1. Introduction

In the past two decades, dual-comb spectroscopy (DCS) has attracted considerable interest in many research fields because of its ability to simultaneously achieve high frequency accuracy, high spectral resolution, and short acquisition times [1–4]. DCS is based on the optical frequency comb (OFC) technique and employs two OFCs with slightly different repetition rates. In DCS, optical responses of materials from the UV to the terahertz frequency regions are down-converted to the radio frequency (RF) region, which enables direct sampling of the time-domain waveform of the light with conventional electronic devices. Thus, DCS techniques enable the simultaneous and independent measurement of amplitudes and phases of light. Owing to this advantage over conventional spectroscopic techniques, various applications of DCS have been proposed [5–15]. For instance, fast optical coherence tomography [5], absolute distance measurement [6], characterization of metamaterials [7], measurement of arbitrary optical waveforms [8], nonlinear DCS spectroscopy [9], broadband atmospheric phase spectroscopy [10], high-speed vibrometry [11], hyperspectral imaging [12], and DCS-based microscopy [13–15] have been demonstrated.

The capability of DCS to measure amplitudes and phases of light is a remarkable feature especially for solid state spectroscopy. We can exploit this capability to realize applications besides those mentioned above; the simultaneous and independent measurement of these two parameters enables direct determination of the complex refractive index, which is a fundamental material parameter. This direct determination by DCS techniques has advantages compared to conventional methods, in which one usually has to employ the Kramers-Kronig relation [16,17] for the determination of complex refractive index. Because a very broad spectrum is required to apply the Kramers-Kronig relation with reasonable accuracy, the simultaneous and independent measurement of amplitudes and phases of light by DCS is a promising alternative. Indeed, there have been a few efforts to use DCS for the determination of complex refractive indices of materials [18].

The complex refractive index of anisotropic media is required to understand optical responses of materials. Almost all natural and artificial materials exhibit optical anisotropy by themselves or induced by external conditions (i.e., intrinsically or extrinsically). For a better understanding of their optical responses, which can be employed in devices, precise information about the anisotropy of the complex refractive index is indispensable. Since polarization spectroscopy is a powerful solution in that regard, we have developed a polarization-sensitive (PS) DCS system by implementing the rotating compensator technique [19]. In addition, a few ideas to extend the applications of DCS to polarization measurement have been proposed [20,21]. However, these methods either contain mechanical moving parts or require a complex optical alignment and lack quantitative analysis [19–21]. Thus, further developement of a powerful PS DCS technique is needed to precisely determine anisotropic complex refractive indices.

In this paper, we propose a PS DCS setup that is based on an electro-optic amplitude modulator (EOM). Because this measurement method directly provides amplitudes and phases in two orthogonal of the electric field for each comb tooth, we can directly derive the anisotropic complex refractive indices. Our proposed method has no mechanically moving component and is a straightforward extension of the typical DCS setup [2–4]. To verify the validity of our approach, we experimentally demonstrate the determination of arbitrary polarization states for each comb tooth, the directions of fast and slow optic axes of the sample, and its birefringence. In the demonstration, we utilized the second harmonic generation of the erbium (Er)-doped-fiber-based mode-locked lasers as light sources, i.e., OFCs with a center wavelength of ∼780 nm, because numerous interesting optical phenomena exist in the range of 780 nm compared to those in the range of 1560 nm. These experimental results reveal the effectiveness of the developed EOM-based PS DCS technique for investigation of the optical anisotropy of materials. This work demonstrates for the first time that DCS is able to measure the birefringence with an accuracy on the order of 10⁻⁵ in the range of 780 nm.

This paper is organized as follows: in Section 2, we explain the principle of the EOM-based PS DCS, the technical details of the experimental setup are briefly explained in Section 3, and three experiments are presented in Section 4. The conclusion is given in Section 5.

2. Measurement principle and data analysis

In this section, we explain the measurement principle behind the PS DCS with an EOM for polarization detection and introduce the analysis method that is required to determine the anisotropic complex refractive indices. In Section 2.1 we elaborate the general configuration of the EOM-based PS DCS system and derive a mathematical expression of the measured interferogram. Section 2.2 treats the case in which an external sinusoidal voltage is applied to the EOM placed in the optical path in front of detector. Owing to the modulation of the polarization state of light induced by the applied voltage, we obtain sidebands for each comb tooth. These sidebands contain the information of amplitudes and phases in two orthogonal directions. The analysis of these sidebands based on the obtained interferogram is explained and we show how to derive the polarization information from them. Section 2.3 briefly discusses the effect of overlapping between different sidebands (more details on the overlapping effect are provided in Appendix A). In Section 2.4, we derive the complex refractive indices of the sample (along its fast and slow optic axes).

2.1 Basic concept of PS DCS with an EOM

A schematic of the EOM-based PS DCS setup is shown in Fig. 1. The x and y axes of the employed coordinate system are indicated in the figure. Two OFCs with slightly different repetition rates (the frequency difference is denoted by Δf_r) are used in this setup and shown with red and blue lines. Before combining the two OFCs by a non-polarizing beam splitter (BS), the OFC beam with repetition rate f_r passes through both a polarizer (P1) and the sample, while the other with repetition rate f_r - Δf_r passes through polarizer P2. After that, the polarization of the combined beam is measured by the subsystem consisting of the first quarter-wave plate (QWP) Q1, the EOM, the second QWP Q2, a polarizer (P3), and a detector. Such a subsystem has so far not been combined with a DCS setup [3]. Because the first comb is used to probe the optical information of the sample, it is labeled “Signal comb (S-comb)”. The second comb is used to measure the amplitudes and phases of the first comb, and is referred to as the “Local comb (L-comb)”.

 

Fig. 1. Simple schematic of the EOM-based PS DCS setup. f_r, f_r - Δf_r: repetition rates of the S-comb and L-comb, respectively. P1, P2, P3: polarizers. BS: beam splitter. Q1, Q2: quarter-wave plates. EOM: electro-optic amplitude modulator. f_EOM: modulation frequency of the EOM. The degrees in parentheses represent the angle of transmission or fast optic axes of each optical component with respect to the x-axis.

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A careful adjustment of the initial polarization angles of the optical beams in Fig. 1 is important for a reliable PS DCS system. Firstly, the transmission axis of P1 is set parallel to the x-axis. Thus, a linearly-polarized S-comb with polarization direction along the x-axis irradiates sample. After the S-comb has passed through the sample, its polarization state differs due to the sample’s optical anisotropy, i.e., birefringence. Therefore, the electric field (E-field) of the S-comb right after the BS has non-zero components along both x and y directions. The time-dependent E-field components of the S-comb in x and y directions are denoted by E_S,x(t) and E_S,y(t), respectively, which can be written as follows:

Secondly, the transmission axis of P2 is also set parallel to the x-axis. Since the L-comb right after the BS is linearly-polarized along the x-axis, we only need to consider the x component of the L-comb's E-field. E_L,x(t) is defined as the time-dependent E-field of the L-comb after the BS in the x-direction,

2.2 Polarization modulation of the interferogram

In this subsection, we consider the case in which an external sinusoidal voltage is applied to the EOM in front of the detector. Under this condition, the retardance induced by the EOM depends on time, resulting in an intensity modulation of the interferogram. Due to this modulation, we can obtain sidebands for each comb tooth. These sidebands contain the information of amplitudes and phases in orthogonal directions, and thus provide information about the polarization.

