## Abstract

A novel spectroscopic Mueller matrix system has been developed and explored for both *fluorescence and elastic scattering* polarimetric measurements from biological tissues. The 4 × 4 Mueller matrix measurement strategy is based on sixteen spectrally resolved (λ = 400 - 800 nm) measurements performed by sequentially generating and analyzing four elliptical polarization states. Eigenvalue calibration of the system ensured high accuracy of Mueller matrix measurement over a broad wavelength range, either for forward or backscattering geometry. The system was explored for quantitative *fluorescence* and *elastic scattering* spectroscopic polarimetric studies on normal and precancerous tissue sections from human uterine cervix. The fluorescence spectroscopic Mueller matrices yielded an interesting diattenuation parameter, exhibiting differences between normal and precancerous tissues.

© 2013 Optical Society of America

## 1. Introduction

Mueller matrix polarimetry has received considerable attention for the characterization of a wide range of optical media for numerous practical applications [1–6]. The Mueller matrix (a 4 × 4 matrix) is the transfer function of an optical system in its interactions with polarized light and the polarization properties of the medium are encoded in its various elements [2,6]. There are three principle characteristics of interest that a Mueller matrix can encode: depolarization, diattenuation and retardance [3,6]. Motivated by the fact that extraction and quantification of these three medium polarization characteristics can potentially serve as useful biological metric, there has been significant recent interest in exploring quantitative polarimetry for biological tissue characterization, assessment and diagnosis [4–14]. However, unlike optically clear medium, quantitative polarimetry in complex turbid medium like biological tissue is severely confounded by multiple scattering effects and simultaneous occurrence of several polarization and scattering events [6]. Multiple scattering within tissue leads to strong depolarization of light, creating a large depolarized source of noise which can inhibit the detection of the weak remaining information-carrying polarization signal. Thus, many traditional polarimetry systems are poorly suited for these applications. Moreover, complicated nature of the polarization effects in tissue, including simultaneous multiple effects in presence of multiple scattering, contribute in a complex interrelated way to the Mueller matrix elements, hindering analysis, quantification and unique interpretation of the tissue polarimetry data [6]. The challenges are thus to maximize measurement sensitivity to obtain polarization signals in presence of strong depolarization, and to decouple the individual contributions of simultaneously occurring polarization effects. A number of researchers are thus pursuing innovative solutions to these challenges [4–6, 10]. Quantitative tissue polarimetry research has therefore two major directions – *(i)* to develop highly sensitive experimental Mueller matrix polarimetry system and *(ii)* to develop novel *inverse* analysis methods for extraction and quantification of the constituent intrinsic medium polarimetry characteristics from the ‘lumped’ system Mueller matrix. With regard to the second issue, various kinds of Mueller matrix decomposition methods (product decompositions including the polar decomposition and its other variants, differential matrix decomposition etc.) have been developed and validated in complex tissue-like turbid medium [14–21]. Some inroads have also been made to address the first issue, to develop highly sensitive experimental Mueller matrix polarimetry systems [6,12, 22–30].

One possible method for improving the sensitivity of the measurement procedure is the use of polarization modulation with synchronous detection [6, 12, 23–27]. Many sensitive detection schemes using polarization modulator have therefore been explored [6]. Snapshot Mueller matrix polarimeter, which employs wavelength polarization coding and decoding for highly sensitive, instantaneous measurement of all the sixteen Mueller matrix elements, is another important development in modulation-based polarimetry [26]. While the modulation based approaches yield the desired high sensitivity (capable of detecting very weak polarization retaining signal transmitted / backscattered from tissue), these are poorly suited for spectroscopic (at multiple wavelengths) applications and applications involving large area imaging, since they usually involve synchronous detection [6]. Yet, in many applications involving tissue characterization, the wavelength dependence of the polarization parameters (depolarization, diattenuation and retardance) or their spatial maps are extremely useful. Therefore there has been growing need to develop alternative (non-modulation-based) approaches for sensitive, spectrally and spatially (imaging) resolved Mueller matrix measurements. Indeed, dc measurement approaches involving sequential measurements with different combinations of polarization state generator (PSG) and polarization state analyzer (PSA) units have been explored for this purpose [22]. For example, polarimetric systems based on this principle using liquid crystal variable retarders have been developed for imaging applications [28, 29].

