# Depolarization Measurement through a Single-Mode Fiber-Based Endoscope for Full Mueller Endoscopic Polarimetric Imaging

^{*}

## Abstract

**:**

## 1. Introduction

_{D}, M

_{R}and M

_{Δ}are, respectively, the Mueller matrices of a diattenuator, a retarder and a depolarizer.

_{1}of the fiber in double path at a wavelength λ

_{1}, and that of the assembly “fiber + sample” (matrix M

_{2}) at a very close wavelength λ

_{2}. For enabling these two measurements, a dichroic mirror highly reflective at λ

_{1}and transparent at λ

_{2}was set at the distal end of the fiber. Thus, two probe beams at λ

_{1}and λ

_{2}, respectively, simultaneously launched in the fiber, were separated by the dichroic filter, the former being directly reflected in the fiber and the latter exiting towards the sample. Part of this beam reflected by the sample was then recoupled into the fiber. The beams at λ

_{1}and λ

_{2}back guided by the fiber and transmitted through the PSA, were separated by a spectral filter, each wavelength being directed towards one intensity detector. Matrices M

_{1}and M

_{2}were deduced from the intensities measured for 16 combinations of the assembly PSG/PSA, at λ

_{1}and λ

_{2}, respectively. Finally, as explained in detail in [25], the Mueller matrix of the sample was calculated from the matrices M

_{1}and M

_{2}. In the implementation of the method reported in [25], we used two CW lasers emitting at λ

_{1}= 634 nm and λ

_{2}= 640 nm, respectively.

_{Δ}extracted from the Lu–Chipman decomposition of M (see Equation (2)), as reminded in more details in Section 2. However, at this point, we must note a profound difference between measures performed by means of a classical free-space Mueller imaging system and those achieved by scanning the sample with a focused beam from a single-mode fiber endoscope. Indeed, when using a free-space Mueller imaging system, the entire imaged area of the sample is illuminated at the same time by a large probe beam. Thus, in the case of spatial depolarizing sample, the light detected by each pixel of the camera may include possible scattered incoherent contributions which had entered the sample at different points of this illuminated area. For that reason, non-zero depolarization can be measured by any pixel of the camera. On the contrary, when a single-mode fiber endoscope is used, the Gaussian beam exiting the fiber at the distal end is focused on a very small area of the sample at a given time. Typically, the mode field diameter of the probe beam in our fiber endoscope (diameter of the core ~3.5 µm) is ~4 µm and, as the magnification of the optical system is ∼1, it produces an enlightened disk of only ∼4 µm in diameter on the sample. As a result, only light entered in this area and backscattered from the same area can be collected by the fiber, with no incoherent contribution. Furthermore, the collected light is coupled in the fundamental LP

_{01}mode of the fiber which is linearly polarized all over its cross section. Thus, the backward beam which is analyzed by the PSA is totally polarized. In other words, at any time, no depolarization can be directly measured through a single-mode fiber endoscope.

## 2. Method and Numerical Simulations

_{0}, S

_{1}, S

_{2}, and S

_{3}denote the four Stokes vector parameters [35]. The degree of polarization (DOP) of this beam represents the fraction of the electromagnetic wave which is fully polarized. It is given by [35]:

_{Δ}of a pure partially depolarizing sample (partial depolarizer) is written in the form:

_{1}, S

_{2}, and S

_{3}of the incident beam. More precisely, M

_{Δ}is the Mueller matrix of a partial depolarizer which reduces the DOP of an incident horizontal or vertical linearly polarized light by a factor $\left|a\right|$, the DOP of an incident 45° or 135° linearly polarized light by a factor $\left|b\right|$, and the DOP of an incident circularly polarized light by a factor $\left|c\right|$ [36].

_{Δ}), the Stokes vector S of an incident beam (Equation (3)) is changed in a new Stokes vector S′:

_{Δ}extracted from the decomposition of the Mueller matrix $\overline{M}$ given by:

_{i}is the Mueller matrix of an elementary non-depolarizing area A

_{i}of A, A being the juxtaposition of the n areas A

_{i}

_{(i = 1, n)}.

- -
- The Mueller matrix of each pixel of an image of the sample (“elementary matrix”) is first measured and registered following the procedure previously reported in [25]. As already stated, since the optical fiber is single-mode, each elementary matrix is a non-depolarizing matrix;
- -
- Then, for each pixel P, the average Mueller matrix over adjacent pixels is calculated, these pixels being those within a floating square window around P, as depicted in Figure 2. The size of this floating window, chosen beforehand, is N = (2n + 1)
^{2}pixels, n being the number of considered rings of pixels around P; - -
- Each average Mueller matrix is decomposed by means of the Lu–Chipman method, in order to extract the associated depolarization matrix M
_{Δ}whose form is given in Equation (5); - -
- The depolarization power Δ of each pixel is finally calculated by means of Equation (8), and a pixelated image of Δ(x,y) is plotted.