In Eq. (16), the parameter C (which represents the relative retardance between the x- and the y-direction) changes by applying an external voltage to the EOM. When a sinusoidal voltage V(t) is applied to the EOM, C changes according to

In order to determine the four expressions $E_{\textrm{S,x}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}$, $E_{\textrm{S,y}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}$, $\phi _{\textrm{S,x}}^\textrm{n} - \phi _{\textrm{L,x}}^\textrm{n}$, and $\phi _{\textrm{S,y}}^\textrm{n} - \phi _{\textrm{L,x}}^\textrm{n}$ independently, we need to choose a proper modulation frequency f_EOM. To explain the procedure for choosing an appropriate f_EOM, a conceptual diagram of $U(t)$ in the RF domain for the case f_EOM= Δf_r/9 is shown in Fig. 2. Each black line represents a beat note between the two modes of the S-comb and the L-comb with index n. These black lines can be always observed irrespective of the EOM modulation, and thus we refer to it as the original signal. The amplitude of the signal at nΔf_r is $({1 + {J_0}({{m_{\textrm{EOM}}}} )} )\cdot E_{\textrm{S,x}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}$ according to Eq. (20). As shown in Fig. 2, because of the modulation of the retardance in the EOM-based PS DCS, several sidebands appear between the original signals in the RF domain. Here, we define the signal amplitudes (A_n,±2k, A_n,±(2k-1)), and phases (Φ_n,±2k, Φ_n,±(2k-1)) of these sidebands as follows:

 

Fig. 2. Conceptual diagram of PS DCS data in the frequency domain for f_EOM= Δf_r /9. Each black line shows an interference signal between two modes of the S-comb and the L-comb with index n. The orange and the green sidebands arise from the modulation of the retardance by the sinusoidal voltage applied to the EOM. The amplitudes (phases) of the signals with frequencies nΔf_r ± f_EOM and nΔf_r ± 3f_EOM are indicators of the amplitudes (phases) in the y direction of the mode n of the S-comb, while those with frequencies nΔf_r ± 2f_EOM and nΔf_r ± 4f_EOM reflect those in the x direction.

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In Fig. 2, the sidebands with frequencies nΔf_r ± 2kf_EOM are indicated with green lines and those with frequencies nΔf_r ± (2k-1)f_EOM are shown with orange lines. According to Eq. (20), the amplitudes (phases) of sidebands with frequencies of nΔf_r ± 2f_EOM and nΔf_r ± 4f_EOM correspond to A_n,±2 (Φ_n,±2) and A_n,±4 (Φ_n,±4), respectively. In the same way, the amplitudes (phases) of sidebands with frequencies of nΔf_r ± f_EOM and nΔf_r ± 3f_EOM correspond to A_n,±1 (Φ_n,±1) and A_n,±3 (Φ_n,±3), respectively. Thus, by analyzing the amplitudes and the phases of these sidebands, we can independently determine the four expressions $E_{\textrm{S,x}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}$, $E_{\textrm{S,y}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}$, $\phi _{\textrm{S,x}}^\textrm{n} - \phi _{\textrm{L,x}}^\textrm{n}$, and $\phi _{\textrm{S,y}}^\textrm{n} - \phi _{\textrm{L,x}}^\textrm{n}$.

In order to quantitatively determine these four expressions of a single S-comb tooth n, the following twelve parameters A_n,±1, A_n,±2, A_n,±3, Φ_n,±1, Φ_n,±2, and Φ_n,±3 (amplitudes and phases of sidebands with frequencies of nΔf_r ± f_EOM, nΔf_r ± 2f_EOM, and nΔf_r ± 3f_EOM) are sufficient if the influence of the higher-order modes is negligible (this condition is discussed in the next subsection). Hereafter, we explain the analytical procedure required to determine these four expressions.

The first step is to determine the modulation depth m_EOM. According to Eq. (21), the following relation is satisfied:

After the evaluation of m_EOM, according to Eq. (21), one can determine $E_{\textrm{S,x}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}$ and $E_{\textrm{S,y}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}$ using the following Eqs. (24) and (25):

2.3 Effect of overlap between the sidebands

In the previous subsection, we only considered the sidebands with frequencies nΔf_r ± 2kf_EOM and nΔf_r ± (2k-1)f_EOM under the restriction k≤ 2. However, as described in Eq. (20), the interferogram has additional higher-order frequency components (k > 2). These components overlap with the lower-order frequency components as shown with the dashed lines in Fig. 2. For example, in case of f_EOM= Δf_r /9, the sidebands of the mode n + 1 with frequencies (n + 1)Δf_r – 8f_EOM, (n + 1)Δf_r – 7f_EOM, (n + 1)Δf_r – 6f_EOM, and (n + 1)Δf_r – 5f_EOM overlap with the sidebands of mode n with frequencies nΔf_r + f_EOM, nΔf_r + 2f_EOM, nΔf_r + 3f_EOM, and nΔf_r + 4f_EOM, respectively. This means that the above relatively simple polarization analysis of the experimentally obtained sidebands contains unavoidable errors. Owing to the fact that the higher-order Bessel functions are usually smaller than the lower-order Bessel functions, we can minimize the above errors as follows:

2.4 Determination of anisotropic complex optical properties

As described in Section 2.2, by analyzing the sideband signals with and without sample, we can obtain the sample-induced changes in the amplitudes and phases of both x and y components of the E-field, i.e., $E_{\textrm{S,x}}^{\textrm{w/,n}}/E_{\textrm{S,x}}^{\textrm{w/o,n}}$, $E_{\textrm{S,y}}^{\textrm{w/,n}}/E_{\textrm{S,y}}^{\textrm{w/o,n}}$, $\phi _{\textrm{S,x}}^{\textrm{w/,n}} - \phi _{\textrm{S,x}}^{\textrm{w/o,n}}$, and $\phi _{\textrm{S,y}}^{\textrm{w/,n}} - \phi _{\textrm{S,y}}^{\textrm{w/o,n}}$. In this subsection we explain how the anisotropic complex optical properties of solids (that is, the complex refractive indices along the optic axes) can be derived from the obtained parameters. For a quantitative data analysis, two procedures are required. One is the determination of the directions of the two optic axes and the other is the determination of the complex refractive indices along the optic axes.

The directions of the optic axes can be determined by comparing the polarization states of the S-comb with and without sample using the Poincaré sphere representation [24–26]. For this analysis, we introduce the Stokes parameters $S_{\textrm{1,n}}^{\textrm{w/}}$, $S_{\textrm{2,n}}^{\textrm{w/}}$, and $S_{\textrm{3,n}}^{\textrm{w/}}$ [27], which represent the polarization state of the n-th mode of the S-comb including the sample. In this study, according to Eqs. (24)–(27), the Stokes parameters can be calculated by the following formulas:

After θ_slow has been determined, we can calculate the complex refractive indices along the optic axes. For this, we rotate the E-field of the S-comb by θ_slow. Then, the E-field components parallel and perpendicular to the slow optic axis $({E_{\textrm{S,}\,\textrm{slow}}^\textrm{n}\exp ({i\phi_{\textrm{S,}\,\textrm{slow}}^\textrm{n}} ),E_{\textrm{S,}\,\textrm{fast}}^\textrm{n}\exp ({i\phi_{\textrm{S,}\,\textrm{fast}}^\textrm{n}} )} )$ are calculated as follows:

Further we define $E_{\textrm{S,}\,\textrm{slow}}^{\textrm{w/,n}}$ ($E_{\textrm{S,}\,\textrm{slow}}^{\textrm{w/o,n}}$), $E_{\textrm{S,}\,\textrm{fast}}^{\textrm{w/,n}}$ ($E_{\textrm{S,}\,\textrm{fast}}^{\textrm{w/o,n}}$), $\phi _{\textrm{S,}\,\textrm{slow}}^{\textrm{w/,n}}$ ($\phi _{\textrm{S,}\,\textrm{slow}}^{\textrm{w/o,n}}$), and $\phi _{\textrm{S,}\,\textrm{fast}}^{\textrm{w/,n}}$ ($\phi _{\textrm{S,}\,\textrm{fast}}^{\textrm{w/o,n}}$) as the amplitudes and the phases E-fields of the S-comb tooth parallel and perpendicular to the slow optic axis with (without) sample. The transmittances parallel and perpendicular to the sample's slow optic axis for the n-th mode of the S-comb can then be calculated by dividing the square of the amplitude of the E-field obtained with sample by that without sample.

