Open Access
Add to BibSonomyAbstract
A method is proposed for extracting the linear birefringence (LB) and linear dichroism (LD) properties of an anisotropic optical sample using reflection-mode optical coherence tomography (OCT) and a hybrid Mueller matrix formalism. To ensure the accuracy of the extracted parameter values, a method is proposed for calibrating and compensating the polarization distortion effect induced by the beam splitters in the OCT system using a composite quarter-waveplate / half-waveplate / quarter-waveplate structure. The validity of the proposed method is confirmed by extracting the LB and LD properties of a quarter-wave plate and a defective polarizer. To the best of the authors’ knowledge, the method proposed in this study represents the first reported attempt to utilize an inverse Mueller matrix formalism and a reflection-mode OCT structure to extract the LB and LD parameters of optically anisotropic samples.
© 2015 Optical Society of America
1. Introduction
Optical coherence tomography (OCT) is a powerful technique for performing the in-depth cross-sectional imaging of scattering-type media [1]. Moreover, recent enhancements to the traditional OCT structure now make possible the incorporation of polarization control into the system such that the anisotropic properties of certain optical materials can be observed. For example, several polarization-sensitive (PS) OCT structures have been proposed for measuring the depth-resolved optical birefringence properties of biological tissues [2–4]. The proposed structures combine the characteristic depth resolution of traditional OCT systems with the polarization sensitivity of scanning polarimetry methods based on Jones calculus. The feasibility of the PS-OCT structures proposed in [2–4] was demonstrated by measuring the phase retardation of a linear birefringent (LB) sample using two orthogonal polarization components at each backscattering or reflection point. However, in general, the polarization of biological tissue is extremely complex and cannot be adequately described using a simple 2 × 2 Jones matrix. Many studies have shown that a complete characterization of optically anisotropic biological samples can be obtained by measuring the Stokes vectors of the light backscattered from the tissue and then calculating the corresponding Mueller matrix. A Mueller OCT system for measuring the full 4 × 4 Mueller matrix of biological tissue was proposed [5, 6]. By utilizing an OCT, the Mueller matrix of the sample can be measured with OCT resolution. However, while the systems proposed in [5, 6] obtain a Mueller representation of the sample, they do not solve the exact parameters related to anisotropic properties. Hence, an analysis on the Stokes-Mueller vectors to extract anisotropic parameters more than LB in an OCT system is needed.
Vitkin et al. [7–10] proposed a method based on the Mueller matrix decomposition method for extracting the linear birefringence (LB), circular birefringence (CB), linear dichroism (LD) and depolarization coefficient of complex turbid media. Pham and Lo [11] proposed a decoupled analytical technique for extracting all nine effective parameters of an anisotropic optical sample such that its LB, CB, LD, circular dichroism (CD), linear depolarization (L-Dep), and circular depolarization (C-Dep) properties can all be fully characterized. However, in methods based on the Mueller matrix decomposition method (e.g., [7–11]), the LB, CB, LD, CD, L-Dep and C-Dep properties are prescribed in a strict sequential order in the analytical model. To resolve this problem, Ossikovski [12] proposed a differential Mueller matrix formalism which allows the anisotropic properties of depolarizing media to be extracted regardless of the sequential order in an independent fashion. Also, Ortega-Quijano and Arce-Diego [13, 14] utilized a differential Mueller matrix method to measure the anisotropic properties of depolarizing media in the backscattering mode. However, the differential matrix formalism is unsuitable for reflective-mode measurements. Additionally, comparative study of differential matrix and extended polar decomposition formalisms for measurement in LB, CB, LD, CD, and depolarization effect was studied by Kumar et al. [15].
Accordingly, the present study proposes an analytical model based on a hybrid Mueller matrix formalism for extracting the LB and LD properties of anisotropic optical samples using a reflection-mode OCT. In addition, a method is proposed for compensating the polarization distortion effect induced by the beam splitters in the OCT structure such that the accuracy of the extracted parameter measurements is improved. The validity of the proposed method is demonstrated numerically and by experimentally determining the linear anisotropic properties of a quarter-wave plate and a defective polarizer.
