1.

Introduction

Light scattering and absorbing are two main processes of the interaction between light and biological tissues. Generally, the scattering in most of biological tissues can degrade the penetration depth and the image contrast using optical methods. Compared with other techniques, polarization imaging can suppress contributions from multiple scattered photons, and improve the imaging quality for superficial tissues. Moreover, multidimensional description-based Stokes vectors or Mueller matrices can expand the potential of tissue characterization.

Since Bickel confirmed the effectiveness to extract useful information from biological materials by combining polarization and light scattering,¹ many polarization parameters (differential polarization,² the degree of polarization,^3,4 Mueller matrix,^5–8 etc.) and polarization scattering models (sphere-birefringence model⁹ and sphere-cylinder-scattering model^10–12) have been put forward to describe the distinctive pathological features. However, for certain biological tissues containing pigment like melanin or chromatophores like chloroplast, the inherent absorbing effect in ambient media or on scatterers make the polarized light propagation in such absorptive media different.

For the absorption effect in ambient media, Li et al. proposed a skin model containing a top isotropic layer, a bottom anisotropic layer, and an absorbing medium to explain the contrast mechanism of polarization imaging for melanoma.¹³ Dmitry et al. proposed an approximate expression to describe the dependence of the degree of residual linear polarization on the absorption of scattering media.¹⁴ Swami et al. studied how the absorption of the scattering medium containing spherical scatterers and absorbing medium affects the depolarization.¹⁵ For absorption on the scatterers, Kienle et al. considered the imaginary part of the scatterers’ complex refractive index for multiple scattering of polarized light propagation in turbid media, and evaluated the whole angle-dependent Mueller matrix by comparing results of polarization sensitive radiative transfer solution with Maxwell theory.¹⁶ Mishchenko et al. employed the numerically exact superposition T-matrix method to demonstrate that the increasing absorption of particles diminishes and nearly extinguishes certain optical effects such as depolarization and coherent backscattering.¹³

Our previous work has developed an optical scattering model called the sphere-cylinder-birefringence model (SCBM) to simulate and simplify complicate and anisotropic biological tissues.¹² Two sets of polarization parameters extracted from the Mueller matrix transformation and Mueller matrix polar decomposition (MMPD) techniques have been used to test this model and explain some pathological changes.^16–21 However, considering the possible absorption in real biological tissues, it is necessary to include the absorbing effect in this model and explore the consequent change of these polarization parameters. This paper studies the absorption in ambient media and the induced impact on polarization parameters; absorption on the scatterers will be discussed in our further research. A Monte Carlo program was used to trace the polarization status of polarized photons scattered by the spheres and cylinders and propagating in a medium including absorption and birefringent effects. Experiments were carried out on forward scattering Mueller matrix measurements of samples containing polystyrene microspheres, well-aligned glass fibers, and absorbing ink solution. Both experiments and simulation results indicate how the absorption effect during the light transmission affects the MMPD parameters.

2.

Theory

2.3.

Experiment Setup

The experiment setup for forward scattering Mueller matrix measurement is shown in Fig. 2. A 3-W LED is used as the light source with its center wavelength of 650 nm. After beam expansion and collimation, the incident light passes through a set of a linear polarizer and quarter wave plate and could be modulated into six types of polarization states: horizontal linear (H), vertical linear (V), 45-deg linear (P), 135-deg linear (M), and right/left circular (R/L). The scattering light exiting from samples is examined by the analyzer, which is also a set of quarter wave plate and linear polarizer, and finally is collected by a lens and recorded by a CCD (Q-imaging Retiga Exi, 12-bit). For each incident polarization state, six components of the forward scattering light are detected through the analyzer. The Mueller matrix can be calculated through the 36 raw images accordingly:

Eq. (8)

$$\begin{gathered} M=\left(\begin{array}{cccc}{m}_{11}& m_{12}& m_{13}& m_{14}\\ m_{21}& m_{22}& m_{23}& m_{24}\\ m_{31}& m_{32}& m_{33}& m_{34}\\ m_{41}& m_{42}& m_{43}& m_{44}\end{array}\right) \\ =\frac{1}{2}\left(\begin{array}{ccc}\mathrm{HH}+\mathrm{HV}+\mathrm{VH}+\mathrm{VV}& \mathrm{HH}+\mathrm{HV}-\mathrm{VH}-\mathrm{VV}& \mathrm{PH}+\mathrm{PV}-\mathrm{MP}-\mathrm{MM}\\ \mathrm{HH}-\mathrm{HV}+\mathrm{VH}-\mathrm{VV}& \mathrm{HH}-\mathrm{HV}-\mathrm{VH}+\mathrm{VV}& \mathrm{PH}-\mathrm{PV}-\mathrm{MH}-\mathrm{MV}\\ \mathrm{HP}-\mathrm{HM}+\mathrm{VP}-\mathrm{VM}& \mathrm{HP}-\mathrm{HM}-\mathrm{VP}+\mathrm{VM}& \mathrm{PP}-\mathrm{PM}-\mathrm{MP}+\mathrm{MM}\\ \mathrm{VR}+\mathrm{HR}-\mathrm{LL}-\mathrm{RL}& \mathrm{VL}+\mathrm{HR}-\mathrm{HL}-\mathrm{VR}& \mathrm{ML}+\mathrm{PR}-\mathrm{PL}-\mathrm{MR}\end{array}\begin{array}{c}\mathrm{RH}+\mathrm{RV}-\mathrm{LH}-\mathrm{LV}\\ \mathrm{RH}-\mathrm{RV}-\mathrm{LH}+\mathrm{LV}\\ \mathrm{RP}-\mathrm{RM}-\mathrm{LP}+\mathrm{LM}\\ \mathrm{RR}+\mathrm{LL}-\mathrm{LR}-\mathrm{RL}\end{array}\right). \end{gathered}$$

In Eq. (8), the first letter of Mueller elements represents input polarization state, and the second letter represents the states of polarization analyzer. In experiments, we define a reference coordinate system with its $x$ axis parallel to the optical platform and $y$ axis perpendicular to it. As shown in Figs. 1 and 2, the surface of sample is set to $x-y$ plane, and the incident light passes through the sample along $z$ axis.

Fig. 1

Schematic of SCBM.

Fig. 2

The forward scattering Mueller matrix measurement setup schematic. LED, light source; L1, L2: lenses; P1, P2: lenses; QW1, QW2: quarter wave plate, the sample position corresponding to Fig. 1 is shown in the dashed frame.

The measured anisotropic and absorbing samples contain polystyrene microspheres and well-aligned glass fibers immersed in absorbing ink solution. The diameter and refractive index of polystyrene microspheres are $1\mu \mathrm{m}$ and 1.59, respectively. The glass fibers, with the diameter of $10\mu \mathrm{m}$ and the refractive index of 1.547, are arranged in alignment and wrapped neatly around the metal frame, which is immersed in the middle of the ink solution. The ink solution is configurable for concentration and is filled in a $5\mathrm{cm}\times 0.5\mathrm{cm}\times 3\mathrm{cm}$ cuvette. Different concentrations of ink solution represent different absorbing coefficients. The MMPD method is used to characterize the polarization properties of samples and investigate the influence of the absorption effect on corresponding parameters.

3.

Results and Discussion

To study the change of full polarization information with the increasing absorption effect, all Mueller matrix elements are simulated using our Monte Carlo polarization scattering simulation program and SBM anisotropic tissue model. Figure 3 shows the spatial distribution spectrum patterns of all Mueller elements. Here, different color curves represent different absorption coefficients. The vertical axes of the subplots represent the analyzed polarized intensity, whose unit is equal to the calculated photon number during the polarized light scattering process. The lateral axis is the spatial location of photon emission, whose value represents the pixel sequence number of the simulated detection plane where the incident spot center is on pixel 100.

Fig. 3

Spatial distribution patterns of Mueller elements under different optical absorptions based on Monte Carlo simulations of SCBM: (a) SCBM with a sphere-cylinder ratio of 10:50; (b) SCBM with a sphere-cylinder ratio of 30:30; and (c) normalized patterns of (a).

