## Numerical compensation of system polarization mode dispersion in polarization-sensitive optical coherence tomography |

Optics Express, Vol. 21, Issue 1, pp. 1163-1180 (2013)

http://dx.doi.org/10.1364/OE.21.001163

Acrobat PDF (1546 KB)

### Abstract

Polarization mode dispersion (PMD), which can be induced by circulators or even moderate lengths of optical fiber, is known to be a dominant source of instrumentation noise in fiber-based PS-OCT systems. In this paper we propose a novel PMD compensation method that measures system PMD using three fixed calibration signals, numerically corrects for these instrument effects and reconstructs an improved sample image. Using a frequency multiplexed PS-OFDI setup, we validate the proposed method by comparing birefringence noise in images of intralipid, muscle, and tendon with and without PMD compensation.

© 2013 OSA

## 1. Introduction

1. M. R. Hee, D. Huang, E. A. Swanson, and J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. B **9**, 903–908 (1992). [CrossRef]

2. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Eye-length measurement by interferometry with partially coherent light,” Opt. Lett. **22**, 934–936 (1997). [CrossRef] [PubMed]

3. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**, 1178–1181 (1991). [CrossRef] [PubMed]

4. A. F. Fercher, K. Mengedoht, and W. Werner, “Two-dimensional birefringence imaging in biological tissue using polarization sensitive optical coherence tomography,” Opt. Lett. **13**, 186–188 (1988). [CrossRef] [PubMed]

5. M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res. **30**, 431451 (2011). [CrossRef]

7. S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, and G. J. Tearney, “Measurement of collagen and smooth muscle cell content in atherosclerotic plaques using polarization-sensitive optical coherence tomography,” J. Am. Coll. Cardiol. **49**, 1474–1481 (2007). [CrossRef] [PubMed]

8. C. E. Saxer, J.F. de Boer, B. H. Park, Y. Zhao, Z. Chen, and J.S. Nelson, “High-speed fiber-based polarization-sensitive optical coherence tomography of in vivo human skin,” Opt. Lett. **25**, 1355–1357 (2000). [CrossRef]

9. E. Z. Zhang and B. J. Vakoc, “Polarimetry noise in fiber-based optical coherence tomography instrumentation,” Opt. Express **19**, 16830–16842 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16830 [CrossRef] [PubMed]

*dB*, significantly degrading image quality. And in many systems, reducing the fiber lengths is inconsistent with practical clinical requirements. In this work, we present a method for compensating system PMD numerically using measurements from calibration signals. Therefore highly sensitive imaging in the presence of significant system PMD can be achieved without compromising SNR or limiting fiber lengths. The approach we present is fully general in the order of PMD that can be compensated and in the location of the PMD within the sample arm. This manuscript is organized as follows. We first show system PMD can be removed by multiplying the detected fringe data by two compensation matrices. Next, we demonstrate how these compensation matrices can be derived directly from measured fringe data. Finally, we demonstrate the application of this method to reduce PMD-induced noise in PS-OCT images of intralipid, tendon, and muscle.

## 2. PS-OCT system model

**R**(

*k*)) without noise and confounding artifacts from the instrument transfer functions (described by

**T**(

_{in}*k*) and

**T**(

_{out}*k*)). In practice, however, the detected signal is associated with both the sample properties and the instrument transmission properties according to where

**M**(

*k*) denotes the Jones vector describing the light immediately prior to the receiver (location D in Fig. 1(b)), and where a single launched polarization state (linear in X) is assumed. Here all transfer functions (Jones matrices) are expressed in normalized form, ignoring an overall amplitude and phase scaling factor.

**M**(

*k*) with direct measurements from the polarization-diverse receiver. Because the receiver measures the real component of the fields, and not the complex signal as needed to construct the Jones vector, there is an additional processing step required to translate the direct measurements to the Jones vector

**M**(

*k*). Multiple approaches have demonstrated how to achieve this, and it is commonplace in OCT processing. In a system without complex-conjugate separation, if it is assumed that there are no signals in either the positive-delay or negative-delay space, then the complex fringe signals can be recovered from the measured real fringes through a mathematical Hilbert transformation. In systems that include complex-conjugate separation by including an acousto-optic frequency shifter, the complex fringes are obtained by simply demodulating the signal about its carrier frequency. This later approach [10

10. S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express **12**, 4822–4828 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4822 [CrossRef] [PubMed]

11. W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express **16**, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096 [CrossRef] [PubMed]

**M**(

*k*) with the measured signal.