3. Experimental setup

In this study, we use two OFCs to probe the anisotropic optical properties of the sample by the EOM-based DCS system. The two OFCs (the S-comb and the L-comb) are generated by two Er-doped-fiber-based mode-locked lasers with a repetition rate f_r of 48 MHz. Both have electro-optic phase modulators for fast servo control and delay lines for roughly tuning f_r. As we set Δf_r to 1073.25 Hz, the observable spectrum bandwidth, which is limited by the Nyquist condition, is about 1.07 THz. (It is possible to choose a smaller Δf_r to obtain a broader spectrum bandwidth as long as Eq. (14) and m_EOM $\le$ ∼3.8 rad are satisfied.) Each laser output is divided into three branches. The first branches are used for detecting the carrier-envelope offset frequencies (f_S,ceo and f_L,ceo) by a f-2f interferometer, and the second branches are used for detecting the beat notes between a 1.54 μm continuous-wave (CW) laser and one of the comb teeth (f_S,beat or f_L,beat). In the third branches, we amplify the pulses by Er-doped fiber amplifiers (EDFA) and convert the center wavelengths from 1550 to 775 nm using a waveguide-type periodically-poled lithium niobate (PPLN).

The S-comb and the L-comb are stabilized by the following method. First, f_S,ceo and f_L,ceo are phase-locked to RF reference signals generated by the function generators via a feedback to the current of the pump laser diodes (LDs). Second, f_S,beat and f_L,beat are also phase-locked to RF reference signals via feedback loop (to the electro-optic phase modulator in the oscillator, a piezo-electronic transducer, and a Peltier element) with different time constants. We controlled f_S,ceo – f_L,ceo and f_S,beat – f_L,beat respectively to be multiple of Δf_r in order to satisfy the condition for the coherent averaging [4]. Because the CW laser is stabilized to an ultra-stable cavity, the relative line width between the S-comb and the L-comb is below 1 Hz, as observed by a spectrum analyzer with resolution limit of 1 Hz. The relative line width of sub-Hertz is narrow enough to enable separation of the sidebands under the present experimental conditions. Note that alternative dual-comb systems based on other stabilization schemes are applicable as long as the line widths of their comb teeth are narrow enough to separate sidebands.

Figure 3 shows the actual EOM-based PS DCS setup used in this study. Here, we divide the S-comb beam into two paths by the beam splitter BS1. The transmitted beam enters the so-called “sample path”, in which the beam passes through a polarizer (P1) and the sample. The other path is called “reference path”, in which the beam is reflected by two mirrors. Then, the two beams of the S-comb and the beam of the L-comb are combined by the two beam-splitters BS2 and BS3 as shown in the figure. Finally, the combined beam passes through the polarization measurement subsystem for detection of amplitudes and phases in two mutually orthogonal directions. To obey the Nyquist condition and avoid aliasing effects, the bandwidth of the OFC has to be restricted [1–4,22]. Therefore, we place an optical band pass filter (BPF) in front of the detector. The RF signal obtained from the detector is electrically filtered by a low-pass filter (LPF). The signal after the LPF is amplified by an amplifier (SA-915 D1, NF Corporation) and sampled by a 14-bit digitizer (PXIe-5122, National Instruments). By dividing the S-comb into two beams, we obtain double peak structures in the interferograms. One peak originates from the interference between the S-comb from the sample path and the L-comb, and the other peak originates from the interference between the S-comb from the reference path and the L-comb. For the polarization modulation of the light inside the EOM (EO-AM-NR-C1, Thorlabs) on the detector side, a sinusoidal voltage with an amplitude of V_pp = 360 V is applied to the EOM by a function generator and a high-voltage amplifier (HVA200, Thorlabs). The modulation frequency is set to f_EOM =Δf_r/9 = 119.25 Hz. As a well-defined sample, we employ a QWP (AQWP05M-980, Thorlabs) attached to a neutral density (ND) filter (NE03B-B, Thorlabs). We set the sample on a motorized rotation stage that is placed on a motorized translation stage. Thus, we can easily change the angle of the sample and we can measure the interferogram with and without the sample by moving the translation stage. All measurements were performed at room temperature and under ambient atmosphere.

 

Fig. 3. Experimental setup of PS DCS with an EOM on the detector side. f_r, f_r –Δf_r: repetition rates of the S-comb and L-comb, respectively. PPLN: periodically-poled lithium niobate, P1, P2, P3: polarizers. BS1, BS2, BS3: beam splitters. S: Sample. Q1, Q2: quarter-wave plates. f_EOM: modulation frequency of EOM on the detector side. BPF: optical bandpass filter. LPF: low-pass filter. PC: personal computer.

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4. Results

In this section, we discuss the experimental results obtained by the EOM-based PS DCS system. In Section 4.1, we present the results on the polarization state of each comb tooth of the S-comb. We clarify that the experimentally obtained polarization states of each comb tooth are consistent with the theory. Section 4.2 focuses on the anisotropic complex refractive index of the sample. In Section 4.3, we evaluate the experimental uncertainty of the sample's complex refractive index obtained by our method. It was possible to measure the birefringence with an accuracy on the order of 10⁻⁵. From the evaluated standard deviation, we conclude that the EOM-based PS DCS is able to accurately determine the birefringence of the sample because of the high polarization sensitivity of this technique and the high stability of the amplitudes and phases in our DCS system. These experimental results prove that the EOM-based PS DCS is reliable.

4.1 Measurement of the polarization state of each comb tooth

The first task is to measure the polarization state of each mode of the S-comb after it has passed through the sample. Since the sample (consisting of QWP and ND filter) possesses birefringence, the polarization state of the S-comb after the sample depends on the angle of the slow optic axis of the sample (which equals that of the QWP, θ_QWP). Thus, we measured the polarization states of each mode of the S-comb for various θ_QWP and compared the results with the theory.

Prior discussing the experimentally observed dependence on θ_QWP, we explain the analytical procedure with the experimental data shown in Fig. 4. This figure plots a representative interferogram measured without sample, where the polarization state of the S-comb is linearly polarized along the x-axis. Hereafter, the interferogram data measured within the period 9/Δf_r is referred to as U(t). U(t) is a periodic function with a period of 9/Δf_r [duration = 9/(1073.25 Hz) ≈ 8.4 ms] in case of f_EOM=Δf_r/9. Thus, we obtained 36 sets of U(t) within a measurement time of ≈ 300 ms and averaged them. The averaged U(t) is denoted by U_ave(t). Figure 4(a) plots U_ave(t) within the time interval 9/Δf_r. We can observe nine relatively large signals, where each peak corresponds to the temporal overlap between the S-comb and the L-comb occurring every 1/Δf_r ≈ 0.93 ms. The intensities of these nine signals differ because of the polarization modulation induced by the EOM. Figure 4(b) shows an enlarged view of the data around 0.16 ms in Fig. 4(a). In Fig. 4(b), two interference signals are observed. The interference signal on the right (left) side corresponds to the interference signal between the L-comb and the S-comb from the reference-(sample-) path as mentioned in Section 3. In our experimental setup, the interference signal from the sample path appears earlier, since the sample-path is shorter than the reference-path. To separately analyze the information of sample or reference paths, we extracted 500 data points around the maximum intensity of each interference signal of either sample- or reference-path [indicated by the dotted square in Fig. 4(b)] and replaced the other data points with zeros. The obtained interferogram containing only data from the sample-path is referred to as U_S(t). The interferogram containing only data from the reference-path is called U_R(t). We retrieved the Fourier transform of these interferograms in the range from the RF to optical frequencies.

 

Fig. 4. (a) Averaged interferogram U_ave(t) on a large time scale and (b) that observed in narrow time range. In the latter, we can confirm the signals from the sample-path (earlier) and that from the reference-path (delayed). (c) Normalized amplitude of the Fourier component of U_S(t) in the optical frequency region. (d) Normalized RF amplitude derived from the Fourier transform of U_S(t). (e) RF amplitude derived using the additional analysis described in Appendix B.

