1.

Introduction

The randomness of tissue structures results in fast depolarization of polarized light propagating in tissues; thus, for many optical imaging methods, such as diffusion optical tomography, polarization effects are usually ignored.^1–6 However, in certain tissues and cell structures (transparent tissues, such as eye tissues, cellular monolayers, mucous membrane, and superficial skin layers), the degree of polarization of transmitted or reflected light remains measurable even when the tissue has a considerable thickness.^2–29 Information about tissue structure can be extracted from the registered depolarization degree of initially polarized light, the transformation of the polarization state, or the appearance of a polarized component in the scattered light.

In regard to the practical implication of the polarization method, its gating ability to select ballistic photons from diffuse ones gives rise to simplified schemes of optical medical tomography compared with time-resolved methods and provides enhanced image contrast and resolution, as well as additional information about the structure, absorption inclusions, and blood supply in tissues.^{8–10,15,23–29}

In addition to the long and successful history of polarimetry as a comprehensive tool for the study of different materials,^30–36, its earlier achievements included characterization of transparent bacterial^37–41 and red blood cell (RBC)⁴² suspensions and examination of clear tissues, such as the eye cornea,^43–46 lens,⁴⁷ and retinal nerve fibers;^48,49 recently, this old research field was significantly driven by innovations in polarization measuring technologies, computing, and perspectives of widespread biomedical applications.^{5–29,50–95} Among the driving forces for improvement of measurement accuracy on the background of multiple scattering and simultaneous existence of different polarization effects are polarization gating in CW and pulsed modes, utilizing the full description of polarized light by amplitude ($2\times 2$, Jones) or intensity ($4\times 4$ or $3\times 3$ Mueller matrices) and the Mueller matrix decomposition technique, application of the concept of Poincaré sphere, and exploiting the polarization-sensitive optical coherence tomography (PS-OCT).

In this review-tutorial paper, we present the fundamentals of polarized light interaction with biological tissues and some of the recent key polarization optical methods that made possible the quantitative studies essential for biomedical diagnostics, as well as examples and a brief review of the cutting-edge PS measuring techniques for early pathology diagnostics. We also demonstrate the substantial improvement of tissue intrinsic polarization properties by elimination of multiple scattering effects at optical clearing.

2.

Fundamentals of Polarized Light Scattering

A polarized light at incidence on an object can be presented as two orthogonal linear polarization components of the incident light field parallel (${\overrightarrow{E}}_{||i}$) and perpendicular (${\overrightarrow{E}}_{\perp i}$) to the scattering plane as is shown in Fig. 1. This figure illustrates the geometry of the scattering of light by an object (a particle representing the elementary component of a tissue or a cell). Here, the incident light beam, ${\overrightarrow{S}}_0$, is parallel to the $z$-axis, and $\theta$ and $\varphi$ are the scattering angles in the scattering plane and in the plane perpendicular to the scattering plane, respectively.⁹⁶ Within the detector plane located at a distance $r$ from the origin along the vector ${\overrightarrow{S}}_1$, two orthogonal polarization components, ${\overrightarrow{E}}_{||s}$ and ${\overrightarrow{E}}_{\perp s}$, of the scattered light create a specific polarization state depending on amplitudes and phase shifts between components.

Fig. 1

Geometry of the scattering of light by a particle located at the origin.⁹⁶ The polarized incident light beam (${\overrightarrow{S}}_0$) is parallel to the $z$-axis. Two orthogonal linear polarization components of the incident light field are presented as vectors ${\overrightarrow{E}}_{||i}$ and ${\overrightarrow{E}}_{\perp i}$ in parallel and perpendicular to the scattering plane, respectively. $\theta$ and $\varphi$ are the scattering angles in the scattering plane and in the plane perpendicular to the scattering plane, respectively. A detector is located at distance $r$ from the origin along the vector ${\overrightarrow{S}}_1$, where two orthogonal polarization components ${\overrightarrow{E}}_{||s}$ and ${\overrightarrow{E}}_{\perp s}$ of scattered light are coming.

The transformation of an arbitrary polarized light (linear, circular, or elliptical) by a scattering particle can be described using a linear relationship between the incident and the scattered field components,^{5,15,22,30–38,96–101}

The complex numbers $S_1$ through $S_4$ are the elements of the amplitude scattering matrix ($S$-matrix) or Jones matrix. They each depend on the scattering and azimuthal angles $\theta$ and $\varphi$, and contain information about the scatterer. Both amplitude and phase must be measured to quantify the amplitude scattering matrix. Polarimetric experimental methods are available to determine the Jones matrix elements of transparent optical materials without sign ambiguity.¹⁰² For scattering tissues, the direct measurements of Jones matrix elements can be done using a two-frequency Zeeman laser, which produces two laser lines with a small frequency separation ($\sim 250\mathrm{kHz}$) and orthogonal linear polarizations,³⁸ or by the OCT technique.^15,103

However, more often, Stokes or Mueller polarimeters based on intensity measurements of polarized light are used.^{5,15,21,22,30–38,87,91,92,96–101,104–108} In that case, light of an arbitrary polarization can be represented by four numbers known as the Stokes parameters, $I,Q,U$, and $V$ ($I^2=Q^2+U^2+V^2$), where $I$ refers to the intensity of the light, and the parameters $Q,U$, and $V$ represent the extent of the horizontal linear, 45 deg linear, and circular polarization, respectively.^22,30,96,107 In polarimetry, the Stokes vector $\mathbf{S}$ of a light beam is constructed based on six flux measurements obtained with different polarization analyzers in front of the detector,

From the Stokes vector, the degree of polarization ($P$), degree of linear polarization ($P_L$), and the degree of circular polarization ($P_C$) are derived as^5,15,22,34

The Stokes vector for a partially polarized beam ($P<1$) can be considered as a superposition of a completely polarized Stokes vector ${\mathbf{S}}_P$ and an unpolarized Stokes vector ${\mathbf{S}}_U$, which are uniquely related to $\mathbf{S}$ as follows:^33,109

The polarized portion of the beam represents a net polarization ellipse traced by the electric field vector as a function of time. The ellipse has a magnitude of the semimajor axis $a$, semiminor axis $b$, orientation of the major axis ${\varphi }_o$ (azimuth of the ellipse) measured counterclockwise from the $x$ axis,

The ellipticity is the ratio of the minor to the major axes of the corresponding electric field polarization ellipse and varies from 0 for linearly polarized light to 1 for circularly polarized light.

In the far field, the polarization of the scattered light from an object (Fig. 1) is described by the Stokes vector ${\mathbf{S}}_s$ connected with the Stokes vector of the incident light ${\mathbf{S}}_i$ by the matrix equation ${\mathbf{S}}_s=M·{\mathbf{S}}_i$, where $M$ is the normalized $4\times 4$ scattering matrix (intensity or Mueller matrix),⁹⁶

Elements of the light-scattering matrix (LSM) depend on the scattering angle $\theta$, the wavelength, and the geometrical and optical parameters of the object. Element $M_{11}(\theta )$ is what is measured when the incident light is unpolarized; its angular dependence is the phase function of the scattered light; it is much less sensitive to chirality and long-range structure than some of the other matrix elements.^38,96 Element $M_{12}$ is obtained by measuring the total scattered intensity for a horizontally linearly polarized incoming beam and subtracting the total scattered intensity for a vertically linearly polarized incoming beam from this; $M_{22}$ displays the ratio of depolarized light to the total scattered light (a good measure of the scatterers’ nonsphericity); $M_{34}$ displays the transformation of the 45 deg obliquely polarized incident light to circularly polarized scattered light (which is a unique characteristic for various tissues and cells); the difference between elements $M_{33}$ and $M_{44}$ is a good measure of the scatterers’ nonsphericity as well.