2. Mueller OCT Structure
Figure 1 presents a schematic illustration of the proposed Mueller OCT structure. As shown, the structure is similar to that of a traditional OCT system other than the use of a thermal light source to enhance the axial resolution and a combined polarizer, quarter waveplate and variable waveplate in both arms to induce polarization effects such that the multiple anisotropic parameters of the optical sample can be measured. As shown, the beam from the thermal light source is split by a non-polarizing beam splitter (NPBS1) into two beams, where one beam is incident upon a fixed mirror while the other beam passes through a second non-polarizing beam splitter (NPBS2) and is again split into two beams, namely a reference beam and a measurement beam. The reference beam passes through a variable wave plate designed to control the polarization state and is then reflected by a mirror mounted on a scanning stage used to carry out path-length scanning. Meanwhile, the measurement beam passes through a glass plate designed to compensate for the dispersion effect and is then incident upon the sample [16]. As shown in Fig. 1, the OCT system additionally includes two composite structures, each comprising two quarter-waveplates and one half-waveplate, designed to compensate the polarization distortion induced by the non-perfect beam splitters. (Note that full details of the compensation process are presented in Section 5.)
During the measurement process, a low-coherence interferometric signal is obtained at Detectors 1 and 2 as the scanning stage is driven at a constant velocity. The detection signal extracted by Detectors 1 and 2 can be expressed analytically as follows:
where d is the penetration depth; Rs(d) is the reflectivity at depth d for various kinds of the reflective interface; ΔL = Lr – Li is the path-length difference of the two arms, in which Li is the path length in the sample arm or the fixed arm. For the sample arm, it includes the optical path within the sample, i.e. Ls = Lt + nd (where n is the refractive index of the sample); k = 2π / λ, in which λ is the central wavelength of the light source; and lc is the coherence length of the thermal light source. Having obtained the signal at Detector 1 by locating the peak of the envelope of the signal as shown in Eq. (1), the thickness and mean refractive index of the sample can be extracted using the method proposed by the current group in [16]. It is noted that the validity of this measurement has be confirmed in [16] and would not be demonstrated in this study.Subsequently, signals obtained by Detector 2 are employed to measure anisotropic properties of the sample by calculating the amplitude of the interferometric signal as shown in Eq. (1), and hence the information of Rs(d) is concerned. To calculate the Mueller matrix of the sample, the quarter-wave plate and polarizer shown in Fig. 1 are rotated to obtain four different polarization states of the light incident on the sample, namely H (horizontal linear polarization), V (vertical linear polarization), P (45° linear polarization), and R (right-circular polarization). In addition, the variable wave plate in the reference arm is adjusted to change the polarization state of the reference beam sequentially to H, V, P, and R, respectively, for each of the four incident lights. Thus, a total of 16 interferometric signals are produced with which to investigate the sample and detected by Detector 2. The 16 elements in the 4x4 Mueller matrix are then computed as [5]
where Mij is the Mueller matrix element of the ith row and jth column, and the left and right symbols in each double polarization state notation represent the polarization states of the measurement light beam and reference light beam, respectively. For example, the notation HV refers to the interferometric signal obtained given a horizontal linear polarized incident beam and a vertical linear polarized reference beam.3. Analytical model to extract LB/LD properties of optical sample
The Mueller matrix of a sample with LB and LD properties has already been derived in differential form in [17, 18]. Furthermore, a macroscopic form of the Mueller matrix containing LB and LD has already been presented in [19]. However, in the OCT system considered in the present study, the measurement beam passes through the sample in both the forward direction and the backward direction. The Mueller matrix of the LB/LD sample for the forward measurement beam, ML,Forward is given in the Appendix of [19]. The Mueller matrix for the backward measurement beam can be obtained from the assumption that ML,Backward = ML(-α, -θd), where α and θd are the orientation angles of the LB and LD properties, respectively. As shown in Fig. 2, in the OCT setup used in the present study, the measurement beam is transmitted through the second beam splitter (NPBS2), forward through the sample, reflected from the mirror, transmitted backward through the sample, and finally reflected from the beam splitter such that it is incident on Detector 2.
