## Complex polarization ratio to determine polarization properties of anisotropic tissue using polarization-sensitive optical coherence tomography

Optics Express, Vol. 17, Issue 16, pp. 13402-13417 (2009)

http://dx.doi.org/10.1364/OE.17.013402

Acrobat PDF (495 KB)

### Abstract

Complex polarization ratio (CPR) in materials with birefringence and biattenuance is shown as a logarithmic spiral in the complex plane. A multi-state Levenberg-Marquardt nonlinear fitting algorithm using the CPR trajectory collected by polarization sensitive optical coherence tomography (PS-OCT) was developed to determine polarization properties of an anisotropic scattering medium. The Levenberg-Marquardt nonlinear fitting algorithm using the CPR trajectory is verified using simulated PS-OCT data with speckle noise. Birefringence and biattenuance of a birefringent film, ex-vivo rodent tail tendon and in-vivo primate retinal nerve fiber layer were determined using measured CPR trajectories and the Levenberg-Marquardt nonlinear fitting algorithm.

© 2009 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**(6), 903–908 (1992). [CrossRef]

2. J. F. de Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. **7**(3), 359–371 (2002). [CrossRef] [PubMed]

8. R. S. Jones, C. L. Darling, J. D. B. Featherstone, and D. Fried, “Remineralization of in vitro dental caries assessed with polarization-sensitive optical coherence tomography,” J. Biomed. Opt. **11**(1), 014016 (2006). [CrossRef] [PubMed]

*(z)|) and vertical (|*

_{h}*(z)|) interference fringe magnitudes and relative phase (∠*

_{v}*(z)) as a function of depth (*

_{vh}*z*) are recorded by controlling polarization elements in source, sample and reference paths of time- or frequency-domain OCT instrumentation [9

9. N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High-sensitivity determination of birefringence in turbid media with enhanced polarization-sensitive optical coherence tomography,” J. Opt. Soc. Am. A **22**(3), 552–560 (2005). [CrossRef]

10. E. Götzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express **13**(25), 10217–10229 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10217. [CrossRef] [PubMed]

*δ*(

*z*)), birefringence (

*Δn*), amplitude attenuation (

*ε*(

*z*)) (or diattenuation (

*d*(

*z*))), biattenuance (

*Δχ*), and optic axis orientation (

*α*) from |

*(z)|, |*

_{h}*(z)| and ∠*

_{v}*(z) [11*

_{vh}11. C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. F. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express **9**(13), 780–790 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-780. [CrossRef] [PubMed]

27. N. J. Kemp, H. N. Zaatari, J. Park, H. G. Rylander III, and T. E. Milner, “Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT),” Opt. Express **13**(12), 4611–4628 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4611. [CrossRef] [PubMed]

*δ*(z)=arctan(|

*(z)|/|*

_{v}*(z)|) [1*

_{h}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**(6), 903–908 (1992). [CrossRef]

11. C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. F. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express **9**(13), 780–790 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-780. [CrossRef] [PubMed]

9. N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High-sensitivity determination of birefringence in turbid media with enhanced polarization-sensitive optical coherence tomography,” J. Opt. Soc. Am. A **22**(3), 552–560 (2005). [CrossRef]

12. S. L. Jiao and L. V. Wang, “Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,” Opt. Lett. **27**(2), 101–103 (2002). [CrossRef]

14. Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, “Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography,” Opt. Lett. **27**(20), 1803–1805 (2002). [CrossRef]

15. 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**(21), 2512–2514 (2004). [CrossRef] [PubMed]

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

15. 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**(21), 2512–2514 (2004). [CrossRef] [PubMed]

16. Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, “Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples,” Appl. Phys. Lett. **85**(15), 3023–3025 (2004). [CrossRef]

17. S. Makita, Y. Yasuno, T. Endo, M. Itoh, and T. Yatagai, “Polarization contrast imaging of biological tissues by polarization-sensitive Fourier-domain optical coherence tomography,” Appl. Opt. **45**(6), 1142–1147 (2006). [CrossRef] [PubMed]

