The interaction of light with a medium provides a noninvasive means for characterization and imaging. In addition to the intensity, phase, coherence, and spectrum variations of scalar light,1 how the vector wave evolves for polarized light interacting with a medium can reveal rich physical properties such as the microscopic structure and anisotropy of the object and material, which are indiscernible to other optical techniques.2^–7 The use of polarization optics has recently expanded rapidly in biomedicine2^,8^–11 and has been applied in, for example, the characterization of complex random media,12^,13 tissue diagnosis,14^–17 identification and staging of cancer,18^,19 glucose sensing,20 advanced fluorescence microscopy,21 and preclinical and clinical applications.9^,22

The polarization state of light can be represented by a four-component vector known as a Stokes vector. The interaction of polarized light with media is fully described by the four-by-four Mueller matrix M, which transforms the Stokes vector of the incident beam to that of the outgoing beam.23 The medium microscopic structure is encoded in the 16 elements of the Mueller matrix. Mueller matrix decomposition facilitates the interpretation of polarized light–medium interactions and links the evolution of the vector light wave to the physical properties of a complex medium. Standard polar decomposition (Lu–Chipman decomposition24) factors the Mueller matrix M into a product of a depolarizer matrix MΔ, a retarder matrix MR, and a diattenuator matrix MD, where the individual matrices describe anisotropic depolarization, phase, and amplitude modulation of polarized light by the medium, providing a straightforward phenomenological interpretation of basic optical properties (depolarization, birefringence, and dichroism) as well as the underlying microstructure of the medium. Variants of polar decomposition25^,26 and differential Mueller matrix decomposition27^,28 have also recently emerged.

The probing polarized light traverses the sample along the forward and backward paths sequentially in the reflection geometry. The anisotropic polarization properties of the sample observed for the probing beam in the forward and backward directions are reciprocal to each other and are not generally the same.29^,30 However, traditional decomposition methods do not account for this unique requirement for decomposing Mueller matrices measured inside a backward geometry—a ubiquitous and often preferred geometry in diverse applications from remote sensing to tissue characterization (especially for in vivo imaging). The adverse consequences of violating reciprocity include contradiction in the recovered properties of media of varying thickness and inferior and often incorrect quantification of complex media. This limits the utility of polarization imaging in reflection geometry and especially hampers the application of polarization optics in in vivo biomedical imaging, such as endoscopic imaging, retinal imaging, and surgery guidance, where reflection is the only feasible geometry.

In this paper, we present reciprocal polarization imaging of complex media after introducing reciprocal polar decomposition of backscattering Mueller matrices, which enforces the reciprocity of the optical wave in its forward and backward paths (see Fig. 1). In contrast to the Lu–Chipman decomposition of the Mueller matrix into the product of MΔMRMD, reciprocal polar decomposition factors the backscattering Mueller matrix into a product of MD#MR#MΔMRMD, with the diattenuation and retardance MD# and MR# in the backward path specified by the reciprocal of their counterparts MD and MR in the forward path. We highlight the importance of reciprocity and demonstrate that reciprocal polarization imaging of complex chiral and anisotropic media (including a birefringence resolution target, fresh tissue sections of varying thickness, a chiral tissue phantom, and unstained gastric cancerous tissue samples) accurately quantifies their anisotropic properties in reflection geometry, whereas traditional approaches encounter difficulties and produce inferior and often incorrect results. In particular, reciprocal polarization imaging provided consistent characterization of complex media of different thicknesses, accurately measured the optical activity and glucose concentration of turbid media in reflection, and discriminated between cancerous and normal tissue with even stronger contrast than forward measurement. Reciprocal polarization imaging enables accurate backscattering polarization measurements and will be instrumental in imaging complex media, opening new applications of polarization optics in reflection geometry.