The polarization elements used in polarimeters, such as polarizers, retarders, and depolarizers, as well as the objects under study, such as tissues and cell layers or suspensions, have three general polarization properties: depolarization, retardance, and diattenuation.^{18–22,30–37,109} A typical object displays some amount of all three properties. Depolarization describes the coupling by an object of incident polarized light into depolarized light in the exiting beam. Depolarization typically occurs when light transmits through or scatters from tissue. Depolarization is intrinsically associated with scattering and a loss of coherence in the polarization state.¹⁰⁹ A depolarization coefficient $\mathrm{\Delta }$ can be defined as the fraction of unpolarized power in the exiting (scattering) beam when the polarized light is incident; $\mathrm{\Delta }$ is generally a function of the incident polarization state.

Retardance is the phase change an object introduces between its two orthogonal polarization eigenstates. For materials with a linear birefringence, which is defined as the refractive index difference between wave components propagating along the slow ($n_1$) and fast ($n_2$) axes, $\mathrm{\Delta }n=n_1–n_2$, light propagation over a distance $d$ introduces a linear retardation $\delta$ expressed in radians as^22,109

A circular retardance appears when two orthogonal polarization eigenstates are counter-rotating circular ones. The origin of the eigenstates is a circular birefringence, which is observed in media lacking any mirror symmetry, like solutions of chiral molecules where only one enantiomer is present.²² When a linearly polarized wave propagates in such a medium along a distance $d$, its polarization remains linear, but it is rotated by an angle $\psi$ called optical rotation or optical activity,

Diattenuation (dichroism) arises when the intensity transmittance of an object is a function of the incident polarization state.¹⁰⁹ The diattenuation $D_A$ of an object is defined in terms of the maximum, $T_{\max }$, and minimum, $T_{\min }$, intensity transmittances,

In terms of the Stokes vector and Mueller matrix (LSM) elements, these basic parameters can be expressed as follows. If the degree of polarization $P$ of a light field remains at unity after transformation by an optical system, this system is nondepolarizing; otherwise, the system is depolarizing, with the depolarization being a function of input polarization,^105,106

In the scattering media with a low absorption, as many tissues are, the origin for dichroism could be anisotropy in the scattering abilities. In that case, the imaginary part $n^{\prime \prime }$ of the complex refractive index of material $n$ will also be determined by light losses associated with scattering,

In nonscattering materials, the imaginary part $n^{\prime \prime }$ is defined only by light losses due to absorption.

In general, all 16 elements of the LSM are nonzero; thus, a minimum of 16 measurements involving both linear and circularly polarized light are required to completely characterize the polarization properties of the sample. For nondepolarizing systems, the number of independent LSM elements cannot be more than seven since both the Mueller matrix and the Jones matrix can represent the system; therefore, in that case, 16 LSM elements are related via nine equations.^98,99,110 Linear and circular diattenuation (dichroism), $D_L$ and $D_C$, can be presented, respectively, as^109,110

3.

Tissue Structure and Anisotropy

Healthy cornea and lens of a human eye are highly transparent for visible light because of their ordered structure and the absence of strongly absorbing bands of endogenous chromophores.^5,6,43–47 The human cornea has a thickness of $\sim 0.5\mathrm{mm}$ with $\sim 90\%$ of the thickness being a stroma. Stroma is composed of several hundred successively stacked layers of lamellae, varying in thickness (0.2 to $0.5\mu \mathrm{m}$ depending on the tissue region)^43,44,111 [see Figs. 2(a)–2(c)]. These lamellae, consisting of collagenous fibrils, are immersed into an amorphous ground (interstitial) substance containing water, glycosaminoglycans, proteins, proteoglycans, and salts. Fibrils in the human cornea have a uniform diameter of $\sim 30.8\pm 0.8\mathrm{nm}$ with a periodicity close to two diameters, i.e., $55.3\pm 4.0\mathrm{nm}$, and a high degree of short-range spatial order—a hexagonal quasi-crystal [see Figs. 2(a) and 2(b)]. Within each lamella, all fibrils are nearly parallel with each other and with the lamella plane. The intermolecular spacing within fibrils is of $1.63\pm 0.10\mathrm{nm}$.¹¹² Therefore, the corneal stroma has at least three levels of structural organization.

Fig. 2

The electron images of [(a) and (b)] the human cornea ($\times \mathrm{32,000}$) and (d) sclera ($\times \mathrm{18,000}$): (a) $\mathbf{K}$ is the keratocyte, (b) magnified image of middle lamella of (a); two circles show a short-range spatial ordering in the form of a hexagonal quasi-crystal; (c) the model of lamellar-fibrillar structure of the corneal stroma; (d) scleral collagen fibrils, which display various diameters; however, locally, they could be quasi-ordered (two circles show a short-range spatial ordering in the form of a hexagonal quasi-crystal).^{43,44,111,113–115}

The sclera is a dense, white, fibrous membrane containing three layers: the episclera, the stroma, and the lamina fusca. The stroma is the thickest layer of the sclera. In the scleral stroma, the collagen fibrils exhibit a wide range of diameters, from 25 to 230 nm.^111,116 The average diameter of the collagen fibrils increases gradually from $\sim 65\mathrm{nm}$ in the innermost part to $\sim 125\mathrm{nm}$ in the outermost part of the sclera. The mean diameter is $\sim 100\mathrm{nm}$ and the mean distance between fibril centers is equal to $\sim 285\mathrm{nm}$.⁵ Collagen intermolecular spacing is similar to that in the cornea, in particular for bovine sclera, $1.61\pm 0.02\mathrm{nm}$.¹¹² The fibrils are arranged in individual bundles parallel to the scleral surface, but more randomly than in the cornea. Within each bundle, the groups of fibrils are separated from each other by large empty lacunae randomly distributed in space [see Fig. 2(d)].¹¹¹ In spite of the scleral collagen fibrils displaying various diameters, locally they could be quasi-ordered with a short-range spatial ordering in the form of a hexagonal quasi-crystal.

The eye lens is also a tissue for which the short-range spatial ordering is of crucial importance.^5,47 A healthy human lens contains $\sim 60\%$ water and 38% proteins and consists of many fiber cells.¹¹⁷ However, the tissue’s predominant dry components are $\alpha -,\beta -$, and $\gamma$–crystallines—structural proteins, which provide $\sim 33\%$ of the total weight of the lens.¹¹⁸ The water-soluble $\alpha$–crystalline is a major component that has a spherical shape with a diameter of $\sim 17\mathrm{nm}$. Age-related biochemical endogenous processes in the organism and exogenous environmental-related stresses induce lens opacity caused by an increase of light scattering and/or pigmentation due to oxidizing and cross-linking of lens proteins. Formed spherical aggregates are variable in size (100 to 250 nm) and occur in clusters that create potential scattering centers.¹¹⁷

One more example of complex anisotropic tissue structure is the retinal nerve fiber layer (RNFL),^46,48,49,119 which is formed by the expansion of the fibers of the optic nerve and comprises bundles of unmyelinated axons that run across the surface of the retina. The two prominent cytoskeletal elements of axons are microtubules and neurofilaments.¹²⁰ Axonal microtubules are long tubular polymers of the protein tubulin with an outer diameter of $\sim 25\mathrm{nm}$, an inner diameter of $\sim 15\mathrm{nm}$, and a length of 10 to $25\mu \mathrm{m}$; neurofilaments are stable protein polymers with a diameter of $\sim 10\mathrm{nm}$.