18. M. Yamanari, S. Makita, V. D. Madjarova, T. Yatagai, and Y. Yasuno, “Fiber-based polarization-sensitive Fourier domain optical coherence tomography using B-scan-oriented polarization modulation method,” Opt. Express **14**(14), 6502–6515 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-14-6502. [CrossRef] [PubMed]

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

20. 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**(5), 300–302 (1999). [CrossRef]

27. N. J. Kemp, H. N. Zaatari, J. Park, H. G. Rylander III, and T. E. Milner, “Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT),” Opt. Express **13**(12), 4611–4628 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4611. [CrossRef] [PubMed]

20. 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**(5), 300–302 (1999). [CrossRef]

21. M. G. Ducros, J. D. Marsack, H. G. Rylander III, S. L. Thomsen, and T. E. Milner, “Primate retina imaging with polarization-sensitive optical coherence tomography,” J. Opt. Soc. Am. A **18**(12), 2945–2956 (2001). [CrossRef]

22. 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**(4), 474–479 (2001). [CrossRef] [PubMed]

23. 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**(19), 2587–2589 (2005). [CrossRef] [PubMed]

24. J. Park, N. J. Kemp, H. N. Zaatari, H. G. Rylander III, and T. E. Milner, “Differential geometry of normalized Stokes vector trajectories in anisotropic media,” J. Opt. Soc. Am. A **23**(3), 679–690 (2006). [CrossRef]

25. H. G. Rylander, N. J. Kemp, J. S. Park, H. N. Zaatari, and T. E. Milner, “Birefringence of the primate retinal nerve fiber layer,” Exp. Eye Res. **81**(1), 81–89 (2005). [CrossRef] [PubMed]

27. N. J. Kemp, H. N. Zaatari, J. Park, H. G. Rylander III, and T. E. Milner, “Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT),” Opt. Express **13**(12), 4611–4628 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4611. [CrossRef] [PubMed]

*in-vivo*primate retinal nerve fiber layer (RNFL).

## 2. Background and Theory

### 2.1 Definition of complex polarization ratio

*) is defined by the ratio of Jones vector components (*

_{yx}C*E*and

_{x}*E*) according to an arbitrary polarization bases (

_{y}*x, y*) as

### 2.2 Characteristics of complex polarization ratio

*h*) and vertical (

*v*) polarization states of light are used as an orthonormal basis set, the origin (

*C*=0) and the point-at-infinity (

_{vh}*C*=∞) in the complex plane are assigned to horizontal (

_{vh}*h*) and vertical (

*v*) polarization states of light, respectively. For this basis set, all linear polarization states of light are located on the real axis of the complex plane with the assignments of linear 45° and −45° polarization states at 1 and −1(

*C*=1 and −1) respectively. Two points (

_{vh}*C*=

_{vh}*j*and −

*j*) on the imaginary axis represent right and left circular polarization states of light respectively. All other points excluding the real axis and the two points on the imaginary axis represent elliptical polarization states (Fig. 1(a)). Interestingly, if the relative magnitude is constant (|

*C*|=|

_{vh}*E*|/|

_{v}*E*|=constant), the locus of points in the complex plane is a circle with a center at the origin. Similarly, if the relative phase is constant (∠

_{h}*C*

_{vh}*θ*=constant), the locus of the points is a straight line passing through the origin (Fig. 1(b)) representing lines of latitude and longitude on the Poincaré sphere respectively. The complex plane used to represent the CPR may be directly mapped geometrically to the Poincaré sphere of unit diameter by stereographic projection (Fig. 1(c)) [30,31].

_{v}−θ_{h}### 2.3 Change of polarization basis vectors

*h, v*) is commonly used in CPR polarization analysis, any two fixed elliptic states (

*a*,

*b*) can be selected as a basis set. Position of CPR in the complex plane depends on the selected basis set, and numerous displays of CPR on the complex plane are possible. An algebraic expression for transformation of the CPR due to a change in basis set is easily derived from Jones vector calculus. An arbitrary state of polarized light expressed by Jones vector (

*E*) with the basis set (

_{xy}*x*,

*y*) can be expressed as a linear combination (

*E*=

_{xy}*E*+

_{a}**ε**_{a}*E*) of basis Jones vectors (

_{b}**ε**_{b}*=[*

**ε**_{a}*f*

_{11}

*f*

_{21}]