## 3. Physical PMD compensation

12. H. Sunnerud, C. Xie, M. Karlsson, R. Samuelsson, and P. A. Andrekson, “A Comparison Between Different PMD Compensation Techniques,” J. Lightwave Technol. **20**, 368–378 (2002). [CrossRef]

**M**(

*k*) describing the optical field immediately prior to the detector is modified as where the first compensator (COMP1) is described by

**C**(

_{in}*k*) and the second compensator (COMP2) is described by

**C**(

_{out}*k*). If the compensators are configured such that

**C**(

_{in}*k*) =

*inv*[

**T**(

_{in}*k*)] and

**C**(

_{out}*k*) =

*inv*[

**T**(

_{out}*k*)], the measured signal becomes associated only with the sample birefringence

**R**(

*k*) as It is important to note that for arbitrary

**R**(

*k*), as is the case for OCT imaging applications, the elements INST1 and INST2 must be compensated independently as described by Eq. (2), and each compensation must take place at the location of its associated optical element. The implications of this requirement on the design of a numerically compensated PS-OCT system are described in the next section.

## 4. Numerical PMD compensation

### 4.1. Applying numerical compensation

11. W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express **16**, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096 [CrossRef] [PubMed]

13. K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express **19**, 552–561 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-552 [CrossRef] [PubMed]

14. M. Yamanari, S. Makita, and Y. Yasuno, “Polarization-sensitive swept-source optical coherence tomography with continuous source polarization modulation,” Opt. Express **16**, 5892–5906 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5892 [CrossRef] [PubMed]

15. W. Y. Oh, B. J. Vakoc, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Single-detector polarization-sensitive optical frequency domain imaging using high-speed intra A-line polarization modulation,” Opt. Lett. **33**, 1330–1332 (2008). [CrossRef] [PubMed]

16. B. Baumann, W. Choi, B. Potsaid, D. Huang, J. S. Duker, and J. G. Fujimoto, “Swept source / Fourier domain polarization sensitive optical coherence tomography with a passive polarization delay unit,” Opt. Express **20**, 10229–10241 (2012). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-10229 [CrossRef]

11. W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express **16**, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096 [CrossRef] [PubMed]

**S**(

*k*) is a 2×2 matrix and each column describes the launched state of polarization (SOP) of the two distinguishable launched signals, and

**M**(

*k*) is now a 2×2 matrix, with each column describing the Jones vector associated with the respective launched states of

**S**(

*k*). For simplicity, we start by assuming we launch orthogonal polarization states, aligned with the laboratory coordinates X and Y, yielding

**S**(

*k*) = [1 0; 0 1]. It is shown at the end of this section that the analysis holds for arbitrary (non-degenerate)

**S**(

*k*). Because we have assumed

**S**(

*k*) to be given by the unity matrix, it can be discarded from Eq. (4).

**M**(

*k*) as where

**C**(

_{in}*k*) and

**C**(

_{out}*k*) are our calibration matrices: Then we are able to remove the influence of

**T**(

_{in}*k*) and

**T**(

_{out}*k*) from the measurement system

**C**(

_{out}*k*) ·

**M**(

*k*) ·

**C**(

_{in}*k*) may differ from actual

**R**(

*k*) by an arbitrary but wavelength-independent rotation [17

17. B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett. **30**, 2587–2589 (2005). [CrossRef] [PubMed]

18. Z. Lu and S. J. Matcher, “Absolute fast axis determination using non-polarization- maintaining fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett. **37**, 1931–1933 (2012). [CrossRef] [PubMed]

18. Z. Lu and S. J. Matcher, “Absolute fast axis determination using non-polarization- maintaining fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett. **37**, 1931–1933 (2012). [CrossRef] [PubMed]

**S**(

*k*) = [1 0; 0 1]. In practice, errors in the implementation of a simultaneous polarization probing system will cause the exact form of

**S**(

*k*) to be unknown and potentially time-varying. However, assumption of identity form can still be used; in this case, deviations from this assumption can be included into the

**T**(

_{in}*k*) matrix, solved for, and calibrated out. Thus, it is not essential that strictly orthogonal polarization states be launched, or that their amplitudes, phases, or group delays be matched. Similarly, many imperfections in the polarization-diverse receiver can be included into

**T**(

_{out}*k*), and removed from the system by this calibration process.

**M**(

*k*) by wavelength-dependent compensation matrices

**C**(

_{in}*k*) and

**C**(

_{out}*k*) according to Eq. (5).