_{δ}set around a central value δ

_{0}for the retardance, and within a predefined range Ψ

_{θ}set around θ

_{0}= 0° regarding the eigenaxes orientation. For each retarder, we computed the corresponding elementary Mueller matrix and then we calculated the average Mueller matrix $\overline{M}$. Afterwards, $\overline{M}$ was decomposed by the Lu–Chipman method in order to extract the pure depolarization matrix from which the depolarization power Δ was finally calculated. In Figure 3, we plot Δ as a function of Ψ

_{δ}and Ψ

_{θ}, for 3 different values of the central retardance δ

_{0}. In Table 1, we report Δ for some particular values of δ

_{0}, Ψ

_{δ}, and Ψ

_{θ}.

- -
- the depolarization power Δ is zero when all the elementary matrices remain the same (Ψ
_{δ}= 0, Ψ_{θ}= 0) whatever the retardance δ_{0}; - -
- Δ increases with both Ψ
_{δ}and Ψ_{θ}. In other words, Δ is higher as the diversity of the elementary matrices is large; - -
- if the orientation of the eigenaxes remains the same for all the retarders (i.e., Ψ
_{θ}= 0), the increase in Δ as a function of Ψ_{δ}remains the same whatever the central retardance δ_{0}; - -
- for given ranges of variations Ψ
_{δ}and Ψ_{θ}, the depolarization power Δ increases with δ_{0}.

_{max}= 2/3~0.67, as demonstrated in the Appendix A. For reaching higher Δ, the variety of the averaged elementary matrices must be increased by involving additional polarimetric effects, namely circular retardance and linear and circular diattenuations. Circular retardance φ induced by the sample between incident right and left circular polarizations results in a rotation of the transmitted polarization state by an angle of −φ/2. Linear diattenuation D

_{L}is given by [19]:

_{LH}and T

_{LV}are, respectively, the transmission coefficients of linear polarizations oriented, respectively, along the orthogonal diattenuation eigenaxes H and V of the sample. There is no linear diattenuation when T

_{LH}= T

_{LV}(D

_{L}= 0) whereas the sample is a perfect linear diattenuator, i.e., an ideal polarizer, when T

_{LV}= 0 or T

_{LH}= 0 (D

_{L}= 1). Similarly, circular diattenuation D

_{c}is given by:

_{CR}and T

_{CL}are the transmission coefficients of incident right and left circular polarizations, respectively.

_{c}= 0 in the following. Then, to simulate pixels exhibiting both linear and circular retardance and linear diattenuation, we numerically built new elementary matrices M, each being the product of three basic matrices, namely, that of a pure linear retarder M

_{LR}, that of a pure rotator M

_{CR}and that of a pure linear diattenuator M

_{LD}, as follows:

_{L}was randomly drawn in a predefined range Ψ

_{DL}= [0; D

_{L}

_{max}] and the orientation of the eigenaxes of diattenuation was set identical to that of the eigenaxes of retardance, as it is generally the case in actual samples. Similarly, for each pixel, the circular retardance φ was drawn in a predefined range Ψ

_{φ}= [0; φ

_{max}]. At this point, we can note that the number of parameters involved in the simulations has been significantly increased. To help the reader, we list these parameters and we remind their physical signification in Table 2.

_{δ}and Ψ

_{θ}, for 4 different pairs of Ψ

_{DL}and Ψ

_{φ}, the central retardance being δ

_{0}= 45° in all cases. Figure 4b,c highlight that the extent of the range of diattenuation has a relatively weak influence on the maximum attainable value of Δ (called Δ

_{max}in the following), obtained when Ψ

_{δ}= 180° and Ψ

_{θ}= 180°. For instance, Δ

_{max}only increases from 0.55 to 0.6 when Ψ

_{DL}increases from 0.1 to 0.99, with Ψ

_{φ}= 90°. On the contrary, by comparing Figure 4a,b,d, we can see that Δ

_{max}significantly increases as Ψ

_{φ}is increased. For example, Ψ

_{DL}being set to 0.1 in all cases, Δ

_{max}reaches 0.37, 0.60 and 0.85 when Ψ

_{φ}is set to 10°, 90° and 180°, respectively. These remarks are supported by the data of Table 3, where Δ

