## Abstract

We propose an efficient direct k-domain interpolation based on spectral phase in swept-source optical coherence tomography (SS-OCT). Both the calibration signal from the Mach-Zehnder interferometer (MZI) and the OCT imaging signal from the Michelson interferometer sharing the same swept source are detected and digitized simultaneously. Sufficient sampling of the OCT imaging signal with uniform k interval are directly interpolated in the k-domain based on the spectral phase derived from MZI calibration signal. Depth profile is then obtained from Fourier transform of the k-domain interpolated data. *In vivo* imaging of human finger skin and nail fold are conducted. Reconstructed images corresponding to different calibration methods are evaluated for comparison. Experimental results demonstrate that improved imaging quality with enhanced resolution and signal-to-noise ratio is realized by the proposed method in contrast to the spectral phase based time-domain interpolation method as well as the intensity based calibration method.

© 2011 OSA

## 1. Introduction

Due to great improvements in A-scan rate and detection sensitivity over time-domain optical coherence tomography (TD-OCT), Fourier-domain optical coherence tomography (FD-OCT) has attracted considerable attention [1–3]. Two implementations of the FD-OCT are the spectral domain optical coherence tomography (SD-OCT) [4] and swept-source optical coherence tomography (SS-OCT) [5] where Fourier transform of the spectral interference signal produces an A-scan. In majority of FD-OCT, spectral interferogram is sampled with non-uniform wave number (k) interval, except in a SS-OCT system using a k-space linear swept source or a SD-OCT using a linear-k spectrometer [6,7]. Therefore, prior to Fourier transformation, non-linear-in-k data should be converted to linear-in-k data, which is known as calibration process. Otherwise axial resolution and ranging accuracy will be degraded [8].

For the purpose of calibration, a Fabry-Perot interferometer (FPI) was employed to generate a linear-in-k comb for fixed k interval sampling of the fringe data [9,10]. However, within the free spectral range (FSR) of the FPI, only one sampling is achieved. Intensity based calibration method based on ‘nearest neighbor check’ algorithm using a band-pass filtered FPI [11] or a Mach-Zehnder interferometer (MZI) [12] was proposed. Such approach generates two linear-in-k points within the FSR of the FPI or MZI. However, the intensity based calibration method might fail to determine the true peak/trough positions due to finite DAQ sampling rate, especially in the case that the MZI with large optical path-length difference (OPD) is required. A hardware based ‘Rt-UKSS’ method was developed for non-uniformly triggering the DAQ card to enable digitizing four linear-in-k points within the FSR of the MZI [13]. But the validity of this hardware based calibration method is limited by the frequency response of the clock-generation circuit. Alternatively, non-uniform discrete Fourier transforms with Vandermonde matrix or with Lomb periodogram were presented to reconstruct SS-OCT image [14,15]. Resampling with convolution prior to FFT was presented and compared [16,17]. The zero-filling interpolation with FFT was proposed for effective resampling with smaller interval for FD-OCT [18]. In SS-OCT, a spectral phase based time-domain interpolation method for calibration was presented and widely applied [19]. The k versus time function *k*(*t*) is fitted in advance or in real-time [20] and then used for linear-in-k time domain spline interpolation (TDSI). The calibration accuracy is mainly dependent on the order of the implemented polynomial fitting, but higher order polynomial fitting means much extensive computing.

In this paper, a direct k-domain spline interpolation (KDSI) method based on the spectral phase is proposed. Compared with the spectral phase based time-domain interpolation method, the proposed method realizes high calibration accuracy with simpler algorithm and hence less computing time. Furthermore, comparison of reconstructed A-scan signals and images corresponding to different calibration methods are also conducted experimentally.

## 2. Method

#### 2.1 Time domain interpolation method based on spectral phase

The time-uniformly acquired N-point non-linear-in-k MZI calibration signal directly from one channel of the DAQ card can be expressed as

*R*represents a scale factor,

*d*is OPD between the two arms in the MZI,

*t*is the time instant of the

_{i}*i*-th sample point, $S({t}_{i})$ is the spectral envelope of the swept source. After FFT of the MZI calibration signal, one half of the frequency transformed signal is filtered and transformed back to the original domain by inverse FFT. The resulted complex calibration signal is then given by

#### 2.2 k-domain interpolation method based on spectral phase

Using the same N-point non-linear-in-k MZI calibration signal from one channel of the DAQ card and complex signal retrieval procedure, the complex calibration signal is also expressed by Eq. (2). The unwrapped spectral phase of this complex MZI calibration signal related to wave-number is given by Eq. (3).