### 4.2. Solving for the compensation matrices

**C**(

_{in}*k*) and

**C**(

_{out}*k*) (see Eq. (5)). In this section, we describe a method for solving for these compensation matrices based on measurements of calibration samples. The basic principle behind this approach is relatively straightforward, but its implementation in practice becomes complicated by the presence of amplitude and phase perturbations dependent on signal depth and by the lack of synchronization between the laser source and data acquisition. In this subsection, we focus on the basic principle, and leave the detailed implementation of numerical PMD compensation including some of the steps necessitated by the PS-OCT empirical system for Section 5.

**M**(

_{1}*k*),

**M**(

_{2}*k*), and

**M**(

_{3}*k*) can be expressed as where we have here explicitly allowed the Jones matrices

**R**(

_{1}*k*),

**R**(

_{2}*k*), and

**R**(

_{3}*k*) to vary with wavelength. Because these are calibrating signals from a known optical path, we assume that the Jones matrices

**R**(

_{1}*k*),

**R**(

_{2}*k*), and

**R**(

_{3}*k*) are known a priori. By combining Eqs. 8a and 8b, and Eqs. 8a and 8c, we arrive the following set of two equations for

**T**(

_{out}*k*): where [

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)] and [

**R**(

_{3}*k*) ·

**R**

_{1}^{−1}(

*k*)] are known a priori, and [

**M**(

_{2}*k*) ·

**M**

_{1}^{−1}(

*k*)] and [

**M**(

_{3}*k*) ·

**M**

_{1}^{−1}(

*k*)] are measured. If the eigen decomposition of [

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)] and [

**R**(

_{3}*k*) ·

**R**

_{1}^{−1}(

*k*)] generate different eigenvectors, it can be shown that Eq. (9) uniquely defines

**T**(

_{out}*k*). They can be solved using numerical solving routines such as Levenberg-Marquardt algorithm [19]. With

**T**(

_{out}*k*),

**T**(

_{in}*k*) can be solved for using any of Eqs. 8a–c.

## 5. Implementation in a frequency-multiplexed PS-OCT system

### 5.1. PS-OCT system design

20. S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, “High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,” Opt. Lett. **28**, 1981–1983 (2003). [CrossRef] [PubMed]

*kHz*over 140

*nm*sweeping range centered at 1270

*nm*and yielded 6

*μm*axial resolution in tissue. The carrier frequencies of two simultaneous polarizations were centered at 25

*MHz*and 75

*MHz*by placing a 50

*MHz*frequency shifter in the reference arm and two identical 25

*MHz*shifters in the sample arm, one oriented to upshift and the other to downshift. The drive signals to each frequency shifter were phase-locked to each other, but not to the digitizer clock. In the sample arm, a fiber coupler was used in place of a circulator to allow calibrated PMD to be placed into the system using polarization-maintaining fiber (PMF) patchcords [9

9. E. Z. Zhang and B. J. Vakoc, “Polarimetry noise in fiber-based optical coherence tomography instrumentation,” Opt. Express **19**, 16830–16842 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16830 [CrossRef] [PubMed]

*mW*, and the back-reflected light was directed to a polarization-diverse receiver.

**16**, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096 [CrossRef] [PubMed]

13. K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express **19**, 552–561 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-552 [CrossRef] [PubMed]

*MHz*(the highest RF frequency in our detection bandwidth), a 1

*cm*mismatch induces 0.02

*radians*of phase shift, and a correspondingly low rotation of polarization.

### 5.2. Microscope design

**C**(

_{in}*k*) and

**C**(

_{out}*k*), we built a specialized microscope that provides three fixed-depth signals as illustrated in Fig. 6. A small fraction of the imaging beam was reflected almost directly backward using a glass beam sampler (approximately 4% reflection). This light was then directed toward beamsplitters, separating the light into three beams directed at three corresponding mirrors. The first beam (signal 1) was not modified by a waveplate, while beams 2 and 3 included quarter-wave plates oriented at 0° and 45°. The beam sampler was implemented at nearly normal incidence to minimize the polarization-dependent loss. Each mirror included z-translation to adjust its fringe frequency. Based on this setup, the assumed functional form for

**R**(

_{1}*k*),

**R**(

_{2}*k*), and

**R**(

_{3}*k*) are given by where

*nm*, and with the laser swept range we can calculate Γ(

*k*).