Reciprocal polarization imaging relies on the reciprocity of optical systems in the absence of a magnetic field. We denote the reciprocal matrix for a Mueller matrix M as M# for the optical system with the incident and outgoing beams interchanged. The reciprocal Mueller matrix can be written as M#=QMTQ, where Q=diag(1,1,−1,1).29 We introduce reciprocal polar decomposition, which decomposes a backscattering Mueller matrix into M=MD#MR#MΔMRMD,(1)where MD, MD# and MR, MR# are pairs of diattenuator and retarder matrices in the forward and backward paths, respectively, and MΔ are the depolarizer matrices. The diattenuator, retarder, and depolarizer describe the modulations of the amplitude, phase, and loss of polarization of polarized light, respectively. The diattenuator and retarder matrices, MD and MR, take the same form as in the Lu–Chipman decomposition.24 The depolarizer matrix, MΔ, accounts for total depolarization over forward and backward paths. Inside an exact backward geometry, the measured Mueller matrix M is exactly its own reciprocal, i.e., M=QMTQ. The depolarizer matrix further belongs to the type of diagonal form31^,32 and reduces to MΔ=MΔd≡diag(d0,d1,d2,d3). The reciprocal polar decomposition of the backscattering Mueller matrix M can be rewritten as QM=MDTMRTMΔd′MRMD,(2)in terms of a symmetric matrix QM, where MΔd′≡QMΔd=diag(d0,d1,−d2,d3). The diattenuator, retarder, and depolarizer matrices MD, MR, and MΔ of the sample are then determined via decomposition of Eq. (2) (see Sec. 5).

Backscattering Mueller matrices are typically measured slightly off the normal direction in reflection geometries to avoid specular reflection. In the reciprocal polar decomposition of experimentally measured backscattering Mueller matrices, we replace QM with [QM+(QM)T]/2 to guarantee symmetry. For comparison, the results from the Lu–Chipman decomposition are also shown. However, the Lu–Chipman decomposition cannot be applied directly to the backscattering Mueller matrix because the incident and outgoing beams are defined in different coordinate systems (see Fig. 1). We replace the backscattering Mueller matrix M with MmirrorM by flipping the coordinate system for the outgoing beam to be the same as that of the incident beam before the Lu–Chipman decomposition, where Mmirror=diag(1,1,−1,−1).33

We first imaged an NBS 1963A birefringence resolution target (R2L2S1B, Thorlabs, Newton, New Jersey, United States) in both backward and forward geometries with our custom polarization imaging system (see Fig. 2, Sec. 5.2, and Appendix A). The target contains a liquid crystal polymer pattern sandwiched between two N-BK7 glass substrates and has minimal diattenuation. The extracted linear retardance and orientation angle as well as depolarization for the target by reciprocal polar decomposition in the backward geometry and by the Lu–Chipman decomposition in both the forward and backward geometries are compared (see Fig. 3). The means and standard deviations of the orientation angle, linear retardance, and depolarization for the birefringent (outlined by a red rectangle) and clear (outlined by a white rectangle) regions of the target measured in the forward and backward geometries are summarized in Table 1. The linear retardance from the Lu–Chipman decomposition of the backscattering Mueller matrix is multiplied by 1/2 in Fig. 3 and Table 1 because it accounts for both forward and backward paths.

The recovered polarization parameters from the reciprocal polar decomposition of the backscattering Mueller matrix are in excellent agreement with those obtained from the Lu–Chipman decomposition of the Mueller matrix measured in the forward geometry other than a stronger depolarization in the former owing to the different detection geometry. The Lu–Chipman decomposition on the backscattering Mueller matrix fails to obtain the correct orientation angle and linear retardance. In particular, the orientation angle obtained via the Lu–Chipman decomposition of the backscattering Mueller matrix is off by 90 deg and contains sporadic artifacts. The edges of the birefringent regions exhibit greater retardance and depolarization in the backscattering measurements [see Figs. 3(b3) and 3(c3)]. This difference originates from edge diffraction, which leads to increased depolarization and greater retardation of light rays that traverse a longer path within the target. By contrast, the edges of the birefringent regions in the forward geometry only show slightly increased depolarization as the edge diffraction rays dominate in the backscattering geometry where the specular reflection is removed, unlike in the forward geometry. The orientation and linear retardance obtained by the Lu–Chipman decomposition of the forward-scattering Mueller matrix and the reciprocal polar decomposition of the backscattering Mueller matrix agree with the data provided by the manufacturer.