As it follows from electron images of human reticular dermis, presented in Fig. 3, this tissue can also be locally considered as a short-range spatially ordered fibrillar system.¹²¹ Anisotropy of light propagation in human skin was proven experimentally.¹²² Skin aging and pathology may dramatically change its structure, i.e., anisotropy. For example, scar tissue typically occupies areas of fibrous tissue that replace normal tissue after injury [Figs. 3(c) and 3(d)]. Another example is benign or malignant tumor, which is abnormally growing tissue sometimes strongly vascularized and having the tendency to spread to other tissue layers, making them more random.

Fig. 3

Area of reticular dermis showing (a) the collagen fibrils in the normal skin and (c) nodule of a hypertrophic scar (electron image, $\times \mathrm{10,000}$); (b) and (d) the close-up view of a region in (a) and (c) at higher magnification power ($\times \mathrm{105,000}$), respectively. Arrows in (b) and (d) show interstitial interfibrillar material; in (b) and (d), the circles show a short-range spatial ordering in the form of a hexagonal quasi-crystal.¹²¹

The bone, tooth enamel, dentin, and cementum, as well as tendon and cartilage belong to hard tissues.¹¹⁹ Most of them are connective tissues, which typically consist of cells rarely distributed in an amorphous mucopolysaccharide matrix in which there are also varying amounts of tissue fibers, mainly collagenous fibers. The apatite or hydroxyapatite (HAP) natural crystals, ${\mathrm{Ca}}_5\mathrm{OH}{({\mathrm{PO}}_4)}_3$, are the major components of some hard tissues. The dental enamel consists of 87 to 95% and bone of 50 to 60% of HAP crystals; the rest is water and proteins. Tooth is a hard body composed of dentin surrounding a sensitive pulp and covered on the crown with enamel [see Fig. 4(a)]. The enamel is the hardest tissue in a body, which is an elastic, white material containing no cells. In particular, it is an ordered array of HAP crystals surrounded by a protein/lipid/water matrix. Fairly well-oriented hexagonal HAP crystals of $\sim 30$- to 40-nm diameter and up to $10\text{-}\mu \mathrm{m}$ length are packed into an organic matrix to form keyhole-shaped interlocking enamel prisms (or rods) with an overall cross-section of 4 to $6\mu \mathrm{m}$. Enamel prisms are roughly perpendicular to the tooth surface. Because of their size, number, and high refractive index, the prisms are the main light scatterers in enamel. The enamel is translucent and grayish white in color, and it has a yellow hue due to the underlying dentin.

Fig. 4

Schematic representation of tooth structure, including dentinal channels: (a) tubules (rectangle), and optical images of human dentinal slab surface showing tubules for two different sections—(b) under some angle and (c) perpendicular to the slab surface (Axio Imager, Carl Zeiss, magnification $20\times$), done by I. V. Fedosov and N. A. Trunina; (d) the close-up view of tubules using electron scanning microscopy of human dentin (presented by R. Vilar): dentinal tubules (1), dense and homogeneous pertubular dentin (2), and less dense and less homogeneous intertubular dentin (3).

Tooth dentin is a complex structure, honeycombed with dentinal tubules, which are shelled organic cylinders with a highly mineralized shell. The tooth dentin is a hard, middle calcified, elastic, yellowish material of the same substance as bone.¹¹⁹ It is the main structural part of a tooth extending from crown to root. The dentin is composed of base material that is pierced by mineralized dentinal tubules 1 to $5\mu \mathrm{m}$ in diameter, which makes it porous [see Fig. 4(c)]. The tubules’ density is in the range of (3.0 to 7.5) $\times {10}^6·{\mathrm{cm}}^{-2}$. They contain organic components and natural HAP crystals of 2.0 to 3.5 nm diameter and up to 100 nm length, which intensively scatter light. Average dentin contains 70% HAP crystals, 20% proteins (e.g., collagen), and 10% water. It is softer than enamel but slightly harder than bone. The tubular microstructure of dentin makes it strongly anisotropic for propagating light.^123–125

The osteon, the basic bone unit with a typical length scale of $\sim 100\mu \mathrm{m}$, is formed by concentric $\sim 10\text{-}\mu \mathrm{m}\text{-}\text{thick}$ lamellae, composed of collagen fibrils that are $\sim 1\mu \mathrm{m}$ wide.¹²⁶ Fibrils in their turn consist of an inhomogeneous matrix of cross-linked type I collagen molecules, $\sim 300\mathrm{nm}$ in length and $\sim 1.5\mathrm{nm}$ in diameter, mineralized with carbonated HAP nanocrystals. Type I collagen and HAP nanocrystals are, therefore, the basic building blocks of bone at the nanoscale (1 to 300 nm).^126–128

In bone and tooth, anisotropy is due to mineralized structures originating from HAP nanocrystals, which play an important role in hard tissue depolarization and birefringence. The demineralization process is the loss of mineral from mineralized tissues, such as bone or tooth.^125–129 In tooth tissue, demineralization may lead to caries, and in bone, to osteoporosis. Evidently, measured depolarization and birefringence could be good markers of these and other pathologies.

The cartilage is a strong, resilient, skeletal tissue. There are a few types of cartilage:¹¹⁹ the hyaline cartilage is the simplest and most common form that consists of a matrix of a polysaccharide-containing protein (trachea, bronchi); the yellow fibrocartilage (elastic cartilage) contains yellow fibers in the matrix (external ear, epiglottis); and the white fibrocartilage contains white fibers in the matrix and occurs in the disks between the vertebrae. Articular cartilage is mainly composed of water, collagen fibrils, proteoglycans, and the chondrocytes with 60 to 85% of water, whereas 50 to 80% and 20 to 35% of the tissue dry weight are collagen and proteoglycans, respectively.^130–133 Chondrocytes contribute $<5\%$ of cartilage volume. Most of the collagen is of type II. The collagen fibril diameter varies from 20 to 200 nm. Articular cartilage consists mostly of three layers.^131–133 In the superficial layer (see Fig. 5), collagen fibrils are mostly oriented parallel to the cartilage surface; high collagen and lower proteoglycan content is characteristic.^130,132,133 In the transitional layer, collagen fibril orientation is more random and has a larger diameter. In the thicker deep layer (radial), collagen fibrils are oriented perpendicular to the cartilage surface and have still a larger diameter than in other layers; chondrocytes tend to align in columns (see Fig. 5, right), and the proteoglycan content is higher than that in the superficial and middle layers.¹³⁰ A thin layer of calcified cartilage separates cartilage and subchondral bone. Between the noncalcified and calcified layers lies a tidemark, which is a line seen in microscopic images at histological staining. Collagen fibers from the deep layer run through the calcified zone and are attached to the subchondral bone.

Fig. 5

The orientation of collagen fibrils in normal articular cartilage, as expounded in the classic arcade model of (a) Benninghoff and (b) cellular variations.¹³³ Collagen fibers arise from the subchondral bone where they are radially oriented, curve over to lie parallel to the cartilage surface in the superficial tangential layer, and then descend back to the underlying bone. The tangential layer is generally less than $100\mu \mathrm{m}$ thick, while the majority of the full thickness of the cartilage (1 mm or so) is occupied by the transitional and radial zones. In the superficial zone, the chondrocytes are ellipsoidal in shape, in transitional zone, they are spheroidal, and in radial, they tend to align in columns.

