## Abstract

Accurate wavelength assignment is important for Fourier domain polarization-sensitive optical coherence tomography. Incorrect wavelength mapping between the orthogonal horizontal (H) and vertical (V) polarization channels leads to broadening the axial point spread function and generating polarization artifacts. To solve the problem, we propose an automatic spectral calibration method by seeking the optimal calibration coefficient between wavenumber *k _{H}* and

*k*. The method first performs a rough calibration to get the relationship between the wavelength

_{V}*λ*and the pixel number

*x*of the CCD for each channel. And then a precise calibration is taken to bring both polarization interferograms in the same

*k*range through the optimal calibration coefficient. The optimal coefficient is automatically obtained by evaluating the cross-correlation of A-line signals. Simulations and experiments are implemented to demonstrate the performance of the proposed method. The results show that, compared to the peaks method, the proposed method is suitable in both Gaussian and non-Gaussian spectrums with a higher calibration accuracy.

© 2017 Optical Society of America

## 1. Introduction

Optical coherence tomography (OCT) is a noninvasive imaging technology based on low coherence interferometry [1]. Originally developed as a time domain technique, in order to obtain a depth profile (A-line) within a sample, its signal is recorded by axially translating the reference mirror. The advent of Fourier domain OCT (FD-OCT) [2] improves imaging speed and sensitivity. FD-OCT doesn’t need the mechanical axial scanning when measuring depth-resolved information. Therefore it has the advantages of higher image acquisition speed and higher sensitivity. Due to the advantages, FD-OCT is transferred to polarization-sensitive OCT (PS-OCT) [3], and the combined technique is Fourier domain PS-OCT (FD-PS-OCT) [4]. FD-PS-OCT is a functional extension of OCT. It can provide additional polarization information of a sample besides the intensity image compared to the traditional OCT. Currently, FD-PS-OCT has been widely applied in biomedical fields such as dermatology [5,6], dentistry [7,8], ophthalmology [9,10] and non-biological fields [11,12].

In FD-OCT, the cross-sectional image of the measured sample is retrieved by inverse Fourier transform of interference signal. Fourier transform links the depth *z* and wavenumber *k* (*k = 2π/λ*, *λ* is the wavelength) space. However, the data detected by CCD is evenly sampled in *λ*. The Fourier transform of the signal detected directly is incorrect [13]. To get the correct cross-sectional image, the signal should be evenly resampled in *k* space and this requires careful wavelength *λ* mapping to the index of the CCD pixels, i.e. spectral calibration. Incorrect wavelength assignment leads to broadening of the coherence peak similar to dispersion in traditional OCT intensity image [13]. As for FD-PS-OCT, wavelength assignment is more important and the calibration accuracy is more critical [14]. FD-PS-OCT can obtain not only the intensity image but also polarization-dependent images such as the retardation image and the optical axis orientation image. To achieve these images, a typical FD-PS-OCT system needs to detect two orthogonal polarization channels which are horizontal (H) and vertical (V) channels. However, the wavelength ranges or the spectral resolutions between two spectrometers are generally not the same, so the A-line signals for the two channels do not overlap exactly. And this reduces the accuracy in calculating the polarization elements [15,16]. Even a slight mismatch of the spectral signals in *k* space will result in nonnegligible polarization artifacts. So besides the wavelength assignment for each of the two channels, an exact pixel-to-pixel mapping between the H and V interference signals is also crucial.

Wojtkowski et al [13] first figured out the importance of the wavelength assignment in FD-OCT. Park et al [16] noted that the improper wavelength mapping could lead to artificial birefringence in PS-OCT. The cumulative effect of phase differences can generate an overall phase retardation that cannot be distinguished from that due to sample birefringence. Mujat et al [17] presented an autocalibration method for wavelength assignment. The method keeps the wavenumber *K _{H}* and the spectrum H unmodified, and changes

*K*and the spectrum V. Through comparing the peaks positions of the corresponding A-line signals of spectrums H and V, the method obtains the percentage to expand or contract

_{V}*K*to make the fringes have the same frequency. For simplicity, the method calls peaks method in this paper. This method is suitable for the Gaussian spectrum because the A-line signal usually has good Gaussian shape. And this is good for getting the percentage of

_{V}*k*. But as for non-Gaussian spectrum, the A-line signal may have more than one peak positions or the highest peak position isn’t at the center of the axial point spread function (PSF). This increases the calibration error.