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Figure 4(c) shows the amplitude of the Fourier component of U_S(t) in the optical frequency range. Figure 4(d) shows the amplitude of the Fourier component of U_S(t) in RF range. The frequency resolution of Fig. 4(d) is determined by f_EOM =Δf_r/9 = 119.2 Hz. The components of the original signal (indicated by the black arrow) are observed every ninth step, i.e., at a frequency interval of Δf_r. Each of these large signals is a beat note between a certain S-comb tooth and the L-comb tooth. In addition to these large components, sidebands appear as well. As described in Section 2.2, these sidebands contain the polarization information of the corresponding large components. For example, the frequency components at nΔf_r ± 2f_EOM = nΔf_r ± 2Δf_r/9 and nΔf_r ± 4f_EOM = nΔf_r ± 4Δf_r/9 (green arrows) contain the information of E_S,x(t), while the information of E_S,y(t) appears at the frequencies of nΔf_r ± f_EOM = nΔf_r ± Δf_r/9 and nΔf_r ± 3f_EOM = nΔf_r ± 3Δf_r/9 (orange arrows).

According to Eq. (20) in Section 2.2, in case that the S-comb is linearly polarized parallel to the x-axis, it is expected that only sidebands at nΔf_r ± 2kf_EOM appear. However, small sidebands intensities at nΔf_r ± (2k-1)f_EOM are clearly observed in Fig. 4(d). We found that the origin of the latter was the residual birefringence of the EOM (see Appendix B). Thus, we further analyzed the experimental data to exclude the effect of the residual birefringence of the EOM from the raw data. Figure 4(e) shows the amplitude of the Fourier component of U_S(t) after this additional analysis step to obtain the actual polarization information. It is clearly found that the intensities of sidebands at nΔf_r ± (2k-1)f_EOM are almost zero while those of sidebands at nΔf_r ± 2kf_EOM are relatively large. This result is consistent with the actual experimental condition (the S-comb is linearly polarized along the x-axis), which means that the additional analysis works well. More details of the additional analysis are provided in Appendix B.

Hereafter, we present the experimental results of the polarization state of each mode of the S-comb after passing through the sample. In this experiment, we rotated θ_QWP from 0 to 270 degrees in steps of 5°. To evaluate the polarization state of the S-comb tooth, we measured U(t) for each θ_QWP as follows: (1) We measured U(t) without the sample for about 300 ms. (2) We inserted the sample into measurement position by moving the translation stage and measured U(t) for about 300 ms. (3) We removed the sample by moving the translation stage again and changed θ_QWP by 5° by rotating the motorized rotation stage. We repeated the above procedure from (1) to (3) 55 times. We averaged the sets of U(t) obtained within the ≈ 300 ms for each θ_QWP to obtain the U_ave(t) for each θ_QWP. Thus in total, we obtained 110 U_ave(t) (55 with and 55 without the sample). We corrected the carrier phase drift among the different U_ave(t) using the interference signals of the reference path according to [4] and [30]. In the following analysis, for all θ_QWP, we use the averaged m_EOM among the θ_QWP derived from Eq. (23).

We first determine the frequency dependence of the three Stokes parameters, which can be derived from Eq. (28) in Section 2.4 and U_ave(t). It is found that the derived Stokes parameters for each θ_QWP are almost independent of the frequency (results not shown here). For Fig. 5, we chose the single comb tooth at 385.034024 THz (n = 17070, corresponding to the absolute mode number 3981558, which is counted from 0 Hz) and plotted the θ_QWP dependences of its three Stokes parameters (S_1,17070, S_2,17070, and S_3,17070). In addition to the experimental data, we also plotted the theoretically calculated curves with the solid curves obtained from [31]

 

Fig. 5. Experimental results of Stokes parameters of the mode at 385.034024 THz. Solid curves are theoretical curves.

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4.2 Determination of the anisotropic complex refractive index of the sample

In the following, we explain the procedure required to determine the anisotropic complex refractive indices along the optic axes of the sample: (1) Determination of the direction of the sample’s slow optic axis by measuring the polarization states with and without sample. (2) Derivation of the phase difference between the E-field components with and without sample, and derivation of the transmittances along the sample’s fast and slow optic axes. (3) Calculation of the complex refractive indices of the sample along its optic axes.

First, we determine the direction of the sample’s slow optic axis with respect to the x-axis, i.e., θ_slow, by comparing the polarization states with and without sample. We derive θ_slow for each θ_QWP by analyzing the data shown in Fig. 5 (data without the sample are not shown in Fig. 5 because they are trivial). We confirmed that θ_slow has no frequency dependence in our case. Thus, we can average θ_slow over the frequency range from 384.85 to 385.35 THz. Figure 6 plots the averaged θ_slow as a function of θ_QWP. The data can be well fitted using the linear function aθ_QWP + θ₀ (the black line in Fig. 6). The obtained fitting parameters are a = 0.998 and θ₀ = −63.5^°. The evaluated Pearson’s correlation coefficient is 0.9999, evidencing that θ_slow is almost linearly proportional to θ_QWP. This indicates that our method can precisely determine the directions of the slow optic axis of optically-anisotropic materials.

 

Fig. 6. θ_QWP dependence of θ_slow. Red symbols represent the experimental data and the black solid line represents the fitting result.

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Second, using the determined θ_slow and Eqs. (29)–(33), we calculate the phase differences between the E-field components with and without the sample, and also the transmittances along the sample’s fast and slow optic axes. Because the birefringence of the sample is determined by a zero-order QWP, we assume that the phase retardance between the fast and slow optic axes is smaller than 2π. In addition, because the refractive index of the sample is assumed to be constant, we fit the abovementioned phase differences (i.e., $\Delta \phi _{\textrm{S,}\,\textrm{slow}}^\textrm{n} = \phi _{\textrm{S,}\,\textrm{slow}}^{\textrm{w/,n}} - \phi _{\textrm{S,}\,\textrm{slow}}^{\textrm{w/o,n}}$ and $\Delta \phi _{\textrm{S,fast}}^\textrm{n} = \phi _{\textrm{S,}\,\textrm{fast}}^{\textrm{w/,n}} - \phi _{\textrm{S,}\,\textrm{fast}}^{\textrm{w/o,n}}$) with linear functions using the same y-intercepts. Then, in order to set the phase at 0 Hz to 0 rad, we added an offset (the same y-intercepts) to these phase differences. The results are referred to as Δϕ_fast and Δϕ_slow.

We note that an unavoidable ambiguity of the y-intercept in the fitting procedures for Δϕ_fast and Δϕ_slow still remains, because of the small number of interferograms used for averaging: the determined y-intercepts of Δϕ_fast and Δϕ_slow might actually differ by multiples of 2π. This fitting uncertainty of the y-intercept primarily leads to an uncertainty in the absolute value of the refractive index. As this study focuses on the assessment of the anisotropy by the EOM-based PS DCS, the knowledge of the absolute value of complex refractive index is not essential.

Figure 7(a) shows Δϕ_fast and Δϕ_slow for θ_slow = 150.2^°. Both phase differences exhibit a clear linear dependence on the frequency in the measured optical frequency range, supporting the assumption that the refractive index is insensitive to the optical frequency. For a quantitative investigation of the phase retardance, we investigate the frequency dependence of the phase retardance plotted in Fig. 7(b). Figure 7(b) evidences that the phase retardance is almost constant π/2 in the whole measured frequency range, although a small oscillation is observed. This oscillation comes from the multiple reflections inside the sample and its influence on the evaluation of the phase retardance is not significant. We emphasize that Δϕ_slow – Δϕ_fast ≈ π/2 is consistent with the specification of the QWP, which is designed to generate a phase retardance of π/2 between mutually orthogonal optic axes. Therefore, these results prove the validity of our analytical method described in Section 2.4.

 

Fig. 7. (a) Phase differences between the E-field components with and without sample along the fast (red) and the slow (blue) optic axes. (b) Phase retardance between the slow and the fast optic axes in (a). (c) Transmittance (for the intensity) along the fast (red) and the slow (blue) optic axes.

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Figure 7(c) shows the transmittances along the fast (red) and the slow (blue) optic axes of the sample, denoted by T_fast and T_slow, respectively. We find that the magnitudes of T_fast and T_slow are almost equal and oscillating. Since the quartz inside the sample possesses birefringence, the multiple reflection inside the quartz causes the oscillation of polarization state of the light with a period of approximately 0.15 THz. This oscillation in polarization leads to the oscillations in transmittances. The averaged values of T_fast and T_slow are both about 0.58. According to the product specifications of the ND filter and the QWP plate, their transmittances are about 0.57 and 0.97, respectively. Therefore, the total expected transmittance of the sample is 0.55. The experimentally determined transmittance and the calculated transmittance are considered consistent.