The tendon is a cord of white fibrous tissue. It usually attaches muscle to bones and consists mostly of parallel, densely packed collagen fibers arranged in parallel bundles interspersed with long, elliptical fibroblasts. In general, tendon fibers are cylindrical in shape with diameters ranging from 20 to 400 nm. The ordered structure of collagen fibers running parallel to a single axis makes tendon a highly optically birefringent tissue. Collagen fibrils are collagen molecules packed into an organized overlapping bundle. In their turn, collagen fibrils associated in a bigger bundle form a collagen or white fiber—the main component of white fibrous tissue. For example, diameters are in the range from $\sim 25$ to 600 nm for tendon collagen fibrils, with bigger sizes for mature tendons, $\sim 25\mathrm{nm}$ for subfibrils, and in the range from 10 to 15 nm for protofibrils (Fig. 6).¹³⁴

Fig. 6

Collagenous fibrillar structure of tendons: collagen triple helices are combined into microfibrils (or protofibrils), then into subfibrils, fibrils, fascicles, and into tendons.¹³⁴

Collagen is a tough, inelastic, fibrous protein.¹³⁵ Collagen in its turn forms white fibers in connective tissue. The tropocollagen or collagen molecule subunit is a rod $\sim 300\mathrm{nm}$ long and 1.5 nm in diameter, made up of three polypeptide strands. There is some covalent cross-linking within the triple helices and a variable amount of covalent cross-linking between tropocollagen helices to form the different types of collagen found in different mature tissues. A distinctive feature of collagen is the regular arrangement of amino acids in each of the three chains of these collagen subunits.

There are a number of collagen types which differ in their composition and structure; type I to type V are the most abundant; types I, II, III, V, VII, and XI are capable of fibril formation; their distribution in tissues is as follows: type I—dermis, bone, cornea, tendon, cartilage, vessel wall, intestine, dentin, uterine wall, fat mashwork; type II—cartilage, notochord, vitreous humor, nucleus pulposus; type III—dermis, intestine, gingiva, heart valve, uterine wall, vessel wall; types IV and VII—basement membranes; type V—cornea, placental membranes, bone, vessel wall, cartilage, gingiva; and type XI—cartilage, intervertebral disc, vitreous humor.¹¹⁹ In optical measurements, the collagen index of refraction is of importance; for example, for type I, it is at the wavelength 850 nm, $n=1.43$ (fully hydrated) and 1.53 (dry).¹³⁵

As follows from the analysis of tissue properties, many biological tissues are structurally anisotropic. Tissue birefringence results primarily from the linear anisotropy of fibrous structures, which forms the extracellular media. The refractive index of a medium is higher (the speed of light is lower here) along the length of fibers than along their cross-section. A specific tissue structure is a system composed of parallel cylinders that create a uniaxial birefringent medium with the optic axis parallel to the cylinder axes. This type of birefringence is called birefringence of form (Fig. 7). A large variety of tissues such as eye cornea, tendon, cartilage, eye sclera, dura mater, muscle, myocardium, artery wall, nerve, retina, bone, teeth, myelin, and so on exhibit birefringence. All these tissues contain uniaxial and/or biaxial birefringent structures. For example, myocardium contains fibers oriented along two different axes. It consists mostly of cardiac muscle fibers arranged in sheets that wind around the ventricles and atria. Since the refractive index along the axis of the cardiac muscle fibers is different from that in the transverse direction, tissue is birefringent.

Fig. 7

Examples of structurally anisotropic models of tissues and tissue components: (a) system of long dielectric cylinders; (b) system of dielectric plates; (c) chiral aggregates of particles; and (d) glucose (chiral molecule) as a tissue component.

Besides anisotropy of form, collagen fibrils as complex molecular structures have intrinsic linear birefringence arising due to anisotropy at the molecular scale and are fundamentally determined by the anisotropic distribution of electrical charge.¹³² The spatial length scale of the structure producing intrinsic birefringence is much smaller than the fibrillary scale at which form birefringence is produced, but both scales are smaller than the wavelength of light. Studies on type I collagen in tendon have led to the consensus that this collagen type displays intense intrinsic positive birefringence due to the quasi-crystalline arrangement of the amino-acid residues that comprise the polypeptide chains of the collagen molecule alpha chains.¹³² A crucial distinguishing feature between form and intrinsic birefringence is that the value of birefringence of form can be reduced by refractive index matching between the background material and the inclusions as intrinsic birefringence is an intrinsic property of the fibril material and cannot be altered unless gross changes in the molecular structure are induced.

Form birefringence arises when the relative optical phase between the orthogonal polarization components is nonzero for forward-scattered light. For linear structures, an increase in optical field phase delay (${\delta }_{\mathrm{oe}}$) is characterized by a difference in the effective refractive index for light polarized along, and perpendicular to, the long axis of the linear structures ($\mathrm{\Delta }{n}_{\mathrm{oe}}$). Phase retardation ${\delta }_{\mathrm{oe}}$ between orthogonal polarization components is proportional to the distance $d$ for light with a wavelength ${\lambda }_0$ traveling through the birefringent medium [see Eq. (10)],^15,132

A structure of parallel dielectric cylinders immersed in isotropic homogeneous ground substance behaves as a positive uniaxial birefringent medium [$\mathrm{\Delta }{n}_{\mathrm{oe}}=(n_e-n_o)>0$] with its optic axis parallel to the cylinder axes [Fig. 7(a)]. Therefore, an incident electrical field directed parallel to the cylinder axes will be called an extraordinary ray (e), and the incident electrical field perpendicular to the cylinder axes will be called an ordinary ray (o). The difference ($n_e-n_o$) between the indices of refraction of the extraordinary and ordinary rays is a measure of the birefringence of a medium. For the Rayleigh limit ($\text{diameter of cylinders}\ll \text{wavelength}$ $\lambda$), the form birefringence is described as¹³⁶

The experimental birefringence values for muscle, coronary artery, myocardium, sclera, skin, cartilage, and tendon are in the range from $1.4\times {10}^{-3}$ to $4.2\times {10}^{-3}$.^5,15,119 For these tissues, it is lowest for the muscle and highest for the tendon. For thermally treated tendon, it is twice as low as for intact tissue.

The diattenuation (linear dichroism) is the difference of attenuation of two waves with orthogonal polarizations traveling in an anisotropic medium, which is described by the difference between the imaginary (losses) parts of the effective indices of refraction for two orthogonal directions. Depending on the relationship between the sizes and the optical properties of the cylinders or plates [see Figs. 7(a) and 7(b)], this difference can take positive or negative values.¹⁵

The magnitude of birefringence and diattenuation is related to the density and other properties of the collagen fibers, whereas the orientation of the fast axis indicates the orientation of the collagen fibers. The densities of collagen fibers in skin and cartilage are not as uniform as in tendon, and the orientation of the collagen fibers is not distributed as orderly as in tendon. Correspondingly, the amplitude and orientation of birefringence of the skin and cartilage are not as uniformly distributed as in tendon.

In addition to linear birefringence and diattenuation, many tissue components show optical activity (circular birefringence) and circular diattenuation.^9,14,15 In complex tissue structures, chiral aggregates of particles may be responsible for tissue optical activity [see Fig. 7(c)]. The molecule’s chirality, which stems from its asymmetric molecular structure [see Fig. 7(d)], also results in a number of effects generically called optical activity. A well-known manifestation of optical activity is the ability to rotate the plane of linearly polarized light about the axis of propagation [see Eqs. (11) and (12)]. The amount of rotation depends on the chiral molecular concentration, the path length through the medium, and the light wavelength. Interest in tissue chirality studies is driven by the attractive possibility of noninvasive in situ optical monitoring of glucose in diabetic patients.^{9,14,15,17,83}

More sophisticated anisotropic tissue models can also be found. For example, the eye cornea can be represented as a system of plane anisotropic layers (plates, i.e., lamellas), each of which is composed of densely packed long cylinders (fibrils) [see Fig. 2(c)] with their optical axes oriented along a spiral.¹³⁷ This fibrillar-lamellar structure of the cornea is responsible for the linear and circular diattenuation and its dependence on the angle between the lamellas.

4.

Tissue Models

4.1.

Single Scattering

4.1.1.