_{max}is reported as a function of Ψ

_{DL}and Ψ

_{φ}. We can notice that, as predictable, Δ

_{max}can now exceed the previous maximum value of 0.67. For instance, for δ

_{0}= 45° and for the largest variety of polarimetric effects in the considered pixels (i.e., Ψ

_{δ}= 180°, Ψ

_{θ}= 180°, Ψ

_{DL}= 1, and Ψ

_{φ}= 180°), Δ

_{max}= 0.88. Δ

_{max}can be even more increased, considering higher δ

_{0}. Thus, with δ

_{0}= 90° and the same Ψ

_{δ}, Ψ

_{θ}, Ψ

_{DL}and Ψ

_{φ}as previously, the calculated Δ

_{max}exceeds 0.99, very close to the maximum possible value of 1. However, because circular retardance is likely to be very low in actual biological samples, values of Δ lower than 0.67 should be expected in experimental measurements.

_{L}and φ) are randomly drawn within a range of some extent (respectively, Ψ

_{δ}, Ψ

_{θ}, Ψ

_{DL}and Ψ

_{φ}), the calculated Δ may change when reiterating a given simulation with a new draw. Thus, we achieved some additional simulations in order to evaluate this uncertainty. For this, we repeated M times (M > 100) a given calculation of Δ with M successive draws of the polarimetric characteristics of the elementary matrices and we determined the uncertainty on Δ as being equal to 0.5(Δ

_{max}− Δ

_{min}), Δ

_{max}, and Δ

_{min}being, respectively, the highest and the lowest values of Δ obtained over the M repeated simulations. We first fixed the central retardance δ

_{0}= 0° and we calculated the maximum uncertainty on Δ versus the number N of averaged matrices, which was obtained in the most unfavorable case, namely, when we considered the largest possible ranges of the retardance (Ψ

_{δ}= 180°), of the orientation of the eigenaxes (Ψ

_{θ}= 180°), of the diattenuation (Ψ

_{DL}= 1), and of the circular retardance (Ψ

_{φ}= 180°). Then, we repeated these calculations with δ

_{0}progressively increased from 0° to 180°, by steps of 10°. Finally, we plot in Figure 5 the highest maximum uncertainty on Δ versus N, noted ± ξ (N), obtained over the different values of δ

_{0}. This curve shows that, whatever δ

_{0}, the maximum uncertainty on Δ decreases as N is increased, i.e., Δ converges towards a limit value. With N = 1000, as used in the above simulations, ξ was found to be lower than 0.02. In view of experimental measurements, we considered a floating window of 5 rings of pixels around the central one (N = 121). In this case, ξ was found to remain lower than 0.07. This value is still acceptable, especially if we remember that it is the highest attainable uncertainty obtained in the most unfavorable case with N = 121, which means that actual ξ will be likely to be significantly lower in practical applications. Obviously, ξ could be reduced with higher N, but at the expense of lower resolution of the image. It is the reason why, in further experimental imaging of Δ, we will choose to not exceed this value of N = 121.

## 3. Material and Experimental Results

#### 3.1. Depolarization Measurement on a Manufactured Sample

_{L}, and the depolarization power Δ of each pixel were extracted. These polarimetric characteristics are plotted in Figure 6a–d, respectively. We can observe that δ, θ, and D

_{L}drastically change from one pixel to the next. However, as expected, Δ is ~zero in most of the pixels and it remains small (lower than 0.3) in the others. These nonzero values are due to the fact that, Spectralon being highly diffusive material, very low intensity is re-coupled into the fiber and then detected, resulting in degraded signal to noise ratio.

_{i,j}(i ≠ j) ~ 0 and coefficients m

_{22}, m

_{33}, and m

_{44}are relatively high but significantly lower than 1. We can notice that the amount of spatial depolarization induced by the Spectralon sample is almost insensitive to the orientation of an incident linear polarization, since m

_{22}~m

_{33}. On the other hand, an incident circular polarization experiences higher depolarization from the Spectralon sample than a linear polarization since coefficient m

_{44}is significantly lower than coefficients m

_{22}and m

_{33}. The depolarization power directly calculated from this matrix $\overline{M}$ is equal to 0.51. It is notably lower than that reported in previous papers, regarding similar Spectralon sample with 99% diffuse reflectance [47,48]. This discrepancy can be explained by the fact that the experimental conditions are significantly different in these papers, in particular the size of the region illuminated at the same time which is much larger than in our measurements.