The optical arrangement shown in Fig. 2 can be modeled using the following Mueller matrix representation:
where MMirror is the Mueller matrix of the mirror; and MT,BS and MR,BS are the Mueller matrixes of the beam splitter in the transmission mode and reflection mode, respectively. General forms of all the Mueller matrices describing the optical components in Eq. (3) are expressed in appendix. Finally, the Mueller matrix of the sample in the OCT system can be reformulated aswhere α and β are the LB orientation angle and phase retardation, respectively, while for P2 = (1-D)/(1 + D), θd and D are the LD orientation angle and diattenuation, respectively. In addition,, and .4. Simulated extraction of LB/LD properties of hybrid sample
For the Mueller OCT structure considered in the present study, the Mueller matrix of the sample has the form shown in Eq. (2). In theory, the LB and LD parameters of the sample can be obtained simply by comparing Eq. (2) with the Mueller matrix given in Eq. (4). However, in practice, Eq. (4) is highly complicated, and is thus not easily solved in closed-form analytical solutions. Consequently, in the present study, the LB/LD properties of the sample (i.e., the unknown parameters in ML,OCT in Eq. (4)) are inversely derived using a Genetic Algorithm (GA) [20–22] based on the Mueller matrix given in Eq. (2) as the target function. Notably, an inspection of Eq. (4) shows that the proposed analytical model of the sample/OCT system theoretically allows the unknown LB and LD parameters to be determined over the full range, i.e., 0<α<180°, 0<β<360°, 0<θd<180°, and 0<D<1, respectively.
The simulations commenced by testing the ability of the proposed model to obtain full-range measurements of the sample parameters. In performing the simulations, the measured Mueller matrix was calculated via a process of inverse differential calculation based on assumed the parameter values for a hypothetical sample. Theoretical input values of the corresponding parameters were inserted into the measured Mueller matrix, and the GA was then used to inversely determine these input values using the proposed analytical model. Finally, the extracted values of the optical parameters were compared with the theoretical values in order to quantify the accuracy of the proposed extraction method.
In conducting the simulations, the default values of the input parameters were specified as follows: α = 30°, β = 60°, θd = 35°, and D = 0.5. In performing each simulation, the considered parameter (α, β, θd or D) was varied over the respective full range, while the remaining three parameters were assigned their default values. Figure 3(a) presents the extracted values of α for input values of α over the range of 0~180° and default parameter settings of β = 60°, θd = 35° and D = 0.5, respectively. Figures 3(b)-3(d) present the equivalent results for parameters β, θd, and D, respectively. In general, the results presented in Fig. 3 show that the proposed analytical model enables all four LB/LD parameters to be measured over the full range.
Fig. 3 Simulation results obtained for α, β, θd and D of hybrid LB/LD sample given theoretical input values of (a) α: 0~180°, β = 60°, θd = 35°, D = 0.5; (b) β: 0~360°, α = 30°, θd = 35°, D = 0.5; (c) θd: 0~180°, α = 30°, β = 60°, D = 0.5; and (d) D: 0~1, α = 30°, β = 60°, θd = 35°.
5. Compensation for polarization distortion caused by non-perfect non-polarizing beam splitters
In the Mueller OCT structure, the polarization distortion caused by the non-perfect beam splitters should be compensated in order to ensure the accuracy of the extracted parameter values. In the present study, the compensation is performed using a composite polarizer component comprising a quarter-waveplate, a half-waveplateand a second quarter-waveplate (Q/H/Q) (see Fig. 1) such that the beams (both reference and measurement) retain their original polarization state and have no phase error. In practice, the phase distortion induced in the transmission mode of the beam splitters is much less than that induced in the reflection mode. Furthermore, the phase distortion caused by the variable wave plate and dispersion compensator can be ignored since this distortion can be easily calibrated. Thus, the compensation process described in the following refers only to the reflection mode of the second non-ideal beam splitter (NPBS 2).
The first step in the compensation process considers the measurement beam transmitted through the beam splitter following its reflection from the mirror mounted on the sample stage (see Fig. 4(a). As shown, the measurement beam passes through a notional “black box” (Black Box 1) containing the beam splitter and the QWP1/HWP1/QWP2 compensator and is then incident on Detector 2. In practice, the black box can be treated as a free-space media and modeled as a unit Mueller matrix by quantitatively adjusting the orientations of QWP1, HWP1 and QWP2 using the analytical model proposed in [23]. In Eq. (5), the following unit matrix of a polarized light from a sample traveling through a beam splitter and then going to a detector in Fig. 4(a) can be treated as the objective function for the GA optimization procedure as
where Hi(βi) means the Mueller matrix of the half-wave plate with the azimuth angle, βi, and R(θi) is the rotational Mueller matrix corresponding to the quarter waveplate, Qi. RBS,i represents the reflection matrix of the corresponding interface in the beam splitter as shown in Fig. 4. All the forms of Mueller matrices can be found in appendix except RBS,i needs to be measured in the actual experiment. Subsequently, a GA is used to inversely derive the optimal values of parameters θ1, θ2 and β1 to achieve a free space compensation in the measurement arm. Finally, the compensation structure consisting of QWP1, HWP1 and QWP2 serves to eliminate polarization distortion in the measurement arm of the Mueller OCT system.Fig. 4 NPBS compensation process for light beams propagating in: (a) measurement arm and (b) reference arm.