Figures 3(a) and 3(b) show two SCBM models with a sphere-cylinder ratio of 10:50 and 30:30, respectively, and Fig. 3(c) shows normalized processed patterns using the same model as Fig. 3(a). We focus on the optical absorption induced polarization characterization for an anisotropic medium. The proportion of cylindrical scatterers is a typical factor to describe the tissue anisotropy according to our previous work,⁴ so we study the influence of the anisotropy of scattering media using the two cases of different sphere-cylinder ratios [Figs. 3(a) and 3(b)]. Furthermore, in Fig. 3(c), by normalization, we remove the scattered intensity change due to absorption and make the polarization-related phenomena clear.

From this figure, we can see that most of the elements can be influenced by the interstitial absorption except for $m_{13}$, $m_{14}$, $m_{31}$, and $m_{41}$, and the tissue model with a high anisotropy will be affected more by comparison of Figs. 3(a) and 3(b). After normalization, we can see that the relative change of nondiagonal elements with the increasing absorption is not so obvious, and the value of diagonal elements on all the spatial locations can be uniformly decreased gradually by absorption. Generally, the additional absorption phenomena in tissue models can cause the scattering intensity change of light with different incident polarization states, implying diattenuation behavior during the light scattering. Also, the apparent change of diagonal elements and the $m_{34}$ and $m_{43}$ indicate the influence of absorption on depolarization and phase retardance processes.

In our previous study, we pointed out that, for SCBM, spherical scatterers only contribute to depolarization, but cylindrical scatterers contribute to diattenuation, depolarization, and partial retardance (the rest comes from birefringence of the medium).²¹ Now we consider the influence of the absorbing coefficient of the surrounding medium by MMPD parameters. Figures 4(a)–4(c) show simulation results, respectively, of the diattenuation, depolarization, and linear retardance with the increasing absorption. In this SCBM, the diameters of spheres and cylinders are 1 and $10\mu \mathrm{m}$, their refractive indices are 1.59 and 1.547. The birefringence of the ambient medium remains $2\times {10}^{-6}$. Considering various tissue anisotropy, we simulated three models marked by different colors in Fig. 4 with different ratios of spherical scattering coefficient and cylindrical scattering coefficient: 0:60, 10:50, and 30:30, respectively. Figures 4(b) and 4(c) clearly point out the negative correlation between the depolarization and the linear retardance parameters with the absorption coefficient. On the contrary, the diattenuation values due to cylinders show a positive correlation with absorption. Moreover, the enhanced diattenuation and the weakened retardance seem more obvious for the scattering model with a higher anisotropy. We use polystyrene microspheres and glass fibers to mimic spherical and cylindrical scatterers for the experimental verification, shown in Figs. 4(d)–4(f). When preparing the polyacrylamide gel, for the stability of the gelation process, we can employ only diluted ink solution to introduce the absorption effect of the surrounding media. As for glass fibers, we wrap them in alignment around a metal frame and then immerse them in the middle of the microsphere solution containing the ink solution. Compared with simulations, the linear retardance is smaller because we cannot simultaneously introduce absorption and birefringence into the surrounding medium in experiments. However, the results still reveal the similar regularity that the increasing absorption enhances the diattenuation but depresses retardance and depolarization.

Fig. 4

MMPD results: (a)–(c) Monte Carlo simulation of SCBM and (d)–(f) experiments of sphere-cylinder scattering model.

For SCBM, we also can modify the tissue anisotropy by the spatial orientation distribution of cylinders. We assume that the orientation of cylindrical scatterers follows a Gaussian distribution function and the FWHM of this function can be used to describe the order of alignment. A bigger FWHM means a more disordered distribution of cylinders, which can significantly reduce the anisotropy of system. So we can simulate the influence on MMPD characters of the absorption coefficient with different orientation distributions of cylinders, as shown in Figs. 5(a)–5(c), where the FWHM of the cylinder distribution function is set to 5 deg, 10 deg, and 20 deg, and their main orientations are along with the direction of $x$ axis shown in Fig. 1. The scattering coefficients of spheres and cylinders are, respectively, 10 and $50{\mathrm{cm}}^{-1}$. Their diameters are, respectively, 1 and $10\mu \mathrm{m}$. The birefringence of ambient medium is set to $2\times {10}^{-6}$. Similarly with Fig. 4, for one thing, these results confirm again that the absorption effect in the ambient medium can surely enhance the diattenuation but reduce depolarization and linear retardance. For another, the influence of the optical absorption effect also varies with different degrees of tissue anisotropy. In Figs. 5(a) and 5(c), the change of diattenuation and linear retardance is clearer for the red line with the smallest FWHM value meaning a more orderly arrangement of cylinders. In order to exclude the distinctiveness of the above simulation results on absorption caused by the main orientation angle of cylinders, we set other three different orientation angles in the following simulations. As shown in Figs. 5(d)–5(f), the red, green, and blue lines, respectively, represent the simulation with the main orientation angle of 0 deg, 30 deg, and 45 deg between the axial direction of cylinders and $x$ axis. It can be seen that the qualitative impact of absorption effects on decomposition parameters remains the same.