Discrete particle model

Biological tissue is an inhomogeneous medium with different levels of organization that include cells, cell organelles, and inclusions, different fiber and tubular/lamellar structures (see Figs. 2Fig. 3Fig. 4Fig. 5–6). In view of the great diversity and structural complexity of tissues, the development of an adequate optical model accounting for the scattering and absorption of light is often the most complex step of a study.^1–5 Many tissues are composed of structures with a wide range of sizes and can be represented as a system of discrete scattering particles. Such models have been used to describe the angular dependence of the intensity and polarization properties of scattered radiation.^1–22,138

Biological media are often modeled as ensembles of homogeneous spherical particles with a refractive index higher than surroundings, since many cells, cell organelles, and biological macromolecules are close in shape to spheres or ellipsoids.^{2,5,96,138,139} A system of noninteracting spherical particles is the simplest tissue model. The Rayleigh and Mie theories or their combinations are basic to calculate tissue scattering properties. In particular, Mie theory rigorously describes the diffraction (elastic scattering) of light by a spherical particle. The advances of this theory account for the structures of the spherical particles, namely, the multilayered spheres and the spheres with radial nonhomogeneity, anisotropy, and optical activity.^96–101

For connective tissue composed from fiber structures, a system of long cylinders is the most appropriate model to describe light scattering. Tendon, cartilage, skin dermis, dura mater, muscular tissue, eye cornea, and sclera belong to this type of tissue formed essentially by collagen fibrils. The solution of the problem of light scattering by a single homogeneous or multilayered cylinder is also well understood.^96–101

At transport in the inhomogeneous (turbid) medium with absorption, a photon (light wave) changes its direction (wave vector) due to reflection, refraction, and diffraction on microscopic internal structures, and can be absorbed by an appropriate molecule on its way. Light scattering means a change in direction of light propagation so that its trajectory is deflected by an angle $\theta$ in the scattering plane and by an azimuthal angle $\varphi$ (0 to $2\pi$) in the perpendicular plane (Fig. 1).⁹⁶

To characterize scattering and absorption efficiency of a medium, scattering (${\mu }_s$) and absorption (${\mu }_a$) coefficients are introduced, which follow from the exponential Bouguer–Beer–Lambert law for light propagation in a tissue layer of thickness $d$.^2,5

Eq. (22)

$$I(d)=I_0\exp (-{\mu }_{t}d),$$

where $I(d)$ is the intensity of transmitted light measured using a distant photodetector with a small aperture (on-line or collimated transmittance), and $I_0$ is the incident light intensity.

Eq. (23)

$${\mu }_t={\mu }_a+{\mu }_s$$

is the interaction or total attenuation coefficient. Equation (22) is valid if scattering is not strong and only the unscattered portion of transmitted light beam (so-called ballistic photons) is detected. Such a regime could be more or less realized for thin tissue layers when absorption is high enough to eliminate multiple scattering events.

In addition to the scattering coefficient, the scattering process is also characterized by a scattering phase function—the function that describes the scattering properties of the medium and is in fact the probability density function for a photon traveling in some direction to be scattered in some new direction $p(\theta ,\varphi )$. Figure 1 illustrates the geometry of scattering of light by a particle, where the incident light beam,${\overrightarrow{S}}_0$, is parallel to the $z$-axis, and $\theta$ and $\varphi$ are the scattering angles in the scattering plane and in the plane perpendicular to the scattering plane, respectively.³⁷ The scattering phase function characterizes an elementary scattering act. If scattering is symmetric relative to the direction of the incident wave, then the phase function depends only on the scattering angle $\theta$ [angle between two directions, new (${\overrightarrow{S}}_1$) and former one (${\overrightarrow{S}}_0$)], $p(\theta )$.

4.1.2.

Rayleigh scattering

If a particle is small with respect to the wavelength of the incident light, its scattering can be described as if it is a single dipole.¹⁴⁰ This Rayleigh theory is applicable under the condition that $m(2\pi a/\lambda )\ll 1$, where $m$ is the relative refractive index of the scatterers, ($2\pi a/\lambda$) is the size parameter, $a$ is the radius of the particle, and $\lambda$ is the wavelength of the incident light in a medium. For this theory, the scattered irradiance strongly increases as $a^6$ and decreases as ${\lambda }^{-4}$ (Fig. 8); the angular distribution of the scattered light is polarization sensitive and isotropic (Fig. 9).

Fig. 8

The wavelength dependence of reduced scattering coefficient, ${\mu }_s^{\prime }={\mu }_s$ $(1-g)$, for human and rat skin; dots present experimental data, and solid and dotted lines represent theoretical dependences from Mie and Rayleigh theories; total means wavelength dependence calculated based on both theories, where coefficients of each theory provide fitting to experimental data (Steven L. Jacques, Ulm, LALS-2014).¹³⁹

Fig. 9

Rayleigh scattering: light distribution in the scattering plane for two orthogonal linear polarization states and unpolarized incident light;¹⁴⁰ 1, electric vector perpendicular to the scattering plane; 2, electric vector parallel to the scattering plane; and 3, unpolarized light.

The angular dependence of the scattered intensity by an ensemble of $N$ randomly distributed particles with a mean distance between particles bigger than the wavelength $\lambda$, unpolarized incident light of intensity $I_0$, and distant detector position $r$ is described by the Rayleigh equation,¹⁴⁰

Eq. (24)

$$I(r,\theta )={(2\pi )}^4\frac{a^6}{{\lambda }^4{r}^2}{\left(\frac{m^2-1}{m^2+2}\right)}^{2}N\frac{1+{\cos }^2\theta }{2}{I}_0,$$

where $m=n_s/n_0$ is the relative index of refraction of the scatterers and ground material, and $N$ is the number of particles in the scattering volume $V$, i.e., $N={\rho }_{p}V$, where ${\rho }_p$ is the scattering particle density.

Figure 9 illustrates the isotropy of Rayleigh scattering for unpolarized incident light and the polarization-insensitive detection (curve 3) and polarization ability of the scattering particles for the particular direction of detection (90 and 270 deg), where only light polarized perpendicular to the scattering plane is scattered (compare curves 1 and 2). An ideal isotropy (circle-shaped phase function) of the scattering is achieved if incident light is linearly polarized perpendicular to the scattering plane (curve 1). For any scattering angle, except $\theta =0$ and 180 deg, interaction of unpolarized light with Rayleigh scatterers provides linear polarization of the scattered light with preferable intensity of the electrical vector oscillating in the plane perpendicular to the scattering plane with polarization degree

Eq. (25)

$$P_L=\frac{I_{\perp }(\theta )-I_{||}(\theta )}{I_{\perp }(\theta )+I_{||}(\theta )}=\frac{{\sin }^2\theta }{1+{\cos }^2\theta }.$$

For the near-infrared (NIR) light and typical biological scatterers with $m=1.05$ to 1.10, the maximum particle radius is $\sim 12$ to 14 nm for the Rayleigh theory to remain valid. The Rayleigh–Gans or Rayleigh–Debye theory addresses the problem of calculating the scattering by a special class of arbitrarily shaped particles; it requires $|m-1|\ll 1$ and $\frac{2\pi a^{\prime }}{\lambda }|m-1|\ll 1$, where $a^{\prime }$ is the largest dimension of the particle. These conditions mean that the electric field inside the particle must be close to that of the incident field and the particle can be viewed as a collection of independent dipoles that are all exposed to the same incident field. A biological cell might be modelled as a sphere of cytoplasm with a higher refractive index ($n_c=1.370$) relative to that of the surrounding interstitial medium ($n_i=1.350$); then $m=1.015$, and for the NIR light, this theory will be valid for the particle dimension up to $a^{\prime }=850$ to 950 nm. This approximation has been applied extensively to calculate light scattering from suspensions of bacteria^37–41 and can be applicable to describe light scattering from cell organelles (mitochondria, lysosomes, peroxisomes, and so on) in tissues due to their small dimensions and refraction.

