## Abstract

Spectral domain phase microscopy for high-sensitive and broad-dynamic-range quantitative phase imaging is presented. The phase retrieval is realized in the depth domain to maintain a high sensitivity, while the phase information obtained in the spectral domain is exploited to extend the dynamic range of optical path difference. Sensitivity advantage of phase retrieved in the depth domain over that in the spectral domain is thoroughly investigated. The performance of the proposed depth domain phase based approach is illustrated by phase imaging of a resolution target and an onion skin.

© 2013 Optical Society of America

## 1. Introduction

Spectral domain optical coherence tomography (SD-OCT) [1–4] provides high-sensitive cross-sectional images of scattering tissues with axial resolution on the scale of several to tens of microns. It detects spectral interferograms as a function of optical frequency and produces A-scan data by applying Fourier transformation to the acquired interference spectrum. Spectral domain phase microscopy (SDPM) [5–9] is a functional extension of SD-OCT. By employing common-path configuration and phase-sensitive measurement, the sensitivity of the axial displacement measurement by SDPM greatly exceeds the axial resolution of the SD-OCT system, allowing for various applications such as material inspection [10], cellular imaging [11, 12], and Doppler flow measurements [13, 14]. In conventional SDPM, the phase due to optical path difference (OPD) between the reference arm and sample arm is extracted in the depth domain. The range of OPD to be measured is restricted to less than half of the source center wavelength due to the well-known $2\pi $ ambiguity. To extend the OPD range of SDPM, different phase unwrapping algorithms have been proposed. Conventional phase unwrapping algorithm [15] requires the corresponding OPD along neighboring sampling points to vary gradually, and fails if multiple wrapping of phase occurs. Synthetic wavelength phase unwrapping techniques [16–18] use two or more wavelengths to increase the OPD range without phase unwrapping at the price of increased phase noises. Instead of phase retrieval in the depth domain, phase retrieval in the spectral domain is proposed to extend the OPD range [19–22]. However, the phase sensitivity is degraded since the relevant signals are distributed over a broadband of wavenumbers in the spectral domain in contrast to localization in depth domain, which leads to a reduced signal-to-noise ratio (SNR). Although by averaging spectral phases over multiple wavenumbers can significantly enhance the phase sensitivity [23], the complicated spectral phase retrieval algorithm is time-consuming and error prone.

In this paper, we introduce a high-sensitive and broad-dynamic-range quantitative phase imaging method for SDPM. The phase retrieval is still realized in the depth domain to maintain a high sensitivity, while the phase information obtained in the spectral domain is further used for removing the phase ambiguity to extend the dynamic range of SDPM. The phase sensitivity in the spectral domain and in the depth domain will be compared theoretically and experimentally. Phase imaging of a resolution target and an onion skin are presented to evaluate the performance of the proposed phase imaging method.

## 2. Theory

#### 2.1 Conventional depth domain phase based SDPM

Excluding autocorrelation and DC terms, the detected interference spectra of the return lights from a sample reflector and a reference reflector can be described as [2, 19, 23]

The peak value in the depth domain will appear at ${z}_{i}={z}_{0}$ if assuming a symmetry distribution of $S({k}_{i}-{k}_{0})$ around its central wavenumber ${k}_{0}$. However, the spectral density of the light source is usually not of an ideal Gaussian form but with asymmetry. The discretely sampled peak value in the depth domain might be at a position inconsistent with ${z}_{0}$. Assuming a deviation of $\delta z$from ${z}_{0}$, the nearest peaked value at sampling point ${z}_{i}={z}_{0}-\delta z$in the depth domain is then given by

To extend the dynamic range of conventional depth domain phase based SDPM, synthetic wavelength based method is introduced. By windowing the signal spectrum into multiple spectra before applying the Fourier transformation, phase information at multiple center wavelengths are obtained and used for phase retrieval corresponding to a longer synthetic central wavelength. The dynamic range is thus extended but the sensitivity is degraded [16–18].

#### 2.2 Spectral domain phase based SDPM

To measure OPD longer than half a central wavelength, spectral domain phase based SDPM [19, 20] is proposed. By Hilbert transformation of Eq. (1), the complex spectral signal is given by

#### 2.3 A comparison of phase sensitivity

Phase sensitivity in SDPM is determined by SNR. A higher SNR means a higher phase sensitivity and hence a lower phase uncertainty. The phase uncertainty in SDPM can be expressed by [5]