### 5.3. Extracting depth-encoded signals **M**_{1}(k), **M**_{2}(k), and **M**_{3}(k) from a single A-line

_{1}

_{2}

_{3}

*MHz*and 75

*MHz*carriers to generate complex fringe signals. Next, the fringes were interpolated to remove laser chirp and multiplied by a dispersion correcting vector. This results in four complex fringes defining the combinations of two input polarizations and two detection polarizations. To separate each of the three calibrating signals, each fringe was Fourier transformed to give A-lines. Each mirror signal appeared as a distinct spectral peak within the A-line. Each spectral peak was isolated by applying a Hanning window of 100

*μm*FWHM width at its center, and the windowed A-lines were inverse Fourier transformed to yield the complex fringe associated with that mirror signal. This was repeated for each of four complex A-lines (two launched polarization states and two detected polarization states), and for each of the three mirror signals. From these 12 complex fringe signals, the wavelength-dependent matrices

**M**(

_{1}*k*),

**M**(

_{2}*k*), and

**M**(

_{3}*k*) were constructed.

### 5.4. Normalizing the measured signals **M**_{1}(k), **M**_{2}(k), and **M**_{3}(k)

_{1}

_{2}

_{3}

**M**(

_{1}*k*),

**M**(

_{2}*k*), and

**M**(

_{3}*k*) as described in Section 5.3 using the system presented in Section 5.1, we must modify our previous model (Eq. (8)) to include effects associated with the phase and amplitude stability of the fringes, and of the depth-dependent phase variation in each fringe. In this section, we present a methodology to normalize out these factors. We start by writing a more generalized model for the measured signals including a set of unknown factors:

*c*

_{1}(

*k*),

*c*

_{2}(

*k*), and

*c*

_{3}(

*k*) include the amplitude and phase variations that are associated with each calibration signal and that originate from the physical properties of the calibration paths (rather than from laser synchronization, for example). Such variations could arise from the wavelength-varying splitting ratio of beamsplitters in each path. Variations induced by the frequency-multiplexing arrangement, or which differ between one frequency channel relative to the other, are included in the matrix

**K**(

*k*). We note that unlike the scalars

*c*

_{1}(

*k*),

*c*

_{2}(

*k*), and

*c*

_{3}(

*k*) which are unique for each mirror signal, the matrix

**K**(

*k*) is assumed to be constant for all mirror signals. Physically,

**K**(

*k*) can result in part from the phase-instability that is present uniquely in each frequency-multiplexed polarization channel, and therefore cannot be included in the scalars factors. This phase instability results in large part from the lack of synchronization between the laser and acquisition clock as described in [21

21. B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express **13**, 5483–6593 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5483 [CrossRef] [PubMed]

*c*

_{1}(

*k*),

*c*

_{2}(

*k*),

*c*

_{3}(

*k*), and

**K**(

*k*) are assumed to be A-line dependent, i.e., to vary from one A-line to the next.

**K**(

*k*) has been cancelled. Here, [

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)] and [

**R**(

_{3}*k*) ·

**R**

_{1}^{−1}(

*k*)] are known a priori, and [

**M**(

_{2}*k*) ·

**M**

_{1}^{−1}(

*k*)] and [

**M**(

_{3}*k*) ·

**M**

_{1}^{−1}(

*k*)] are measured, but the scalars

*c*

_{1}(

*k*),

*c*

_{2}(

*k*), and

*c*

_{3}(

*k*) are not known. Our goal is to either calculate these scalars, or equivalently to derive a new left-hand side of Eq. (12) that is independent of the scalars and allows

**T**(

_{out}*k*) to be solved for. We note that

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)]. Therefore, each has the same eigenvalues upon eigen decomposition [22

22. J. Gentle, *Numerical Linear Algebra for Applications in Statistics* (Springer, 1998), chap. 2. [CrossRef]

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)] is written by where

**V**(

_{21}*k*) contains the eigenvectors and

**D**(

_{21}*k*) is a diagonal matrix containing the eigenvalues, then the eigendecomposition of [

**M**(

_{2}*k*) ·

**M**

_{1}^{−1}(

*k*)] (absent the scalar factors) can be written by where again

**U**(

_{21}*k*) contains the eigenvectors,

**d**(

_{21}*k*) is a diagonal matrix containing the eigenvalues, and

**D**(

_{21}*k*) and

**d**(

_{21}*k*) differ only because of the presence of the scalar factors

*c*

_{1}(

*k*),

*c*

_{2}(

*k*), and

*c*

_{3}(

*k*). We can therefore calculate the lhs (left-hand side) of Eq. (12a) by first calculating the eigendecomposition of [

**M**(

_{2}**k**) ·

**M**

_{1}^{−}

**(**

^{1}**k**)] to get

**U**(

_{21}*k*), and then using the eigenvalues matrix

**D**(

_{21}*k*): A similar process provides the lhs of Eq. (12b). With values for the lhs of Eq. (12a) and (12b), we can now solve this set of equations for

**T**(

_{out}*k*) in the manner described in Section 4.2.