4.1.3.

Mie scattering

Mie or Lorenz-Mie scattering theory relates to scattering by comparatively large spherical particles, which are of the order of the wavelength, and is based on an exact solution of Maxwell’s electromagnetic field equations for a homogeneous sphere.^2,5,96,140 Typically, tissues contain both types of scatterers, small and large (for instance, cell components and collagen fibers of connective tissues, see Fig. 8). Mie theory operates with the following relevant particle parameters: radius $a$ and complex refractive indices of its material $n_s({\lambda }_0)$ and dielectric host (ground material) $n_0({\lambda }_0)$,

Eq. (26)

$$n_{s,0}({\lambda }_0)=n_{s,0}^{\prime }({\lambda }_0)+i{n}_{s,0}^{\prime \prime }({\lambda }_0),$$

where ${\lambda }_0$ is the wavelength in vacuum. The imaginary part of the complex refractive index of the material is responsible for light losses due to absorption. Mie theory yields the scattering $(Q_{\text{sca}}={\sigma }_{\mathrm{sca}}/\pi a^2)$ and absorption $(Q_{\text{abs}}={\sigma }_{\mathrm{abs}}/\pi a^2)$ efficiencies and the phase function from which the absorption and scattering cross-sections (${\sigma }_{\mathrm{sca}}$ and ${\sigma }_{\mathrm{abs}}$) and the scattering anisotropy factor $g\equiv <\cos \theta >$ are calculated,

Eq. (27)

$${\sigma }_{\mathrm{sca}}^{\mathrm{Mie}}=\frac{{\lambda }_0^2}{2\pi n_0^2}\sum _{n=1}^{\infty }(2n+1)({|a_n|}^2+{|b_n|}^2),$$

Eq. (28)

$${\sigma }_{\mathrm{abs}}^{\mathrm{Mie}}=\frac{{\lambda }_0^2}{2\pi n_0^2}\sum _{n=1}^{\infty }(2n+1)[\mathrm{Re}(a_n+b_n)-({|a_n|}^2+{|b_n|}^2)],$$

Eq. (29)

$$p^{\text{Mie}}(\theta )=\frac{{\lambda }_0^2}{2\pi n_0^2{\sigma }_{\mathrm{sca}}^{\mathrm{Mie}}}({|S_1|}^2+{|S_2|}^2),$$

Eq. (30)

$$g^{\text{Mie}}=\frac{{\lambda }_0^2}{\pi n_0^2{\sigma }_{\mathrm{sca}}^{\mathrm{Mie}}}[\sum _{n=1}^{\infty }\frac{2n+1}{n(n+1)}\mathrm{Re}(a_n{b}_n^{*})+\sum _{n=1}^{\infty }\frac{n(n+2)}{n+1}\mathrm{Re}(a_n{a}_{n+1}^{*}+b_n{b}_{n+1}^{*})],$$

where $a_n$ and $b_n$ are Mie coefficients, which are functions of the relative complex refractive index of particles ($m$) and parameter $\alpha =2\pi n_{0}a/{\lambda }_0$; an asterisk indicates that the complex conjugate is to be taken.

Eq. (31)

$$a_n=\frac{{\psi }_n(\alpha ){\psi }_n^{\prime }(m\alpha )-m{\psi }_n(m\alpha ){\psi }_n^{\prime }(\alpha )}{\zeta (\alpha ){\psi }_n^{\prime }(m\alpha )-m{\psi }_n(m\alpha ){\zeta }_n^{\prime }(\alpha )},$$

Eq. (32)

$$b_n=\frac{m{\psi }^{\prime }(m\alpha ){\psi }_n(\alpha )-{\psi }_n(m\alpha ){\psi }_n^{\prime }(\alpha )}{m{\psi }_n^{\prime }(m\alpha ){\zeta }_n(\alpha )-{\psi }_n(m\alpha ){\zeta }_n^{\prime }(\alpha )}.$$

${\psi }_n$ and ${\zeta }_n$ are the Riccati–Bessel functions; $S_1$ and $S_2$ are functions of the polar scattering angle and can be obtained from the Mie theory as

Eq. (33)

$$S_1(\theta )=\sum _{n=1}^{\infty }\frac{2n+1}{n(n+1)}[a_n{\pi }_n(\cos \theta )+b_n{\tau }_n(\cos \theta )],$$

Eq. (34)

$$S_2(\theta )=\sum _{n=1}^{\infty }\frac{2n+1}{n(n+1)}[b_n{\pi }_n(\cos \theta )+a_n{\tau }_n(\cos \theta )].$$

The parameters ${\pi }_n$ and ${\tau }_n$ can be given as

Eq. (35)

$${\pi }_n(\cos \theta )=\frac{1}{\sin \theta }{P}_n^1(\cos \theta ),$$

Eq. (36)

$${\tau }_n(\cos \theta )=\frac{d}{d\theta }{P}_n^1(\cos \theta ),$$

where $P_n^1(\cos \theta )$ is the associated Legendre polynomial; the following recursive relationships are used to calculate ${\pi }_n$ and ${\tau }_n$,

Eq. (37)

$${\pi }_n=\frac{2n-1}{n-1}{\pi }_{n-1}\cos \theta -\frac{n}{n-1}{\pi }_{n-2},{\tau }_n=n{\pi }_n\cos \theta -(n+1){\pi }_{n-1},$$

and the initial values are

Eq. (38)

$$\{\begin{array}{ll}{\pi }_1=1,& {\pi }_2=\cos \theta ,\\ {\tau }_1=\cos \theta ,& {\tau }_2=3\cos 2\theta .\end{array}$$

From the expressions for ${\sigma }_{\mathrm{sca}}$ (${\mathrm{cm}}^2$) and ${\sigma }_{\mathrm{abs}}$ (${\mathrm{cm}}^2$), scattering and absorption coefficients can be calculated as ${\mu }_s={\rho }_p{\sigma }_{\mathrm{sca}}$ and ${\mu }_a={\rho }_M{\sigma }_{\mathrm{abs}}$, where ${\rho }_p$ is the scattering particle density and ${\rho }_M$ is the number of absorbing molecules per unit volume (${\mathrm{cm}}^{-3}$).

Mie theory predicts that scattering introduced by spherical micrometer-sized particles is strongest and highly anisotropic if the particle radius and wavelength are of the same order of magnitude (Figs. 9 and 10). The scattering coefficient decays with the wavelength much more slowly than for Rayleigh scattering, i.e., ${\lambda }^{-b}$, with $b$ in the range of 1 to 3. The scattering coefficient increases strongly with the elevation of the relative index of refraction $m$. In contrast, for the matched refractive indices of scatterers and background material, the scattering coefficient goes to zero as scattering anisotropy factor approaches 1 (extremely high scattering directness).

Fig. 10

The polar diagrams of the calculated angular-dependent intensity distributions (Mie phase functions) for five single isotropic spheres with the diameter in the range of those that are typical for cell organelles, from $2a=0.1$ to $2.00\mu \mathrm{m}$, ${\lambda }_0=500\mathrm{nm}$, and $m=1.5$.¹⁴⁰

Similar to the Rayleigh equation [Eq. (24)] for unpolarized incident light of intensity $I_0$ and distant detector position $r$, the angular dependence of intensity of the light scattered by an ensemble of $N$ randomly distributed Mie particles with a mean distance between particles bigger than the wavelength $\lambda$ is described by

Eq. (39)

$$I(\theta )=p^{\text{Mie}}(\theta )·N·I_0.$$

Mie theory is strictly applicable only to particles of particular regular shapes, but the results are still useful if the shape is irregular.