In this paper, we propose a calibration method for FD-PS-OCT by automatically seeking the optimal calibration coefficient of wavenumber *k*. First, a rough calibration is applied to each of the two polarization channels H and V. Then a more precise calibration is performed to bring both spectrums in the same *k* range. The precise calibration keeps one spectrum unmodified and modifies the other one through the optimal calibration coefficient. Compared to the peaks method [17], the proposed method uses cross-correlation to evaluate the calibration results and is insensitive to spectrum shape. It is suitable in both Gaussian and non-Gaussian spectrums.

## 2. Method

By measuring the polarization state of light encoded by sample information, PS-OCT is capable of measuring the polarization properties of samples. In FD-PS-OCT, the spectral interferograms are typically detected in two orthogonal polarization channels, H and V. And they can be expressed as *I _{H}* and

*I*, respectively:

_{V}*I*is dc and autocorrelation terms of interferogram H and

_{H0}*I*is dc and autocorrelation terms of interferogram V;

_{V0}*S*(

*k*) is the spectrum of the light source;

*R*is the reflectivity of the reference mirror and

_{r}*R*is the

_{sn}*n*th reflectivity of the sample;

*Δz*is the optical path difference (OPD) between the reference and the

_{n}*n*th interface within the sample;

*δ*is the retardation from the sample surface to the

_{n}*n*th interface;

*θ*is the fast optical axis orientation of the

_{n}*n*th interface.

An inverse Fourier transform of the two spectral interferograms H and V is performed. Then A-line signals of the measured sample are obtained in forms of A_{H,V}(z)exp[iΦ_{H,V}(z)]. A_{H,V}(z) is amplitude and Φ_{H,V}(z) is phase. And the depth-resolved reflectivity *R*(*z*), retardation *δ*(*z*) and fast axis orientation *θ*(*z*) can be calculated as [18]:

*z*is the depth coordinate. The ranges of

*δ*(

*z*) and

*θ*(

*z*) are [0°, 90°] and [0°, 180°], respectively.

From Eqs. (3) and (4), in order to calculate polarization parameters in high accuracy, the corresponding peaks of A-line signals of H and V should be overlapped. However, mismatch of wavelength assignment leads to the result that the peaks corresponding to the same reflection site for the two signals are non-overlapping, as shown in Fig. 1. The depth position of the coherence peak is related to the frequency of the interferogram in *k* space. Therefore the two polarization interferograms should have the same frequency, i.e. the same *k* mapping.

To solve the problem descried above, we propose a calibration method. The method performs spectral calibration by seeking the optimal calibration coefficient between wavenumber *k _{H}* and

*k*. There are two main steps: rough calibration and precise calibration. In the first step, a rough calibration is applied to each of the two polarization channels H and V. In this paper, this is achieved by characteristic wavelength method [19]. The method requires several characteristic wavelengths to get the relationship between the wavelength

_{V}*λ*and the pixel number

*x*of the CCD. Through the rough calibration, the wavelength range for each of the two channels H and V is obtained. However, the two ranges are generally not exactly the same. So in the second step, a more precise calibration is performed. This step seeks out the optimal calibration coefficient to bring the spectral interferograms in the same

*k*range.

The following are the details of the precise calibration. First, a mirror is used as the sample. Due to the mirror has no birefringence, we adjust the wave plate in the sample arm to 22.5° to provide an equal intensity for the orthogonal channels. The spectral interferograms *I _{H}* and

*I*are then detected by CCD. If the experiment system is well adjusted, the amplitude difference between the two A-line signals is very small, even negligible. Generally, the amplitudes of the two channels can be considered the same. Then a preprocess step is performed on the detected interferograms to remove the fixed noise. The step consists in first taking an average over 1024 A-lines, then subtracting the average from the raw data. After that, a global interpolation [20] is performed to the interferograms. The detected interferograms are linear in wavelength

_{V}*λ*but not for wavenumber

*k*. To get a depth profile, the detected interferograms must be resampled to equal sampling intervals in

*k*space. Therefore, wavenumber

*k*and

_{H}*k*should be interpolated into even space first. For a higher accuracy, four times interpolation is performed. And the interferograms

_{V}*I*and

_{H}*I*are resampled evenly by the wavenumbers obtained after the four times interpolation in

_{V}*k*space [13,20].