**T**(

_{out}*k*) into any of Eqs. 11 allows us to solve for the product

**T**(

_{in}*k*) ·

**K**(

*k*). While it might appear that we need to uniquely resolve

**T**(

_{in}*k*) from

**K**(

*k*), it can be appreciated from the form of Eq. (11) that we can simply include

**K**(

*k*) into the definition of

**T**(

_{in}*k*). We therefore require the product

**T**(

_{in}*k*) ·

**K**(

*k*) since the signal fringes (e.g., from the sample) will be modified by this product in the same manner as the calibrating signals (from the mirrors). However, by defining

**T**(

_{in}*k*) to include

**K**(

*k*), we impose the A-line to A-line variability of

**K**(

*k*) onto the product

**T**(

_{in}*k*) ·

**K**(

*k*). Thus, even for a stable system where the optical transfer function of the block elements INST1 and INST2 remain fixed, the effective optical transfer function for INST1 when including the phase instability described by

**K**(

*k*) varies from one A-line to the next. By contrast, the transfer function of INST2 (denoted by

**T**(

_{out}*k*)) is A-line stable. Thus, the solution for

**T**(

_{out}*k*) provided by solving Eq. (12) can be used repeatedly across A-lines for as long as the optical transfer function from the sample to receiver part is stable. In our experiment,

**T**(

_{out}*k*) was calculated from one A-line measurement within a frame, and used for all A-lines within the frame. It was thus updated every 1024 a-lines. However, the product

**T**(

_{in}*k*) ·

**K**(

*k*) must be recalculated for each A-line using the solution of

**T**(

_{out}*k*) and one of Eqs. 11. As a result, three signals are required to solve for the combination of

**T**(

_{out}*k*) and

**T**(

_{in}*k*) ·

**K**(

*k*), but once

**T**(

_{out}*k*) is known, one of the three signals must remain to be used to recalculate

**T**(

_{in}*k*) ·

**K**(

*k*). Finally, the compensation matrices

**C**(

_{in}*k*) and

**C**(

_{out}*k*) are calculated straightforwardly as

**R**(

_{1}*k*),

**R**(

_{2}*k*), and

**R**(

_{3}*k*), the sensitivity of the approach to errors in the assumed and actual forms depend critically on whether these errors affect the resulting eigenvalues (

**D**(

_{21}*k*) and

**D**(

_{31}*k*)) or the eigenvectors (

**V**(

_{21}*k*) and

**V**(

_{31}*k*)). Errors that affect only the eigenvalues are effectively removed, since the measured data is normalized to the eigenvalues of the assumed form; it is critical therefore only that the eigenvalues between the lhs and rhs match, and not that they accurately describe the true transfer denoted by [

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)] and [

**R**(

_{3}*k*) ·

**R**

_{1}^{−1}(

*k*)]. This flexibility however does not extend to the errors that affect the eigenvectors, as those are not normalized (set equal) on the lhs and rhs of Eq. (12). The implication of this is that in our system, for example, it is important that our expressions for

**R**(

_{1}*k*),

**R**(

_{2}*k*), and

**R**(

_{3}*k*) accurately describe the orientation of the waveplates, but need not accurately describe the magnitude of their retardance. This allows us, for example, to neglect any wavelength-dependence of the retardance of these waveplates. However, we note that the influence of the beamsampler and BBS are not included in the model of calibration signals (Eq. (8)), and it is possible that the performance could be improved if the model is extended to explicitly describe their transmission properties.

### 5.5. Forcing **T**_{out}(k) to be continuous across wavelength

_{out}

**T**(

_{out}*k*) which includes an eigendecomposition of the measured data [

**M**(

_{2}*k*) ·

**M**

_{1}^{−1}(

*k*)] and [

**M**(

_{3}*k*) ·

**M**

_{1}^{−1}(

*k*)]. This eigendecomposition is applied for each wavelength, resulting in a set of eigenvectors

**U**(

_{21}*k*) and

**U**(

_{31}*k*). There is an order ambiguity, however, in how these eigenvector matrices are constructed; their columns (and associated eigenvalues in the matrices

**d**(

_{21}*k*) and

**d**(

_{31}*k*)) can be swapped. If the eigenvectors are not forced to be oriented within these matrices in a manner that is continuous across wavelength, then this discontinuity will be transferred to the solution of

**T**(

_{out}*k*) and finally to the compensated fringes.