*,*

^{T}*=[*

**ε**_{b}*f*

_{12}

*f*

_{22}]

*) corresponding to the basis set (*

^{T}*a*,

*b*) where

*f*are complex numbers representing the basis transformation. If

_{ij}*E*is written as

_{xy}*x*,

*y*) and (

*a*,

*b*) is computed by taking the ratios in both sides of Eq. (2), and gives

### 2.4 CPR trajectory

*a*and an orthogonal state

*b*. The orthonormal basis pair is represented by (

*a*,

*b*). We consider this basis set and examine the CPR trajectory corresponding to light propagation in the birefringent tissue. After forward-scattered light propagates a distance (

*z*) through tissue, the CPR with an arbitrary polarization basis set (

*a*,

*b*) is given by

*δ*(

*z*) and

*ε*(

*z*) are the depth-resolved phase retardation and amplitude attenuation, respectively [24

24. J. Park, N. J. Kemp, H. N. Zaatari, H. G. Rylander III, and T. E. Milner, “Differential geometry of normalized Stokes vector trajectories in anisotropic media,” J. Opt. Soc. Am. A **23**(3), 679–690 (2006). [CrossRef]

*b*,

*a*depth-resolved trajectory of the CPR in the (

*a*,

*b*) basis complex plane is generated by rotation

*δ*(

*z*) and attenuation

*ε*(

*z*) (Fig. 2). Considering only

*δ*(

*z*), the trajectory of the CPR uniformly rotates around the origin and forms a circular arc (Fig. 2(a)). Diattenuation (or biattenuance [27

**13**(12), 4611–4628 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4611. [CrossRef] [PubMed]

*ε*(

*z*) exponentially collapsing the trajectory for increasing depths (

*z*). Therefore, trajectory in the complex plane with both

*δ*(

*z*) and

*ε*(

*z*) becomes a logarithmic spiral converging toward the origin in the (

*a*,

*b*) basis set (Fig. 2(b).

## 3. Methods

### 3.1 Generation of simulated PS-OCT data

*M*incident polarization states were uniformly distributed on a meridian of the Poincaré sphere. A Jones matrix formalism was used to transform each incident polarization state due to light propagation through the material [14

14. Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, “Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography,” Opt. Lett. **27**(20), 1803–1805 (2002). [CrossRef]

15. 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**(21), 2512–2514 (2004). [CrossRef] [PubMed]

*z*) for the

*M*polarization states.

*σ*).

### 3.2 Acquisition of tissue specimen PS-OCT data

_{2}O

_{3}laser source (λ

_{0}=830 nm, λ

_{FWHM}=55 nm) was employed to record PS-OCT data of specimens including birefringent film,

*ex-vivo*rat tail tendon and

*in-vivo*primate retinal nerve fiber layer (RNFL). PS-OCT data of specimens in the

*M*polarization states were acquired by controlling the polarization state of light incident on tissue specimens using a liquid-crystal variable retarder positioned in the sample arm of the interferometer. Details of the PS-OCT system were described previously [9

9. N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High-sensitivity determination of birefringence in turbid media with enhanced polarization-sensitive optical coherence tomography,” J. Opt. Soc. Am. A **22**(3), 552–560 (2005). [CrossRef]

### 3.3 Basis transformation

*(z)|) and vertical (|*

_{h}*(z)|) interference fringe magnitudes and relative phase (∠*

_{v}*(z))) were converted to CPRs (*

_{vh}*C*(z)) as a function of depth (

_{vh}*z*) using Eq. (1). Generally, when the

*C*(z) is displayed in the (

_{vh}*h*,

*v*) based complex plane, the trajectory of

*C*(z) is not uniformly rotated around the origin nor exponentially converging toward the origin because the real basis set ((

_{vh}*x*,

*y*)) of CPRs in the specimen is not matched with the basis set ((

*h*,

*v*)) for displaying the CPRs. The mismatch of basis sets also complicates determination of

*δ*(

*z*) and

*ε*(

*z*) in a specimen using a Levenberg-Marquardt nonlinear fitting algorithm. A trajectory of the normalized Stokes vectors on the Poincaré sphere gives us geometrical knowledge about the determination of the basis set in a specimen. Trajectory of Stokes vectors with respect to