In this paper, we present a novel spectral Mueller matrix system for both *fluorescence and elastic scattering* polarimetric measurements from depolarizing scattering medium such as tissues. In this approach, four required elliptical polarization states are sequentially generated and analyzed by using a PSG unit (comprising of a fixed linear polarizer with its axis oriented at horizontal position followed by a rotatable achromatic quarter wave retarder) and a PSA unit (similar arrangement of fixed linear polarizer with its axis oriented at vertical position and rotatable achromatic quarter wave retarder, but positioned in a reverse order). Sixteen spectrally resolved measurements are combined to construct the sample Mueller matrix. Eigenvalue calibration [28–30] of the system was performed to yield the exact values of the system PSG and PSA matrices over a broad wavelength range (λ = 400 - 800 nm), which also ensured high accuracy of Mueller matrix measurement. With some modifications, the system was successfully used to record fluorescence spectroscopic Mueller matrices in addition to elastic scattering spectral Mueller matrices. To the best of our knowledge, this is the first report on fluorescence spectroscopic Mueller matrix measurement from biological tissue. Initial exploration of such measurements yielded interesting differences in both the fluorescence and elastic scattering spectral diattenuation parameters (derived from respective Mueller matrices) between normal and precancerous tissue sections of human uterine cervix.

The paper is organized as follows. In Section 2, we provide specifics of the signal analysis strategy for construction of sample Mueller matrix. The Eigenvalue calibration method is briefly summarized in this section. A very brief account of the polar decomposition of Mueller matrix (which has been used to analyze the Mueller matrices) is also provided in this section. Section 3 describes the experimental Mueller matrix polarimetry systems. Both the elastic scattering and fluorescence spectroscopic Mueller matrix systems for either forward or backward scattering geometry are presented in this section. The results of the eigenvalue calibration of the spectral Mueller matrix system, and the results of initial exploration of fluorescence and elastic scattering spectral Mueller matrix measurements and analysis on human cervical tissue sections, are presented in Section 4. The paper concludes with a discussion on the salient advantages of the developed spectral Mueller matrix system, and on the potential of fluorescence spectroscopic Mueller matrix polarimetry for tissue characterization.

## 2. Theory

#### 2.1. Strategy for constructing 4 × 4 spectral Mueller matrix

Our Mueller matrix measurement strategy is based on sixteen measurements performed by sequentially generating and analyzing four elliptical polarization states using a polarization state generator (PSG, comprising of a fixed linear polarizer with its axis oriented at horizontal position, followed by a rotatable quarter wave retarder) and a polarization state analyzer (PSA, similar arrangement of fixed linear polarizer with its axis oriented at vertical position and rotatable quarter wave retarder, but positioned in a reverse order) unit respectively. For the combination of a horizontal linear polarizer and a linear retarder (having retardance $\delta $and orientation angle of fast axis $\theta i$with respect to the horizontal axis) in the PSG unit, the output Stokes vector can be written as [6]

The four required elliptical polarization states can thus be generated by sequentially changing the orientation angle of the retarder (θ_{i}^{1}, θ_{i}^{2,} θ_{i}^{3,} θ_{i}^{4}) and the corresponding PSG output can be represented by **W**, a 4 × 4 matrix whose column vectors are the four generated Stokes vectors $W(\theta i)$ (of Eq. (1)) incident on the sample. Similarly, after sample interactions, the PSA results can also be described by a transformation matrix for the combination of a general linear retarder (retardance$\delta $, orientation angle$\theta o$) and a fixed linear polarizer (vertical) as

Note that the PSA unit is followed by an intensity based detector, which records the total intensity falling on it (the first element of the Stokes vector coming out of the PSA system). Thus, only the first row of the matrix of Eq. (2), is required to represent the PSA unit basis states

_{o}

^{1}, θ

_{o}

^{2,}θ

_{o}

^{3,}θ

_{o}

^{4}). The corresponding 4 × 4 analysis matrix

**A**can be formed by writing the four basis states ($A(\theta o)$ of Eq. (3) for four different values of θ

_{o}) as the row vectors [28–30].

The sixteen intensity measurements required for the construction of full Mueller matrix can be grouped into the measurement matrix ** M_{i}**, which is related to PSG/PSA matrices

**and**

*W***, as well as the sample Mueller matrix**

*A***by [6, 28]**

*M***is the 16 × 16 matrix given by the Kronecker product of**

*Q***with the transpose of**

*A***:Once the exact forms of the system**

*W***W**and

**A**matrices are known, all the sixteen elements of the sample Mueller matrix (${M}^{vec}$ written in column vector form) can be determined from the sixteen measurements (${M}_{i}^{vec}$) using Eq. (5). The 4 × 4 sample Mueller matrix can then be obtained by rearranging the elements of${M}^{vec}$.

