior of ∆ well reflects the polarization modulation by SLM: +45° linearly polarized light for ∆ = 0
rad, right-handed circularly polarized light for ∆ = π/2 rad, +135° linearly polarized light for ∆ = π rad, left-handed
circularly polarized light for ∆ = 3π/2 rad, and +225° linearly polarized light for ∆ = 2π rad, respectively. The +45°
linearly polarized light for ∆ = 0 rad is equivalent to the +225° linearly polarized light for ∆ = 2π rad although there is alarge difference of SLM grayscale value between them. In this way, the SLM could cover the optical phase change over2π.
To evaluate ∆ and ψ in SLM more precisely, we extracted the data of ∆ and ψ at 192.000 THz from Figs. 2(a) and2(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 ∆ and ψ were good agreement with the specification value of them. Whenthe measurement accuracy of ∆ and ψ (∆RMSE and ψRMSE) is defined as a root mean square error (RMSE) between themeasured value and the specification value in ∆ and ψ, ∆RMSE and ψRMSE were 1.6×10−1 rad and 1.4×10−2 rad,respectively. The reason why ∆RMSE is an order of magnitude more than ψRMSE might be due to polarization dependenceof complexed reflectance or transmittance in detection optics such as BS and PBS.

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Fig. 2. Static characterization of ∆ and ψ in SLM. 3D plots of (a) ∆ and (b) ψ with respect to optical frequency and SLM
grayscale value. (c) ∆ and (d) ψ at 192.000 THz with respect to SLM grayscale value.

We next investigated the temporal stability of ∆ and ψ in SLM when its SLM grayscale value was set to be a fixedvalue of 500. We here define the standard deviation of ∆ and ψ (∆STD and ψSTD) as their temporal stability. Red plots in
Figs. 3(a) and 3(b) show ∆STD and ψSTD at 192.000 THz with respect to the accumulation time, respectively. For
comparison, we also measured ∆STD and ψSTD at 192.000 THz as indicated by blue plots in Figs. 3(a) and 3(b) when agold flat mirror was set as a sample. ∆STD and ψSTD at 192.000 THz was decreased in them when the accumulation time
was increased. Also, ∆STD and ψSTD in SLM was higher than those in the mirror, implying a little instability of ∆ and ψ in
SLM. ∆STD at an accumulation time of 1 s for SLM was reasonably coincides with the specification value of phasestability in SLM (< 0.002π rad).

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Figure 3 Temporal stability of (a) ∆ and (b) ψ with respect to accumulation time in SLM.

3.2 Dynamic characterization of SLM
We next investigated characteristics of temporal response in SLM. To this end, we measured the transient change of∆ 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. ∆ 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 ∆ and ψ with respect to opticalfrequency and elapsed time. ∆ 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.

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Fig. 4. Dynamics characterization of ∆ and ψ in SLM. 3D plots of (a) ∆ and (b) ψ with respect to optical frequency andelapsed time. Temporal response of ∆ and ψ at 192.000 THz within the time range of (c) 0 to 3.5 s and (d) 1.89 s to 2.51 s.
(e) Trajectory of polarization change on the Poincare sphere.

To evaluate the temporal dynamics of ∆ and ψ in SLM more precisely, we extracted the data of ∆ 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 plotsof ∆ and ψ in the vicinity of a rising edge of ∆ in Fig. 4(c). Somewhat blunt response of ∆ to the step-function-like input
was confirmed by a sufficient number of sampling points in DCSP. From ∆ 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). Here,
Poincare parameters (S0, S1, S2, and S3) are given as follows

S0 = E x 2 + E y 2 ,(1)

S1 = Ex 2 − E y 2 ,(2)S2 = 2Ex E y cos Δ,(3)

S3 = 2Ex E y sin Δ,(4)

where Ex and Ey are x-polarization and y-polarization components of optical electric field. When S0 is normalized, Eqs.
(1) to (4) are transformed as follows

S0 = 1,

Ex 2 − E y 2

1− tan 2 ψ
S1 = 2=,Ex + E y 2 1+ tan 2 ψ
S2 =2Ex E y cos Δ
Ex + E y22=

2 tanψ cos Δ,
1+ tan 2 ψ

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(5)

(6)

(7)

S3 =

2Ex E y sin Δ
Ex + E y22=

2 tanψ sin Δ.
1+ tan 2 ψ(8)Plots on the Poincare sphere clearly indicate a straight trajectory from +S2 to +S3, which corresponds to a + 45˚ linearlypolarized light to a right-handed circularly-polarized light. A little deviation from the straight trajectory will be due to thephase stability in DCSP rather than inherent dynamic response of SLM.
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 grayscalevalue. 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 toeach other because SLM controls Δ and ψ by changing the alignment direction of liquid crystal molecules with theexternal 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 theprinciple 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 agreementwith 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.

4. DISCUSSION
We discuss the reason for hysteresis property of dynamic response in SLM. Under no external electric field, cylindricalmolecules 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, along axis of liquid-crystalline molecules is aligned along the axis of the external electric filed depending on its fieldstrength. SLM controls the polarization of incident light via the birefringence of liquid-crystalline molecules caused bytheir aligned direction depending on the applied external electric field. When the external electric field is increased as astep 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 astep 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

Δv =f rep1 ( f rep1 + Δf rep )2Δf rep≈

f rep12
2Δf rep.

(9)

For example, in the present setup (frep1 ≈ 100 MHz, Δfrep ≈ 1 kHz), Δν is estimated to be 5 THz, corresponding to awavelength range of 1550 ± 20 nm. In other words, even though OFC has sufficiently broad bandwidth of optical
spectrum, the effective spectral bandwidth is limited to avoid the aliasing effect as indicated by Eq. (9). This wavelength
range may be still insufficient for some applications of DCSP. From Eq. (9), increase of Δfrep leads to reduction of Δν
whereas Δν is quadratic of frep1. Therefore, increase of Δfrep and Δν can be achieved by increasing both Δfrep and frep1.
Fortunately, increase of frep1, corresponding to decrease of spectral resolution, is acceptable to DCSP because most of
samples in DCSP does not indicate so sharp spectral features. Furthermore, increase of frep1 without the change of total
power and spectral bandwidth leads to increase of mode power in OFC and hence increase of signal-to-noise ratio in
DCSP, enabling no signal-accumulation measurement at an acquisition rate of Δfrep. There are a few candidates forhigher-frep or GHz-spacing OFCs: electro-optic OFC [28, 29], quantum-cascade-laser OFC [30, 31], and microcomb [32,33]. Such OFCs were effectively applied for higher-Δfrep DCS [29, 31, 33] together with nonlinear spectral broadening.

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For example, in the case of DCSP with frep1 of 10 GHz and Δfrep of 500 kHz, Δν is increased up to 100 THz,corresponding to a wavelength range of 1550 ± 400 nm. In this way, there is still room for enhancement of the dataacquisition rate and the spectral bandwidth by selection of OFCs.

5. DISCUSSION
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 stillinsufficient 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.

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