The recovered anisotropic properties should be consistent between complex media of varying thicknesses. Beef tissue, which is abundant in muscle, is a typical birefringent biological medium. We imaged fresh beef sections in serial cuts with thicknesses of 100 and 300μm under identical experimental conditions. The extracted tissue birefringence orientation angle, linear retardance, and depolarization by the Lu–Chipman and reciprocal polar decomposition in the backward geometry are compared with the ground truth from those of the 100-μm tissue section measured by the Lu–Chipman decomposition in the forward geometry (see Fig. 4). Satisfactory recovery should reveal consistent orientation angles, approximately three times greater linear retardance, and greater depolarization for the 300-μm section than for the 100-μm section. The linear retardance from the Lu–Chipman decomposition of the backscattering Mueller matrix was multiplied by 1/2 in Fig. 4 because it accounts for forward and backward paths.

For the 100-μm tissue section, the orientation angle from the reciprocal polar decomposition is in closer agreement than that computed by the Lu–Chipman decomposition of the backscattering Mueller matrix with the orientation angle measured in the forward geometry [mean squared error: 117 versus 216 and correlation coefficient: 0.265 versus 0.206; see Figs. 4(a1)–4(a3)]. Owing to their geometric differences, light depolarization is greater in the backward geometry, which entails large angle scattering, than in the forward geometry. Tissue depolarization is distorted in the Lu–Chipman decomposition of the backscattering Mueller matrices [see Figs. 4(c1)–4(c3)]. The images of tissue linear retardance and depolarization from reciprocal polar decomposition are much sharper than those obtained by the Lu–Chipman decomposition of the common backscattering Mueller matrix (sharpness: 5.01×103 versus 4.71×103 for linear retardance, 3.64×103 versus 1.89×103 for depolarization).

Importantly, the Lu–Chipman decomposition in the backward geometry of the 300-μm tissue section produces incorrect orientation angles and retardance. By contrast, reciprocal polar decomposition correctly yields consistent orientation angles and approximately three times greater linear retardance for the 300-μm section than for the 100-μm section [see Figs. 4(a4)–4(a6) and 4(b4)–4(b6)]. The depolarization is greater for thicker sections [see Fig. 4(c4)–4(c6)]. The images of tissue depolarization from reciprocal polar decomposition are much sharper than those obtained by the Lu–Chipman decomposition of the backscattering Mueller matrix (sharpness: 7.32×103 versus 6.32×102). Results on 50-μm and 200-μm fresh beef tissue sections are provided in the Supplementary Material.

Optical activity resulting from chirality is ubiquitous. Owing to the different refractive indices for left-handed and right-handed circularly polarized light, optical activity causes the polarized light to rotate. The measurement of optical rotation during reflection is particularly challenging because optical rotations in the forward and backward paths tend to cancel each other in reflection geometry.34 However, reciprocal polarization imaging can realize accurate recovery of the optical rotation of chiral media in reflection after breaking such cancellation.

For highly scattering chiral media, the cancellation of optical rotations in the forward and backward paths can be broken by keeping the collection optics at an angle away from the exact backscattering direction, which introduces anisotropic depolarization (d1≠−d2).30 For example, a backward scattering geometry when the angle was set at ∼30deg breaks such cancellation with anisotropic depolarization (d1=0.76, d2=−0.49) for a chiral tissue emulating turbid medium.13 The glucose concentration of a highly scattering chiral medium (scattering coefficient μs=0.6mm−1, anisotropy factor g=0.91, glucose concentration 5 M, and thickness 10 mm) can be correctly obtained via reciprocal polarization imaging, whereas the Lu–Chipman decomposition fails from the backscattering Mueller matrix (see Appendix B). The glucose concentration obtained by reciprocal polar decomposition was 4.91 M, which is in excellent agreement with the 4.82 M obtained by the Lu–Chipman decomposition from forward geometry and the ground truth (5 M), whereas the Lu–Chipman decomposition yielded 1.06 M in reflection.