Hence M-point k-uniform positions are calibrated from the unwrapped spectral phase as

As depicted in Fig. 1 , in time-domain interpolation method based on spectral phase, a polynomial $t(k)=a+b\cdot k+c\cdot {k}^{2}+d\cdot {k}^{3}+e\cdot {k}^{4}\cdots $is generally adopted to fit the N-point calibrated $k[{t}_{i}]$based on Eq. (4). The M-point fractional time indexes $t(k[j])$ corresponding to uniform distributed $k[j]$ are then determined from the fitted polynomial. The M-point linear-in-k OCT imaging signal $\{{I}_{OCT}(k[j]),\text{\hspace{1em}}j=1\cdots M\}$are thus interpolated in time-domain at these fractional time indexed positions $t(k[j])$ using the acquired N-point non-linear-in-k OCT imaging signal ${I}_{OCT}[{t}_{i}]$ along the time axis shown in Fig. 1. Actually, due to a finite order polynomial fitting method the obtained polynomial results must be deviated with the recovered spectral phase data. In the proposed k-domain interpolation method the polynomial fitting process is totally omitted, hence the errors related to the fitting process can be removed. Higher k-linear calibration accuracy is expected. Meanwhile, without polynomial fitting process the interpolation is conducted directly in the k-domain along the wave-number axis as shown in Fig. 1, less computing time can be achieved with the proposed method.

## 3. Experiments and results

The schematic of the high-speed SS-OCT system in this study is shown in Fig. 2 . The OCT imaging signal is formed by a balanced detection based Michelson interferometer comprised of a reference arm, a sample arm. The MZI calibration signal is formed by a balanced detection based MZI in the calibration arm. The OPD in the calibration MZI is set to be 6 mm. The swept laser source (HSL2000, Santec Inc.) operating at 20 kHz sweeping rate covers 110 nm wavelength sweeping range around 1310 nm central wavelength with 10 mW average output power. 10% of the output power of the swept source is fed into the MZI of which the spectral interference signal is sampled by one input channel of the DAQ card (PCI-5122, National Instrument Inc.). 90% of the output power of the swept source is sent into the interferometer through a wideband circulator. The frequency sweeping laser light is directed into sample and reference arm through a 50/50 fiber coupler. The reference arm light is projected by a fiber collimator and single lens onto a mirror. The sample beam is scanned by an X-Y galvo-mirror scanner, and then focused by an achromatic lens with focal length of 60 mm. The lateral resolution is 12 μm according to the optics used in the sample arm. After combined and split by the same coupler, the OCT imaging signal is redirected to the balanced photon detector (BPD) by the circulator to realize balanced detection, and sampled by another input channel of the DAQ card. The sweep trigger generated by the swept source is used as the line-trigger signal for the trigger channel of the DAQ card.

To investigate the phase error by unwrapping the spectral phase, phase unwrapping is done under case of different jump thresholds and with localized phase noise. Phase unwrapping is realized by adding multiples of $\text{2}\pi $ when absolute jump between consecutive wrapped phase data is greater than or equal to the jump threshold. As shown in Fig. 3(a)
, the two spectral phases corresponding to red and blue curves are unwrapped with jump threshold of *π* and $1.5\pi $, respectively. At the both ends of the sweeping range with low laser output, the phase data corresponding to low quality MZI signal may be mistaken as a jump point. Phase error will be accumulated if an inappropriate jump threshold is chosen as shown in the labeled square region A of Fig. 3(a). Except the areas around both ends of the sweeping range, no additional phase errors are introduced due to high quality of the MZI calibration signal. In the region B of the Fig. 3(a), where under both the two thresholds the unwrapped spectral phase corresponding to high laser output will be reconstructed correctly. In practice the unwrapped spectral phase corresponding to the high laser output can be used for calibration, which guaranteed a constant spectral range. If some white noise is added into the MZI calibration signal at the beginning small signal region, the noise will not change the whole spectral phase jump distribution at the region. The unwrapped phase will just produce errors at the noise region as shown in the region C of the Fig. 3(b), and the noise induced phase error will not be conducted to the high laser output region confirmed by the region D of the Fig. 3(b). Therefore, from the results shown in Fig. 3, the accumulated spectral error does not affect the unwrapped spectral phase range explicitly.