Finally, we determine the complex refractive indices along the fast and the slow optic axes of the sample by using Eq. (34) and the results for Δϕ_fast, Δϕ_slow, T_fast, and T_slow provided in Fig. 7. The sample thickness d is 3.62 mm, according to the product specifications. Figures 8(a) and 8(b) show the evaluated real (N) and the imaginary (K) part of the anisotropic complex refractive index of the sample, respectively. The red (blue) line corresponds to the complex refractive index along the fast (slow) axis. It is found that the real parts of the complex refractive indices of the two optic axes are well distinguishable over the whole frequency range. Because the value of K is mostly based on the value of the transmittance, K oscillates at the same rate as the oscillations observed in the transmittance. The subtle difference between their values can be detected owing to the high polarization sensitivity of our technique and the high stability of the DCS system.

 

Fig. 8. (a) The real and (b) the imaginary part of the anisotropic complex refractive index of the sample.

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4.3 Precision of the obtained anisotropic complex refractive index

To clarify the reliability of the presented measurement technique, we need to clarify the experimental uncertainty of the anisotropic complex refractive index obtained by the EOM-based PS DCS. To evaluate the measurement error of our system, we determine the standard deviations of N and K (denoted by σ_N and σ_K, respectively). In general, in order to derive the standard deviation of a given parameter, at least three values obtained under the same experimental condition are required. Therefore, we divided the totally accumulated data within one measurement cycle (≈ 300 ms) into several groups, where each group contains 5 sets of U(t). Each group was averaged using the corresponding 5 sets of U(t), and we refer to this average over 5 data sets as U′_ave(t).

In the following, we use the same m_EOM as that used in Section 4.2. To determine the precision of the complex refractive index, we use the procedure outlined in Section 4.2, but replace U_ave(t) by U′_ave(t). First, we calculate θ_slow by comparing the Stokes parameters with and without sample. The Stokes parameters obtained with and without sample are averaged among the Stokes parameters derived from several U′_ave(t) obtained for each θ_QWP. Then, using Eq. (29) and θ_slow, we calculate several pairs comprising data (amplitudes and phases of the E-field components along the optic axes) obtained with and without sample. After that, we can calculate the mean values and the standard deviations of phase differences and transmittances between the E-field components with and without sample along both optic axes for each U′_ave(t). To determine the y-intercept, we use the mean values of the phase differences calculated above. Then we add the same value to the phase differences of the E-field components along the sample’s optic axes to derive Δϕ_fast and Δϕ_slow. By using Eq. (34) and Δϕ_fast, Δϕ_slow, T_fast, and T_slow, N and K can be calculated. Finally, we can evaluate the mean values and the standard deviations of N and K for the slow and fast optic axes. For evaluating the standard deviations of N and K, we adopted the evaluation method based on error propagation, which has been proposed by Tripathi et al. [29].

Figures 9(a) and 9(b) show the mean values of N and K, respectively, for the single comb tooth at 385.034024 THz as a function of θ_slow. The standard deviations of these N and K values are shown in Figs. 9(c) and 9(d), respectively. We note that large deviations are observed at around 0, 90, and 180° in Figs. 9(a)–9(d). Under these angles, the polarization direction of the S-comb is parallel to the one of the two optic axes. In such cases, the complex refractive index along the optic axis parallel to the y-axis cannot be precisely evaluated, because of the absence of a y-component in the E-field of the S-comb. However, excluding the above singular angles, the mean values of N and K are almost insensitive to θ_slow. The small fluctuation of these mean values has two possible reasons: (1) Because of the rotation of the sample, the position on the sample probed by the OFC beam is different for each θ_slow. Since the values of N and K weakly depend on the sample’s position, N and K depend on the angle of the sample. (2) The ambiguity of the y-intercept in the fitting procedure for Δϕ_fast and Δϕ_slow causes a fluctuation of N and K. There may be other reasons as well, but the above are considered to mainly contribute to the fluctuation. The data evidences that σ_N and σ_K are also insensitive to θ_slow and they are on the order of 10⁻⁶ (note that the ambiguity of multiples of 2π in Δϕ_fast and Δϕ_slow still remains). In Fig. 10, the derived difference between the N along the fast and slow optic axes (that is, the birefringence ΔN) is plotted as a function of θ_slow. The evaluated ΔN is insensitive to θ_slow except at the singular angles and is 5.49(55)×10⁻⁵ especially for θ_slow = 150.2^°. Thus, it is clearly found that ΔN is an order of magnitude larger than σ_N.

 

Fig. 9. (a) N and (b) K of the sample along the fast (red) and slow (blue) axes at 385.034024 THz. (c) The standard deviations of the mean of N and (d) that of K for various θ_slow along the fast (red) and slow (blue) axes.

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Fig. 10. Birefringence of the sample for 385.034024 THz measured at various θ_slow.

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5. Conclusion

In this study, we developed a PS DCS technique that employs the polarization modulation induced by an EOM to probe sample-induced changes in the polarization of the OFC. Our EOM-based PS DCS achieves a high polarization sensitivity in addition to general advantages of DCS such as high precision, high frequency resolution, and short acquisition times. Using the developed system, we demonstrated three experiments using a QWP attached to a ND filter as a sample: (1) The polarization states of each comb tooth of the S-comb were measured for various angles of the sample. The measured polarization information shows a good agreement with the theory. (2) The anisotropic complex refractive indices of the sample were determined by the following procedure: (a) The direction of the sample’s slow optic axis was determined from the changes in the polarization states with and without sample. (b) Phase differences between the E-field components obtained with and without sample were calculated. The transmittances along the sample’s optic axes were evaluated. (c) From the derived phase differences and the transmittances, the complex refractive indices of the sample were determined along its two optic axes, leading to a precise value of the birefringence. (3) Finally, we estimated the experimental uncertainty of the anisotropic complex refractive index obtained by this method. In conclusion, the EOM-based PS DCS is able to accurately determine the birefringence of a given sample, because of the EOM-based modulation technique for measuring the polarization, and due to the high stability in amplitudes and phases obtained in our DCS system. We demonstrated the first DCS measurement of birefringence near 780 nm.

Appendix A: Effect of higher-order Bessel functions on polarization analysis

The interferogram signal U(t) in Eq. (20) contains signals with amplitudes proportional to Bessel functions with orders higher than four. These sidebands can decrease the accuracy of the proposed simple analysis of the polarization state of each S-comb tooth. In addition, as the number of sidebands (2 l) increases, the measurement time becomes longer [the latter is proportional to (2l + 1)]. Thus, it is advantageous to choose a l that is as small as possible. In the following, we discuss the minimum number of sidebands required to correctly determine the polarization state of each comb tooth.