*δ*(

*z*) and

*ε*(

*z*) of the specimen is a spiral which rotates around and converges toward a point that represents the optic axis of the specimen [24

24. J. Park, N. J. Kemp, H. N. Zaatari, H. G. Rylander III, and T. E. Milner, “Differential geometry of normalized Stokes vector trajectories in anisotropic media,” J. Opt. Soc. Am. A **23**(3), 679–690 (2006). [CrossRef]

26. N. J. Kemp, H. N. Zaatari, J. Park, H. G. Rylander III, and T. E. Milner, “Depth-resolved optic axis orientation in multiple layered anisotropic tissues measured with enhanced polarization-sensitive optical coherence tomography (EPS-OCT),” Opt. Express **13**(12), 4507–4518 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4507. [CrossRef] [PubMed]

*h*,

*v*) basis set is expressed as a complex number (

*C*_

_{vh}*=*

_{oa}*r*exp(

*jθ*)), the CPR representing optic axis in the (

*x*,

*y*) basis set is zero (

*C*=0).

_{yx _oa}### 3.4 Levenberg-Marquardt nonlinear fitting algorithm

*δ*), double-pass amplitude attenuation (2

*ε*), CPR of optic axis (

*C*), noise-free CPRs at the surface of specimen in (

_{vh_oa}*h*,

*v*) basis set (

*C*(0)) by minimizing a residual function (

_{vh}*). The residual function specifies goodness of fit between depth-resolved CPR (*

**R**_{o}*C*(z)) in (

_{vh}*h*,

*v*) basis set measured by PS-OCT and noise-free CPR (

*C*(z)) in an arbitrary basis set (

_{yz}*x*,

*y*).

*[*

**T**_{yx}*C*] represents basis transformation of

_{vh}*C*from (

_{vh}*h*,

*v*) to (

*x*,

*y*) basis sets. Multiple incident polarization states of light were applied as input to the Levenberg-Marquardt nonlinear fitting algorithm for suppressing noise and increasing polarimetric signal to noise ratio (PSNR) in PS-OCT data [9

**22**(3), 552–560 (2005). [CrossRef]

*) was computed as an algebraic sum of single-state residual function (*

**R**_{M}*) over the M incident polarization states.*

**R**_{o}*δ*, 2

*ε*,

*C*, and noise-free CPRs at the surface of the specimen in (

_{vh_oa}*h, v*) basis set (

*C*

_{vh(m)}(0)) at the

*M*states were determined by minimizing a multi-state residual function (

*) [9*

**R**_{M}**22**(3), 552–560 (2005). [CrossRef]

## 4. Results

### 4.1 Mapping of trajectory by basis transformation

*C*(z)) is displayed in (

_{vh}*h*,

*v*) basis complex plane (Fig. 3(a)). Similar to Fig. 2(a), the trajectory is generated by double-pass phase retardation (2

*δ*(

*z*)=360°), but a CPR representing an optic axis (

*C*=0.3exp(

_{vh_oa}*j*45°)) is assumed. Although double-pass amplitude attenuation (2

*ε*(

*z*)) is usually considered in biological tissues, only

*δ*(

*z*) is considered to observe skewing of

*C*(z). The

_{vh}*C*in the (

_{vh_oa}*h*,

*v*) basis complex plane is not the origin of the trajectory of

*C*(z), which indicates the

_{vh}*C*(z) cannot rotate around and converge to the

_{vh}*C*. Therefore, accurate estimates of

_{vhvh_oa}*δ*(

*z*) and

*ε*(

*z*) are not determined using an incorrect basis set. By the basis transformation procedure mentioned in Methods section, complex number variables are computed. The trajectory of

*C*(z) is transformed to a trajectory of CPRs (

_{vh}*C*(z) in (

_{yx}*x*,

*y*) basis complex plane (Fig. 3(b)). A CPR representing optic axis (

*C*) in (

_{yx_oa}*x*,

*y*) basis complex is zero and the origin of trajectory of

*C*(z). The

_{vh}*δ*(

*z*) and

*ε*(

*z*) can be determined by uniform rotation around the origin and exponential convergence toward the origin of

*C*(

_{yx}*z*) the (

*x*,

*y*) basis set.