We finally imaged an unstained gastric stage III cancerous tissue section of thickness 30μm. The extracted linear retardance, orientation angle, depolarization, and linear depolarization anisotropy for the gastric tissue section by reciprocal polar decomposition in the backward geometry and by the Lu–Chipman in the forward geometry are compared, together with its Hematoxylin and Eosin H&E-stained histological image (see Fig. 5). The recovered linear retardance and orientation angle from the reciprocal polar decomposition in the backward geometry agree with those obtained from the Lu–Chipman decompositions in the forward geometry other than the stronger depolarization and linear depolarization anisotropy in the former owing to their different geometries. Notably, the linear retardance, depolarization, and depolarization anisotropy obtained via reciprocal polar decomposition exhibit greater consistency in the data distribution (tighter clustering around the median) and much stronger contrast than those obtained via the Lu–Chipman decomposition in the forward geometry [see Figs. 5(c)–5(j)]. The cancerous region has less linear retardation, depolarization, and depolarization anisotropy than the normal region does, which is attributed to the disruption of collagen organization in cancer.35 Triple (linear retardance, depolarization, and depolarization anisotropy) from reciprocal polar decomposition in reflection provides a stronger contrast between cancerous and normal gastric tissue and can better differentiate them than the same parameters from Lu–Chipman decomposition in forward geometry [see Figs. 5(k)–5(p)].

Mueller matrix decomposition facilitates the interpretation of the polarized light–medium interaction and reveals the physical properties of a complex medium from the evolution of the vector light wave. The fundamental difference between polarization imaging in forward or backscattering geometry is that the anisotropic property of the medium observed by the probing beam along the forward path and backward path is not identical but reciprocal in the latter. Unfortunately, traditional approaches such as the Lu–Chipman decomposition do not consider this distinctive nature of backscattering polarimetry. Like polarization imaging, which factors the forward-scattering Mueller matrix into a product of three matrices describing the medium depolarization, birefringence, and dichroism with the Lu–Chipman decomposition, reciprocal polarization imaging achieves this feat by the reciprocal polar decomposition of Mueller matrices measured in the backward scattering geometry. It factors the backscattering Mueller matrix into a product of MD#MR#MΔMRMD, with the diattenuation and retardance along the backward path specified by the reciprocal of their counterparts along the forward path based on the reciprocity of the optical wave in the forward and backward paths. Despite being a phenomenological theory, the reciprocity assumed in reciprocal polar decomposition strictly holds when single scattering dominates and is justified in multiple scattering situations as long as the angular distributions of the incident photons and the bounced-back photons are close to each other surrounding the optical axis at the plane where photons are backscattered.

Compared with forward-scattering Mueller matrices, backscattering Mueller matrices have a reduced degree of freedom of 10. These 10 deg of freedom exactly map to 10 polarization parameters: the diattenuation vector D, the total retardance R (i.e., the linear retardance δ, its orientation θ, and the optical rotation α), and the depolarization Δ factors d0, d1, d2, and d3. Reciprocal polarization imaging allows straightforward interpretation of the polarization measurement in the backward geometry and determines the same set of 10 polarization parameters typically extracted from Mueller matrices measured in the forward geometry. It presents a significant advantage because the complexity (degree of freedom) is reduced, and the backward geometry is often preferred or the only feasible approach for bulk samples and is more convenient in many applications.

We have demonstrated reciprocal polarization imaging with various complex media and have shown the superiority of reciprocal polar decomposition over the Lu–Chipman decomposition of backscattering Mueller matrices. The polarization properties of complex media determined by the reciprocal polar decomposition of the backscattering Mueller matrices are in excellent agreement with those obtained by polarization imaging of the same sample in forward geometry. In particular, the anisotropic properties are correctly recovered via reciprocal polar decomposition and are consistent for complex media of varying thickness. Optical rotation is correctly recovered by reciprocal polarization imaging, whereas the Lu–Chipman decomposition produces erroneous results in reflection. The values obtained via reciprocal polar decomposition are also more consistent in terms of the data distribution (tighter clustering around the median) and even exhibit a stronger contrast between cancerous and normal tissue than forward measurements do. Furthermore, other popular traditional approaches, such as differential and symmetric decomposition, even produce inferior results to those of Lu–Chipman decomposition in backward scattering geometry (see the Supplementary Material).

