## 1.

## Introduction

Polarization-sensitive optical coherence tomography (PS-OCT) is a functional extension of OCT and has been used extensively to image birefringent biological tissues.^{1} Recently PS-OCT with continuous polarization modulation (CPM-PS-OCT) has been reported,^{2}^{,}^{3} in which CPM is required to obtain frequency-shifted OCT signals with respect to the modulation frequency. An electro-optic modulator (EOM) is typically necessary to modulate the polarization state of the sample illuminating light continuously. The phase modulation of the EOM is $\phi =A\text{\hspace{0.17em}}\mathrm{sin}({\omega}_{m}t)$, where ${\omega}_{m}$ is the angular frequency of CPM. It is customary to choose the EOM phase modulation amplitude of $A={A}_{0}=2.405\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{radians}$ in CPM-PS-OCT, so that the zero’th-order of the Bessel function of the first kind is evaluated to be zero, namely, ${J}_{0}({A}_{0})=0$.^{2}

To achieve this condition, the EOM must be carefully calibrated to ensure that $A={A}_{0}$. For this calibration, Jiao et al.^{2} proposed an approach, in which the ratio of the intensities of the $2{\omega}_{m}$ and the $4{\omega}_{m}$ harmonics of the EOM modulation frequency in both detection channels was measured and ensured to be ${J}_{2}({A}_{0})/{J}_{4}({A}_{0})=6.667$. However, it is extremely difficult to use this calibration method when a resonant EOM is used since the detection of the harmonics is typically limited by the data acquisition speed. This is because the resonant frequency ${\omega}_{m}$ is typically set between $1/3$ and $1/2$ of the data acquisition speed to optimize the system measurement depth range. Todorović et al.^{4} pointed out that a phase modulation amplitude other than ${A}_{0}$ could in principle be used provided that the EOM does not saturate at any point. However, theoretical and experimental studies on this have not been presented to date. In this letter, we present theoretical and experimental analysis on the use of different phase modulation amplitude for CPM-PS-OCT measurements, and show that the optic axis orientation and phase retardance of a sample can be calculated without prior knowledge of the actual value of the phase modulation amplitude provided that the amplitude is in a reasonable range. This eliminates the need for the EOM calibration process and allows the use of any phase modulation amplitude that does not saturate the EOM. These conclusions are validated by measurements on a three-quarter waveplate (TQWP) and equine tendon using a CPM-PS-OCT system.

## 2.

## System and Theory

The CPM-PS-OCT system used in this work has been described in our previous paper.^{5} Briefly, the light source was a 10 kHz wavelength sweeping laser (HSL-2000, Santec) which sweeps over 128 nm across a center wavelength of 1.3 µm. The light is polarized by a linear polarizer and then modulated continuously by a broadband waveguide EOM (PC-B3-00-SFAP-SFA-130-U, EOSpace). The modulated light is split into the reference and sample arm and recombined and detected. The theoretical description and data processing procedures of the system have been described in details elsewhere.^{2}^{,}^{3} When the EOM amplitude is adjusted to a value of ${A}_{0}$ the depth-resolved Jones matrices of the combined system fibers and sample are algebraically calculated from the experimental data,

## (1)

$${J}_{\text{measured}}=\left[\begin{array}{cc}-{\tilde{I}}_{h0}^{*}-\frac{{\tilde{I}}_{h1}^{*}}{{J}_{1}({A}_{0})}& {\tilde{I}}_{h0}^{*}-\frac{{\tilde{I}}_{h1}^{*}}{{J}_{1}({A}_{0})}\\ -{\tilde{I}}_{v0}^{*}-\frac{{\tilde{I}}_{v1}^{*}}{{J}_{1}({A}_{0})}& {\tilde{I}}_{v0}^{*}-\frac{{\tilde{I}}_{v1}^{*}}{{J}_{1}({A}_{0})}\end{array}\right],$$^{3}However, when the amplitude is set to a general value $A$, we use the Jones matrix-based analysis of PS detection in CPM-PS-OCT