To achieve the highest phase sensitivity, the phase information should be retrieved at their peak values in both domains. Assume a Gaussian distribution $S({k}_{i})=\mathrm{exp}(-\frac{4\mathrm{ln}2{({k}_{i}-{k}_{0})}^{2}}{\Delta {k}^{2}})$ and FWHM bandwidth of $\Delta k$ for the light source, the peak values of signal in the depth domain and spectral domain can be obtained from Eqs. (2) and (5) to be ${I}_{\mathrm{max}}({z}_{i})=\frac{\Delta k\sqrt{\pi {R}_{R}{R}_{S}}}{2\delta k\sqrt{\mathrm{ln}2}}$and ${\tilde{I}}_{\mathrm{max}}({k}_{i})=2\sqrt{{R}_{R}{R}_{S}}$, respectively. Here $\delta k$ is the spectral resolution of the spectrometer in the SDPM. The peak signal ratio between depth domain and spectral domain is given by

According to Parseval's theorem of discrete Fourier transformation, the noise variance ratio between depth domain and spectral domain is given by

$N$ is the number of discrete wavenumbers in the spectral domain. The ratio of SNR between the depth domain and the spectral domain is thus deduced to beSimulation is further conducted to investigate the phase sensitivity in both domains. The parameters used for simulation are consistent with the experimental settings described later. A Gaussian light source with a central wavelength of 835nm and FWHM bandwidth of 45nm is adopted. The spectral resolution is set to be 0.0674 nm. The number of discretely sampling points *N* in both domains is 2048. The OPD between the sample reflector and the reference reflector is set to be 210 μm. Figures 1(a) and 1(b) demonstrate the calculated depth distribution and spectral distribution from an ideal normalized interference signal based on Eqs. (2) and (5), respectively. We can see that the signal distribution in the depth domain is densely concentrated at the location corresponding to OPD while the signal distribution in the spectral domain is broadly distributed over wavenumbers determined by the power spectral density of the light source. Similarly, the calculated depth distribution and spectral distribution from an additive white Gaussian noise for SNR of 70 dB in related to the normalized interference signal are presented in Figs. 1(c) and 1(d), respectively. The calculated SNR distributions in the depth domain and in the spectral domain are shown in Figs. 1(e) and 1(f), respectively. The peak SNR in the depth domain is determined to be 87.50 dB while the peak SNR in the spectral domain is 70 dB. An enhancement of 17.50 dB is resulted in the depth domain, leading to 7.50 fold higher phase sensitivity based on Eq. (9). This simulated result is coincident with the theoretical value of 7.74 determined by Eq. (13).

#### 2.4 The proposed depth domain phase based SDPM

As discussed previously, the conventional depth domain phase based SDPM has the advantage in phase sensitivity and the disadvantage in limited dynamic range. To achieve high dynamic range while keeping high sensitivity, we propose a new depth domain phase retrieval method, in which phases in both domains are fully exploited.

As discussed in section 2.3, to achieve the highest phase sensitivity, the phase information should be retrieved at the peak values in depth domain. Due to asymmetry of spectral density distribution of the light source and discrete sampling, the peak value of the response profile in depth domain usually cannot be sampled. We take the sampling point ${z}_{i}={z}_{0}-\delta z$ with a maximum value as the estimated peak point. At the estimated peak point, the corresponding wrapped phase is ${\phi}_{w}({z}_{0}-\delta z)$. Now we introduce how to retrieve $\phi \text{(}{z}_{0}-\delta z\text{)}$ based on the spectral phase $\phi \text{(}{k}_{i}\text{)}$ and the wrapped phase ${\phi}_{w}({z}_{0}-\delta z)$.

The spectral phase at the central wavenumber $\phi \text{(}{k}_{0})$ can be determined by Eq. (8), which is then used to quantify an estimated phase of $\phi ({z}_{0}-\delta z)$ in the depth domain as

## 3. Experiment

As shown in Fig. 2, the proposed phase retrieval method is experimentally evaluated by the SDPM based on an existing SD-OCT system described in our previous papers [24, 25]. The light source is a super luminescent diode with central wavelength of 835 nm and FWHM bandwidth of 45nm. To realize a common-path configuration for the SDPM, the reference arm of the SD-OCT system is disconnected and the light in the sample arm is focused by the objective lens with a focal length of 30mm on the top surface of a coverslip. Back reflected light from the bottom surface of the coverslip acts as the reference light. The interference signal from the SDPM is detected by a high-speed 2048 pixel spectrometer with a resolution of 0.0674 nanometer running at the highest A-scan rate of 29,000 lines/s. Data processing is performed by program written in MATLAB. The computer CPU is an Intel Core(TM) i7-2600k.