To evaluate the merit of the spectral phase based k-domain interpolation method in contrast to the spectral phase based time-domain interpolation method, a mirror located at different depths of 0.52 mm, 2.13 mm, 4.2 mm, and 6.1 mm as the sample are measured using the SS-OCT system. The non-linear-in-k sampled data of each A-scan is comprised of 2048 points and k-uniform interpolated into 2048 points as well. Spline interpolation is adopted in both TDSI and KDSI methods. Figure 4
shows the reconstructed A-scans corresponding to time-domain spline interpolation method (blue) and k-domain spline interpolation method (red), respectively. Point spread functions (PSF) reconstructed by TDSI with 6-order polynomial fitting method and the zoomed view corresponding to the black box region is shown in the Fig. 4(a) and 4(b). The resulting parameters of the fitted 6-order polynomial are given by *a* = 1.36, *b* = −0.811, *c* = −0.000127, *d* = −4.85*10^{−08}, *e* = −1.19*10^{−11}, *f* = −1.66*10^{−15}, *g* = −7.89*10^{−20}. PSFs reconstructed by TDSI with 16-order polynomial fitting and the corresponding zoomed view are also illustrated in Fig. 4(c) and 4(d) for comparison. As shown in Fig. 4(b), due to limited fitting accuracy of the spectral phase by 6-order polynomial fitting, the reconstructed PSF at depth of 4.2 mm is broadened comparing to that by k-domain spline interpolation method. With 16-order polynomial for improved fitting accuracy, the reconstructed PSF by the time-domain spline interpolation methods are nearly identical to that using k-domain spline interpolation as shown in Fig. 4(d). To compare the processing time of different calibration methods, the spectral interferogram data is post processed by a custom-designed Matlab^{®} program and not processed real-time. Processing time for each A-scan by k-uniform linear interpolation takes approximately 6 ms measured with a PC of Intel Core2 2.4 GHz CPU and 2 GB RAM, while it takes about 9 ms by the time-domain method with 6-order polynomial fitting. This indicates a high-quality reconstruction and 30% reduction of computing time of the proposed method in contrast to the existing spectral phase based time-domain interpolation method. According to Fig. 4(a) and 4(c), the SNR of the SS-OCT system is calibrated as 50 dB experimentally.

If the order of the polynomial fitting for the k-t relationship approaches infinity, the calibration accuracy of the TDSI method can be approached to that of the proposed KDSI method. The theoretical limit of the proposed KDSI method is determined by the linearity dependence between the recovered spectral phase and the emitted wave-number from the swept source, which should be ensured by a well packaged MZI calibration box.

To compare the practical axial resolution of the reconstructed axial line signal of the SS-OCT system based on different calibration methods, full-width-half-maximum (FWHM) values of the point spread functions at different depths are illustrated in Fig. 5 . As shown in the Fig. 5, k-domain spline interpolation, time-domain spline interpolation with 6-order polynomial fitting, and time-domain spline interpolation with 16-order polynomial fitting are compared. With center wavelength of 1310 nm and FWHM bandwidth of 75 nm determined by the fiber components employed in the SS-OCT system, a theoretical axial resolution of 10.1 μm is calculated and depicted at all the measured depths as a reference shown in Fig. 5. Because the reconstructed point spread functions by KDSI is totally prevented from the polynomial fitting induced error, the axial resolution maintains the value around 10 μm. On the contrary the axial resolution reconstructed by TDSI both with 6-order and 16-order decreases gradually with larger imaging depth.