Note that the choice of the four sets of orientation angles of the retarders (θ_{i} and θ_{o}) in the PSG and the PSA units are important for recording stable sample Mueller matrix. Specifically, optimum selection of the basis states of **W** and **A** matrices are required to have minimal propagation of errors in the determined Mueller matrix. Theoretical optimization of θ_{i} and θ_{o} values was therefore performed. One basic criterion for choosing the polarization basis states, is to maximize the value of the determinant of the 16 × 16 matrix *Q*, since *Q* is required to be invertible [6]. Note that in practice, the invertability of the matrix *Q* is not binary (i.e., either invertible or not), but rather presents a continuum in which some matrices are more invertible than others. Optimization of *Q* was therefore performed by maximizing its determinant for varying orientation angles of the quarter wave retarders (achromatic retarders with δ = π/2).Thus, optimized orientation angles for the two retarders were found to be 35°, 70°, 105° and 140°. For simplicity, the orientation angles of the retarders at the PSG and the PSA unit were kept the same (θ_{i} = θ_{o}) in the optimization process. A more general criterion for having minimal error propagation has previously been suggested to be based on optimization of the so-called ‘condition number’, defined as the ratio of the smallest to the largest singular values of the individual square matrices ** W** and

**[29]. The above chosen angles were therefore verified further with this more rigorous approach based on singular value decomposition [29]. Note that the actual forms of the**

*A***and**

*W***matrices of the experimental set-up might differ from the theoretical ones derived above. Specifically, they might also show some wavelength variations, even though the quarter wave retarders used in the experimental set-up are achromatic (over λ = 400 – 800 nm). The experimentally determined**

*A***and**

*W***matrices (determined via Eigenvalue calibration method, discussed subsequently) were thus tested to meet the optimization criteria discussed above.**

*A*#### 2.2. Eigenvalue calibration method

Calibration of the spectral Mueller matrix polarimeter is an essential issue to ensure accuracy of acquiring sample Mueller matrices. This is particularly important for measurement over a broad wavelength range, since the actual polarization state generator (** W**) and the analyzer (

**) matrices of the experimental system may differ from the theoretical ones, and additionally they might also exhibit wavelength response. Fortunately, there exists a method to address this issue, the so-called Eigenvalue calibration method (ECM), which can determine the exact nature of the system**

*A***and**

*W***matrices and their wavelength response [28–30]. This involves measurements on a set of chosen reference or calibrating samples. The details of this method can be found elsewhere [28, 30]. Here, we provide an outline of the various steps of this method. Usually the calibration is done on samples having Mueller matrices (**

*A***) of the form of diattenuating retarder. Let us call these set of measurements as**

*M***. The ECM method requires another set of measurements without any sample (blank), denoted by**

*B***The essential steps of ECM are as follows.**

*B*._{o}The above sets of measurements will yield

From Eq. (7), another two sets of matrices**and**

*C***are constructed such that one of them is independent of**

*C*^{/}**and the other is independent of**

*A***Clearly, the matrices**

*W***,**

*C***and**

*C*^{/}**has the same eigenvalues. Thus the eigenvalues of the Mueller matrix**

*M***can be determined from the eigenvalues of either of the matrices**

*M***and**

*C***The actual Mueller matrix**

*C*^{/}.**M**of the reference sample can then be constructed using the determined four set of eigenvalues, through a series of algebraic manipulation, as detailed in references [28, 30].

Once the Mueller matrix **M** of the reference sample is determined, in the next step, the system ** W** and

**matrices are determined from Eq. (8) using**

*A***M**. First the generator

**matrix is determined by solving the following Eq.**

*W***,**a linear operator

**K**is built such a way that its only eigenvector associated with null eigenvalue is

**satisfying the Eq.**

*W (*

*– K W*_{16}_{× 1}

**[28, 30]. Note that the eigenvalues of**

*= 0)***are all different from zero except**

*K*_{$\lambda 1$}, which should actually be null and practically as close to zero as possible ($0\approx \lambda 1<<\lambda 2<\lambda \mathrm{3......}<\lambda 16$). Thus, the obtained eigenvector (

*W*_{16}_{× 1}) corresponding to the smallest eigenvalue of the matrix

**, is written back in 4 × 4 matrix form to obtain the system polarization state generator matrix**

*K***. Once**

*W***is determined, the polarization state analyzer matrix**

*W***is eventually calculated as**

*A*The ECM was used to determine the exact nature of the system ** W** and

**matrices and their wavelength response. This was performed using four sets of measurements for two different types of calibrating reference samples, Glen Thompson linear polarizer (as diattenuator,**

*A**d*= 1, over λ = 400 – 800 nm) and quarter waveplate (δ = π/2 at 632.8 nm, as retarder). Measurements from the linear polarizer were taken for two different orientations of polarization axis - 28°, 73°, and from the quarter waveplate for two different orientations of the fast axis - 23°, 68° respectively. These sets of orientation angles of the diattenuator and retarder have previously been shown to be optimum for eigenvalue calibration. Since, ECM provides the actual experimental