Actual biological tissue models are more complex than a monodispersive system of distant spherical particles or even randomly shaped ones. Sometimes, a mixture of large particles contributing high scattering anisotropy and small particles with increased scattering toward shorter wavelengths may be a good approximation to describe tissue scattering properties (see Fig. 8).¹³⁹

The scattering anisotropy factor (mean cosine of the scattering angle $\theta$, see Fig. 1) is expressed as that corresponding to scattering symmetry and assumption of random distribution of particles in a medium,^96,98

Eq. (40)

$$g\equiv \langle \cos \theta \rangle =2\pi {\int }_0^{\pi }{M}_{11}(\theta )\cos \theta \sin \theta d\theta ,2\pi {\int }_0^{\pi }{M}_{11}(\theta )\sin \theta d\theta =1,$$

where $M_{11}(\theta )$ is the first element of LSM [see Eq. (9)].

In addition to theoretical Mie phase function [see Eq. (30)], several semi-empirical approximations for the scattering phase function have been used. One of the most often exploited is the Henyey-Greenstein (HG) phase function,

Eq. (41)

$$p^{\mathrm{HG}}(\theta )=\hspace{0.17em}\frac{1}{4\pi }·\frac{1-g^2}{{(1+g^2-2g\cos \theta )}^{3/2}}.$$

The transport mean free path (TMFP), $l_{\mathrm{tr}}$, of a photon traveling in a scattering medium characterizes the distance within which the direction of light propagation becomes totally random after many sequential scattering events,^{2,5,8,11,15,52,54}

Eq. (42)

$$l_{\mathrm{tr}}=\frac{1}{{\mu }_a+{\mu }_s^{\prime }}\cong \frac{1}{{\mu }_s^{\prime }},$$

where ${\mu }_s^{\prime }=(1-g){\mu }_s$ is the reduced scattering coefficient, and g is the scattering anisotropy factor defined by Eqs. (37) and (40).

4.2.

Quasi-Ordered Tissue Model

Many tissues composed from optically soft particles (refractive index mismatch of scatterers and surrounding medium is small) and thin enough, such as cornea, eye lens, mucosa, epithelia, can be approximated as single-scattering systems. For example, healthy tissues of eye cornea and lens are highly transparent for visible light because of their ordered structure and the absence of strongly absorbing chromophores.⁵ However, scattering is an important feature of light propagation in eye tissues. Typical eye tissue models are long, round dielectric cylinders (corneal and scleral collagen fibers) [Fig. 7(a)] or spherical particles (lens protein structures) having a refractive index $n_s$; they are randomly (sclera, opaque lens) or regularly (transparent cornea and lens) distributed in the isotropic base matter with a refractive index $n_0<n_s$.

All these tissues can be considered as densely packed disperse systems in which light propagation can be analyzed using the radial distribution function $g(r)$. This function statistically describes the spatial arrangement of particles in the system (Fig. 11). The $g(r)$-function is the ratio of the local number density of the particles at a distance $r$ from a reference particle placed at $r=0$ to the bulk number density of particles.⁴³ For a corneal model as a system of interacting fibrils, it expresses the relative probability of finding two fibril centers separated by a distance $r$; thus, $g(r)$ must vanish for values of $r\le 2a$ ($a$ is the radius of a fibril; fibrils cannot approach each other closer than touching).

Fig. 11

(a) A short-range spatially ordered fibrillar system around fixed fibril is shown schematically by two successive rings, characterizing the probability of spatial distribution of neighbor fibrils. (b) The close-up view of collagen fibrils from electron microscopy image of human cornea presented in Fig. 2; diagram of radial distribution function $g(r)$ that is proportional to the probability of particle displacement $r$ at a certain distance from an arbitrarily fixed fibril.¹⁴³

The $g(r)$-function of scattering centers for a certain tissue is calculated from tissue electron micrographs (see Fig. 2). The $g(r)$-function was first found for the rabbit cornea by McCally and Farrell⁴³ In Fig. 12(a), a typical experimental $g(r)$-function for one of the cornea regions is shown, $g(r)=0$ for $r\le 25\mathrm{nm}$, which is consistent with a fibril radius of $14\pm 2\mathrm{nm}$. The first peak in the distribution gives the most probable separation distance between fibrils, which is $\sim 50\mathrm{nm}$. $g(r)$ is essentially unity for $r\ge 170\mathrm{nm}$, indicating that the fibril positions are correlated over no more than a few of their nearest neighbors. Therefore, a short-range spatial order exists in the system and should be accounted for the optical model.

Fig. 12

Experimental histograms of radial distribution function $g(r)$ obtained from electron micrographs of (a) rabbit cornea (700 particles)⁴³ and (b) human sclera (100 particles)^5,111 [see Figs. 2(b) and 2(d)]. For rabbit cornea, fibril mean diameter is $\sim 28\mathrm{nm}$, and it is $\sim 100\mathrm{nm}$ for human sclera; for a fibril placed at $r=0$, the nearest most probable particle position is at $r\approx 50\mathrm{nm}$ for rabbit cornea and at $r\approx 285\mathrm{nm}$ for human sclera.

Similar calculations for several regions of the human eye sclera are illustrated in Fig. 12(b). Electron micrographs from Ref. 111 [see Fig. 2(d)], averaged for 100 fibril centers, were processed.⁵ The obtained results present evidence of the presence of a short-range spatial order in the sclera, although the degree of order is less pronounced than in the cornea, because of polydispersity of fibrillary structure. $g(r)=0$ for $r\le 100\mathrm{nm}$, which is consistent with the mean fibril diameter of $\sim 100\mathrm{nm}$ derived from the electron micrograph. The first peak in the distribution gives the most probable separation distance, which is $\sim 285\mathrm{nm}$. The value of $g(r)$ is essentially unity for $r\ge 750\mathrm{nm}$, indicating a short-range spatial order in the system.

For an isotropic system of $N$ identical interacting long cylinders, the scattered intensity is defined as^5,43

Eq. (43)

$$\langle I\rangle ={|E_0|}^{2}N{S}_2(\theta ),$$

where $E_0$ is the scattering amplitude of an isolated particle.

Eq. (44)

$$S_2(\theta )=\{1+8\pi a^2{\rho }_c{\int }_0^R[g(r)-1]J_0(\frac{2\pi a}{\lambda }r\sin \frac{\theta }{2}\left)\mathrm{d}r\right\}$$

is the structure factor; $a$ is the radius of the cylinder face; $\rho$ is the mean density of the cylinder faces; ${\rho }_c=f_c/\pi a^2$, where $f_c$ is the surface fraction of the cylinders’ faces; $J_0$ is the zero-order Bessel function; $R$ is the distance for which $g(r)\to 1$; $\theta$ is the scattering angle.

The structure factor $S_2(\theta )$ describes complex interference of the scattered fields from individual fibrils distributed in two-dimensional space. With no spatial correlation of fibrils, when $g(r)\to 1$, $S_2(\theta )\to 1$; thus, no interference of individual scattered fields exists, and in Eq. (43), intensities of light scattered by individual fibrils rather than fields are summarized. For an isotropic system of identical spherical particles in a single-scattering approximation,^5,104,143

Eq. (45)

$$\langle I\rangle ={|E_0|}^{2}N{S}_3(\theta ),$$

Eq. (46)

$$S_3(\theta )=\{1+4\pi {\rho }_p{\int }_0^R{r}^2[g(r)-1]\frac{\sin qr}{qr}\mathrm{d}r\},$$

where $q=\frac{4\pi }{\lambda }\sin \frac{\theta }{2}$, ${\rho }_p$ is the mean density of particles, and $R$ is the distance for which $g(r)\to 1$. Quantity $S_3(\theta )$ is the three-dimensional (3-D) structure factor. This factor describes the alteration of the angular dependence of the scattered intensity that appears with a higher particle concentration due to complex interference of the scattered fields from individual spheres distributed in 3-D space. Similar to the fibril system, at no spatial correlation of spherical particles, $g(r)\to 1$, $S_3(\theta )\to 1$; no interference of scattered fields and intensities of light scattered by individual spheres are summarized.