*C*

_{vh}_{(M=6)}(z)) with six incident polarization states (

*M*=6) are plotted in (

*h*,

*v*) basis complex plane (Fig. 4(a)). Each trajectory is generated with 2

*δ*(

*z*)=60°, 2

*ε*(

*z*)=6.0° and optic axis (

*C*=0.3exp(

_{vh_oa}*j*45°)). Even though

*C*is a common optic axis in all trajectories, identical

_{vh_oa}*δ*(

*z*) and

*ε*(

*z*) in each state are not observed in (

*h*,

*v*) basis complex plane. Multi-state trajectories of PS-OCT data (

*C*

_{yx}_{(M=6)}(z) in the (

*x*,

*y*) basis complex plane are transformed from

*C*

_{yx}_{(m=6)}(z) using basis transformation method. Identical

*δ*(

*z*) and

*ε*(

*z*) are observed corresponding to the optic axis (

*C*=0) (Fig. 4(b)).

_{yx_0a}### 4.2 Determination of polarization properties using simulated CPR

*M*=6) of speckle-noise corrupted CPRs (

*c*

_{vh}_{(M=6)}(z) in (

*h*,

*v*) basis complex plane are applied to verify the ability of a multi-state Levenberg-Marquardt nonlinear algorithm to determine

*δ*and

*ε*. Polarization values (2

*δ*(

*z*)=60°, 2

*ε*(

*z*)=6.0° and

*C*=0.3exp(

_{vh_0a}*j*45°)) similar to those used in Fig. 4(a) are used to generate noise-free CPRs (

*C*

_{vh}_{(M=6)}(z). Speckle noise with standard deviation (

*σ*=3°) is added into each noise-free CPRs. Figure 5 shows the performance of the multi-state Levenberg-Marquardt nonlinear algorithm. The (

*C*

_{vh}_{(M=6)}(z) and fitted CPRs (

*C*

_{vh}_{(M=6)}

_{_fit}(z), black trajectories) are depicted. A fitted CPR (

*C*=0.29exp(

_{vh_oa_ fit}*j*45.4°), black dot) of optic axis, each CPR at the surface of specimen (

*C*

_{vh}_{(M=6)_fit}(0)) in the (

*h*,

*v*) basis complex plane, double-pass retardation (2

*δ*(z)=60.3°), and double-pass attenuation (2

_{_fit}*ε*(z)=6.02°) are computed by minimizing the multi-state residual function (

_{_fit}*) including estimation of basis transformation.*

**R**_{M}### 4.3 Determination of polarization properties using birefringent film

### 4.5 Determination of polarization properties using in-vivo primate RNFL

*In-vivo*primate RNFL was imaged to obtain the phase retardation (

*δ*(

_{RNFL}*z*)) in a thick region (1mm inferior to the center of the optic nerve head, thickness

*z*=150.0µm) and a thin region (1mm nasal to the center of the optic nerve head, thickness

*z*=53.5µm). Speckle noise-corrupted CPRs of the primate RNFL in the two regions were determined and displayed in the (

*h*,

*v*) complex plane (Fig. 7 (a–b)). The Levenberg-Marquardt nonlinear fitting algorithm with six incident polarization states was used to estimate the phase retardation, optic axis, and initial CPRs in each state by minimizing the multi-state residual function. Fitted trajectories (black trajectories) were obtained from estimated parameters, and overlapped on the noisy trajectories (color trajectories). The CPR of the optic axis (black dot) is almost coincident with the linear horizontal state (

*h*basis). Phase retardations in the inferior and nasal regions were determined as

*δ*(

_{RNFL}*z*=150.0µm)=28.2° and

*δ*(z=53.5µm)=2.78°, respectively. Birefringence in the inferior (

_{RNFL}*Δn*=4.33×10

_{RNFL}^{−4}or 18.8°/100µm) and nasal region (Δ

*n*=1.20×10

_{RNFL}^{−4}or 5.20°/100µm) were easily determined from the fitted phase retardation values. These values closely match those determined using Stokes vectors on the Poincaré sphere [9