**and**

*W***matrices, without any prior modeling of the PSG and PSA units, many effects which can modify**

*A***and**

*W***values (misalignment in orientation or configurations, positioning artifacts, beam divergence, non-ideal optical characteristics of the polarizing optics etc.), are automatically taken care off. Moreover, the ratio of the smallest to the largest eigenvalue (λ**

*A*_{min}/λ

_{max}) of the 16 × 16 matrix

**, is an indicator of the accuracy or sensitivity of the system, which was thus determined for our spectral Mueller matrix polarimeter. The performance of the system was further tested by quantitatively determining the medium polarization parameters, diattenuation d, linear retardance δ (and their wavelength response) of standard optical elements. The results of the eigenvalue calibration and the initial exploration of the spectral Mueller matrix polarimeter to record spectral Mueller matrices from biological tissues are presented subsequently, in Section 4.**

*K*#### 2.3. Inverse analysis of Mueller matrix using polar decomposition

Polar decomposition of Mueller matrix, originally developed by Lu and Chipman [15], has been widely explored for inverse analysis of Mueller matrices to extract the constituent polarization properties from any unknown system. This approach involves sequential factorization of a given Mueller matrix **M** into the product of three ‘basis’ matrices

**M**contains information on the effects of linear and circular diattenuation,

_{D}**M**

_{R}accounts for linear and circular retardance (or optical rotation) effects, and

**M**includes the effect of any depolarization present in the medium [6, 15]. This product decomposition and its other variants have recently been validated in complex scattering media such as tissues, which exhibit simultaneously polarization and scattering effects [15–21]. Once decomposed, the four individual polarization medium properties, namely, diattenuation (

_{Δ}*d*), depolarization coefficients (

*Δ*), linear retardance (

*δ*), and circular retardance or optical rotation (

*ψ*, circular retardance = 2 × optical rotation), can be determined from the corresponding basis matrices as [6, 15]

## 3. Experimental methods

We have developed two separate experimental systems based on the measurement scheme described in Section 2.1. A schematic of the first one, for angle resolved elastic scattering spectral Mueller matrices, is shown in Fig. 1(a). The second one is capable of Mueller matrix measurement in the exact backscattering configuration (Fig. 1(b)). In addition to recording elastic scattering spectral Mueller matrices, this set-up was also used to record fluorescence spectroscopic Mueller matrices in the exact backscattering configuration (results are presented subsequently in Section 4). Either of the systems comprise of a Xe lamp (USB4000, Ocean Optics, USA) as excitation source (for elastic scattering spectral measurements), a polarization state generator (PSG) and a polarization state analyzer (PSA) unit to generate and analyze the required polarization states, coupled to a spectrometer for spectrally resolved (wavelength λ = 400 - 800 nm) signal detection. The PSG unit comprises of a fixed linear polarizer (P_{1}, LPVIS100, Thorlabs, USA) with its axis oriented at horizontal position, followed by a rotatable achromatic quarter wave retarder (Q_{1}, AQWP05M-600, Thorlabs, USA) mounted on a computer controlled rotational mount (PRM1/M-27E, Thorlabs, USA). The retarders used in the system are achromatic quarter waveplates over λ = 400 – 800 nm, has a retardance accuracy of ~λ/60. Note that these sets of achromatic retarders used in the PSG and the PSA units, are different from the other type of quarter wave retarder (WPQ10M-633, δ = π/2, at λ = 632.8 nm, retardance accuracy of ~λ/300) used in the eigenvalue calibration as a reference sample (discussed in the section 2.2).

The sample-scattered light collected and collimated using an assembly of lenses, then passes through the PSA unit, and is finally recorded using a spectrometer. The PSA unit essentially consists of a similar arrangement of fixed linear polarizer (P_{2}, oriented at vertical position) and a rotatable achromatic quarter wave retarder (Q_{2}), but positioned in a reverse order. A series of sixteen measurements are performed by sequentially changing the orientation of the fast axis of the quarter wave retarders of the PSG unit and that of the PSA unit, to the four optimized angles 35°, 70°, 105° and 140° (as discussed in Section 2.1). The spectra corresponding to the sixteen combinations of the *PSG* and *PSA* are recorded using a fiber optic spectrometer (HR2000, Ocean Optics, USA) in set-up - 1 and using a CCD spectrometer (Shamrock imaging spectrograph, SR-303i-A, ANDOR technology, USA) in set-up-2.

In order to record fluorescence spectroscopic Mueller matrices, the 405 nm line of a diode laser (PE.BDL.405.50, Pegasus, Shanghai, China) is used as excitation source in set-up 2 (exact backscattering configuration). The fluorescence spectra (λ_{em} = 450 – 800 nm) are recorded for sixteen combinations of the *PSG* and *PSA*. The set-up was completely automated using Labview so that sixteen spectrally resolved measurements can be performed with relative ease.