Interestingly, symmetric decomposition leads to inferior or incorrect results in reflection, although it appears to be similar to reciprocal polar decomposition. This is attributed to the fact that reciprocity is not being enforced (and is typically violated) in symmetric decomposition, and reciprocity is the key for correct decomposition in the backscattering geometry. Finally, we note that the accuracy of recovered media polarization parameters deteriorates from polarization imaging when light depolarization (Δ) is elevated.36 Depolarization anisotropy may more clearly reveal structural features for media with high depolarization. The code for reciprocal polarization imaging is publicly available.37

Polarization imaging with Mueller matrix decomposition is a powerful means to quantify the diattenuation, retardance, and depolarization of complex media, uncovering the underlying microstructure and anisotropy invisible to other optical methods. The measurement of spatially resolved polarization properties has a wide array of applications in the noninvasive characterization and diagnosis of complex random media,12^,13 from biological cells and tissue14^–17 to the ocean and atmosphere.6^,38 However, such measurements have traditionally been limited to the transmission geometry, partly owing to existing methods encountering difficulties and producing inferior and often incorrect results in reflection geometry, although reflection geometry is ubiquitous and often preferred. Reciprocal polarization imaging enables accurate polarization measurements in reflection geometry and has potential for broad applications in backscattering polarization optics ranging from remote sensing to biomedicine. Reciprocal polarization imaging can especially be of greatest utility in in vivo biomedical imaging, for example, polarimetric endoscopic imaging,39 retinal imaging,40 and surgery guidance,41 where reflection is the only feasible geometry.

The decomposition of Eq. (2) involves two steps. First, the diattenuation matrix can be written as MD=(1DTDmD),(3)in which mD=1−D2I+(1−1−D2)D^D^T, where I is a 3×3 identity matrix, and D^ is the unit vector for the diattenuation vector D of length D. The diattenuation matrix of the form Eq. (3) satisfies (1−D2)MD−1=GMDG,(4)when the Mueller matrix M does not represent a perfect analyzer (D<1),24 where G≡diag(1,−1,−1,−1) is the Minkowski metric matrix. Equations (2) and (4) yield (QMG)(MDG)=(1−D2)MDTN,(5)with N≡MRTMΔd′MR=(d00T0n).(6)

Expanding Eq. (5) after substituting Eqs. (3) and (6) for MD and N, respectively, leads to QMG(1D)=d0(1−D2)(1D).(7)

This equation has one unique positive eigenvalue [=d0(1−D2)] and an associated eigenvector,26 providing the diattenuation vector D satisfying |D|=D<1. With the determination of the diattenuation vector D, the diattenuator matrix MD is obtained.

Second, Eq. (2) reduces to MRTMΔd′MR=MD−1QMMD−1,(8)and equivalently mRTdiag(d1,−d2,d3)mR=n,(9)for the bottom right 3×3 submatrices mR and n of the matrices MR and N. An orthogonal decomposition of n can then be used to determine mR and di(i=1,2,3). In addition, the order and sign of the eigenvectors should be determined with a priori information, if available, regarding the depolarization properties of the medium. The convention of ordering the eigenvectors, which may be multiplied by ±1, to have a minimal total retardance is adopted in the absence of a priori information, similar to the Lu–Chipman and symmetric decompositions.24^,42 The reciprocal polar decomposition of the backscattering Mueller matrix M into products (1) and (2) is then completely obtained.