In case of f_EOM= Δf_r/(2l + 1), where l is an integer ≥3, the sidebands with frequencies nΔf_r ± f_EOM and nΔf_r ± 3f_EOM overlap with those with frequencies (n ± 1)Δf_r ${\mp} $ 2lf_EOM and (n ± 1)Δf_r ${\mp} $ 2(l-1)f_EOM, respectively. Thus, A′_n,±3/A′_n,±1, which is the apparent amplitude ratio (i.e., that including the effect of the overlap) of the sideband at nΔf_r ± 3f_EOM to that at nΔf_r ± f_EOM is expressed as follows according to Eq. (20):

(36)$$\begin{array}{l} \frac{{{{A'}_{\textrm{n,3}}}}}{{{{A'}_{\textrm{n,1}}}}}\\ = \frac{\sqrt {\begin{array}{l} {J_\textrm{3}}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}} \right)^2} + {J_{\textrm{2}\left( {\textrm{l - 1}} \right)}}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,x}}^{\textrm{n + 1}}E_{\textrm{L,x}}^{\textrm{n + 1}}} \right)^2}\\ + 2{J_\textrm{3}}({m_{\textrm{EOM}}}){J_{2\left( {\textrm{l - 1}} \right)}}({m_{\textrm{EOM}}})E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}E_{\textrm{S,x}}^{\textrm{n + 1}}E_{\textrm{L,x}}^{\textrm{n + 1}}\sin \left[ - \left( {2l + 1} \right){\varphi _{\textrm{EOM}}} \right. \\ \left.+ \left( {\phi _{{\textrm{S,x}}}^{\textrm{n + 1}} - \phi _{{\textrm{L,x}}}^{\textrm{n + 1}} - \phi _{{\textrm{S,y}}}^\textrm{n} + \phi _{{\textrm{L,x}}}^\textrm{n}} \right) \right] \end{array} }}{\sqrt {\begin{array}{l} {J_1}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}} \right)^2} + {J_{\textrm{2l}}}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,x}}^{\textrm{n + 1}}E_{\textrm{L,x}}^{\textrm{n + 1}}} \right)^2}\\ + 2{J_\textrm{1}}({m_{\textrm{EOM}}}){J_{\textrm{2l}}}({m_{\textrm{EOM}}})E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}E_{\textrm{S,x}}^{\textrm{n + 1}}E_{\textrm{L,x}}^{\textrm{n + 1}}\sin \left[ - \left( {2l + 1} \right){\varphi _{\textrm{EOM}}} \right. \\ \left.+ \left( {\phi _{{\textrm{S,x}}}^{\textrm{n + 1}} - \phi _{{\textrm{L,x}}}^{\textrm{n + 1}} - \phi _{{\textrm{S,y}}}^\textrm{n} + \phi _{{\textrm{L,x}}}^\textrm{n}} \right) \right] \end{array} }}\\ = \frac{{{J_\textrm{3}}\left( {{{m'}_{\textrm{EOM}}}} \right)}}{{{J_\textrm{1}}\left( {{{m'}_{\textrm{EOM}}}} \right)}} \end{array}$$

(37)$$\begin{array}{l} \frac{{{{A'}_{\textrm{n, - 3}}}}}{{{{A'}_{\textrm{n, - 1}}}}}\\ = \frac{\sqrt {\begin{array}{l} {J_\textrm{3}}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}} \right)^2} + {J_{\textrm{2}\left( {\textrm{l - 1}} \right)}}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,x}}^{\textrm{n - 1}}E_{\textrm{L,x}}^{\textrm{n - 1}}} \right)^2}\\ - 2{J_\textrm{3}}({m_{\textrm{EOM}}}){J_{2\left( {\textrm{l - 1}} \right)}}({m_{\textrm{EOM}}})E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}E_{\textrm{S,x}}^{\textrm{n - 1}}E_{\textrm{L,x}}^{\textrm{n - 1}}\sin \left[ \left( {2l + 1} \right){\varphi _{\textrm{EOM}}} \right. \\ \left.+ \left( {\phi _{{\textrm{S,x}}}^{\textrm{n - 1}} - \phi _{{\textrm{L,x}}}^{\textrm{n - 1}} - \phi _{{\textrm{S,y}}}^\textrm{n} + \phi _{{\textrm{L,x}}}^\textrm{n}} \right) \right] \end{array} }}{\sqrt {\begin{array}{l} {J_1}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}} \right)^2} + {J_{\textrm{2l}}}{({m_{\textrm{EOM}}})^2}{\left( {E_{\textrm{S,x}}^{\textrm{n - 1}}E_{\textrm{L,x}}^{\textrm{n - 1}}} \right)^2}\\ - 2{J_\textrm{1}}({m_{\textrm{EOM}}}){J_{2\textrm{l}}}({m_{\textrm{EOM}}})E_{\textrm{S,y}}^\textrm{n}E_{\textrm{L,x}}^\textrm{n}E_{\textrm{S,x}}^{\textrm{n - 1}}E_{\textrm{L,x}}^{\textrm{n - 1}}\sin \left[ \left( {2l + 1} \right){\varphi _{\textrm{EOM}}} \right. \\ \left.+ \left( {\phi _{{\textrm{S,x}}}^{\textrm{n - 1}} - \phi _{{\textrm{L,x}}}^{\textrm{n - 1}} - \phi _{{\textrm{S,y}}}^\textrm{n} + \phi _{{\textrm{L,x}}}^\textrm{n}} \right) \right] \end{array} }}\\ = \frac{{{J_\textrm{3}}\left( {{{m'}_{\textrm{EOM}}}} \right)}}{{{J_\textrm{1}}\left( {{{m'}_{\textrm{EOM}}}} \right)}} \end{array}$$

Here, m′_EOM, which is experimentally determined, is the modulation depth including the effect of the overlap between the sidebands. We can combine Eqs. (36) and (37) as shown below.

(38)$$\frac{{\left( {\frac{{{{A^{\prime}}_{\textrm{n,3}}}}}{{{{A^{\prime}}_{\textrm{n,1}}}}} + \frac{{{{A^{\prime}}_{\textrm{n, - 3}}}}}{{{{A^{\prime}}_{\textrm{n, - 1}}}}}} \right)}}{2} = \frac{{{J_\textrm{3}}({{{m^{\prime}}_{\textrm{EOM}}}} )}}{{{J_\textrm{1}}({{{m^{\prime}}_{\textrm{EOM}}}} )}}$$

In the range where m′_EOM is smaller than ≈ 3.8 rad, the ratio of J₃ to J₁ increases with m′_EOM. Because the impact of the overlap on A′_n,±3/A′_n,±1 decreases with increasing l, the value of m′_EOM approaches the actual modulation depth m_EOM (i.e., that determined by the modes without overlap as explained in Section 2) as l increases.

Figure 11 shows the frequency dependence of m′_EOM for three different values of l: (a) 3, (b) 4, and (c) 5. The employed values of f_r and Δf_r are the same as those described in Section 4. For each figure, we first measured the interferograms U(t) without the reference path within (a) 7/Δf_r, (b) 9/Δf_r, and (c) 11/Δf_r. We repeated data acquisition 100 times and averaged the U(t) with same acquisition times. The polarization state of the S-comb was linearly polarized along the x-axis. Then, we calculated m′_EOM for each comb tooth of the S-comb using Eq. (38) in the same way as described in Section 2.2. m′_EOM has a strong frequency dependence in case of l = 3 [Fig. 11(a)], while m′_EOM is almost constant in cases of l = 4 [Fig. 11(b)] and l = 5 [Fig. 11(c)]. In addition, the values of the frequency-averaged m′_EOM in Figs. 11(b) and 11(c) are both about 108°. The observed l dependence of m′_EOM evidences that m′_EOM initially becomes smaller and then converges as l increases. Based on this tendency, we claim that the values of m′_EOM in Figs. 11(b) and 11(c) are very close to the actual modulation depth of m_EOM. In conclusion, a value of l = 4, i.e. f_EOM=Δf_r/9, is enough to correctly analyze the data. In this case, the shortest measurement time is 9/Δf_r.

 

Fig. 11. Experimental results of m′_EOM when we set f_EOM at (a) Δf_r/7, (b) Δf_r/9, and (c) Δf_r/11.

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Appendix B: Analysis including the residual birefringence of the EOM

For precise analysis of the polarization state of light measured with the present EOM-based PS DCS setup, it is necessary to consider the residual birefringence of the EOM. In Eq. (17), we assumed that the retardance C induced by the EOM is proportional to the external voltage V(t) applied to the EOM. However, the actual device has a non-zero retardance at V(t) = 0 as shown in Fig. 12. This offset retardance at V(t) = 0 is called residual birefringence of the EOM.

 

Fig. 12. Voltage dependence of the retardance C induced by the EOM. V₀ (V_π): Value of the voltage applied to the EOM required to induce a retardance of zero (π).