Note that an important advantage of constructing Mueller matrix according to this scheme is that, this is independent of the source and detector polarization responses since the two linear polarizers P_{1} and P_{2} are always fixed at horizontal and vertical polarization states respectively. This is important particularly, for spectrally resolved (employing a spectrometer) Mueller matrix measurements, since usually spectrometers are associated with complex polarization response (owing to the presence of grating). Never-the-less, eigenvalue calibration was performed following the process discussed in Section 2.2, to yield the exact nature of the system polarization state generator $W(\lambda )$ and polarization state analyzer $A(\lambda )$ matrices. These along with the sixteen spectrally resolved measurements (${M}_{i}(\lambda )$) were used to construct the spectral Mueller matrices $M(\lambda )$ (Section 2.1 Eqs. (4)-(6)). For constructing the elastic scattering spectral Mueller matrices, the $W(\lambda )$and$A(\lambda )$matrices for varying excitation and emission wavelengths are used (λ_{ex}, λ_{em} = 400 - 800 nm). For constructing fluorescence spectroscopic Mueller matrices (in set-up 2), generator matrix $W(\lambda )$ for a fixed excitation wavelength (λ_{ex} = 405 nm) and analyzer matrix $A(\lambda )$for varying emission wavelengths (λ_{em} = 450 – 800 nm) are used.

The tissues were histopathologically characterized (CIN or dysplasia Grade I, II and III) biopsy samples of human cervical tissues, obtained from G.S.V.M. Medical College and Hospital, Kanpur, India. The unstained tissue sections (thickness ~5μm, lateral dimension ~4 mm × 6 mm) were prepared on glass slides. The standard method employed was tissue dehydration, embedding in wax, vertical sectioning (containing both epithelial and connective tissue regions) under a rotary microtome and subsequent de-waxing. The normal counterparts were obtained from the adjacent normal area of the resected tissues.

## 4. Results and discussion

#### 4.1. Results of eigenvalue calibration

For eigenvalue calibration, we used separately a linear polarizer (for two different orientations of polarization axis - 28°, 73°) as diattenuator calibration sample and a quarter waveplate (at 632.8 nm, for two different orientations of the fast axis - 23°, 68°) as the retarder calibrating samples. The derived system polarization state generator $W(\lambda )$and polarization state analyzer$A(\lambda )$ matrices and their wavelength variation are shown in Fig. 2(a) and 2(b) respectively.

In order to evaluate the overall system performance of the spectral Mueller matrix set-up, the Mueller matrices corresponding to the series of calibrating samples and those without any sample (blank) were determined using the $W(\lambda )$and$A(\lambda )$matrices. A useful way to test the accuracy of the system is to look at the expected null elements (which are expected to vanish) of the recorded Mueller matrix of any reference sample [28, 29]. The off-diagonal elements of Mueller matrix for blank (no sample) and the expected null elements of the calibrating diattenuator (polarizer) and retarder (quarter waveplate) were indeed found to be quite small (~0.01).Typical illustrative example of such measurement is shown in Fig. 3(a), where the wavelength variations of the expected null elements (M_{12}, M_{13} and M_{14} elements) of the Mueller matrix for a quarter wave retarder (orientation angle of fast axis from the horizontal −23°) are shown. The inset shows the element M_{44}, which ideally should vanish at λ = 632.8 nm (M_{44} = cos δ, δ = π/2 at 632.8 nm). The small value of the M_{44} (~0.01) element at 632.8 nm, and the other elements over the entire spectral range (~0.01), ensured reasonably high accuracy of the spectral Mueller matrix system.

In order to further illustrate the efficacy of the Mueller matrix system for quantitative characterization of the spectral polarimetric properties, the wavelength variation of the polarization parameters, diattenuation *d* (λ), linear retardance *δ (*λ*)*, optical rotation *ψ* (λ) and depolarization coefficients *Δ* (λ), were determined from polar decomposition (following Section 2.3, Eqs. (11) and (12)) of Mueller matrices recorded from the calibrating diattenuator and the retarder samples. Typical wavelength variation of the extracted *d* (λ) and *δ (*λ*)* for the calibrating linear polarizer (orientation of polarization axis - 28°) and the quarter wave retarder (orientation of fast axis - 23°) sample are shown in Fig. 3(b). As can be seen from the figure, the derived values are in agreement with that expected for an ideal polarizer (*d* ~1) over the spectral range 500 nm – 800 nm. The values for the other derived polarization parameters (*δ (*λ*)*,*ψ* (λ) and *Δ* (λ)), which are expected to be zero for an ideal polarizer, were also quite negligible (≤ 0.01). Similarly, the determined value for linear retardance *δ =* 1.57 radian at 632.8 nm reasonably agrees with the expected value (*δ* = π/2), and the corresponding wavelength variation also exhibits the expected 1/λ behavior. Finally, as discussed in Section 2.2, the ratio of the smallest to the largest eigenvalue (λ_{min}/λ_{max}) of the 16 × 16 matrix ** K**, is an indicator of the accuracy or sensitivity of the system. The wavelength variation of λ