The presented models can be used to describe unpolarized and polarized light interactions with particle media showing a single scattering. Figure 13 illustrates the angular dependences of the scattered intensity for systems of small spherical particles ($a=20\mathrm{nm}$) with volume fraction $f=0.1$ and large spherical particles ($a=500\mathrm{nm}$) with volume fraction $f=0.4$. The incident wave is linearly polarized parallel with or perpendicular to the scattering plane (see Fig. 1). The dotted lines show angular light distributions for the independent (randomly distributed) particles and the solid lines are that for the ordered particles, for which the interference of the scattered fields plays a significant role. The overall scattering is suppressed due to interference between the out-of-phase scattered fields from individual particles. For the small particle system, such suppression is quite isotropic for both polarization states, and for the large particle system, is mostly suppressed in the forward directed scattering. As a result, both particle systems became more transparent for the incident light (see a big portion of ballistic photons, schematically shown in Fig. 13). It is important to note that for big particle systems, the results for the scattered intensity are presented in the logarithmic scale. Data presented in Fig. 13 allow for evaluation of unpolarized light (50% mixture of two orthogonal linear polarized light beams) interaction with a closely packed particle system that leads to creation of a partly polarized light at scattering as it follows from data in Fig. 9. It is well seen that for the ordered small particles, the polarization ability at the scattering angles around 90 deg is stronger than for random (independent) particles.

Fig. 13

The calculated angular dependences of the scattered intensity for systems of [(a) and (b)] small spherical particles ($a=20\mathrm{nm}$) with volume fraction $f=0.1$ and [(c) and (d)] large spherical particles ($a=500\mathrm{nm}$) with volume fraction $f=0.4$; the incident wave is linearly polarized [(a) and (c)] parallel with or [(b) and (d)] perpendicular to the scattering plane (see Fig. 1); dotted lines show calculations for independent particles; wavelength, 633 nm; relative refraction index, $m=1.105$ (calculated by I. L. Maksimova).⁵

The interference between elementary scattered fields is able to modulate the transformation of the arbitrary polarized incident light into another state of polarization at light scattering that is described in frames of amplitude ($2\times 2$, Jones) or intensity ($4\times 4$, Mueller) matrices.^5,15

4.3.

Light Scattering Matrix

As is already presented in Sec. 2, the polarization of the scattered light from an object (Fig. 1) is fully described by the Stokes vectors of the incident and scattered light and $4\times 4$ Mueller matrix [intensity or LSM, see Eq. (9)].⁹⁶ Elements of the LSM depend on the scattering angle $\theta$, the wavelength, the shape, and the optical properties of the object. For scattering by a collection of randomly oriented particles, there are 10 independent LSM elements. The LSM for macroscopically isotropic and symmetric media has a well-known block-diagonal structure.¹⁰⁷

Eq. (47)

$$M(\theta )=\left[\begin{array}{cccc}{M}_{11}(\theta )& M_{12}(\theta )& 0& 0\\ M_{12}(\theta )& M_{22}(\theta )& 0& 0\\ 0& 0& M_{33}(\theta )& M_{34}(\theta )\\ 0& 0& -M_{34}(\theta )& M_{44}(\theta )\end{array}\right].$$

It follows that only eight LSM elements are nonzero and only six of these are independent. Moreover, there are special relationships for two specific scattering angles, 0 and $\pi$,⁹⁸

Eq. (48)

$$M_{22}(0)=M_{33}(0),M_{22}(\pi )=-M_{33}(\pi ),M_{12}(0)=M_{34}(0)=M_{12}(\pi )=M_{34}(\pi )=0,M_{44}(\pi )=M_{11}(\pi )-2M_{22}(\pi ).$$

Rotationally symmetric particles have an additional property,⁹⁸

Eq. (49)

$$M_{44}(0)=2M_{22}(0)-M_{11}(0).$$

The structure of the LSM further simplifies for spherically symmetric particles, which are homogeneous or radially inhomogeneous (composed of isotropic materials with a refractive index depending only on the distance from the particle center), because in this case,⁹⁸

Eq. (50)

$$M_{11}(\theta )\equiv M_{22}(\theta ),M_{33}(\theta )\equiv M_{44}(\theta ).$$

Mie theory is an exact solution of Maxwell’s electromagnetic field equations for a homogeneous sphere.^96,141 For spherically symmetric particles made from optically inactive materials, the Mueller matrix is given by Eqs. (47) and (50). Mie theory has been extended to arbitrary coated spheres and to arbitrary cylinders.^38,98,99 In this theory, the electromagnetic fields of the incident, internal, and scattered waves are each expanded in a series.¹⁴¹ A linear transformation can be made between the fields in each of these regions. This approach can also be used for nonspherical particles such as spheroids.^98,99 The linear transformation is called the transition matrix (T-matrix).

For systems of spherical and cylindrical particles, when a single-scattering approximation is valid, the angular intensity characteristics are described by Eqs. (43)–(46). The structure factor modifying these characteristics depends only on the spatial particle arrangement, and it is independent of the state of light polarization. Therefore, for systems of identical particles, the angular dependences of all LSM elements are multiplied by the same quantity, accounting for interference interaction [see Eq. (46) for spherical particles].

Eq. (51)

$$M_{ij}(\theta )=M_{ij}^0(\theta )N{S}_3(\theta ),$$

where $M_{ij}^0(\theta )$ are the LSM elements for an isolated particle. Consequently, the LSM for the system of monodispersive interacting particles coincides with that of the isolated particle [see Eq. (47)] if normalization to the magnitude of its first element is done.¹⁴⁴

Unlike for monodispersive systems, in differently sized densely packed particle systems, the normalization to the first element does not eliminate the impact of the structure factor on the angular dependences of the matrix elements. In the simplest case of a bimodal system of particles, expressions analogous to Eqs. (44) and (46) can be found using four structural functions instead of one (see Figs. 11 and 12): $g_{11}(r),g_{22}(r),g_{12}(r)$, and $g_{21}(r)$.¹⁰⁴ These functions characterize the interaction between particles of similar and different sizes. In tissues, a bimodal system formed by a great number of equally sized small particles, and a minor fraction of coarse ones, provides a good model of pathological tissue, e.g., a cataract eye lens or enlargement of a cell nucleus at cancer.

Figure 14 shows results of numerical modeling for a binary mixture of spherical particles with two different diameters and volume fractions.¹⁰⁴ It is seen that the normalized LSM elements of a dense binary mixture are substantially altered due to the interference interaction.

Fig. 14

Calculated angular dependences of LSM elements for a binary mixture of spherical particles:¹⁰⁴ particle diameters: $2a_1=60\mathrm{nm}$ and $2a_2=500\mathrm{nm}$; volume fractions: $f_1=0.3$ and $f_2=0.02$; relative index of refraction, $m=1.07$ for both types of particles; $\lambda =633\mathrm{nm}$; solid lines demonstrate the impact of particle interactions, and dashed lines indicate neglecting of cooperative effects.

So far, Stokes vectors have been defined for the case of a monochromatic plane wave, and the Mueller matrix for a single scattering. However, this approach has to be generalized to more complicated cases. For a partially polarized [see Eq. (6)] quasi-monochromatic wave, the following inequality is valid:^96,109

Eq. (52)

$$I^2\ge Q^2+U^2+V^2.$$

When Mueller matrices from an ensemble of particles differing in size, orientation, morphology, or optical properties are added incoherently, many of the above-mentioned equalities became inequalities.¹⁴⁵