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To consider the practical case, we define C_mod as the retardance including the effect of residual birefringence:

(39)$${C_{\bmod}}({V(t),{f_{\textrm{opt}}}} )= {\pi}\frac{{V(t)}}{{{V_{\pi}}({{f_{\textrm{opt}}}} )- {V_\textrm{0}}({{f_{\textrm{opt}}}} )}} - {\pi}\frac{{{V_\textrm{0}}({{f_{\textrm{opt}}}} )}}{{{V_{\pi}}({{f_{\textrm{opt}}}} )- {V_\textrm{0}}({{f_{\textrm{opt}}}} )}},$$

where V₀ is the voltage that results in C = 0 (see Fig. 12) and V(t) is defined as shown below.

(40)$$V(t) = \frac{{{V_{\textrm{pp}}}}}{2}\sin ({2{\pi}{f_{\textrm{EOM}}}t + {\varphi_{\textrm{EOM}}}} )+ {V_{\textrm{off}}}$$

Here, V_off is a small offset voltage that is added to the applied sinusoidal voltage. By substituting Eq. (40) into Eq. (39), C_mod can be rewritten as follows:

(41)$${C_{\bmod}}(t,{f_{\textrm{opt}}}) = {m_{\textrm{EOM,mod}}}({f_{\textrm{opt}}}) \cdot \sin ({2{\pi}{f_{\textrm{EOM}}}t + {\varphi_{\textrm{EOM}}}} )+ \delta ({f_{\textrm{opt}}}), $$

where m_EOM,mod(f_opt) and δ(f_opt) are defined as follows:

(42)$$\begin{aligned}{m_{\textrm{EOM,mod}}}({f_{\textrm{opt}}}) &\equiv \frac{{{V_{\textrm{pp}}}}}{{{V_{\pi}}({f_{\textrm{opt}}}) - {V_0}({f_{\textrm{opt}}})}} \cdot \frac{{\pi}}{2}\\ \delta ({f_{\textrm{opt}}}) &\equiv {\pi} \cdot \frac{{{V_{\textrm{off}}} - {V_0}({f_{\textrm{opt}}})}}{{{V_{\pi}}({f_{\textrm{opt}}}) - {V_0}({f_{\textrm{opt}}})}} \end{aligned}$$

Equation (41) shows that C_mod(t, f_opt) equals δ(f_opt) under the condition V_pp= 0 [that is, V(t) = 0]. Thus, δ(f_opt) is proportional to the residual birefringence of the EOM.

Because the presence of a residual birefringence of the EOM alters the polarization state of the combined OFC beam irrespective of the sample, the mathematical description of the interferogram U(t) has to be modified. In order to rewrite Eq. (20), we define α and β as follows (hereafter, we omit the argument f_opt of m_EOM,mod andδ for the sake of a short notation):

(43)$$\alpha \equiv 2\sum\limits_{k = 1}^\infty {{J_{\textrm{2k}}}} ({{m_{\textrm{EOM, mod}}}} )\cos ({2{\pi} \cdot 2k{f_{\textrm{EOM}}}t + 2k{\varphi_{\textrm{EOM}}}} ),$$

(44)$$\beta \equiv 2\sum\limits_{k = 1}^\infty {{J_{\textrm{2k - 1}}}} ({{m_{\textrm{EOM,mod}}}} )\sin [{2{\pi} \cdot ({2k - 1} ){f_{\textrm{EOM}}}t + ({2k - 1} ){\varphi_{\textrm{EOM}}}} ].$$

α and β include the frequency components with frequencies 2kf_EOM and (2k-1) f_EOM, respectively. Then, cos C(t) and sin C(t) can be expressed in terms of α, β, δ, and J₀.

(45)$$\begin{aligned} \cos C(t) &= ({{J_0}({m_{\textrm{EOM,mod}}}) + \alpha} )\cdot \cos \delta - \beta \cdot \sin \delta \\ \sin C(t) &= \beta \cdot \cos \delta + ({{J_0}({m_{\textrm{EOM,mod}}}) + \alpha} )\cdot \sin \delta \end{aligned}$$

By substituting Eq. (45) into Eq. (16), we obtain

(46)$$\begin{aligned}U(t) &\propto \sum\limits_n {\{{({1 + {J_0}({{m_{\textrm{EOM,mod}}}} )\cos \delta} )}} \cdot E_{\textrm{S,x}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}\cos ({2{\pi}n\Delta {f_\textrm{r}}t + \phi_{_{\textrm{S,x}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n}} )\\ &- {J_0}({{m_{\textrm{EOM,mod}}}} )\cdot \sin \delta \cdot E_{\textrm{S,y}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}\cos ({2{\pi}n\Delta {f_\textrm{r}}t + \phi_{_{\textrm{S,y}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n}} )\\ &+ ({\alpha \cdot \cos \delta - \beta \cdot \sin \delta} )\cdot E_{\textrm{S,x}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}\cos ({2{\pi}n\Delta {f_\textrm{r}}t + \phi_{_{\textrm{S,x}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n}} )\\ &{ - ({\beta \cdot \cos \delta + \alpha \cdot \sin \delta} )\cdot E_{\textrm{S,y}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}\cos ({2{\pi}n\Delta {f_\textrm{r}}t + \phi_{_{\textrm{S,y}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n}} )} \}. \end{aligned}$$

As the last two terms in Eq. (46) contain α and β, the last two terms constitute the sideband signals. This means that, for the polarization analysis, we only need to consider the last two terms in Eq. (46). We define U′(t) as a new function consisting of the last two terms in Eq. (46) as follows:

(47)$$U^{\prime}(t) = \sum\limits_n {\left\{ \begin{array}{c} ({\alpha \cdot \cos \delta - \beta \cdot \sin \delta} )\cdot E_{\textrm{S,x}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}\cos ({2{\pi}n\Delta {f_\textrm{r}}t + \phi_{_{\textrm{S,x}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n}} )\\ - ({\beta \cdot \cos \delta + \alpha \cdot \sin \delta} )\cdot E_{\textrm{S,y}}^\textrm{n} \cdot E_{\textrm{L,x}}^\textrm{n}\cos ({2{\pi}n\Delta {f_\textrm{r}}t + \phi_{_{\textrm{S,y}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n}} )\end{array} \right\}} .$$

For further simplification, we introduce the four parameters ${E_{\textrm{S,x}}^\textrm{n}}^{\prime}$, ${E_{\textrm{S,y}}^\textrm{n}}^{\prime}$, ${\phi_{\textrm{S,x}}^\textrm{n}}^{\prime}$, and ${\phi_{\textrm{S,y}}^\textrm{n}}^{\prime}$, which satisfy the following relation:

(48)$$\left( {\begin{array}{{c}} {E_{\textrm{S,x}}^\textrm{n}}^{\prime}\exp (i{\phi_{\textrm{S,x}}^\textrm{n}}^{\prime})\\ {E_{\textrm{S,y}}^\textrm{n}}^{\prime}\exp (i{\phi_{\textrm{S,y}}^\textrm{n}}^{\prime}) \end{array}} \right) = {R^{ - 1}}(\delta )\left( {\begin{array}{{c}} {E_{\textrm{S,x}}^\textrm{n}}\exp ({i\phi_{\textrm{S,x}}^\textrm{n}}) \\ E_{\textrm{S,y}}^\textrm{n}\exp (i{\phi_\textrm{S,y}}^\textrm{n}) \end{array}} \right),$$

where the rotation matrix R(δ) reads

(49)$$R(\delta) = \left({\begin{array}{{cc}} {\cos \delta} &{\sin \delta} \\ {- \sin \delta} &{\cos \delta} \end{array}} \right).$$

Then, Eq. (47) can be rewritten as

(50)$$U^{\prime}(t) = \sum\limits_n {\left\{\begin{array}{l} \alpha \cdot {E_\textrm{S,x}^\textrm{n}}^{\prime} \cdot {E_\textrm{L,x}^\textrm{n}}\cos (2{\pi}n\Delta {f_\textrm{r}}t + {\phi_\textrm{S,x}^\textrm{n}}^{\prime} - {\phi_\textrm{L,x}^\textrm{n}} ) \\ - \beta \cdot {E_\textrm{S,y}^\textrm{n}}^{\prime} \cdot {E_\textrm{L,x}^\textrm{n}}\cos (2{\pi}n\Delta {f_\textrm{r}}t + {\phi_\textrm{S,y}^\textrm{n}}^{\prime} - {\phi_\textrm{L,x}^\textrm{n}})\end{array} \right\}} .$$