_{min}/λ

_{max}is shown in Fig. 3(c). The considerably small magnitudes of the parameter λ

_{min}/λ

_{max}(~10

^{−3}) confirm high accuracy of the system in acquiring Mueller matrices over a broad wavelength range.

The results of Eigenvalue calibration presented above are for the Mueller matrix system operating in the exact backscattering configuration (set-up 2, Fig. 1(b)). Similar calibration studies were also conducted on the system operating in the forward detection geometry (set-up 1, Fig. 1(a), results not shown here), which also yielded similar accuracy of spectral Mueller matrix measurements. Following successful calibration (results presented above and other results on a series of calibrating samples), the developed set-ups were initially explored for biomedical tissue polarimetry - for quantitative fluorescence and elastic scattering spectral Mueller matrix polarimetry on normal and precancerous human cervical tissues, as illustrated below.

#### 4.2. Results of fluorescence and elastic scattering spectroscopic Mueller matrices from tissue sections

Polarization properties of auto-fluorescence (fluorescence from endogenous tissue fluorophores) have previously been explored for discriminating between normal and cancerous tissues [4]. These studies have shown that measurement of depolarization of auto fluorescence from tissue can provide additional diagnostic information [31]. These studies were based on conventional measurement of fluorescence polarization anisotropy (ratio of the polarized fluorescence component to the total intensity, a measure of depolarization) with linear polarization excitation [32]. The fluorescence polarization anisotropy is known to arise due to several causes like, the random orientation of the fluorophore molecules, rotational diffusion of fluorophores, the radiation less energy transfer among fluorophores etc., and accordingly this carries useful information on the fluorophore and its local environment [32]. In tissue, depolarization of fluorescence is additionally influenced by multiple scattering effects, encoding the signature of the medium scattering properties in the measured fluorescence polarization anisotropy [4, 31]. Note that when the emitting fluorophores exhibit all the three basic polarimetry effects, namely, diattenuation, depolarization and retardance, the resulting fluorescence polarization anisotropy determined using the conventional two orthogonal linear polarization state measurements, should reflect ‘lumped’ polarization effects. The analysis / interpretation get considerably simplified in case of isotropic fluorescence emission from randomly oriented fluorophores, where the fluorescence polarization anisotropy can be interpreted as mainly due to the depolarization effects (arising due to randomization of the excitation and the emission dipoles) [32, 33]. However, when the fluorescence emission takes place from fluorophores having anisotropically oriented / organized molecular structures, the other polarimetry characteristics may also have significant contributions [33]. In such situation, measurement and analysis of the complete 4 × 4 fluorescence spectral Mueller matrix may yield additional structural information on the organization, orientation of the fluorophore molecules. Note that collagen is known to be the major fluorophore in connective tissue (stroma), which possess such anisotropically oriented / organized molecular structures [34]. The anisotropic structure primarily arises due to the cross-linking between individual collagen molecules, which results in formation of micro-fibrils and fibers. It is well recognized that progression of epithelial precancers and cancers involves defective interactions between epithelial cells and the underlying stroma, and alterations in stromal biology may precede and stimulate neoplastic progression in pre-invasive disease [35]. Quantification of intrinsic fluorescence polarimetry characteristics (from fluorescence Mueller matrix) of collagen present in stroma may thus prove to be useful for probing defective epithelial-stromal interactions, as signature of precancer.