The second step of the compensation process considers the reference beam reflected from the beam splitter (see Fig. 4(b)). The reference beam again passes through a notional black box (Black Box 2) containing the beam splitter, a QWP3/HWP2/QWP4 polarization controller and the QWP1/HWP1/QWP2 polarization controller described above. Black Box 2 can also be treated as a free-space media via the suitable adjustment of the individual orientations of components QWP3, HWP2 and QWP4 using the method described in [23]. (Note that in performing the adjustment process, the orientations of QWP1, HWP1 and QWP2 in Black Box 1 are set to the positions determined in the prior step.) From an inspection of Fig. 4(b), the objective function for the GA optimization procedure is obtained as
It is noted that the optimal parameter values obtained from Eq. (5) are taken as constants in the optimization procedure described in Eq. (6). Similarly, a GA is used to inversely derive the optimal values of parameters θ3, θ4 and β2 to achieve a free-space compensation in the reference arm. Finally, the compensation structure consisting of QWP3, HWP2, QWP4, QWP1, HWP1 and QWP2 serves to eliminate polarization distortion in the reference arm of the Mueller OCT system.
It is found that in the traditional PS-OCT, the reasonable result is still achieved without NPBS compensation process because the measurement is much simpler and only two signals are processed to calculate the LB parameters. However, 16 signals are employed in this study to calculate the whole Mueller matrix for extracting multiple properties, LB and LD. Thus, multiple errors exist in 16 elements of the Mueller matrix and finally make the unaccountable error for extracting corresponding parameters. Therefore, the distortion of the polarized light caused by NPBS needs to be compensated in advanced for this proposed Mueller OCT system.
6. Experimental setup and results
The validity of the proposed measurement method was evaluated by extracting the linear anisotropic parameters of two optical samples, namely a quarter wave-plate and a defective (non-ideal) polarizer. (Note that the experiments considered only the anisotropic properties of the two samples. That is, the thickness and mean refractive index were not considered.)
6.1 Measurement system
In implementing the Mueller OCT structure shown in Fig. 1, the light source was provided by a halogen lamp (Model R150-BM1, TECHNIQUIP Co., USA) with a color temperature of 3250 K. Note that a halogen illumination source was specifically chosen since it has a broad spectral bandwidth and therefore provides an improved axial resolution. The light from the halogen lamp was filtered through a light guide such that only the visible wavelengths were remained. Hence, the light entering the Mueller OCT system had a spectral bandwidth of Δλ = 210 nm and a central wavelength of λ0 = 603 nm. The photo-detectors (Detector 1 and Detector 2) were acquired from New Focus Corporation, USA, (Model 2001) and had an operating wavelength range of 300~1050 nm and a maximum conversion gain of 9.4 × 106 V/W. The scanning stage was purchased from Sigma Koki Co, Ltd., Japan (Model SGSP 20-85). During the scanning experiments, the stage was moved with a constant velocity of 1000 μm/second to induce a Doppler shift frequency for modulation purposes [16]. In the present experiments, the displacement in the reference arm was determined from the oscilloscope used to display the detected interferometric signal (i.e., not from the stage controller itself), and hence the velocity must be known with a high degree of accuracy. Accordingly, the scanning stage was controlled using a Mark-204MS stage controller (Sigma Koki Co., Ltd; Japan) via the GPIP interface of a PC.
When using a broadband light source in a PS-OCT system, the reference arm and sample arm must be perfectly symmetrical in order to prevent the resolution from being degraded by dispersion mismatches [24, 25]. In the present OCT setup, the dispersion effect was compensated by means of two D263T glass plates, as shown in Fig. 1. Additionally, available wave plate (New Focus Corporation, USA, Model 5540) was used to adjust the polarization of the beam in the reference arm in order to obtain the state of polarization (SOP) required for measurement purposes. Finally, the interferometric signals observed by the photo-detector were recorded by an oscilloscope (LeCroy Corporation; USA; Model: Wave Runner 6050) with a maximum sampling rate of 5 GS/sec and a bandwidth of 500 MHz.