#### 3.2. Depolarization Measurement on a Biological Sample

## 4. Conclusions

- -
- calculating a new matrix $\overline{M}$ as the normalized sum of the Mueller matrices of all pixels included in a floating window centered on P;
- -
- performing the polar decomposition of $\overline{M}$ by the Lu–Chipman method;
- -
- calculating Δ from the depolarization matrix resulting from this decomposition.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{max}attainable with a parallel series of pure retarders with varied retardances δ

_{i}in a range of extend Ψ

_{δ}around a central value δ

_{0}, and with the orientation of their eigenaxes θ

_{i}taken in the range of maximal extend Ψ

_{θ}= π.

_{i}(1 ≤ i ≤ n) with their retardance δ

_{i}taken in a range of extend Ψ

_{δ}around a central value δ

_{0}, and with the orientation of their eigenaxes θ

_{i}taken in a range of extend Ψ

_{θ}around a central value θ

_{0}, so that: δ

_{0}− Ψ

_{δ}/2 ≤ δ

_{i}≤ δ

_{0}+ Ψ

_{δ}/2 and θ

_{0}− Ψ

_{θ}/2 ≤ θ

_{i}≤ θ

_{0}+ Ψ

_{θ}/2.

_{i}being noted M

_{Ri}, the Mueller matrix of the n retarders in parallel is the mean Mueller matrix of the matrices M

_{Ri}(1 ≤ i ≤ n), noted $\overline{M}$:

_{max}of the matrix $\overline{M}$ will be obtained with the largest diversity of the orientations of the eigenaxes, i.e., with Ψ

_{θ}= π. Let us calculate the non-diagonal coefficients of $\overline{M}$ (i.e., $\overline{D},\overline{E},\overline{F},\overline{G},\overline{H}$ and $\overline{I}$) in this case where Ψ

_{θ}= π.

_{i}and θ

_{i}is uniform and since we consider a large number of retarders, we can write:

_{max}is reached when ${\delta}_{0}=\pm \frac{\pi}{2}$, whatever Ψ

_{δ}. This maximum value is $\frac{2}{3}$.

_{δ}.