In order to explore this possibility, fluorescence spectral Mueller matrices (with 405 nm excitation) were recorded from human cervical tissue sections (vertically sectioned tissue slices, thickness ~5 μm) having different grades of precancers. The spectral Mueller matrices were recorded from the tissue sections in the exact backscattering configuration using set-up 2. Figure 4 shows typical fluorescence spectral Mueller matrix recorded from the connective tissue region (stroma) of a Grade I precancerous tissue section. Several observations are at place. The M_{11}(λ) element of the Mueller matrix represent the unpolarized fluorescence spectra. As noted, this is characterized by broad spectral feature with a peak around 500 nm, which is characteristic of collagen emission with 405 nm excitation [34, 36]. The spectral intensities of the diagonal elements, M_{22}(λ),M_{33}(λ) and M_{44}(λ) are considerably weaker as compared to M_{11}(λ). Among these, the M_{44}(λ) element appears to be the weakest. The considerably weak magnitude of M_{44}(λ) is characteristics of fluorescence emission from molecules exhibiting very weak optical activity (circular anisotropy) [33]. In general, the diagonal elements of a pure depolarization Mueller matrix represent the polarization preserving components for horizontal/ vertical, + 45° / −45°linear polarization and circular polarizations (left and right) respectively [6]. Low magnitudes of all these elements thus underscore the overall strong depolarizing nature of fluorescence emission from tissue. Indeed, the values of depolarization coefficient *Δ*(λ) (derived using Eq. (12b) of Section 2.3) were found to be quite high (*Δ* ≥ 0.9) over the entire spectral range (not shown here). Such high values of Δ are primarily due to the intrinsic causes of depolarization of fluorescence (random orientation, rotational diffusion and radiation-less energy transfers among fluorophores etc.) [32]. Even though, multiple scattering is not expected to be a significant issue for thin tissue sections (thickness ~5 μm), some additional contribution of depolarization of fluorescence due to scattering at the excitation and the emission wavelengths, is unavoidable [4, 31]. Indeed, the corresponding elastic scattering spectroscopic Mueller matrices recorded from the same tissue sections showed non-zero value of *Δ* (not shown here), indicating non-negligible contribution of scattering-induced depolarization effect for the backscattering collection geometry. Quantitative description of scattering-induced depolarization in tissue would necessitate use of vector radiative transfer theory or its other variants [4–6, 10, 37]. However, since the main finding and emphasis of the present study is the observation of an interesting fluorescence spectral diattenuation effect (as illustrated in greater details below), such quantitative modeling of scattering-induced depolarization has not been attempted here.

Next, the off-diagonal elements, M_{23}(λ) / M_{32}(λ), M_{24}(λ) / M_{42}(λ),and M_{34}(λ) / M_{43}(λ) also show low intensities. While the elements M_{34}(M_{43}) and M_{24}(M_{42}) represent linear retardance-*δ* effects for horizontal/ vertical and + 45° /-45° linear polarizations respectively,

Since, the fluorescence process is associated with strong de-phasing (randomization of phase), the resulting emission does not exhibit any appreciable phase retardation effects [33], and accordingly, the derived values of *δ* (λ) and *ψ(*λ*)* (Eq. (12c) and (12d)) were also found to be quite negligible (*δ,ψ ≤* 0.01 radian, within the experimental errors of the system). Interestingly, the elements M_{12}(λ) / M_{21}(λ),and M_{13}(λ) / M_{31}(λ) show appreciable intensity values and they also preserve the spectral characteristics of the fluorescence emission. The magnitudes of the remaining M_{41}(λ) / M_{41}(λ) elements, on the other hand, are relatively weaker. Analogous to the retardation effects, while the elements M_{12}(M_{21}) and M_{13} (M_{31}) represent linear diattenuation effect for horizontal/ vertical and + 45° /-45° linear polarizations respectively, the elements M_{41}(M_{41}) represent circular diattenuation (between left and right circular polarization states). Accordingly, the values for the diattenuation parameter *d*(λ) (derived using Eq. (12a) of Section 2.3) were found to be appreciable (subsequent results shown in Fig. 5), with major contribution arising from linear diattenuation (negligible circular diattenuation contribution).

Some qualitative understanding on the origin of such interesting fluorescence spectral diattenuation effects can be gained from a phenomenological approach of modeling fluorescence Mueller matrix as a sequential product of Mueller matrices for each of the contributing processes in the resulting emission as [33]

Here, the matrix **M _{A}** includes the polarizing transfer function of the absorption process at the excitation wavelength (due to the ground molecular state), the matrix

**M**accounts for the effect of ground state to the excited state transformation and the remaining matrix

_{G→E}**M**includes the effects of the excited state on the resulting fluorescence emission [33]. In case of isotropic molecules, where the dipolar approximation is valid,

_{EM}**M**of Eq. (13) can be assumed to be a diattenuating depolarizer matrix [32, 33]. Whereas, the depolarization effects included in

_{G→E}**M**arise from various processes such as the random orientation of the fluorophore molecules, rotational diffusion of fluorophore, radiation less energy transfer among fluorophores etc., the diattenuation effect is characteristic of dipolar absorption / emission and has geometric origin [32, 33]. Note that this diattenuation effect encoded in the M

_{G→E}_{12}/ M

_{21}elements of the matrix

**M**is maximum at a scattering angle of θ = 90° and vanishes at exact forward or backward scattering angles (θ = 0° / 180°, as should be the case for our experimental geometry of backscattering Mueller matrix measurement - θ = 180°) [33]. On the other hand, the matrices