**22**(3), 552–560 (2005). [CrossRef]

25. H. G. Rylander, N. J. Kemp, J. S. Park, H. N. Zaatari, and T. E. Milner, “Birefringence of the primate retinal nerve fiber layer,” Exp. Eye Res. **81**(1), 81–89 (2005). [CrossRef] [PubMed]

## 5. Discussion

### 5.1 Complexity of nonlinear fitting algorithms using CPR and Stokes vectors

### 5.2 Processing time of two nonlinear fitting algorithms using CPR and Stokes vector

*δ*(

*z*)=10°, 30°, 50°, 70° and 90°) and high speckle noise with standard deviation (σ=5°) were applied to generate the speckle noise-corrupted CPRs and Stokes vectors. For the Stokes vector calculation, as the retardation increases a monotonic reduction of the relative processing time is observed. However, for CPR as the retardation increases a more rapid reduction of the relative processing time is observed. Difference of the relative processing time apparently increases, and an over 20% time difference is observed when the retardation is 90° (Fig. 8(b)). Therefore, the nonlinear algorithm using CPRs is faster by approximately 25% than the algorithm using Stokes vectors. Processing times with respect to different speckle noise (σ=1°, 2°, 3°, 4° and 5°) at a phase retardation (

*δ*(

*z*)=30°) are also studied and displayed in Fig. 8(c). The processing time in the Stokes vectors is less affected than that in CPRs by magnitude of speckle noise, and a 10% time difference is observed when the speckle nose is 5° (Fig. 8(d)). As speckle noise is reduced, difference between processing times increases. In other words, determination of polarimetric properties is increasingly faster as the speckle noise is reduced in the nonlinear fitting algorithm using CPRs compared to Stokes vectors.

### 5.3 Accuracy of fitted phase retardation in nonlinear fitting algorithms using CPR and Stokes vector

*s*) of the variance of the residual function with

^{2}*n − p*degrees of freedom and

*n*×

*p*Jacobian matrix (

*). The n refers to number of CPRs or Stokes vectors, and the*

**J***p*to number of variables. A

*p*×

*p*product matrix (

*) given by the inverse of*

**P****was used to acquire confidence interval of each estimated polarization property. The confidence interval (**

*J*^{T}×J*CI*) was determined by [32

32. A. R. Gallant, “Nonlinear-Regression,” Am. Stat. **29**(2), 73–81 (1975). [CrossRef]

*t*is a value of the t-distribution, and

*P*is the ith diagonal element of the matrix (

_{ii}*).*

**P***δ*(

*z*)=10°, 30°, 50°, 70° and 90°) at a fixed optic axis (

*C*=0.3exp(

_{vh_oa}*j*45°)) and relatively large speckle noise with standard deviation (

*σ*=5°). Confidence intervals increase when phase retardation increases in both algorithms. The confidence interval in CPR is slightly larger than that for Stokes vectors, which suggests that the algorithm using Stokes vector is statistically more accurate than that using CPRs. But the difference in accuracy between the two algorithms may be negligible because the confidence intervals represented by green error bars are too small compared to estimated phase retardation in both algorithms (Fig. 9(b)). Similarly, 95% confidence intervals were observed by changing magnitude of speckle noise (

*σ*=1°, 2°, 3°, 4° and 5°) at a fixed phase retardation (

*δ*(

*z*)=30°) (Fig. 9(c)). The confidence intervals linearly increase with increased speckle noise, and difference between the confidence intervals of the two algorithms increases as the speckle noise increases. Although the algorithm using Stokes vectors is slightly more accurate than that using CPRs, the confidence intervals are substantially smaller than the fitted phase retardation (Fig. 9(d)).