## References

- Jacques, S.; Ramella-Roman, J.; Lee, K. Imaging skin pathology with polarized light. J. Biomed. Opt.
**2002**, 7, 329–340. [Google Scholar] [CrossRef] [PubMed] - Ghosh, N.; Vitkin, I.A. Tissue polarimetry: Concepts, challenges, applications, and outlook. J. Biomed. Opt.
**2011**, 16, 110801. [Google Scholar] [CrossRef] [PubMed] - Vitkin, A.; Ghosh, N.; De Martino, A. Tissue Polarimetry. In Photonics: Biomedical Photonics, Spectroscopy and Microscopy; John Wiley and Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Tuchin, V.V. Polarized light interaction with tissues. J. Biomed Opt.
**2016**, 21, 071114. [Google Scholar] [CrossRef] [PubMed] - Yasui, T.; Tohno, Y.; Araki, T. Determination of collagen fiber orientation in human tissue by use of polarization measurement of molecular second-harmonic-generation light. Appl. Opt.
**2004**, 43, 2861–2867. [Google Scholar] [CrossRef] - Bancelin, S.; Nazac, A.; Ibrahim, B.H.; Dokládal, P.; Decencière, E.; Teig, B.; Haddad, H.; Fernandez, H.; Schanne-Klein, M.-C.; De Martino, A. Determination of collagen fiber orientation in histological slides using Mueller microscopy and validation by second harmonic generation imaging. Opt. Express
**2014**, 22, 22561–22574. [Google Scholar] [CrossRef] - Chipman, R.A. Polarimetry. In Handbook of Optics Volume II; McGraw-Hill: New York, NY, USA, 1995. [Google Scholar]
- Wood, M.F.G.; Ghosh, N.; Wallenburg, M.A.; Li, S.-H.; Weizel, R.D.; Wilson, B.C.; Li, R.-K.; Vitkin, I.A. Polarization birefringence measurements for characterizing the myocardium, including healthy, infarcted, and stem-cell-regenerated tissues. J. Biomed. Opt.
**2010**, 15, 047009. [Google Scholar] - Pierangelo, A.; Benali, A.; Antonelli, M.R.; Novikova, T.; Validire, P.; Gayet, B.; De Martino, A. Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging. Opt. Express
**2011**, 19, 1582–1593. [Google Scholar] [CrossRef] - Dubreuil, M.; Babilotte, P.; Martin, L.; Sevrain, D.; Rivet, S.; Le Grand, Y.; Le Brun, G.; Turlin, B.; Le Jeune, B. Mueller matrix polarimetry for improved liver fibrosis diagnosis. Opt. Lett.
**2012**, 37, 1061–1063. [Google Scholar] [CrossRef] - Jagtap, J.; Chandel, S.; Das, N.; Soni, J.; Chatterjee, S.; Pradhan, A.; Ghosh, N. Quantitative Mueller matrix fluorescence spectroscopy for precancer detection. Opt. Lett.
**2014**, 39, 243–246. [Google Scholar] [CrossRef] - Qi, J.; Alson, D.S. Mueller polarimetric imaging for surgical and diagnostic applications: A review. J. Biophotonics
**2017**, 10, 950–982. [Google Scholar] [CrossRef] - Kupinski, M.; Boffety, M.; Goudail, F.; Ossikovski, R.; Pierangelo, A.; Rehbinder, J.; Vizet, J.; Novikova, T. Polarimetric measurement utility for pre-cancer detection from uterine cervix specimens. Biomed. Opt. Express
**2018**, 9, 5691–5702. [Google Scholar] [CrossRef] [PubMed] - Goldstein, D.H. Mueller Matrix Polarimetry. In Polarized Light, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Morio, J.; Goudail, F. Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices. Opt. Lett.
**2004**, 29, 2234–2236. [Google Scholar] [CrossRef] [PubMed] - Ossikovski, R. Analysis of depolarizing Mueller matrices through a symmetric decomposition. J. Opt. Soc. Am. A
**2009**, 26, 1109–1118. [Google Scholar] [CrossRef] [PubMed] - Vizet, J.; Ossikovski, R. Symmetric decomposition of experimental depolarizing Mueller matrices in the degenerate case. Appl. Opt.
**2018**, 57, 1159–1167. [Google Scholar] [CrossRef] [PubMed] - Gil, J.J.; San José, I. Arbitrary decomposition of a Mueller matrix. Opt. Lett.
**2019**, 44, 5715–5718. [Google Scholar] [CrossRef] - Lu, S.Y.; Chipman, R.A. Interpretation of Mueller matrices based on polar decomposition. JOSA A
**1996**, 13, 1106–1113. [Google Scholar] [CrossRef] - Desroches, J.; Pagnoux, D.; Louradour, F.; Barthélémy, A. Fiber-optic device for endoscopic polarization imaging. Opt. Lett.
**2009**, 34, 3409–3411. [Google Scholar] [CrossRef] - Fade, J.; Alouini, M. Depolarization remote sensing by orthogonality breaking. Phys. Rev. Lett.
**2012**, 109, 043901. [Google Scholar] [CrossRef] - Vizet, J.; Brevier, J.; Desroches, J.; Barthélémy, A.; Louradour, F.; Pagnoux, D. One shot endoscopic polarization measurement device based on a spectrally encoded polarization states generator. Opt. Express
**2015**, 23, 16439–16448. [Google Scholar] [CrossRef] - Forward, S.; Gribble, A.; Alali, S.; Lindenmaier, A.A.; Vitkin, I.A. Flexible polarimetric probe for 3 × 3 Mueller matrix measurements of biological tissue. Sci. Rep.
**2017**, 7, 11958. [Google Scholar] [CrossRef] - Fu, Y.; Huang, Z.; He, H.; Ma, H.; Wu, J. Flexible 3 × 3 Mueller matrix endoscope prototype for cancer detection. IEEE Trans. Instrum. Meas.
**2018**, 67, 1700–1712. [Google Scholar] [CrossRef] - Vizet, J.; Manhas, S.; Tran, J.; Validire, P.; Benali, A.; Garcia-Caurel, E.; Pierangelo, A.; De Martino, A.; Pagnoux, D. Optical fiber-based full Mueller polarimeter for endoscopic imaging using a two-wavelength simultaneous measurement method. J. Biomed. Opt.
