## Abstract

We present an analysis of the structural image information acquired with polarization sensitive optical coherence tomography (PS-OCT). In PS-OCT a total of four channels of data are acquired: two orthogonal polarization state components for each of two incident polarization states by which the sample is interrogated. Up to recently, the structural information of the sample was obtained by incoherent summation of these four channels. The four channels can be represented as a Jones matrix for each data point acquired from a sample. We show that the Signal to Noise ratio of the structural information can be improved by 2.3 dB by taking advantage of the structure of this Jones matrix, imposed by the propagation and scattering properties of the sample. We demonstrate that the Jones Matrices are all in the shape of an SU(2) matrix, which is key to understanding the coherent composition of the structural image in PS-OCT and the 2.3 dB SNR improvement. We also discuss a global phase of the Jones matrix in signal multiplexed PS-OCT.

© 2014 Optical Society of America

## 1. Introduction

Optical Coherence Tomography (OCT) is an interferometric technique that can noninvasively acquire micro-meter resolution, three-dimensional images of biological tissue. Ever since its early realization [1], OCT has significantly progressed, from technical advances like Spectral Domain OCT [2–4] and Optical Frequency Domain Imaging [5–7], to functional extensions such as Doppler OCT [8–10] and Polarization Sensitive OCT (PS-OCT) [11,12].

Besides providing structural information, PS-OCT is able to measure birefringence of tissue. Analysis of birefringence helps understand tissue’s physical and physiological property and may contribute to better clinical diagnosis [13–15]. In many PS-OCT implementations signal multiplexing is employed to detect the reflected polarization states of two different incident states in a single depth scan (A-line). This can be achieved either by a “frequency shifting” technique [16,17] or a “passive delay” technique [18,19]. In the recently developed “passive delay” technique in fiber based PS-OCT, two orthogonal input states multiplexed in depth are generated simultaneously. Therefore for each measurement the two detectors receive four interrelated signals per depth point in a depth scan. These four interrelated signals can be represented as a 2x2 complex field matrix, equivalent to a Jones matrix [18,19]. Conventionally, the intensity image is constructed by adding the intensity of the four channels, also known as incoherent addition or intensity sum method. On average the Signal to Noise Ratio (SNR) of each individual channel is lower by 6 dB since the power returning from the sample arm is divided over four detection channels, while the shot noise background remains constant. Combining the intensities of the four detection channels in a single image still leads to a 6 dB loss in SNR, since on average the signal intensity increases by a factor of four, but the background noise also increases by a factor of four.

To overcome this SNR penalty Ju et al. [20] proposed to sum the fields of all four channels coherently after aligning the complex phase angle. However, this method does not give a consistent signal related to the tissue reflectivity. It can fluctuate based on the particular distribution of the backscattered fields over the two orthogonal polarization channels.

In this paper we developed a comprehensive solution to this problem based on the SU(2) properties of the detected field matrix. With a fiber based passive delay Polarization Sensitive Optical Frequency Domain Imaging (PS-OFDI) setup we demonstrate that, if the four detection channels maintain phase correlation, the SNR loss can be recovered by a coherent image composition method in which the signal vectors are constructively composed while the noise vectors are not. Based on a Jones Matrix framework, we derive an expression for the image intensity based on the determinant of a 2x2 complex field matrix. We compared the method with the conventional intensity sum method. Improvements on SNR are demonstrated both by simulations and experimentally. The effect is evaluated by assessing both the measured OCT A-line profile and an image of a biological sample.

## 2. Mathematic framework

The analysis is presented in the framework of the Jones formalism [21,22]. In our approach we assume there is no diattenuation in the sample or the measurement system. The evolution of the sample arm beam in a fiber based OCT system is illustrated in Fig. 1.

The input state ${E}_{in}$ travels through the inbound fiber based light path and penetrates a certain depth into the sample (together modeled as Jones Matrix ${J}_{in}$), gets scattered (modeled as Scattering Matrix $S$), and the backscattered light escapes the sample and travels through the outbound fiber based light path (together modeled as Jones Matrix ${J}_{out}$) to a beam combiner to interfere the sample arm light with the reference arm light. After a polarizing beam splitter two detectors detect the orthogonally polarized magnitude and phase of the interference.

The detected fields are

where ${E}_{in,out}$ are field matrices and${J}_{all}$ is a Jones matrix that describes the overall transformation matrix of the entire process, given by the product of three matrices,The phase term ${e}^{i\psi}$ in Eq. (1) represents the combined influence of the fluctuation of reference arm length (which is assumed to be constant on the time scale of a single A-line acquisition), and the randomness of the position of the scatter within an image voxel for a single A-line. Based on the latter, the phase term ${e}^{i\psi}$ will vary as a position of depth, but is assumed to be constant at a particular depth for two polarization states.