Thereafter, the optimal calibration coefficient and the corresponding wavenumber are calculated. A series of calibration coefficients $P=\left[{P}_{1},{P}_{2},\mathrm{...},{P}_{i},\mathrm{...},{P}_{n-1},{P}_{n}\right]$ are preset. The coefficient is chosen as follows: first we compare *k _{V0}* and

*k*to get a basic coefficient${P}_{basic}={k}_{V0}/{k}_{H0}$, where

_{H0}*k*and

_{V0}*k*are the center wavenumbers of

_{H0}*k*and

_{V}*k*, respectively; then the range of the coefficient is set to $\left[0.9{P}_{basic},\text{\hspace{0.05em}}1.1{P}_{basic}\right]$. Coefficients P can enlarge or narrow the range of

_{H}*k*

_{H}. We keep wavenumber

*k*and spectrum

_{V}*I*unchanged. A new

_{V}*k*is calculated based on ${k}_{H}^{\text{'}}={k}_{H0}+{P}_{i}\cdot \left({k}_{H}-{k}_{H0}\right)$, where the symbol ‘·’ represents multiplication. Then spectrum

_{H}*I*is resampled by the new ${k}_{H}^{\text{'}}$, described as

_{H}Fourier transform of ${I}_{H}\left({k}_{H}^{\text{'}}\right)$ is performed and the A-line signals are obtained. And After post-processing such as zero-padding, interpolation and Fourier transform, the A-line signals of the interferograms *I _{V}* are also acquired.

The next step is to cross-correlate the A-line signals of spectrums H and V and obtain a correlation value C(*P _{i}*). Each

*P*corresponds to a correlation value by repeating the above-mentioned process. The two A-line signals should have the same shapes and positions in theory. So when the two signals have the maximum correlation, the wavenumber

_{i}*k*

_{H}is calibrated. In other words, the optimal calibration coefficient P

_{o}is obtained when C(

*P*) is maximum, i.e.$Max\left(C\left({P}_{i}\right),{P}_{i}\in P\right)=C\left({P}_{\text{o}}\right)$ . The corresponding ${k}_{H}^{\text{'}}$ can be acquired as

_{i}After the calibration described above, the corresponding peaks of H and V are generally overlapped. However, the fringe patterns may not overlap perfectly in *k* space [17]. This indicates that the spectral interferograms are shifted with respect to each other and this will lead to polarization artifacts. Thus the calibration still needs to be done. We use a rectangular window to filter the signals associated with the mirror sample in *z* space of the two channels. The center of the window coincides with the center of the peak of the A-line signal for each channel. The width of the window should exceed 2 times of the full width at half maximum (FWHM) of the PSF to ensure that all the signals associated with the mirror sample are filtered. The lengths of the two filtered signals should be the same. Then the Fourier transform is performed on the filtered signals. Thus the contour curves of the spectral interferograms are obtained in *k* space. We cross-correlate the two contour signals. When they have the maximum correlation, the pixel shift *N _{shift}* in ${k}_{H}^{\text{'}}$ (

*k*keeps unmodified) is obtained. Then the wavenumber shift

_{V}*k*is calculated as:

_{shift}*N*is the number of pixels of CCD.

Finally, the calibrated ${k}_{H}^{\text{'}\text{'}}$ is expressed as:

When measuring other samples, the wave plate in the sample arm is reset to 45°. We remove the mirror and place the measured sample at the sample position. The two orthogonal polarization components of the sample will be obtained. And then the spectrum H is resampled by the calibrated ${k}_{H}^{\text{'}\text{'}}$ obtained through the proposed calibration method. The spectrum V and the calibrated spectrum H are then used to analysis the information of the sample.

The whole calibration procedure of the proposed method is illustrated in Fig. 2.