The performance advantage of the proposed method over intensity based method is also investigated. Because a relative large OPD setting of 6 mm for the calibration MZI, the intensity based method cannot always accurately acquire the true peak/trough positions due to finite DAQ sampling rate, leading to a reduced calibration precision. Figures 6(a) and 6(b) present the reconstructed PSFs corresponding to a mirror located at depth of 2.13 mm using the spectral phase based k-domain spline interpolation method (red) and the intensity based method using ‘nearest neighbor check’ algorithm (blue), respectively. As shown in Fig. 6(a), axial resolution is improved from 11.3 μm to 10.3 μm due to improved precision of calibration realized by the proposed method in contrast to that by the intensity based method. What is more, as demonstrated in Fig. 6(b), the noise floor is evidently decreased and 15-dB SNR enhancement is achieved by the proposed method in comparing with that by the intensity based method. Figure 6(c) gives the reconstructed PSFs versus four different imaging depths at 0.52 mm, 2.13 mm, 4.2 mm, and 6.1 mm, respectively. Poor quality reconstruction without calibration is also presented in Fig. 6(c), demonstrating the strong requirement for calibration of the swept source. Sub-Nyquist-sampling in the intensity based method is evident in Fig. 6(c), where PSF peaks corresponding to imaging depth deeper than 3 mm are rolled back to incorrect positions labeled by asterisk marks in Fig. 6(c) as a result of aliasing, which induces artifacts and increases the noise level in this region. On the other hand, linear-in-k sampling in the proposed method is not limited by finite peak/trough points in the MZI calibration signal, and an imaging depth of ~8 mm is achieved due to sufficient high linear-in-k sampling rate of the proposed method.

*In vivo* imaging of human finger skin and nail fold are conducted. Image reconstruction is then done by intensity based method, spectral phase based time-domain spline interpolation method and spectral phase based k-domain spline interpolation method, respectively. Linear-in-k sampled OCT imaging data of each A-scan is 2048 points. In both TDSI and KDSI methods, linear-in-k interpolated points are intentionally set to be 750, equal to the number of the effective linear-in-k points determined by the total number of the peak and trough points of the MZI calibration signal. These k-uniform data are then zero padded to 1024 points for FFT-based reconstruction. 6-order polynomial fitting is adopted in the spectral phase based time-domain interpolation method. Figure 7
reveals the OCT images reconstructed by different methods. The image data was converted to logarithm scale before mapping to the gray scale. Images of human finger skin located closer to the zero OPD position are shown in Figs. 7(a), 7(b) and 7(c) corresponding to intensity based calibration method, spectral phase based time-domain spline interpolation method and spectral phase based k-domain spline interpolation method, respectively. In comparison, images of nail fold located relative far away from the zero OPD position are also presented in Fig. 7(d), 7(e) and 7(f) corresponding to the three methods. With increased modulation frequency in OCT imaging signal corresponding to large OPD, the image quality degradation is most severe in the intensity based method, where strong artifacts appear in Fig. 7(d). While the best image quality reconstruction is achieved by the proposed method, as demonstrated in Fig. 7(c) where epidermis (E), sweat duct (SD) and dermis (D) can be clearly differentiated, as well in Fig. 5(f) where detailed structures such as epidermis (E), cuticle (C), nail plate (NP), nail bed (NB), and dermis (D) can be clearly identified. Comparing the Fig. 7(b) and 7(c), except some subtle difference in terms of the resolution and contrast, the quality of the image reconstructed by TDSI is approached to that by KDSI in the case of the near ZPD object position. However, some spurious shadow appears at the large OPD region in the Fig. 7(e), while the image reconstructed by the proposed KDSI method remains high quality as shown in Fig. 7(f). Therefore, the spectral phase based time-domain interpolation method with 6-order polynomial fitting offers image quality better than the intensity based calibration method but not as good as the proposed method. Higher order polynomial fitting could be implemented for better image quality reconstruction comparable to the proposed method but requires more extensive computing.

## 4. Conclusion

A direct k-domain interpolation method based on the spectral phase of MZI calibration signal is proposed and its performance advantage is experimentally confirmed. The proposed method achieves higher calibration accuracy compared to that offered by the spectral phase based time-domain interpolation method with 6-order polynomial fitting, while reduces the computing time by 30% due to its simpler algorithm. The axial resolution of the reconstructed signal nearly keeps constant within the whole imaging range. In comparison to the intensity based calibration method, the proposed method improves the axial resolution from 11.3 μm to 10.3 μm and achieves 15 dB SNR enhancements due to its improved precision in calibration. In contrast to the intensity based calibration method which is limited by finite peak/trough points in the MZI calibration signal, an imaging depth range of ~8 mm can be realized by the proposed method due to its high efficiency of linear-in-k sampling. *In vivo* images reconstructed with best quality are demonstrated by the proposed method in comparing with existing methods.

## Acknowledgment

This work was supported by National High Technology Research and Development Program of China (2006AA02Z4E0), Natural Science Foundation of China (60978037, 60878057) and Natural Science Foundation of Zhejiang Province of China (Y2091019).

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