For passive delay PS-OCT using a Polarization Delay Unit (PDU), two orthogonal input vector states ${E}_{in}^{1}$ and ${E}_{in}^{2}$ are launched [18,19], with a certain delay,

In OCT we are interested in the backscattered light. In our model to analyze the properties of the Jones matrix ${J}_{all}$ we assume that the scattering particle is small with respect to the wavelength and that the refractive index contrast is small with respect to the surrounding medium. Under these assumptions, the backscattering is polarization independent and the Jones matrix for scattering in the exact backward direction is given by (See Bohren and Huffman [23], pp. 158-159),

Since${J}_{out}$, ${J}_{in}$ and $R$ are all members of the SU(2) group, their product is also a SU(2) group member, denoted as $J$

with the following properties imposed by the SU(2) form,and ${a}^{2}+{b}^{2}=1$.From Eqs. (1)-(8) we have${J}_{all}=\alpha \cdot J$, and

The detected fields including noise are now given by

The conventional image composition method calculates the intensities from each entry and adds them up. The composite signal is given by

Ju *et al*. [20] proposed to sum the fields of all four channels coherently after aligning the complex phase angle. Starting from Eq. (9), the signal is then defined according to Ju *et al*. as $I={\alpha}^{2}{\left(2\left|a\right|+2\left|b\right|\right)}^{2}=4{\alpha}^{2}\left({a}^{2}+{b}^{2}+2\left|a\right|\left|b\right|\right)$.

Using ${a}^{2}+{b}^{2}=1$ gives$I=4{\alpha}^{2}+8{\alpha}^{2}\left|a\right|\left|b\right|$. The value of $\left|a\right|\left|b\right|$ can vary between 0 and 0.5, and the measured signal for a fixed reflectivity $\alpha $ can vary between $I=4{\alpha}^{2}$ and $I=8{\alpha}^{2}$ depending on the particular distribution of the backscattered fields over the two orthogonal polarization channels.

Here we propose to define the image intensity as twice the absolute value of the determinant of the field matrix$E$. Governed by the overall SU(2) structure of the field matrix, the composite signal is then given by

This definition retains the signal as $2{\alpha}^{2}$ when the signal dominates (i.e.$\alpha >>\left|\eta \right|$). However, when the noise dominates,

In order to evaluate the magnitude of the noise (Eq. (14)), we assumed that the probability distribution of the noise in each channel${\eta}_{1H}$,${\eta}_{1V}$,${\eta}_{2H}$ and ${\eta}_{2V}$ is an independent complex Gaussian distribution with equal averaged power. We could not find an exact expression for Eq. (14) and therefore performed a numerical integration by Mathematica^{®} 9, which resulted in ${I}_{\mathrm{det}}=(2.36\pm 0.02)\cdot \u3008{\left|\eta \right|}^{2}\u3009$. Compared to the conventional image composition which gave a noise of $4\u3008{\left|{\eta}_{}\right|}^{2}\u3009$, the noise is 2.3 dB lower. In other words, 2.3 dB reduction of the noise background of the composite image can be achieved by defining the image intensity as twice the absolute value of the determinant of the field matrix$E$.

The noise in *I*_{conv} and *I*_{det} behave quite differently. In the conventional method, the noise is always additive. In the determinant method, the intensity is given by taking the absolute value of the sum of complex phasors. For small signals this means that noise phasors are added to a signal phasor that have a comparable length. On average, the intensity (the length of the summed phasors) will be larger than the length of the signal phasor. For very large signals the noise phasors are much smaller than the signal phasor, and on average the intensity will become equal to the length of the signal phasor. Thus, the noise is not simply additive as in the conventional method.

We can even further exploit the SU(2) properties of the matrix $J$. The determinant of an SU(2) matrix is real (see Eq. (7)). However the determinant of the OCT signal matrix ${E}_{out}$ (Eq. (9)) is complex,

## 3. Results

To verify this approach, we conducted a numerical simulation with MATLAB^{®} 2012. We first modeled four signal fields as ${E}_{H1}$, ${E}_{H2}$, ${E}_{V1}$ and ${E}_{V2}$ following the definition in Eq. (9). Thus, ${\alpha}^{2}$ is determined by the average chosen power of the signal per channel, and the variables a and b are randomly chosen such that $a{a}^{*}+b{b}^{*}=1$. We then defined four complex Gaussian noise terms as described in Eq. (10). Each of them has a power chosen as 0 dB. With these definitions, an average signal of 0 dB per channel, and a noise of 0 dB per channel results in the conventional case in ${I}_{signal\text{\hspace{0.17em}}without\text{\hspace{0.17em}}noise}=6\text{\hspace{0.17em}}dB$, ${I}_{noise}=6\text{\hspace{0.17em}}dB$ and ${I}_{conv}=9\text{\hspace{0.17em}}dB$. The vector sum of noise and signal fields simulate measured signals of a passive delay PS-OCT system. The conventional intensity sum method and our method are utilized to calculate the intensity of the composed signals. We vary the power of the signals from −20 dB to 20 dB to represent zero and high SNR cases and those in between. In each case 1.000.000 independent realizations are simulated. Figure 2 shows the composed intensity given by the conventional and determinant methods as a function of average signal power per channel. Furthermore, the noise levels of each method are shown by horizontal lines and a line with slope one shows the signal in the absence of noise. It clearly indicates that with our method signal fields are constructively composed while the noises are not. For example, when the signal field dominates there is nearly no difference between the intensities determined by the two different methods. When the noise dominates, our determinant method gives a noise floor 2.3 dB lower than that given by the conventional method. This agrees with our numerical integration very well.