**U**(

_{21}*k*) and

**U**(

_{31}*k*) to ensure continuity before Eq. (15) is applied. To do this, we stepped across wavelengths and analyzed the change in the first column eigenvector of

**U**(

_{21}*k*) (or

**U**(

_{31}*k*)). Specifically, we calculated the Stokes vector associated with each column, and then looked for the angular displacement between it and that of an adjacent wavelength. A large angular shift indicates a discontinuity in eigenvectors, and triggers a swapping of columns to eliminate this discontinuity.

## 6. Results

*ps*per path) and high-PMD (0.08

*ps*per path) circulators.

### 6.1. Mirror sample

*ps*of differential group delay (single-pass). A neutral density (ND) filter was placed before the sample mirror to prevent saturation from the mirror signal.

*é*sphere representation [23

23. J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett. **24**, 300–302 (1999). [CrossRef]

### 6.2. Intralipid sample

14. M. Yamanari, S. Makita, and Y. Yasuno, “Polarization-sensitive swept-source optical coherence tomography with continuous source polarization modulation,” Opt. Express **16**, 5892–5906 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5892 [CrossRef] [PubMed]

27. E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express **17**, 22704–22717 (2009). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22704 [CrossRef]

*ps*(single-pass) patchcord was inserted into the bi-directional portion of the sample arm. For comparison, Figure 8(b) presents the uncompensated local birefringence image without patchcord. These images demonstrate that numerical PMD compensation (Fig. 8(d)) provides an image qualitatively equivalent to that obtained by physically eliminating the dominant source of PMD (Fig. 8(b)). Quantitatively, the mean local birefringence measured across the topmost 500

*μ*m of the intralipid over 1024 A-lines was 0.35

*deg*/

*μm*with the 0.08

*ps*patchcord and without PMD compensation (Fig. 8(c)), 0.0076

*deg*/

*μm*with the 0.08

*ps*patchcord and with PMD compensation (Fig. 8(d)), and 0.0090

*deg*/

*μm*without the patchcord and without PMD compensation (Fig. 8(b)).

### 6.3. Tendon and muscle sample

*ps*and 0.08

*ps*patchcord respectively and without PMD compensation are presented. In Fig. 9(d) and 9(f), the images with the 0.04

*ps*and 0.08

*ps*patchcords and with numerical PMD compensation are presented. A similar noise reduction is achieved by numerical PMD compensation with these tissue samples as was demonstrated in intralipid.

## 7. Conclusions

## Appendix: Why three calibration signals are required?

**T**(

_{in}*k*) and

**T**(

_{out}*k*). In this appendix, analytic method is derived by ignoring amplitude and phase variation on measured calibration signals. We note, however, that given measured data contaminated by noise and by amplitude/phase variation, a more generalized model and numerical methods for solving for

**T**(

_{in}*k*) and

**T**(

_{out}*k*) are implemented in Section 5.4.

**T**(

_{in}*k*) and

**T**(

_{out}*k*) must be measured and compensated independently before recovering the sample property

**R**(

*k*). Therefore it is obvious to prove that only one calibration signal (for example, a mirror signal) is not enough to resolve both

**T**(

_{in}*k*) and

**T**(

_{out}*k*), as only the combined effect [

**T**(

_{out}*k*) ·

**T**(

_{in}*k*)] can be obtained.

**T**(

_{in}*k*) and

**T**(

_{out}*k*). We start with Eq. (9a) to solve for

**T**(

_{out}*k*) by combing Eqs. 8a and 8b where [

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)] is known a priori from calibration signals, and [

**M**(

_{2}*k*) ·

**M**

_{1}^{−1}(

*k*)] is measured. Note that [

**M**(

_{2}*k*) ·

**M**

_{1}^{−1}(

*k*)] is a similarity transformation of [

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)], their eigenvalues are equal upon eigendecomposition [22

22. J. Gentle, *Numerical Linear Algebra for Applications in Statistics* (Springer, 1998), chap. 2. [CrossRef]

**U**and

_{21}**V**are eigenvectors of [

_{21}**M**(

_{2}*k*) ·

**M**

_{1}^{−1}(

*k*)] and [

**R**(

_{2}*k*) ·

**R**

_{1}^{−1}(

*k*)], respectively, while

**D**is their eigenvalue matrix. Equation (17) can be rearranged to write

**D**is a diagonal matrix, it can be shown that a necessary and sufficient condition for this equation to be solved is that [