## 3. Experiments and discussions

#### 3.1 Simulations

Simulations were implemented using Matlab to test the proposed calibration method. Two simulations were carried out. In the first simulation, the simulated light source was non-Gaussian spectrum with a central wavelength of 835 nm and a full width at half maximum (FWHM) of 45 nm. The power spectrum of the light source is illustrated in Fig. 3(a). Figure 3(b) shows the horizontal polarization spectrum H and the vertical polarization spectrum V of the simulated source detected by two spectrometers. The number of pixels was set to be 1024 for each CCD. There existed mismatch of wavelength assignment. In Fig. 3(b), the dashed line is spectrum H and the solid line represents spectrum V. In theory, H and V should be overlapped. However, due to different spectral resolutions, misalignment of spectrometers or light with different incident angles illuminated on the gratings of the two spectrometers, the spectrums H and V do not overlap perfectly. The sample was simulated as a one-dimensional birefringence object which had three interfaces. The optical path difference *Δz _{n}* between the reference and the three interfaces were set to 200 μm, 700 μm, 1200 μm, respectively. The retardation

*δ*were set to 0.4969 rad, 0.8065 rad, 1.1282 rad and the fast axis orientation

*θ*were set to 1.2226 rad, 0.7083 rad, 0.4501 rad, respectively, for each of the three interferograms. All the

*δ*and

*θ*data were generated randomly by Matlab. A random noise with a sensitivity of 20dB was added to the simulated spectrums.

The simulated orthogonal polarization interferograms H and V are showed in Fig. 4(a). It shows that H and V don’t overlap because of the misalignment of the wavelength. And the corresponding A-line signals are also misaligned, as shown in Fig. 1. Figure 1 is the A-line signals of Fig. 4(a). The misalignment error of the A-line signals increases with depth (Fig. 1). At the third interface position, the misalignment number of pixels is about 38 between the two A-line signals, corresponding to an OPD of 135.3 μm. If there is even a slight mismatch error between the two cameras, the phase artifact can be obvious. For example, for a central wavelength 835nm and with an OPD *δz* = 1 mm, if there is a mismatch error Δ*λ* = 1 nm in wavelength, the phase difference can be$\Delta \phi =\Delta k\cdot \delta z=2\pi \cdot \Delta \lambda /{\lambda}^{2}\cdot \delta z=9.0117\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{rad}$. Figure 4(b) shows the spectrum H is resampled by${k}_{H}^{\text{'}}$and the two fringe patterns do not overlap perfectly. However, when spectrum H is resampled by${k}_{H}^{\text{'}\text{'}}$, the fringe patterns overlap perfectly, as shown in Fig. 4(c). The result is consistent with the description in section 2. Figure 4(d) is the corresponding wavenumber. After the calibration, the wavenumber *k _{H}* is resampled to ${k}_{H}^{\text{'}\text{'}}$, which is similar with the wavenumber

*k*. The graph in the lower left corner is the larger view of the region enclosed by the dashed box. The trend could be seen clearly.

_{V}Figure 5 gives the A-line signals obtained through different calibration methods. ‘H’ and ‘V’ represent the A-line signals before calibration; ‘peaks’ is the calibration result of the H polarization using the peaks method and ‘cor’ represents the calibration result by use of the proposed method; the symbol ‘△’ is the simulated depth positions. Figures 5(b)-5(d) are the details with enlarged scale for the regions enclosed by the dashed rectangles in Fig. 5(a). From Fig. 5, it can be seen that the A-line signals of H and V do not overlap if there is no calibration; the A-line signal of H is more or less overlapped with V whether using the peaks method or the proposed method. However, the corresponding peaks are more perfectly overlapped by using the method we proposed, as shown obviously in Fig. 5(c). The shape of A-line signal of H is more similar to the shape of V and it is particularly obvious at the arrow. The results indicate that the proposed method is superior to the peaks method.