**2016**, 21, 071106. [Google Scholar] [CrossRef] [PubMed] - Buckley, C.; Fabert, M.; Kinet, D.; Kucikas, V.; Pagnoux, D. Design of an endomicroscope including a resonant fiber-based microprobe dedicated to endoscopic polarimetric imaging for medical diagnosis. Biomed. Opt. Express
**2020**, 11, 7032–7052. [Google Scholar] [CrossRef] - Van Eeckhout, A.; Garcia-Caurel, E.; Ossikovski, R.; Lizana, A.; Rodriguez, C.; González-Arnay, E.; Campos, J. Depolarization metric spaces for biological tissues classification. J. Biophotonics
**2020**, 13, e202000083. [Google Scholar] [CrossRef] [PubMed] - Rehbinder, J.; Vizet, J.; Park, J.; Ossikovsk, R.; Vanel, J.C.; Nazac, A.; Pierangelo, A. Depolarization imaging for fast and non-invasive monitoring of cervical microstructure remodeling in vivo during pregnancy. Sci. Rep.
**2022**, 12, 1232. [Google Scholar] [CrossRef] [PubMed] - Du, E.; He, H.; Zeng, N.; Sun, M.; Guo, Y.; Wu, S.; Liu, S.; Ma, H. Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues. J. Biomed. Opt.
**2014**, 19, 076013. [Google Scholar] [CrossRef] [PubMed] - Zhou, X.; Maloufi, S.; Louie, D.C.; Zhang, N.; Liu, Q.; Lee, T.K.; Tang, S. Investigating the depolarization property of skin tissue by degree of polarization uniformity contrast using polarization-sensitive optical coherence tomography. Biomed. Opt. Express
**2021**, 12, 5073–5088. [Google Scholar] [CrossRef] - Ushenko, V.A.; Hogan, B.T.; Dubolazov, A.; Piavchenko, G.; Kuznetsov, S.; Ushenko, A.G.; Ushenko, Y.O.; Gorsky, M.; Bykov, A.; Meglinski, I. 3D Mueller matrix mapping of layered distributions of depolarisation degree for analysis of prostate adenoma and carcinoma diffuse tissues. Sci. Rep.
**2021**, 11, 5162. [Google Scholar] [CrossRef] - Novikova, T.; Pierangelo, A.; de Martino, A.; Benali, A.; Validire, P. Polarimetric imaging for cancer diagnosis and staging. Opt. Photonics News
**2012**, 13, 26–33. [Google Scholar] [CrossRef] - Ivanov, D.; Dremin, V.; Borisova, E.; Bykov, A.; Novikova, T.; Meglinski, I.; Ossikovski, R. Polarization and depolarization metrics as optical markers in support to histopathology of ex vivo colon tissue. Biomed Opt. Express
**2021**, 12, 4560–4572. [Google Scholar] [CrossRef] - Gil, J.J.; Bernabeu, E. A Depolarization Criterion in Mueller Matrices. Opt. Acta Int. J. Opt.
**1985**, 32, 259–261. [Google Scholar] - Goldstein, D.H. Stokes Polarization Parameters. In Polarized Light, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Chipman, R.A. Depolarization. In Polarization: Measurement, Analysis, and Remote Sensing II; Proceedings of SPIE 3754; SPIE: Bellingham, WA, USA, 1999; pp. 14–20. [Google Scholar]
- Cloud, S.R. Group theory and polarization algebra. Optik
**1986**, 75, 26–36. [Google Scholar] - Ortega-Quijano, N.; Fanjul-Vélez, F.; Arce-Diego, J.L. Physically meaningful depolarization metric based on the differential Mueller matrix. Opt. Lett.
**2015**, 40, 3280–3283. [Google Scholar] [CrossRef] [PubMed] - Van de Hulst, H.C. Light Scattering by Small Particles; John Wiley and Sons: New York, NY, USA, 1957. [Google Scholar]
- Götzinger, E.; Pircher, M.; Geitzenauer, W.; Ahlers, C.; Baumann, B.; Michels, S.; Schmidt-Erfurth, U.; Hitzenberger, C.K. Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography. Opt. Express
**2008**, 16, 16410–16422. [Google Scholar] [CrossRef] - Götzinger, E.; Pircher, M.; Baumann, B.; Ahlers, C.; Geitzenauer, W.; Schmidt-Erfurth, U.; Hitzenberger, C.K. Three-dimensional polarization sensitive OCT imaging and interactive display of the human retina. Opt. Express
**2009**, 17, 4151–4165. [Google Scholar] [CrossRef] - Ghosh, N.; Wood, M.F.; Vitkin, I.A. Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence. J. Biomed. Opt.
**2008**, 13, 044036. [Google Scholar] [CrossRef] - Voss, K.J.; Zhang, H. Bidirectional reflectance of dry and submerged Labsphere Spectralon plaque. Appl. Opt.
**2006**, 45, 7924–7927. [Google Scholar] [CrossRef] - Bhandari, A.A.; Hamre, B.; Frette, Ø.; Zhao, L.; Stamnes, J.J.; Kildemo, M. Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light. Appl. Opt.
**2011**, 50, 2431–2442. [Google Scholar] [CrossRef] - Svensen, Ø.; Kildemo, M.; Maria, J.; Stamnes, J.J.; Frette, Ø. Mueller matrix measurements and modeling pertaining to Spectralon white reflectance standards. Opt. Express
**2012**, 20, 15045–15053. [Google Scholar] [CrossRef] - Kildemo, M.; Maria, J.; Ellingsen, P.G.; Aas, L.M.S. Parametric model of the Mueller matrix of a Spectralon white reflectance standard deduced by polar decomposition techniques. Opt. Express
**2013**, 21, 18509–18524. [Google Scholar] [CrossRef] - Dupont, J.; Orlik, X.; Ceolato, R.; Dartigalongue, T. Spectralon spatial depolarization: Towards an intrinsic characterization using a novel phase shift distribution analysis. Opt. Express
**2017**, 25, 9544–9555. [Google Scholar] [CrossRef] [PubMed] - Sanz, J.M.; Extremiana, C.; Saiz, J.M. Comprehensive polarimetric analysis of Spectralon white reflectance standard in a wide visible range. Appl. Opt.
**2013**, 52, 6051–6062. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Schematic representation of an endomicroscope designed for achieving remote in situ Mueller polarimetric imaging by means of the two-wavelength differential method [26].