By substituting Eqs. (43) and (44) into Eq. (50), we obtain

(51)$$\begin{aligned} U'(t)& = \sum\limits_n \left( \sum\limits_{k = 1}^\infty {{J_{\textrm{2k}}}} \left( {{m_{\textrm{EOM,mod}}}} \right) \cdot E{{_{\textrm{S,x}}^\textrm{n}}^\prime } \cdot E_{\textrm{L,x}}^\textrm{n} \right.\\ & \left.\quad \cdot \left\{ \begin{array}{l} \cos \left[ {2\mathrm{\pi }\left( {n\mathrm{\Delta }{f_\textrm{r}} + 2k{f_{\textrm{EOM}}}} \right)t + 2k \cdot {\varphi _{\textrm{EOM}}} + \phi {{_{{\textrm{S,x}}}^\textrm{n}}^\prime } - \phi _{{\textrm{L,x}}}^\textrm{n}} \right]\\ + \cos \left[ {2\mathrm{\pi }\left( {n\mathrm{\Delta }{f_\textrm{r}} - 2k{f_{\textrm{EOM}}}} \right)t - 2k \cdot {\varphi _{\textrm{EOM}}} + \phi {{_{{\textrm{S,x}}}^\textrm{n}}^\prime } - \phi _{{\textrm{L,x}}}^\textrm{n}} \right] \end{array} \right\} \right. \\ &\quad \left. - \sum\limits_{k = 1}^\infty {{J_{\textrm{2k - 1}}}} \left( {{m_{\textrm{EOM,mod}}}} \right) \cdot E{{_{\textrm{S,y}}^\textrm{n}}^\prime } \cdot E_{\textrm{L,x}}^\textrm{n} \right. \\ & \left. \quad \cdot \left\{ \begin{array}{l} \sin \left[ {2\mathrm{\pi }\left( {n\mathrm{\Delta }{f_\textrm{r}} + \left( {2k - 1} \right){f_{\textrm{EOM}}}} \right)t + \left( {2k - 1} \right) \cdot {\varphi _{\textrm{EOM}}} + \phi {{_{{\textrm{S,y}}}^\textrm{n}}^\prime } - \phi _{{\textrm{L,x}}}^\textrm{n}} \right]\\ - \sin \left[ {2\mathrm{\pi }\left( {n\mathrm{\Delta }{f_\textrm{r}} - \left( {2k - 1} \right){f_{\textrm{EOM}}}} \right)t - \left( {2k - 1} \right) \cdot {\varphi _{\textrm{EOM}}} + \phi {{_{{\textrm{S,y}}}^\textrm{n}}^\prime } - \phi _{{\textrm{L,x}}}^\textrm{n}} \right] \end{array} \right\} \right) \end{aligned}$$

Note that Eq. (51) has the same format as the last two terms in Eq. (20). Therefore, we can determine the four parameters ${E_\textrm{S,x}^\textrm{n}}^{\prime}$, ${E_\textrm{S,y}^\textrm{n}}^{\prime}$, ${\phi_\textrm{S,x}^\textrm{n}}^{\prime}$, and ${\phi_\textrm{S,y}^\textrm{n}}^{\prime}$ by analyzing the sidebands in the same way as described in Section 2 [see Eqs. (21)–(27)].

From the determined ${E_\textrm{S,x}^\textrm{n}}^{\prime}$, ${E_\textrm{S,y}^\textrm{n}}^{\prime}$, ${\phi_\textrm{S,x}^\textrm{n}}^{\prime}$, and ${\phi_\textrm{S,y}^\textrm{n}}^{\prime}$, we can derive ${E_\textrm{S,x}^\textrm{n}}$, ${E_\textrm{S,y}^\textrm{n}}$, ${\phi_\textrm{S,x}^\textrm{n}}$, and ${\phi_\textrm{S,y}^\textrm{n}}$ using

(52)$$\left( {\begin{array}{{c}} {E_{\textrm{S,x}}^\textrm{n}\exp ({i\phi_{\textrm{S,x}}^\textrm{n}} )}\\ {E_{\textrm{S,y}}^\textrm{n}\exp ({i\phi_{\textrm{S,x}}^\textrm{n}} )} \end{array}} \right) = R(\delta )\left( {\begin{array}{{c}} {E{{_{\textrm{S,x}}^\textrm{n}}^\prime}\exp ({i\phi {{_{\textrm{S,x}}^\textrm{n}}^\prime}} )}\\ {E{{_{\textrm{S,y}}^\textrm{n}}^\prime}\exp ({i\phi {{_{\textrm{S,y}}^\textrm{n}}^\prime}} )} \end{array}} \right).$$

In order to use Eq. (52), it is necessary to evaluate δ. In practice, we can determine δ by the following procedure: first, set the polarization state of the S-comb to linear polarization parallel to the $x$-axis. In this case we have $E_{\textrm{S,y}}^\textrm{n} = 0$ and by using Eqs. (43) and (44), the measured U′(t) described by Eq. (47) reduces to

(53)$$U^{\prime}(t) = \sum\limits_n \sum\limits_{k = 1}^\infty \left[ \begin{array}{l} {J_{\textrm{2k}}}({{m_{\textrm{EOM,mod}}}} )\cos \delta \cdot E_{\textrm{S,x}}^\textrm{n} \\ \quad \cdot E_{\textrm{L,x}}^\textrm{n}\left\{ \begin{array}{l} \cos [{2{\pi}({n\Delta {f_\textrm{r}} + 2k{f_{\textrm{EOM}}}} )t + \phi_{_{\textrm{S,x}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n} + 2k{\varphi_{\textrm{EOM}}}} ]\\ + \cos [{2{\pi}({n\Delta {f_\textrm{r}} - 2k{f_{\textrm{EOM}}}} )t + \phi_{_{\textrm{S,x}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n} - 2k{\varphi_{\textrm{EOM}}}} ]\end{array} \right\}\\ - {J_{\textrm{2k - 1}}}({{m_{\textrm{EOM,mod}}}} )\sin \delta \cdot E_{\textrm{S,x}}^\textrm{n} \\ \quad \cdot E_{\textrm{L,x}}^\textrm{n}\left\{ \begin{array}{l} \sin [{2{\pi}({n\Delta {f_\textrm{r}} + ({2k - 1} ){f_{\textrm{EOM}}}} )t + \phi_{_{\textrm{S,x}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n} + ({2k - 1} ){\varphi_{\textrm{EOM}}}} ]\\ - \sin [{2{\pi}({n\Delta {f_\textrm{r}} - ({2k - 1} ){f_{\textrm{EOM}}}} )t + \phi_{_{\textrm{S,x}}}^\textrm{n} - \phi_{_{\textrm{L,x}}}^\textrm{n} - ({2k - 1} ){\varphi_{\textrm{EOM}}}} ]\end{array} \right\} \end{array} \right]$$

Second, calculate (A_n,1+ A_n,-1)/(A_n,2+ A_n,-2) according to Eq. (54).

(54)$$\frac{{{A_{\textrm{n,1}}} + {A_{\textrm{n, - 1}}}}}{{{A_{\textrm{n,2}}} + {A_{\textrm{n, - 2}}}}} = \frac{{{J_1}({{m_{\textrm{EOM,mod}}}} )\sin \delta}}{{{J_2}({{m_{\textrm{EOM,mod}}}} )\cos \delta}}$$

Finally, derive δ using

(55)$$\delta = \textrm{ta}{\textrm{n}^{ - 1}}\left( {\frac{{{A_{\textrm{n,1}}} + {A_{\textrm{n, - 1}}}}}{{{A_{\textrm{n,2}}} + {A_{\textrm{n, - 2}}}}} \cdot \frac{{{J_2}({{m_{\textrm{EOM,mod}}}} )}}{{{J_1}({{m_{\textrm{EOM,mod}}}} )}}} \right).$$

Funding

Japan Society for the Promotion of Science (JP18H02040, JP18J21480).

Disclosures

The authors declare no conflicts of interest.

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