The errors in retardation and fast axis orientation at the preset reflected interfaces (the symbol ’△’ position in Fig. 5) are shown in Fig. 6. In this paper, the errors are absolute errors (AE, i.e. deviations between the calculated (or measured) and the real values,$\text{AE}=\left|{x}_{measured}-{x}_{real}\right|$). It can sufficiently indicate the accuracy of the method. At the first interface (near the zero OPD) in Fig. 6, the retardation error of the proposed method is smaller than the peaks method while the fast axis orientation errors are almost the same. However, as the depth increases, at the second or the third interface, both of the absolute errors are smaller when using the proposed method. At the second interface position, the absolute errors in retardation of the proposed method and the peaks method are 0.005 rad and 0.025 rad, respectively; while the absolute errors in fast axis orientation are 0.015 rad and 0.033 rad, respectively. At the third interface, it can be obviously seen the fast axis orientation deviation is up to 0.091 rad using the peaks method. However, after using the proposed method, the fast axis orientation deviation is only 0.029 rad and the accuracy increases by more than 50%.

Figure 7 shows the mean absolute deviations and the standard deviations of the absolute errors in retardation and fast axis orientation achieved through the Monte Carlo simulation. The mean absolute deviation (MAD) is the average distance between each absolute error value and the mean. And the standard deviation (SD) is used to quantify the dispersion of the absolute errors. The simulations were performed 20 times. Figure 7(a) is the retardation error. The mean absolute deviations between calculated and preset retardation are all on the order of 10^{−2} for the two calibration methods. But obviously, the deviations of the proposed method are smaller than that of the peaks method. At the third interface position, the mean absolute deviation is 0.0017 rad for the peaks method while it is 0.0005 rad for the proposed method. The deviation is reduced by ~70.6%. Figure 7(b) is the fast axis orientation error. At the closer position to the zero OPD, the mean absolute deviations are almost the same for the two calibration methods. As the OPD increases, the deviations become larger. Especially, at the third interface, the mean absolute deviations are 0.092 rad and 0.030 rad for the peaks method and the proposed method, respectively. The accuracy increases by ~67.4%. Figure 7 also shows the standard deviations are all on the order of 10^{−4} for each calibration method. The simulation results indicate the proposed method is more precise in calibration for the non-Gaussian spectrum and it has a high robustness.

In the first simulation, the proposed method shows better overlap of the A-line signals and more precise in calculating the polarization parameters. This is probably because the shape of the light source spectrum is non-Gaussian, and the axial point spread function is not Gaussian distribution, even not symmetrical. So the peak may not position at the center of the point spread function and that will cause bigger deviations when using the peaks method.

In the second simulation, the simulated light source was Gaussian spectrum and other parameters were the same. The data was obtained via the Monte Carlo simulation. We also performed 20 repeated simulations. The simulation results are illustrated in Fig. 8. Figure 8(a) is the simulated Gaussian light source and Fig. 8(b) is the spectrums of the simulated light source shown in two CCD pixel spaces. Due to the misalignment of wavelength, the two spectrums do not overlap. Figure 8(c) shows the mean absolute deviations and the standard deviations of the absolute errors in retardation. It can be seen that the mean absolute deviations obtained through the proposed method are smaller than that through the peaks method. The mean absolute deviations and the standard deviations in fast axis orientation are plotted in Fig. 8(d). The standard deviations are on the order of 10^{−4} for both methods. The mean absolute deviations become larger with the depth and this is similar as the non-Gaussian spectrum. The maximum mean absolute deviation is ~0.041 rad for the peaks method while it is ~0.026 rad for the proposed method. The accuracy improves by ~36.6%. The simulation results show the proposed method is also better in calibration for the Gaussian spectrum.

#### 3.2 Experiments

To further demonstrate the performance of our method, we measured a quarter wave plate (Union Optic, WPF4210, China) with a retardation of π/2 at the central wavelength of 840 nm. The quarter wave plate is marked as the third quarter wave plate to distinguish it from the wave plates in the system shown in Fig. 9. The data was measured by the FD-PS-OCT system [18,21] shown in Fig. 9.

The light from a super luminescent diode (Superlum, S840-B-I-20) at 840 nm with an FWHM of 50 nm was vertically polarized by a linear polarizer. The polarized light was divided into a reference beam and a sample beam by a non-polarizing beam splitter (NPBS). The reference light passed through a quarter wave plate (QWP1) with its fast axis oriented at 22.5° and illuminated on a reference mirror (M). In the sample arm, the polarized light passed through another quarter wave plate (QWP2) oriented at 45° and became a circularly polarized light. Then the light was delivered to the sample by a galvanometer scanner. The light backscattered from the sample interfered with the light reflected from the reference mirror at the NPBS. And then the interferogram was split into horizontal (H) and vertical (V) channels by a polarizing beam splitter (PBS). The two orthogonal beams illuminated onto the same grating with different incident angles and were imaged side by side onto a single linear CCD with 2048 pixels (Atmel Aviiva M2 CL 2014). Each channel used half of the CCD. The A-line acquisition speed was 20 kHz.