**Figure 2.**Pixels (green) in which Mueller matrices are averaged in order to calculate the depolarization power associated to the central pixel P. (

**a**) considering 1 ring of pixels surrounding P (9 averaged Mueller matrices); (

**b**) considering 2 rings of pixels surrounding P (25 averaged Mueller matrices).

**Figure 3.**Depolarization power Δ of the average Mueller matrix of 1000 elementary Mueller matrices of neighboring pure retarders, calculated as a function of the variation range of the retardance Ψ

_{δ}and of the variation range of the orientation of the eigenaxes Ψ

_{θ}of these retarders. Three central values δ

_{0}of the retardance are considered: (

**a**) δ

_{0}= 45°; (

**b**) δ

_{0}= 90°; (

**c**) δ

_{0}= 135°.

**Figure 4.**Depolarization power Δ of the average Mueller matrix of 1000 elementary Mueller matrices, each of them being the product of 3 matrices: the matrix of a pure linear retarder, the matrix of a pure circular retarder, and the matrix of a pure linear diattenuator. Δ is calculated as a function of the variation range of the retardance Ψ

_{δ}and of the variation range of the orientation of the eigenaxes Ψ

_{θ}of the linear retarders. The central value of the linear retardance is δ

_{0}= 45°. The diattenuation of each pixel is randomly drawn within the range [0; Ψ

_{DL}] and the circular retardation of each pixel is randomly drawn within the range [0; Ψ

_{φ}]. (

**a**) Ψ

_{DL}= 0.1 and Ψ

_{φ}= 10°; (

**b**) Ψ

_{DL}= 0.1 and Ψ

_{φ}= 90°; (

**c**) Ψ

_{DL}= 0.99 and Ψ

_{φ}= 90°; (

**d**) Ψ

_{DL}= 0.1 and Ψ

_{φ}= 180°.

**Figure 5.**Highest uncertainty on the depolarization power Δ calculated versus the number of averaged elementary matrices, obtained using the largest possible ranges of variations of the polarimetric characteristics of these matrices, and whatever the central value of the retardance.

**Figure 6.**(

**a**) linear retardance δ; (

**b**) orientation of the eigenaxes of retardance θ; (

**c**) linear diattenuation D

_{L}; (

**d**) and depolarization power Δ, directly calculated for each pixel of the 126 × 126 Mueller image of a 500 µm × 500 µm area of a Spectralon sample, from the Lu–Chipman decomposition of the corresponding elementary Mueller matrices measured with the TWDM-based endoscopic Mueller polarimeter.

**Figure 7.**Images of depolarization power of a Spectralon sample (

**a**–

**c**) and associated histograms (

**d**–

**f**) calculated by the method of averaging Mueller matrices of neighboring pixels, when considering a floating square window around each pixel containing: (

**a**,

**d**) 9 pixels; (

**b**,

**e**) 49 pixels; (

**c**,

**f**) 121 pixels.

**Figure 8.**(

**a**) linear retardance δ; (

**b**) orientation of the eigenaxes of retardance θ; (

**c**) linear diattenuation D

_{L}; (

**d**) and depolarization power Δ, directly calculated for each pixel of the 126 × 126 Mueller image of a 500 µm × 500 µm area of a rat tail tendon sample, from the Lu–Chipman decomposition of the corresponding elementary Mueller matrices measured with the TWDM-based endoscopic Mueller polarimeter.