## 6. Conclusion

*ex-vivo*rat tail tendon and

*in-vivo*primate RNFL are determined. The Levenberg-Marquardt nonlinear fitting algorithm using CPR has less complexity, faster processing time than that using Stokes vectors [9

**22**(3), 552–560 (2005). [CrossRef]

25. H. G. Rylander, N. J. Kemp, J. S. Park, H. N. Zaatari, and T. E. Milner, “Birefringence of the primate retinal nerve fiber layer,” Exp. Eye Res. **81**(1), 81–89 (2005). [CrossRef] [PubMed]

## Acknowledgements

## 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 and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. |

3. | C. K. Hitzenberger, E. Götzinger, and M. Pircher, “Birefringence properties of the human cornea measured with polarization sensitive optical coherence tomography,” Bull. Soc. Belge Ophtalmol. |

4. | B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography,” Invest. Ophthalmol. Vis. Sci. |

5. | M. C. Pierce, R. L. Sheridan, B. Hyle Park, B. Cense, and J. F. de Boer, “Collagen denaturation can be quantified in burned human skin using polarization-sensitive optical coherence tomography,” Burns |

6. | N. A. Patel, J. Zoeller, D. L. Stamper, J. G. Fujimoto, and M. E. Brezinski, “Monitoring osteoarthritis in the rat model using optical coherence tomography,” IEEE Trans. Med. Imaging |

7. | S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, “Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT),” Int. J. Cardiol. |

8. | R. S. Jones, C. L. Darling, J. D. B. Featherstone, and D. Fried, “Remineralization of in vitro dental caries assessed with polarization-sensitive optical coherence tomography,” J. Biomed. Opt. |

9. | N. J. Kemp, J. Park, H. N. Zaatari, H. G. Rylander, and T. E. Milner, “High-sensitivity determination of birefringence in turbid media with enhanced polarization-sensitive optical coherence tomography,” J. Opt. Soc. Am. A |

10. | E. Götzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express |

11. | C. K. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. F. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express |

12. | S. L. Jiao and L. V. Wang, “Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,” Opt. Lett. |

13. | S. L. Jiao, M. Todorović, G. Stoica, and L. V. Wang, “Fiber-based polarization-sensitive Mueller matrix optical coherence tomography with continuous source polarization modulation,” Appl. Opt. |

14. | Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, “Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography,” Opt. Lett. |

15. | 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. |

16. | Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, “Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples,” Appl. Phys. Lett. |

17. | S. Makita, Y. Yasuno, T. Endo, M. Itoh, and T. Yatagai, “Polarization contrast imaging of biological tissues by polarization-sensitive Fourier-domain optical coherence tomography,” Appl. Opt. |

18. | M. Yamanari, S. Makita, V. D. Madjarova, T. Yatagai, and Y. Yasuno, “Fiber-based polarization-sensitive Fourier domain optical coherence tomography using B-scan-oriented polarization modulation method,” Opt. Express |

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

20. | 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. |

21. | M. G. Ducros, J. D. Marsack, H. G. Rylander III, S. L. Thomsen, and T. E. Milner, “Primate retina imaging with polarization-sensitive optical coherence tomography,” J. Opt. Soc. Am. A |

22. | 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. |

23. | 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. |

24. | J. Park, N. J. Kemp, H. N. Zaatari, H. G. Rylander III, and T. E. Milner, “Differential geometry of normalized Stokes vector trajectories in anisotropic media,” J. Opt. Soc. Am. A |

25. | H. G. Rylander, N. J. Kemp, J. S. Park, H. N. Zaatari, and T. E. Milner, “Birefringence of the primate retinal nerve fiber layer,” Exp. Eye Res. |

26. | N. J. Kemp, H. N. Zaatari, J. Park, H. G. Rylander III, and T. E. Milner, “Depth-resolved optic axis orientation in multiple layered anisotropic tissues measured with enhanced polarization-sensitive optical coherence tomography (EPS-OCT),” Opt. Express |

27. | N. J. Kemp, H. N. Zaatari, J. Park, H. G. Rylander III, and T. E. Milner, “Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT),” Opt. Express |

28. | B. P. Lathi, |

29. | J. D. Paliouras, |

30. | R. M. A. Azzam and N. M. Bashara, |

31. | C. Brosseau, |

32. | A. R. Gallant, “Nonlinear-Regression,” Am. Stat. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(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: April 7, 2009

Revised Manuscript: June 30, 2009

Manuscript Accepted: July 1, 2009

Published: July 20, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Jesung Park, Nate J. Kemp, H. Grady Rylander, and Thomas E. Milner, "Complex polarization ratio to determine polarization properties of anisotropic tissue using polarization-sensitive optical coherence tomography," Opt. Express **17**, 13402-13417 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13402

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### References

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