We note that the degeneracy of the eigenvalues d1, −d2, and d3 will cause either partial or complete loss of the determination of the retardance parameters, as an arbitrary rotation can be applied to the degenerate eigenvectors. One potential degeneracy in reciprocal polar decomposition arises for a depolarizer matrix MΔd=diag(d0,d1,−d1,d3), which is fortunately rare and results in indeterminate circular retardance.

The retarder matrix MR can further be expressed as an ordered product of a circular retarder of retardance (optical rotation) α and a linear retarder of retardance δ with its fast axis oriented at θ.13^,43 Their values are given by δ=cos−1[(MR11+MR22)2+(MR12−MR21)2−1],(10)θ=1/2atan2(MR31,−MR32),(11)α=1/2atan2(MR12−MR21,MR11+MR22).(12)

Furthermore, the total retardance R and the depolarization coefficient Δ are given by R=cos−1[1/2tr(MR)−1],(13)Δ=1−1/3[tr|MΔMΔ00|−1].(14)

We also define the linear depolarization anisotropy as A≡(|d1|−|d2|)/(|d1|+|d2|).(15)

Here, Mij is the ij’th element of the Mueller matrix M, where i,j=0,1,2,3. For the Lu–Chipman decomposition, we compute A via Eq. (15) using the diagonal elements of its depolarization matrix.

The polarization imaging system is shown in Fig. 2. Collimated light (λ=633nm) passes through a polarization state generator consisting of a rotating polarizer and a rotating quarter-wave plate and illuminates the sample after reflection by a beam splitter (BS, Bp245b1, Thorlabs). The backscattered light is reflected by a mirror (M) and passes through a second rotating quarter-wave plate and a focusing lens (L2) before being recorded by a polarization camera (BFS-U3-51S5P-C, FLIR, Wilsonville, Oregon, United States). The second rotating quarter-wave plate and the linear polarizers along the 0-, 45-, 90-, and 135-deg directions of the polarization camera form the polarization state analyzer. The Mueller matrix for the sample is measured in both the backward (sample at S1) and forward (sample at S2 and an additional mirror at S1) geometries after the removal of the stray light contributions and calibration of the Mueller matrix of the imaging system itself (see Appendix A). In the backward geometry, the system images light backscattered by the bulk sample, and the specular reflection from the sample surface is removed by slightly tilting the sample surface (∼5deg). This Mueller imaging system has a field of view of 1.8cm×2.0cm. The acquisition of one complete Mueller matrix of a sample takes 1 min.

Silverside beef rich in muscle fibers was selected for the experiment. One 1cm×1cm×1cm of tissue was mounted in an OCT embedding compound (Sakura, Nagano, Japan) and kept at −80°C for 10 h. Tissue sections were cut via a cryostat microtome (HM525, Thermo Fisher, Waltham, Massachusetts, United States) and mounted on a coverslip.

Paraffin-embedded gastric tissue obtained from Shanghai Qutdo Biotech Company was cut via a microtome (RM2455, Leica, Wetzlar, Germany) and sliced onto glass slides at 42°C. The sliced pieces were placed on a dryer meter (HI1210, Leica) at 60°C for 2 h. The mounted slices were sequentially washed with xylene I for 5 min, xylene II for 5 min, 100% ethanol for 5 min, 90% ethanol for 5 min, 80% ethanol for 3 min, 70% ethanol for 3 min, and phosphate-buffered saline for 5 min. The slices were then covered with a coverslip with neutral resin. The protocol was approved by the Shanghai Qutdo Biotech Company Ethics Committee. All experiments were performed in compliance with the relevant regulations, and all patients provided written informed consent.

The polarization state generator (consisting of a rotating polarizer and a rotating quarter wave plate) generates 0-, 45-, and 90-deg linear polarization states and one right circular polarization state. A polarization camera (BFS-U3-51S5P-C, FLIR)44 measures the polarized light intensity after passing through a quarter-wave plate (QW2) to obtain the Stokes vector S of the polarized light. The measurement accuracy45 is optimized by setting the angle of QW2 at θ1, θ2 in sequence satisfying |θ1−θ2|=45deg. The Stokes vector S is then expressed as S=[(I0+I1+I2+I3+I4+I5+I6+I7)/4I0−I2I5−I7(I1−I3−I4+I6)/2],(16)where I0,I1,I2,I3 and I4,I5,I6,I7 are the intensities of linearly polarized light along the 0-, 45-, 90-, and 135-deg directions for θ1=0deg and θ2=45deg, respectively.