Fig. 5

MMPD results: (a)–(c) Monte Carlo simulation of SCBM with different FWHM of cylinder orientation distribution function while the main orientation remains the direction of $x$ axis. (d)–(f) Monte Carlo simulation of SCBM with different main orientation angles of cylinders while the FWHM of the orientation distribution function remains 5 deg.

Next, we consider another anisotropic factor in our SCBM tissue model, i.e., the birefringence effect in the ambient medium. As shown in Fig. 6, we change the absorption coefficient in our SCBM models with different birefringence of 0, $2\times {10}^{-6}$, $4\times {10}^{-6}$ colored with red, green, blue markers in sequence. The diameters of spheres and cylinders are, respectively, 1 and $10\mu \mathrm{m}$, and the scattering coefficient of spheres and cylinders are, respectively, 10 and $50{\mathrm{cm}}^{-1}$. The FWHM of cylinder orientation distribution function remains 5 deg. Here the red line with $\mathrm{\Delta }n=0$ shows the linear retardance totally produced by cylinder scattering; the retardance difference among different color results originates from the birefringence induced retardance. Consistent with the above figures, the optical absorption can cause decreased retardance, however, the parallel colored lines in Fig. 6 imply that the influence of absorption on retardance parameter is independent with the birefringence effect in the surrounding medium, and the decrease of the shown retardance curves is due to the depressed scattering process by absorption.

Fig. 6

MMPD results of SCBM with different birefringence in ambient medium.

Combined with the above three figures, we can find that the additional optical absorption do have impact on polarization processes and apparently depress the depolarization and linear retardance, however, enhance the biattenuation. By comparison, the influence of absorption varies with the degree of scattering anisotropy in the simulated tissue models and SCBM models with more cylindrical elements or a highly ordered orientation seem sensitive to the optical absorption. The existence of birefringence and the main orientation angle do not affect how MMPD parameters change with the absorption coefficient, and conversely the optical absorption cannot change the contribution of birefringence on tissue anisotropy.

The following discussions try to explain the impact of absorption effect in ambient medium on scattering induced diattenuation, depolarization, and linear retardance. First, for diattenuation, we start from its definition as

In SCBM, when the orientation of the cylinder is set along $x$ axis, we can approximate $D$ as the following expression considering the value of matrix element $m_{13}$ and $m_{14}$ are much smaller than that of $m_{12}$:

Here “HI” represents the intensity of scattered light in the case of the horizontal polarized incident light, and “VI” represents the intensity of scattered light in the case of the vertical polarized incident light. Using one set of the above data as an example, in which the scattering coefficients of spheres and cylinders are 10 and $50{\mathrm{cm}}^{-1}$, the FWHM of cylinders orientation distribution function is 5 deg, and the birefringence of ambient medium is $2\times {10}^{-6}$, we investigate how the $|\mathrm{HI}-\mathrm{VI}|/(\mathrm{HI}+\mathrm{VI})$ vary with the absorption coefficient. As shown in Fig. 7(a), $|\mathrm{HI}-\mathrm{VI}|/(\mathrm{HI}+\mathrm{VI})$ show the same changing trend as the red line in Fig. 5(a), which confirms that $D$ reflects the scattering difference for two orthogonal linear polarization incidences in this model, like Fig. 7(b). Theoretically for isotropic scattering media without irregular scatterers and optical anisotropy, the scattered photon behaviors of orthogonal linear polarization incidence light are equal. However, the existence of anisotropic factors in the tissue model, like cylindrical scatterers, can cause polarization response during scattering sensitive to the incident polarization status. As shown in Fig. 7(c), we can see that the phase function of cylinder depends on the polarization state of incident light. When the scattering angle ranges from 0 deg to 45 deg, the horizontal polarized light has nearly the same phase function as the vertical polarized light. While when the scattering angle ranges from 45 deg to 90 deg, the phase function of the horizontal polarized light is clearly bigger than that of the vertical polarized light, thus indicating a bigger lateral scattering possibility for horizontal polarized incident light. Correspondingly during the scattering process, the horizontal polarized incident light may undergo more lateral scattering steps and has a longer optical path when scattered by cylindrical scatterers. As a consequence, the scattered behavior of horizontal polarized light will be more affected than the vertical polarized light by the absorption effect in the ambient medium, which leads to the enhanced difference of HI and VI and finally produces a bigger diattenuation.