#### 3.1 Coverslip

To illustrate the enhanced phase sensitivity of the proposed depth domain phase based SDPM over that achieved by the spectral domain phase based SDPM, 1024 A-scans of a stationary coverslip with nominal optical thickness of 210 ± 35 µm are performed. The OPDs obtained by the spectral domain phase based SDPM and the proposed depth domain phase based SDPM are shown in Fig. 3(a). Evidently, the OPD measured by the proposed depth domain phase based SDPM is much more stable than that measured by the spectral domain phase based SDPM. The probability distribution of the measured OPD by spectral domain phase based SDPM and by the proposed depth domain phase based SDPM are shown in Figs. 3(b) and 3(d), respectively. The standard variation (STD) of the former one is 161.81 pm while the STD of the latter one is 21.41 pm. The phase sensitivity of the proposed depth domain phase based SDPM is 7.56 fold higher than that of the spectral domain phase based SDPM, which is in good agreement with the theoretical expectation according to Eq. (13).

The phase sensitivity in the spectral domain is degraded since the relevant signals are distributed over a broadband of wavenumbers in contrast to localization in depth domain, which leads to a reduced SNR. On the other hand, the averaging of spectral phases over multiple wavenumbers of our previous reported multiple spectral phases averaged SDPM [23] can again enhance the SNR. To compare the proposed depth domain phase based SDPM with our previous reported multiple spectral phases averaged SDPM, the OPDs obtained from the same experimental data by the multiple spectral phases averaged SDPM is also shown in Fig. 3(a). The probability distribution of the measured OPD by multiple spectral phases averaged SDPM is presented in Fig. 3(c). The STD of the measured OPD is 20.73 pm, which is comparable to that obtained by the proposed depth domain phase based SDPM. Therefore, phase sensitivities in multiple spectral phases averaged SDPM and the proposed depth domain phase based SDPM are similar. However, the previously reported multiple spectral phases averaged SDPM must retrieve and choose the spectral phases deliberately before averaging. Spectral phases corresponding to 343 wavenumbers are selected for the averaging procedure here, which is time-consuming. The processing time taken by the multiple spectral phases averaged SDPM is 31.64 s while that by the proposed depth domain phase based SDPM is only 0.47 s. Hence the proposed method is much more efficient than the previously reported multiple spectral phases averaged approach.

#### 3.2 Resolution target

To validate the ability of the proposed depth domain phase based SDPM for broad-dynamic-range OPD measurement, a resolution target (1951 USAF, Newport) is adopted as the sample. The sample configuration is shown in Fig. 4(a) where an optical fiber with diameter of 245 ± 5 µm is placed at one end between the coverslip and the resolution target to form a wedge realizing a large dynamic range of OPDs. Interference spectra of the return lights from the top surface of the coverslip and the topological surface of the resolution target are collected and used to measure the OPDs between these two surfaces. Figure 4(b) demonstrates the quantitative phase obtained by conventional depth domain phase based SDPM, where the measurement range is limited by half of the central wavenumber and the patterns on the resolution target are corrupted by phase ambiguity. Figure 4(c) shows the quantitative phase image obtained by the proposed depth domain phase based SDPM, where the wedge-shaped OPD-variations are well reconstructed. The patterns on the resolution target can be clearly resolved from the phase image. The height of the pattern is measured to be 120 nm, which is consistent with the reported value in the literatures [11, 12]. The quantitative phase image obtained by the spectral domain phase based SDPM is also shown in Fig. 4(d) for comparison. However, due to the large dynamic range of OPD, it is hard to notice the difference between the phase images demonstrated in Figs. 4(c) and Fig. 4(d).

#### 3.3 Single layer onion skin

To demonstrate the feasibility of the proposed depth domain phase based SDPM for biological sample, phase imaging of a single layer of onion skin is done. Figure 5(a) is the image recorded by an optical microscope showing the onion skin cells. The reconstructed phase image is presented in Fig. 5(b), where ellipse shape of onion skin cell (about 30 μm to 125 μm in lateral directions and 30 μm in depth direction) is clearly resolved.

## 4. Conclusion

By fully exploitation of phase information in the depth domain and the spectral domain, SDPM capable of high-sensitive and broad-dynamic-range quantitative phase imaging is presented. Sensitivity advantage of phase retrieved in the depth domain over that in the spectral domain is confirmed both theoretically and experimentally. Phase ambiguity in the depth domain is overcome by the proposed depth domain phase retrieval method, in which the unwrapped phase in the spectral domain and the location of the sample point in the depth domain is used to extend the dynamic range of SDPM. The performance of the proposed depth domain phase based SDPM with extended dynamic range is illustrated by phase imaging of a resolution target and an onion skin. Further application of this technique could be the study of dynamical phenomenon in biological sample, such as optical measurement of nerve activations.

## Acknowledgments

The authors would like to acknowledge the financial supports from the Chinese Natural Science Foundation (61275196, 61335003, 61327007) and Zhejiang Province Science and Technology Grant (2012C33031).

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