**U**

_{21}^{−1}·

**T**·

_{out}**V**] is diagonal. With this requirement, we can write a general form for [

_{21}**U**

_{21}^{−1}·

**T**·

_{out}**V**as with

_{21}]*α*

_{11}and

*α*

_{22}can be any complex value. The analytic solution for

**T**becomes any linear combination of two orthogonal bases

_{out}**T**=

_{out}*α*

_{11}

**A**+

_{1}*α*

_{22}

**A**, where

_{2}**A**and

_{1}**A**can be calculated by

_{2}**T**(

_{out}*k*). In order to completely specify

**T**(

_{out}*k*), a third calibration signal

**R**(

_{3}*k*) must be added in. We can obtain another equation about

**T**(

_{out}*k*) as Eq. (9b) Following the same steps, we arrive a similar form for

**T**as Eq. (19) where

_{out}**U**and

_{31}**V**are eigenvectors of [

_{31}**M**(

_{3}*k*) ·

**M**

_{1}^{−1}(

*k*)] and [

**R**(

_{3}*k*) ·

**R**

_{1}^{−1}(

*k*)], respectively, and

*β*

_{11}and

*β*

_{22}an be any complex value. Again,

**T**becomes any linear combination of two orthogonal bases

_{out}**T**=

_{out}*β*

_{11}

**B**+

_{1}*β*

_{22}

**B**, where

_{2}**B**and

_{1}**B**can be calculated by Combing Eq. (19) and Eq. (21), it turns out to solve

_{2}**T**=

_{out}*α*

_{11}

**A**+

_{1}*α*

_{22}

**A**=

_{2}*β*

_{11}

**B**+

_{1}*β*

_{22}

**B**by linear algebra

_{2}*ij*) denotes the

*ith*row,

*jth*column element of the 2 × 2 matrix. By solving this equation for the constants

*α*

_{11},

*α*

_{22},

*β*

_{11}and

*β*

_{22},

**T**can then be constructed through either

_{out}**T**=

_{out}*α*

_{11}

**A**+

_{1}*α*

_{22}

**A**or

_{2}**T**=

_{out}*β*

_{11}

**B**+

_{1}*β*

_{22}

**B**. With

_{2}**T**(

_{out}*k*),

**T**(

_{in}*k*) can be easily solved using any calibration signal in Eq. (8).

## Acknowledgment

## References and links

1. | M. R. Hee, D. Huang, E. A. Swanson, and J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. B |

2. | J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Eye-length measurement by interferometry with partially coherent light,” Opt. Lett. |

3. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science |

4. | A. F. Fercher, K. Mengedoht, and W. Werner, “Two-dimensional birefringence imaging in biological tissue using polarization sensitive optical coherence tomography,” Opt. Lett. |

5. | M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res. |

6. | B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt. |

7. | S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, and G. J. Tearney, “Measurement of collagen and smooth muscle cell content in atherosclerotic plaques using polarization-sensitive optical coherence tomography,” J. Am. Coll. Cardiol. |

8. | C. E. Saxer, J.F. de Boer, B. H. Park, Y. Zhao, Z. Chen, and J.S. Nelson, “High-speed fiber-based polarization-sensitive optical coherence tomography of in vivo human skin,” Opt. Lett. |

9. | E. Z. Zhang and B. J. Vakoc, “Polarimetry noise in fiber-based optical coherence tomography instrumentation,” Opt. Express |

10. | S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express |

11. | W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express |

12. | H. Sunnerud, C. Xie, M. Karlsson, R. Samuelsson, and P. A. Andrekson, “A Comparison Between Different PMD Compensation Techniques,” J. Lightwave Technol. |

13. | K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express |

14. | M. Yamanari, S. Makita, and Y. Yasuno, “Polarization-sensitive swept-source optical coherence tomography with continuous source polarization modulation,” Opt. Express |

15. | W. Y. Oh, B. J. Vakoc, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Single-detector polarization-sensitive optical frequency domain imaging using high-speed intra A-line polarization modulation,” Opt. Lett. |

16. | B. Baumann, W. Choi, B. Potsaid, D. Huang, J. S. Duker, and J. G. Fujimoto, “Swept source / Fourier domain polarization sensitive optical coherence tomography with a passive polarization delay unit,” Opt. Express |

17. | B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett. |

18. | Z. Lu and S. J. Matcher, “Absolute fast axis determination using non-polarization- maintaining fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett. |

19. | |

20. | S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, “High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,” Opt. Lett. |