^{2}

^{,}

^{3}to deduce a general expression for ${J}_{\text{measured}}$,

## (2)

$${J}_{\text{measured}}\phantom{\rule{0ex}{0ex}}=\left(\begin{array}{cc}-{\tilde{I}}_{h0}^{*}-\frac{{\tilde{I}}_{h1}^{*}}{{J}_{1}(A)}[1+{J}_{0}(A)]& {\tilde{I}}_{h0}^{*}-\frac{{\tilde{I}}_{h1}^{*}}{{J}_{1}(A)}[1-{J}_{0}(A)]\\ -{\tilde{I}}_{v0}^{*}-\frac{{\tilde{I}}_{v1}^{*}}{{J}_{1}(A)}[1+{J}_{0}(A)]& {\tilde{I}}_{v0}^{*}-\frac{{\tilde{I}}_{v1}^{*}}{{J}_{1}(A)}[1-{J}_{0}(A)]\end{array}\right),\phantom{\rule{0ex}{0ex}}$$Both Eqs. (1) and (2) show, however, that the value of $A$ must be known in order to recover the desired Jones matrices from the measurement data ${\tilde{I}}_{h0}^{*}$, etc. This requires that the EOM must be carefully calibrated. In practice, it is also necessary to compensate the fiber-induced birefringence in the sample arm fiber. To do this, the Jones matrix at the sample surface ${J}_{\mathrm{sur}}$ is used as a reference matrix to calculate the birefringence in the sample. The double-pass phase retardance $\eta $ and fast-axis orientation $\theta $ of the sample can then be obtained from the matrix diagonalization of the following equation (Ref. 6),

## (3)

$${J}_{c,m}={J}_{\text{measured}}{J}_{\mathrm{sur}}^{-1}={J}_{U}\left(\begin{array}{cc}{p}_{1}{e}^{i\eta /2}& 0\\ 0& {p}_{1}{e}^{-i\eta /2}\end{array}\right){J}_{U}^{-1},$$## (4)

$${J}_{\mathrm{sig}}=\left(\begin{array}{cc}{\tilde{I}}_{h0}^{*}& {\tilde{I}}_{h1}^{*}\\ {\tilde{I}}_{v0}^{*}& {\tilde{I}}_{v1}^{*}\end{array}\right),\phantom{\rule[-0.0ex]{1em}{0.0ex}}{J}_{i}=\left(\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right),\phantom{\rule{0ex}{0ex}}{J}_{B1}(A)=\frac{1}{2}\left(\begin{array}{cc}1+\frac{1}{{J}_{1}(A)}& \frac{1}{{J}_{1}(A)}-1\\ \frac{1}{{J}_{1}(A)}-1& 1+\frac{1}{{J}_{1}(A)}\end{array}\right),\phantom{\rule{0ex}{0ex}}{J}_{B0}(A)=\left(\begin{array}{cc}1+\frac{{J}_{0}(A)}{2}& -\frac{{J}_{0}(A)}{2}\\ \frac{{J}_{0}(A)}{2}& 1-\frac{{J}_{0}(A)}{2}\end{array}\right).$$## (5)

$${J}_{c,m}={J}_{\text{measured}}{J}_{\mathrm{sur}}^{-1}\phantom{\rule{0ex}{0ex}}=({J}_{\mathrm{sig}}{J}_{i}{J}_{B1}(A){J}_{B0}(A)){({J}_{\mathrm{sig}\_\mathrm{sur}}{J}_{i}{J}_{B1}(A){J}_{B0}(A))}^{-1}\phantom{\rule{0ex}{0ex}}={J}_{\mathrm{sig}}{J}_{\mathrm{sig}\_\mathrm{sur}}^{-1},$$## 3.