6.2 Test on a Quarter-wave plate
The performance of the Mueller OCT structure was investigated initially using a zero-order quarter-waveplate (Thorlabs, AQWP05-630) with pure LB properties. In accordance with the experimental procedure described in Section 3, the intensity of the detection signal was obtained for 16 combinations of the polarization states in the reference and measurement arms given different orientations α of the quarter-wave plate so as to construct the Mueller matrix given in Eq. (2). Thus, calibrating the sample in successive measurement tests is complex and time consuming. Accordingly, in the present study, the sample properties were measured only for orientation angles over the range of α = 0 ~20°. For each orientation of the sample, the LB principal axis angle α, LB phase retardation β, LD principal angle θd, and LD diattenuation D, were inversely derived using the GA, as described in Section 3. Table 1 and Fig. 5 show the extracted results for the four parameters over the considered range of α = 0~20°.
Table 1. Experimental results for zero-order quarter-wave plate
Fig. 5 Experimental results for zero-order quarter-wave plate: (a) LB orientation angle, α; (b) LB phase retardation, β; (c) LD orientation angle, θd; and (d) LD diattenuation, D.
It is observed that a good agreement exists between the input value and the extracted value of the LB orientation angle, α. Moreover, the extracted value of the phase retardation is close to 90° for all sample orientations. In other words, the basic validity of the proposed measurement method is confirmed. By using the Fresnel equation to calculate different transmissions (Te and To) induced by birefringence properties (ne and no), the theoretical diattenuation of this quarter-wave plate can be found as 0.0013 according to D = (To - Te) / (To + Te). As expected, the LD principal axis angle θd varies randomly with increasing α, while D maintains an approximately constant value close to zero.
6.3 Test on a defective polarizer
The validity of the Mueller OCT structure was further evaluated using a defective polarizer (LLC2-82-18S, OPTIMAX Co.) with slight LB properties. Table 2 and Fig. 6 show the extracted results for α, β, θd, and D. As shown, a good agreement exists between the extracted value of the LD principal axis angle, θd, and the input value. Moreover, the extracted value of the diattenuation (D) is close to 1 (i.e., 0.9058), as shown in Fig. 6(d). It is observed that the measured LB phase retardation has a value of around 14.45°. Furthermore, the LB principal axis angle, α, deviates from the LD orientation angle, θd, by around 35°. In other words, the principal axis of the LB property of the defective polarizer does not coincide with that of the LD property.
Table 2. Experimental results for defective polarizer
Fig. 6 Experimental results for defective polarizer: (a) LB orientation angle, α; (b) LB phase retardation, β; (c) LD orientation angle, θd; and (d) LD diattenuation, D.
For comparison purposes, the LB and LD parameters of the defective polarizer were also extracted using the Stokes polarimetry method developed by the current group in [19] based on a laser source with a wavelength of 633 nm. The corresponding results for α, β, θd, and D are shown in Table 3 and Fig. 7. It is seen that the extracted value of D is 0.9866, while the measured value of β is around 10.87°. It is noted that these values are in good qualitative agreement with those obtained using the Mueller OCT structure (i.e., 0.9058 and 14.45°, respectively). The slight difference between the two sets of results can be attributed to a difference in the central wavelengths of the two light sources used in the Mueller OCT system and the Stokes polarimetry system, respectively (i.e., 603 nm and 633 nm). As for the OCT structure, it is seen in Figs. 7(a) and 7(c) that the LB principal axis angle α in the Stokes polarimetry system deviates by approximately 35° from the LD principal angle θd. In general, the results presented in Fig. 7 confirm the feasibility and accuracy of the proposed Mueller OCT structure.
Table 3. Experimental results obtained by Stokes polarimeter system for defective polarizer
Fig. 7 Experimental results obtained by Stokes polarimeter system for optical parameters of defective polarizer: (a) LB orientation angle, α; (b) LB phase retardation, β; (c) LD orientation angle, θd; and (d) LD diattenuation, D.
An inspection of Figs. 5 and 6 shows that the extracted values have standard deviations as α: 1.13° (with the degree of accuracy at 1.54°), β: 0.30°, θd: 1.50° (with the degree of accuracy at 0.97°), and D: 0.0059. The experimental errors observed in the present study can be attributed to two main factors. Firstly, the dispersion effect from the sample affects both the intensity and the width of the detected interferometric signals, and thus for the broad bandwidth light source used in the Mueller OCT system, an effective dispersion compensation scheme is required. Secondly, since the measurement was obtained according to the 16 signals caused by 16 combinations of the polarization states, the signals are not easily calibrated in successive measurements because of the variation of the light source or the mechanical vibration of the scanning stage. Thus, an inevitable variation in the intensity of the interferometric signal occurs to produce the experimental error. It is noted that the more stable light source and scanning stage with less vibration are required for the elimination of experimental errors. Also, the intensity of the interferometric signal should be calibrated carefully.