Fig. 7

(a) and (b) The function of $|\mathrm{HI}-\mathrm{VI}|/(\mathrm{HI}+\mathrm{VI})$ and HI/VI with absorption coefficient and (c) the scattering phase function of cylinder.

In order to explain the impact of absorption on scattering induced depolarization and linear retardance, we analyze the photon behaviors in SCBM with various absorption extents shown in Fig. 8. As we can see, the enhanced absorption depresses the multiscattered photons more and thus results in decreased total scattering numbers. For the absorption effect in the surrounding media, the absorption can cause additional light attenuation between one and the next scattering event. So the multiple scattered photons mean more absorption probability and show a clearly declined proportion with the increase of the absorption setting value. For the SCBM tissue model and the forward detection scheme, not only is the depolarization phenomena related to the multiple scattering, but also the phase retardance is contributed by the scattering of anisotropic cylinders, so the depressed multiple scattering by absorption can produce a lower depolarization and retardance.

Fig. 8

The distribution of the scattering number of the forward scattering photons.

Finally, we pay attention to the cylindrical scatterers with the special orientation angles not on the surface plane, as shown in Fig. 9. It can be seen that the change trend of diattenuation and depolarization items [Figs. 9(a) and 9(b)] seems similar to Fig. 5. So, the discussion about the influence of optical absorption on these two processes is still valid even for the fibrous microstructures not perpendicular to the incident light and not on the surface plane. However, as shown in Fig. 9(c), the phase retardance increases along with the increasing absorption. These simulation results seem different from the case of cylinders perpendicular to incident light, but actually, we can provide an explanation based on our previous research.²⁴ Concretely, the total retardance depends on the coupling of two parts of retardances, respectively, from birefringence and cylindrical scatterers, closely related to the interception angle between the orientation of birefringence and cylinders. Based on the above results and discussions in this paper, the cylinder induced retardance can be decreased due to the reduced scattering times by absorption. But for the case of Figs. 9(a)–9(c), the main orientation angle of cylinders is on $x-z$ plane and the interception angle from the birefringence axis ($x$ axis) is equal to 90 deg, where the retardance induced by birefringence and cylinders will cancel each other first, as demonstrated in Ref. 24. It means that even less contribution from cylinders could restrain such canceling out between different retardance sources and cause the enhanced total retardance, which can explain the increased simulated retardance in Fig. 9(c) in this paper with the increasing optical absorption.

Fig. 9

MMPD results: (a)–(c) Monte Carlo simulation of SCBM with different orientation angles on $x-z$ plane and (d) schematic diagram of SCBM model.

4.

Conclusion

In this paper, we focus on the influence of absorption effect on Mueller matrix. By Monte Carlo simulation on our previously proposed SCBM and forward scattering experiments on phantoms containing polystyrene microspheres, well-aligned glass fibers, and ink solution, we analyze the changing trend of Mueller matrix and MMPD parameters with the absorption coefficient. Both simulation and experiment results can indicate that optical absorption in the medium can cause an enhanced diattenuation evaluation and reduce the scattering induced depolarization and linear retardance. The increasing diattenuation can be explained by the different scattering phase functions of cylindrical scatterers for horizontal polarized incident light and vertical polarized incident light. The weakened depolarization and linear retardance should be due to the decrease of multiple scattering photons. However, the linear retardance caused by the intrinsic birefringence in the medium is insensitive to the absorption effect. These simulations and experiments imply that optical absorption affects the change regularity of polarization state of polarized light in tissues, and also show possible potential of polarization parameters used in tissue characterization involving absorption effect.