## Experimental Results

A single-plate TQWP (WPF410, CryLight) was used as a test target to validate the theoretical analysis in previous section using our system. The phase modulation amplitude, $A$, of the EOM used was first calibrated by using a generalized analysis based upon previous reports^{7} and the measured half-wave voltage of the EOM was 9.5 V. Therefore, the different values of $A$ could be set manually by slowly adjusting the drive voltage. It should be noted that this calibration method is limited for the specific EOM used (i.e., waveguide based) and thus has limitation in terms of generality and practicality. In this measurement, $A$ was set from $\sim 3.20$ to 0 radians with a 0.23 radians increment (equivalent to 2 V peak-to-peak drive voltage). The sample Jones matrix was calculated without knowledge of the actual value of $A$ by using Eq. (5) and the results are shown in Fig. 1, in which relative fast-axis orientation was calculated using the method proposed in Ref. 6. The TQWP was kept untouched during the measurement. Figure 1 shows the measured phase retardance (left) and optic-axis orientation (right) along with the standard deviation of each measurement as functions of $A$. Several tens of A-scans were collected at each $A$ during the experiment and the results were used to calculate the standard deviation of the measurement. It is clear that the measured retardance and orientation are essentially the same when $A$ varies from $\sim 0.65$ to 3.20 radians, suggesting that the measurement of phase retardance and orientation is indeed independent of $A$. The systematic measurement error and the standard deviation increase when $A$ falls below $\sim 0.65\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{radian}$. This is because the phase retardance error depends on the effective signal-to-noise ratio (SNR) of the measurement, which is inversely proportional to $[1+1/{J}_{1}^{2}(A)]$ (Ref. 8), as shown in Fig. 2(a). Our system sensitivity was measured to be $\sim 107\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ near zero optical path difference. Improving the system sensitivity can possibly extend the effective range of $A$ that yields accurate measurements. It should be stressed that the measured phase retardance and orientation in Fig. 1 do change with the value of $A$ when $A$ reduces from $\sim 0.65\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{radian}$, although Eq. (2) shows in principle the Jones matrix of the sample can be calculated with any values of $A$ and the calculated value is independent of $A$. This does not indicate that the measurement results depend on the value of $A$ because the change of the measurement results is due to the decreasing of the system effective SNR leading to the significant increasing of the measurementuncertainty.

A birefringent biological tissue, equine tendon, was also used as a test target. The sample surface was oriented orthogonal to the incident beam and the results are shown in Fig. 3. It is clearly seen that the phase retardance and orientation images are essentially the same when the phase modulation amplitude $A$ varies from 2.405 to $\sim 0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{radians}$. For a quantitative analysis, the averaged profile of the phase retardance image was obtained by lateral averaging 100 $A$-scans in the image at each value of $A$ and shown in Fig. 3 (bottom). It is clear that the reasonable image quality can be obtained for $A>\sim 0.5$ radian, requiring a peak-to-peak AC drive voltage of $\sim 4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{V}$. This voltage could easily be provided by a typical function generator itself or an analog output card with no auxiliary amplifier being required.

## 4.

## Conclusions

In conclusion, both theory and experiment have demonstrated that the phase modulation amplitude of the EOM has limited effect on the measurement in CPM-PS-OCT over a large range of values and furthermore that using surface calibration to remove fiber birefringence effects also removes the need to explicitly know the value of $A$. In actual applications, the practical upper limit for $A$ is set by the required drive voltage which depends on electronic constraints and also on the type of modulator (especially crystal versus waveguide). The limit for small values of the amplitude depends on the system sensitivity and sample optical properties since the system will show large measurement uncertainty and systematic errors when the generated first-order OCT signal is close to/below the system noise floor. The effective SNR of the system is proportional to the square of the first-order Bessel function of the first kind evaluated at $A$. Therefore, the optimal working point of the EOM in CPM-PS-OCT is at a phase modulation amplitude whose first-order Bessel function of the first kind is maximized i.e., $A=1.841$ as shown in Fig. 2(b). This could allow simpler EOM drive electronics and improved SNR compared with the current widely used value of $A=2.405$.

## Acknowledgments

The authors acknowledge the contributions of M Yamanari and Y Yasuno from Tsukuba University in developing the system. This research was supported by EPSRC Grant EP/F020422.

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