21. | B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express |

22. | J. Gentle, |

23. | J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett. |

24. | S. Jiao, G. Yao, and L. V. Wang, “Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue measured with optical coherence tomography,” Appl. Opt. |

25. | B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,” Opt. Lett. |

26. | S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express |

27. | E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express |

**OCIS Codes**

(060.2420) Fiber optics and optical communications : Fibers, polarization-maintaining

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: August 28, 2012

Revised Manuscript: November 13, 2012

Manuscript Accepted: November 19, 2012

Published: January 10, 2013

**Virtual Issues**

Vol. 8, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Ellen Ziyi Zhang, Wang-Yuhl Oh, Martin L. Villiger, Liang Chen, Brett E. Bouma, and Benjamin J. Vakoc, "Numerical compensation of system polarization mode dispersion in polarization-sensitive optical coherence tomography," Opt. Express **21**, 1163-1180 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-1163

Sort: Year | Journal | Reset

### References

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- J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Eye-length measurement by interferometry with partially coherent light,” Opt. Lett.22, 934–936 (1997). [CrossRef] [PubMed]
- D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254, 1178–1181 (1991). [CrossRef] [PubMed]
- A. F. Fercher, K. Mengedoht, and W. Werner, “Two-dimensional birefringence imaging in biological tissue using polarization sensitive optical coherence tomography,” Opt. Lett.13, 186–188 (1988). [CrossRef] [PubMed]
- M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res.30, 431451 (2011). [CrossRef]
- B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001). [CrossRef]
- S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, and G. J. Tearney, “Measurement of collagen and smooth muscle cell content in atherosclerotic plaques using polarization-sensitive optical coherence tomography,” J. Am. Coll. Cardiol.49, 1474–1481 (2007). [CrossRef] [PubMed]
- C. E. Saxer, J.F. de Boer, B. H. Park, Y. Zhao, Z. Chen, and J.S. Nelson, “High-speed fiber-based polarization-sensitive optical coherence tomography of in vivo human skin,” Opt. Lett.25, 1355–1357 (2000). [CrossRef]
- E. Z. Zhang and B. J. Vakoc, “Polarimetry noise in fiber-based optical coherence tomography instrumentation,” Opt. Express19, 16830–16842 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16830 [CrossRef] [PubMed]
- S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express12, 4822–4828 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4822 [CrossRef] [PubMed]
- W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express16, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096 [CrossRef] [PubMed]
- H. Sunnerud, C. Xie, M. Karlsson, R. Samuelsson, and P. A. Andrekson, “A Comparison Between Different PMD Compensation Techniques,” J. Lightwave Technol.20, 368–378 (2002). [CrossRef]
- K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express19, 552–561 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-552 [CrossRef] [PubMed]
- M. Yamanari, S. Makita, and Y. Yasuno, “Polarization-sensitive swept-source optical coherence tomography with continuous source polarization modulation,” Opt. Express16, 5892–5906 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5892 [CrossRef] [PubMed]
- W. Y. Oh, B. J. Vakoc, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Single-detector polarization-sensitive optical frequency domain imaging using high-speed intra A-line polarization modulation,” Opt. Lett.33, 1330–1332 (2008). [CrossRef] [PubMed]
- B. Baumann, W. Choi, B. Potsaid, D. Huang, J. S. Duker, and J. G. Fujimoto, “Swept source / Fourier domain polarization sensitive optical coherence tomography with a passive polarization delay unit,” Opt. Express20, 10229–10241 (2012). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-10229 [CrossRef]
- B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett.30, 2587–2589 (2005). [CrossRef] [PubMed]
- Z. Lu and S. J. Matcher, “Absolute fast axis determination using non-polarization- maintaining fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett.37, 1931–1933 (2012). [CrossRef] [PubMed]
- http://www.mathworks.com/help/toolbox/optim/ug/fsolve.html
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- B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express13, 5483–6593 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5483 [CrossRef] [PubMed]
- J. Gentle, Numerical Linear Algebra for Applications in Statistics (Springer, 1998), chap. 2. [CrossRef]
- J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett.24, 300–302 (1999). [CrossRef]
- S. Jiao, G. Yao, and L. V. Wang, “Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue measured with optical coherence tomography,” Appl. Opt.39, 6318–6324 (2000). [CrossRef]
- B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,” Opt. Lett.29, 2512–2514 (2004). [CrossRef] [PubMed]
- S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express18, 854–876 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-854 [CrossRef] [PubMed]
- E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express17, 22704–22717 (2009). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22704 [CrossRef]

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