First, a mirror was tested for the precise calibration as described in section 2. In order to get equal intensity in the horizontal and vertical channel, the quarter wave plate in the sample arm (QWP2) was turned to 22.5°. When the spectral data of the mirror was obtained, the quarter wave plate (QWP2) was reset to 45° to provide a circularly polarized light. Then the third quarter wave plate (Union Optic, WPF4210, China) was placed in the sample position.

In a first measurement series, the quarter wave plate was placed at a fixed OPD of 0.45 mm; the fast axis orientation was turned from 0° to 170° in steps of 10°; at each set orientation, the retardation and the fast axis orientation were measured and each measurement was repeated 5 times. The raw data was firstly calibrated by the peaks method and the proposed method, respectively, and then used to calculate the retardation and fast axis orientation. Figures 10(a) and 10(b) are the calculation errors in retardation and fast axis orientation, respectively. In Fig. 10(a), the mean absolute deviations are both below 0.09 rad for the two methods. When using the peaks method to calibration, the largest mean absolute deviation is 0.0904 rad and the largest SD is 0.0037 rad. When using the proposed method, the two values are 0.0892 rad and 0.0026 rad, respectively. The proposed method is slightly better than the peaks method. In Fig. 10(b), the proposed method is obviously more precise than the peaks method. The largest mean absolute deviation is 0.0352 rad and the largest SD is 0.0074 rad for the proposed method while they are 0.0661 rad and 0.0177 rad for the peaks method.

In second measurement series, we changed the OPD between the sample and the reference mirror by shifting the sample along the direction of the incident light; the fast axis orientation of the quarter wave plate was set to a fixed value of 45°; at each depth, the retardation and the fast axis orientation were measured. We performed 5 repeated measurements at each depth. Figure 11 shows the errors obtained by use of the two calibration methods. It can be seen that the mean absolute deviations in retardation are similar, but the proposed method is slightly better than the peaks method. As for the deviations in fast axis orientation, totally, there is a trend that the deviations increase with the OPD, i.e. depth. This may because the sensitivity decays with depth leading to a lower quality in imaging and larger errors. The largest MAD and SD in fast axis orientation are 0.0601 rad and 0.0196 rad, respectively, for the peaks method. And for the proposed method, these two values are 0.0303 rad and 0.0095 rad, respectively, which are smaller than the peaks method. Compared with the peaks method, the biggest increase of the accuracy could be ~50%. The experiments show that the proposed method is more precise in wavelength alignment and has a better performance at different depths.

The calibration method proposed in this paper is achieved by seeking the optimal calibration coefficient to map the wavelength. Compared to the peaks method presented by Mujat et al [17], the proposed method is insensitive to the spectral shape. It uses the maximum value of the cross-correlation of A-line signals as the evaluation function. The errors in seeking the peak positions in the peaks method could be avoided. Thus the proposed method improves the calibration accuracy. The simulations and experiments both show the ability.

## 4. Conclusion

In conclusion, we have proposed a spectral calibration method by automatically seeking the optimal calibration coefficient of wavenumber *k* in PS-OCT. First, a rough calibration is applied to each of the polarization components H and V. Then a more precise calibration is performed. The precise calibration keeps one spectrum unmodified and modifies the other one through the optimal coefficient. The optimal coefficient is obtained by using cross-correlation to evaluate the calibration results. Compared to the peaks method, the proposed method extends the adaptation which is suitable both in Gaussian and non-Gaussian spectrums. And it has a higher calibration accuracy. The simulation and experiment results demonstrate the biggest increase of the accuracy in fast axis orientation could be ~50%. The feasibility and practicality of the method have been proved.

## Funding

Innovation Action Plan of Science and Technology Commission of Shanghai Municipality (15441905600), the Open Fund of Key Laboratory of Optoelectronic Information Processing of University in Guangxi (KFJJ2016-04).

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