**Figure 9.**Histograms of the linear retardance (

**a**,

**c**) and of the orientations of the eigenaxes of retardance (

**b**,

**d**) in sub-area 1 (

**a**,

**b**) and in sub-area 2 (

**c**,

**d**) of the polarimetric images of Figure 8.

**Figure 10.**Depolarization power of a rat tail tendon sample calculated by the method of averaging Mueller matrices of neighboring pixels, when considering a floating square window around each pixel of: (

**a**) 9 pixels; (

**b**) 49 pixels; (

**c**) 121 pixels.

Ψ_{θ} (°) | Ψ_{δ} (°) | δ_{0} = 45° | δ_{0} = 90° | δ_{0} = 135° | ||
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | ||

0 | 90 | 0.07 | 0.07 | 0.07 | ||

0 | 180 | Δ→ | 0.24 | 0.24 | 0.24 | |

90 | 90 | 0.17 | 0.42 | 0.63 | ||

180 | 0 | 0.20 | 0.67 | 0.67 | ||

180 | 90 | 0.24 | 0.67 | 0.67 |

**Table 2.**Parameters involved in the calculation of the depolarization power when linear retardance, circular retardance, and linear diattenuation occur in each pixel.

δ | Retardance of a given pixel |

δ_{0} | Central retardance of the N considered pixels |

Ψ_{δ} | Extend of the range of the retardances around δ_{0} for the N considered pixels |

θ | Orientation of the eigen axis of a given pixel |

θ_{0} | Central orientation of the eigen axes of the N considered pixels (θ_{0} = 0) |

Ψ_{θ} | Extend of the range of the orientations of the eigen axes around θ_{0} for the N considered pixels |

D_{L} | Linear diattenuation of a given pixel |

Ψ_{DL} | Extend of the range of the linear diattenuations, from 0 to the maximum value D_{Lmax}, for the N considered pixels (i.e., Ψ_{DL} = D_{Lmax}) |

φ | Circular retardance of a given pixel |

Ψ_{φ} | Extend of the range of the circular retardances, from 0 to the maximum value φ_{max}, for the N considered pixels (i.e., Ψ_{φ} = φ_{max}) |

Δ | Depolarization power calculated for the N considered pixels, for given δ_{0}, Ψ_{δ}, Ψ_{θ}, Ψ_{DL} and Ψ_{φ} |

Δ_{max} | Maximum attainable depolarization power calculated for the N considered pixels, for given δ_{0}, Ψ_{DL} and Ψ_{φ} (Ψ_{δ} = 180°, Ψ_{θ} = 180°) |

**Table 3.**Maximum attainable depolarization power Δ

_{max}calculated as a function of the variation range of the linear diattenuation Ψ

_{DL}and of the variation range of the circular retardance Ψ

_{φ}, in 1000 neighboring pixels (δ

_{0}= 45°).

Ψ_{DL} → | 0 | 0.1 | 0.5 | 0.8 | 0.99 | |||||

Ψ_{φ} (°)↓ | ||||||||||

0 | 0.37 | 0.37 | 0.38 | 0.40 | 0.43 | |||||

60 | 0.45 | 0.46 | 0.47 | 0.50 | 0.51 | |||||

120 | 0.65 | 0.65 | 0.66 | 0.69 | 0.71 | |||||

180 | 0.84 | 0.85 | 0.85 | 0.86 | 0.88 |

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## Share and Cite

**MDPI and ACS Style**

Buckley, C.; Fabert, M.; Pagnoux, D. Depolarization Measurement through a Single-Mode Fiber-Based Endoscope for Full Mueller Endoscopic Polarimetric Imaging. *Photonics* **2023**, *10*, 387.
https://doi.org/10.3390/photonics10040387

**AMA Style**

Buckley C, Fabert M, Pagnoux D. Depolarization Measurement through a Single-Mode Fiber-Based Endoscope for Full Mueller Endoscopic Polarimetric Imaging. *Photonics*. 2023; 10(4):387.
https://doi.org/10.3390/photonics10040387

**Chicago/Turabian Style**

Buckley, Colman, Marc Fabert, and Dominique Pagnoux. 2023. "Depolarization Measurement through a Single-Mode Fiber-Based Endoscope for Full Mueller Endoscopic Polarimetric Imaging" *Photonics* 10, no. 4: 387.
https://doi.org/10.3390/photonics10040387