Figure 3(a) and 3(b) show a further comparison between the two methods. Here the signal plus noise over noise are shown as a function of signal strength, where the noise is defined as the composed signal with −50 dB input signal power per channel. Figure 3(b) shows how much more the intensities rise above the noise floor as a function of signal strength using the determinant methods over the conventional method. Even at signal levels equal to the noise a gain of 0.6 dB is realized.

A polarization sensitive OFDI setup was constructed to validate the noise reduction effect of the new algorithm (Fig. 4). We use a mirror as a dummy sample to create the interference signals. The signals were then processed according to the two different algorithms and the results are compared in Fig. 5. It is clear that with the determinant method the interference peak is virtually unaffected but the noise floor is indeed suppressed by 2.3 dB.

Further validation is performed by analyzing a B-scan image acquired from in vivo human skin. An endoscopic catheter described in our early work [25] was placed between two fingers to image the skin. The noise reduction effect is clearly visible in Fig. 6. Image SNR is distinctly improved when the determinant method is applied. The conventional method (Fig. 6(a)) shows a slight reduction in the speckle appearance compared to the determinant method (Fig. 6(b)). We attribute this to two possible reasons; 1) Polarization diverse incoherent combination reduces speckle in Fig. 6(a), and 2) the lower noise level for Fig. 6(b), as the presence of noise on a weak speckle signal will smooth out the minima of the speckle distribution and therefore reduce the speckle appearance in areas with low signal.

## 4. Discussion

The general SU(2) format of the Jones matrix plays a key role in the noise reduction algorithm. The absolute value of the determinant of $J$, according to Eq. (7) and (8), is 1. It allows using the determinant of the detected field matrix $E$ to reconstruct OCT intensity signals. This procedure can be regarded as a “filter” - “true” PS-OCT signal pass through unaffectedly while noise does not. In fact any component in $E$ that does not follow the SU(2) format would be excluded to a certain extent. The method works equally well in the presence of tissue birefringence, since tissue birefringence is also an SU(2) transformation of the measured field matrix. The 2.3 dB reduction is specifically associated with independent Gaussian noises in the four channels. In common OCT setups shot noise, detector thermal noise, data acquisition noise and laser source relative intensity noise are present [26,27]. The former three noise components meet this criterion while the laser source relative intensity noises in four E fields are correlated. Thus this filter cannot well eliminate them from signals. On the other hand, greater than 2.3 dB reduction can be achieved if the noise fields deviate further from the SU(2) format. This is evident in our measurement where excess reflection in the delayed polarization state overlap with the undelayed image, disturbing it in a similar way as noise (Fig. 6).

The concept global phase correction in PS-OCT was introduced by the Tsukuba University research group to address field averaging [20,24]. It does not alter the determination of tissue birefringence, which is encoded in the relative phases between the four E field components and not in the global phase. The global phase correction rotates the complex field vectors of the matrix $E$ as a whole. The Tsukuba University research group proposed a correction method in which a reference pixel with the highest SNR within the averaging kernel was chosen as reference and the four $E$ field vectors of each neighboring pixels were aligned accordingly. Then localized field averaging involving these pixels was performed to improve the quality of PS-OCT. We hypothesize that the global phase can be determined for each pixel individually based on the phase of the determinant of the field matrix. It should be notice that the relative phase between the four E field components still varies across depth locations due to birefringence and other factors, therefore field averaging still should take place locally.

## 5. Conclusion

In this study we address the image quality degeneration issue in signal multiplexed polarization sensitive Optical Coherence Tomography and suggest a coherent signal composition technique to mitigate it. We demonstrate that the field matrix follows a structure governed by the SU(2) group. By computing the absolute value of the determinant of the field matrix, instead of the conventional intensity sum, a 2.3 dB noise reduction is achieved. The method was validated by imaging a mirror and in biological tissue.

## Acknowledgments

We gratefully acknowledge financial support from LaserLaB Amsterdam, Laserlab-Europe (EC-GA 284464) the Netherlands Organization for Scientific Research (NWO) with a Vici (JFdB) and an NWO-Groot grant (JFdB). This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO).

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