7. Conclusions and discussions
A method has been proposed for extracting the LB and LD properties of anisotropic optical samples using a reflection-mode Mueller OCT system, an analytical model based on a hybrid Mueller matrix formalism, and a GA. In addition, a compensation scheme based on a composite quarter-waveplate / half-waveplate / quarter-waveplate structure has been proposed for compensating the polarization distortion effect induced by the beam splitters in the OCT structure. In addition, the simulated and experimental results have confirmed the practical feasibility of the proposed approach.
In the present study, the Mueller OCT structure has been applied to a quarter-waveplate sample with LB properties only and to a defective polarizer with both LB and LD properties. In a future study, the proposed method will be extended to the measurement of samples having not only combined LB/LD properties, but also combined circular birefringence (CB)/circular dichroism (CD) properties. Furthermore, the application of the proposed method to scattering optical media will also be addressed.
Appendix
General forms of all the Mueller matrices describing the optical components in this study are expressed as
Therefore,Acknowledgments
The authors gratefully acknowledge the financial support provided to this study by the Ministry of Science and Technology of Taiwan under Grant No. NSC 102-2221-E-006-043 -MY2 and MOST 104-3113-E-006-002.
References and links
1. 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(5035), 1178–1181 (1991). [CrossRef] [PubMed]
2. 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]
3. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22(12), 934–936 (1997). [CrossRef] [PubMed]
4. C. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. 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). [CrossRef] [PubMed]
5. G. Yao and L. V. Wang, “Two-dimensional depth-resolved Mueller matrix characterization of biological tissue by optical coherence tomography,” Opt. Lett. 24(8), 537–539 (1999). [CrossRef] [PubMed]
6. 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(34), 6318–6324 (2000). [CrossRef] [PubMed]
7. N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophoton. 2(3), 145–156 (2009). [CrossRef] [PubMed]
8. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009). [CrossRef]
9. X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010). [CrossRef] [PubMed]
10. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011). [CrossRef] [PubMed]
11. T. T. H. Pham and Y. L. Lo, “Extraction of effective parameters of anisotropic optical materials using a decoupled analytical method,” J. Biomed. Opt. 17(2), 025006 (2012). [CrossRef] [PubMed]
12. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36(12), 2330–2332 (2011). [CrossRef] [PubMed]
13. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36(13), 2429–2431 (2011). [CrossRef] [PubMed]
14. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition for direction reversal: application to samples measured in reflection and backscattering,” Opt. Express 19(15), 14348–14353 (2011). [CrossRef] [PubMed]
15. S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012). [CrossRef] [PubMed]
16. C. C. Liao, Y. L. Lo, and C. Y. Yeh, “Measurement of Multiple Optical Parameters of Birefrigent Sample Using Polarization-Sensitive Optical Coherence Tomography,” IEEE/OSA. J. Lightwave Technol. 27(5), 483–493 (2009). [CrossRef]
17. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. 68(12), 1756–1767 (1978). [CrossRef]
18. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized light in optics and spectroscopy, Academic Press, Inc., (1990).
19. C. C. Liao and Y. L. Lo, “Extraction of anisotropic parameters of turbid media using hybrid model comprising differential- and decomposition-based Mueller matrices,” Opt. Express 21(14), 16831–16853 (2013). [CrossRef] [PubMed]
20. Z. Michalewicz, Genetic Algorithm + Data structure = Evolution Programs, Springer-Verlag, New York (1994).
21. H. C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm and two thermally modulated intensity spectra,” IEEE/OSA. J. Lightwave Technol. 23(6), 2158–2168 (2005). [CrossRef]
22. T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” IEEE/OSA. J. Lightwave Technol. 25(3), 946–951 (2007). [CrossRef]
23. Y. C. Liu, Y. L. Lo, and C. C. Liao, “Compensation of Non-Ideal Beam Splitter Polarization Distortion Effect in Michelson Interferometer,” J. Opt. Soc. Am. A (submitted) (2015).
24. A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001). [CrossRef] [PubMed]
25. C. K. Hitzenberger, A. Baumgartner, W. Drexler, and A. F. Fercher, “Dispersion effects in partial coherence interferometry: Implications for intraocular ranging,” J. Biomed. Opt. 4(1), 144–151 (1999). [CrossRef] [PubMed]