The calibration of the polarization imaging system followed the steps outlined in Fig. 6. First, the Stokes vectors S0deg, S45deg, S90deg, and Scircular for the four input polarization states were measured via configuration (a). Second, the Mueller matrices of mirrors M1 and M2, MM1 and MM2, were measured via configuration (b). Third, the reflection and transmission Mueller matrices of the BS beam splitters, R and T, were measured in configurations (c) and (d), respectively. The obtained values are given below {S0deg=[10.9830.002−0.025]TS45deg=[0.9840.0190.9650.020]TS90deg=[0.978−0.9520.003−0.021]TScircular=[0.9720.030.0610.947]T,(17)Sin=[S0degS90degS45degScircular],(18)MM1=[10.1410.013−0.0050.1350.9860.0010.033−0.0200.059−0.997−0.028−0.032−0.0280.068−0.999],(19)MM2=[1−0.053−0.027−0.009−0.0520.9956−0.025−0.0210.038−0.002−0.9420.371−0.002−0.073−0.380−0.884],(20)R=[10.366−0.011−0.0240.3440.998−0.078−0.038−0.031−0.013−0.851−0.316−0.033−0.0340.281−0.814],(21)T=[1−0.3950.015−0.001−0.3780.9970.006−0.0230.025−0.0500.9070.1030.032−0.001−0.0450.874].(22)

The measured Mueller matrix Mout is given by Mout=MM2TMSampleRSin,(23)in the backscattering geometry and is given by Mout=MM2MSampleTMM1RSin,(24)in the forward geometry, which contains an additional mirror M1 placed at the S1 position. The Mueller matrix MSample for the sample is then solved from Mout.

Experimentally measured Mueller matrices for a chiral highly scattering turbid medium (scattering coefficient μs=0.6mm−1, anisotropy factor g=0.91, glucose concentration of 5 M, and thickness of 10 mm) with forward and backward scattering geometries reported by Manhas et al.13 were analyzed to demonstrate the applicability of reciprocal polarization imaging for the recovery of the optical rotation of multiple scattering chiral media. This medium is nonbirefringent, and the apparent linear retardance is attributed to light scattering.13 The linear retardance from the Lu–Chipman decomposition of the backscattering Mueller matrix is multiplied by 1/2 in Table 2 because it accounts for both forward and backward paths. The reciprocal polar decomposition shows an increase in the linear retardance δ and optical rotation α at a similar rate in the backward versus forward scattering geometries, which is consistent with the larger linear retardance and optical rotation in the backward scattering geometry owing to the increased path lengths for the detected photons. By contrast, the Lu–Chipman decomposition of the backscattering Mueller matrix yields an erroneous optical rotation in the backscattering geometry (see Table 2).

In addition, the total optical path of the backscattered photons in the chiral turbid medium is estimated to be 44.6 mm by simulating the detection conditions (∼30deg away from the exact backscattering direction and ∼20-deg collection angle) via the electric field Monte Carlo method.46 The glucose concentration C is computed by C=α/[α]λT/d ([α]λT, where α and d are the specific rotation, optical rotation, and optical path), with [α]λT=45.6deg·mL·g−1·dm−1 taken from the literature47^,48 at the wavelength λ=633nm. The glucose concentration C obtained by reciprocal polar decomposition is 4.91 M (α=0.157rad, d=22.3mm), which is very close to the value of 4.82 M (α=0.069rad, d=10mm) obtained from the Lu–Chipman decomposition of the Mueller matrix measured in the forward geometry and the ground truth (5 M). The Lu–Chipman decomposition of the backscattering Mueller matrix failed, and an incorrect glucose concentration C of 1.06 M was obtained via the same estimated total optical path.

Biographies of the authors are not available.