_{G→E}**M**and

_{A}**M**may possess intrinsic diattenuation and retardance properties of the ground molecular state and the excited state respectively [33].These matrices can be simplified further to represent as pure diattenuator matrices, if one assumes that fluorescence process is associated with strong randomization of the phase information. In such situation, these matrices would carry information on the anisotropic (linear and circular) absorption and emission properties of fluorophores arising from their anisotropically oriented / organized molecular structures [33]. Note that in addition to these, for fluorophores embedded in a scattering medium (like tissue), there would be scattering-induced depolarization effects also, which could be incorporated by an additional depolarization matrix. Never-the-less, the resulting fluorescence Mueller matrix

_{EM}**M**will possess diattenuation property in addition to depolarization. The observed appreciable intensities of M

_{F}_{12}(λ) / M

_{21}(λ),and M

_{13}(λ) / M

_{31}(λ) elements of the fluorescence Mueller matrices (and accordingly significant value of linear diattenuation) from tissue are thus manifestation of the intrinsic linear anisotropy (linear dichroism) of the emitting fluorophores (collagen). The weak magnitudes of M

_{41}(λ) / M

_{41}(λ), on the other hand, indicate that the emitting molecules exhibit very little / or no circular anisotropy (circular dichroism).

In Fig. 5(a), we show the spectral variation of the fluorescence diattenuation parameter *d _{FL}*(λ)for normal and precancerous tissue sections. Results are once again shown for the connective tissue region of the tissue sections. Here, the mean values of

*d*(λ) for normal (three samples) and precancerous (five samples of pooled Grade I, Grade II and Grade III tissues) tissues are plotted. As previously mentioned, significant magnitude of

_{FL}*d*(λ) and its spectral variation (which resembles characteristic fluorescence spectra of collagen) clearly arise due to the (linear) anisotropic orientation / organization of collagen structures in either the normal or precancerous connective tissues. Importantly, the magnitude of

_{FL}*d*(λ) is observed to be reduced for the precancerous tissue specimens. This might arise due to the fact that progression of precancer is accompanied by destruction of the collagen cross-links, resulting in a loss of the anisotropic organization [35]. In order to comprehend this further, the elastic scattering spectral Mueller matrices were also recorded in the same geometry from identical connective tissue regions of the cervical tissue sections. The corresponding mean values of the derived spectral diattenuation parameter

_{FL}*d*(λ) for normal and precancerous connective tissues are shown in Fig. 5(b). Elastic scattering of light from anisotropically organized scattering structures is also expected to yield diattenuation effects, as is evident from the Fig. Once again, the qualitative trends in the fluorescence and elastic scattering diattenuation are similar, the values for

_{ES}*d*(λ) decreases in precancer, indicating randomization of the fibrous collagen structures [35].

## 5. Conclusions

A novel spectral Mueller matrix system has been developed, which is capable of acquiring both *fluorescence and elastic scattering* Mueller matrices over a broad wavelength range. The performance of the system has been calibrated using Eigenvalue calibration method that also yielded the exact values of the system polarization state generator and polarization state analyzer matrices over the entire wavelength range. The accuracy of the system for measuring sample Mueller matrices were tested on various calibrating reference samples. Following successful evaluation, the system was used to acquire *fluorescence* and *elastic scattering* spectroscopic Mueller matrices from biological tissue sections. Initial exploration of the system for quantitative fluorescence spectral polarimetric studies on normal and precancerous human cervical tissue sections has yielded promising results. Specifically, the polar decomposition analysis on fluorescence Mueller matrix from connective tissue region yielded an interesting spectral diattenuation effect, which was attributed to the anisotropic molecular structure of collagen. Note that the results presented here are on thin tissue sections. The potential of this fluorescence spectral diattenuation parameter for probing precancerous alterations in connective tissue however, needs to be rigorously investigated from intact tissues. Our current studies are thus directed towards – (i) development of a more encompassing approach for the analysis / interpretation of fluorescence Mueller matrix and (ii) comprehensive evaluation and exploration of the quantitative fluorescence Mueller matrix polarimetry (and the derived intrinsic fluorescence polarimetry characteristics) for the detection of precancerous changes from intact tissues. The novel spectral Mueller matrix polarimetry system is also being explored for other interdisciplinary applications [38].

## Acknowledgments

This work was supported by Indian Institute of Science Education and Research (IISER) - Kolkata, an autonomous teaching and research institute funded by the Ministry of Human Resource Development (MHRD), Govt. of India. The authors thank Prof. Asima Pradhan, Department of Physics, IIT Kanpur, India for fruitful discussions and Dr. Asha Agarwal, G.S.V.M. Medical College and Hospital, Kanpur, India for providing the histopathologically characterized tissue samples.

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