CLASSIFYING OPTICAL PROPERTIES OF SURFACE-AND BULK-SCATTERING BIOLOGICAL LAYERS WITH POLARIZATION SINGULAR STATES

The results of singular approach usage in the tasks of description and classi ̄cation of appearance of optical anisotropy of di®erent types of phase-inhomogeneous biological layers (surfacescattering, optically thin and optically thick) have been presented. The characteristic values of the fourth Stokes vector parameter (S4 1⁄4 0 — linear polarization — (L-state); S4 1⁄4 1 — circular polarization — ( C-state)) have been chosen as the main analytical tool describing polarization-singular states. The value of S4 has been determined by the value of phase shift between the orthogonal components of amplitude in the point of biological layer laser image and therefore is azimuthally stable. Hence, statistic moments of the ̄rst to the fourth orders characterizing the distribution of the amount of characteristic values S4 1⁄4 0;S4 1⁄4 1 have been used for de ̄nition and di®erentiation of optical properties of di®erent types of biological layers — surface scattering, optically thin and optically thick human skin.


Introduction
By tradition, the processes of transforming optical radiation of phase-inhomogeneous objects and media are considered, as a rule, in a statistic approach (theory of radiation transfer, 1À5 Monte-Carlo modeling 6,7 ).Among the most spread traditional methods for studying the scattered light ¯elds, one can separate the following independent directions: \scalar" (photometry and spectrophotometry) 8À10 and \vector" (polarization nephelometry and Muellermatrix optics).11À15 Using these approaches, interrelations between the sets of statistic moments of the 1st to the 4th orders have been found. 16,17ompared with traditional statistic investigations, new optical approaches for describing the structures of ¯elds in the case of scattered coherent radiation have been formed in recent 10À15 years.The complex coordinate distribution of azimuths and ellipticities values, formed due to statistic interference of partial coherent waves, is typical for such ¯elds.Hereinafter, we will call such ¯elds \polarization-inhomogeneous ones".The main feature of this approach is the analysis of particular (boundary) polarization states to determine the whole structure of coordinate distributions for azimuths and ellipticities of polarization.The so-called polarization singularities are commonly used as these states 18À22 : . states with linear polarization when the direction of rotation for the electric ¯eld vector is inde¯nite, the so-called L-states; .circularly polarized states when the azimuth of polarization for the electric ¯eld vector is inde¯nite, the so-called C-states.
It is shown in works 23À27 that having the information about the coordinate networks of such points, it is possible to determine the peculiarities of the ¯eld's topological structure.Still, the data that has been obtained by now are generally of theoretical nature.It is noteworthy that practically all the variety of biological objects possesses the optical anisotropy, which reveals in formation of Land C-states networks.
Presently, there are no systematic data about polarization-singular structure of laser ¯elds scattered by biological layers of di®erent morphological structure and optical thickness. 27Besides, formation of networks of linearly and circularly polarized states is azimuthally dependent to changes of both probing beam polarization plane and investigated sample orientation. 28Therefore, there is a topical task of elaboration of universal technique of diagnostic and di®erentiation of optical properties of di®erent biological layers within the singular approach with the help of di®erent azimuthally stable parameters.
We have chosen the characteristic values of the fourth Stokes vector parameter, which universally describe all polarization-singular states as such parameter.
Another important circumstance is that the value of S 4 is determined by the value of phase shift between the orthogonal components of amplitude in point of laser image of biological layer and, therefore, is azimuthally stable.That is why, the distributions S 4 ¼ 0ð ¼ 0Þ and S 4 ¼ AE1ð ¼ AE0; 5Þ can be used as universal and azimuthally stable characteristic of optical properties of biological layers of di®erent morphological structures.
This work is directed at investigation of possibility of de¯nition and di®erentiation of optical properties of biological layers of di®erent typessurface-scattered, optically thin (single scattered) and optically thick (multiple scattered, depolarized) in the framework of statistic description of distributions S 4 ¼ 0 and S 4 ¼ AE1.

Model Conceptions and Analytical Relations
As a basis for analytical description of the processes providing the formation of polarization-inhomogeneous images for various types of biological phaseinhomogeneous layers (PhIL), we have used the model concepts developed in the works 27,28 : 1. Four main types of tissuesconnective, muscular, epithelial and neuralrepresent the variety of human biological tissues.2. Optically, morphological structure of any biological tissue type is considered as a superposition of phase-inhomogeneous surface (rough) and volume (isotropic-anisotropic) scattered components.3. Surface-scattering PhIL (attenuation coe±cient 0:1) is a rough surface (super¯cial layer of the skin epithelium) consisting of the ensemble of quasi plane, randomly tilted micro-areas with optical dimensions l >group 1.The optical properties of a local micro-area of such surface are described by the following matrix operator fRg: where r x ð; nÞ; r y ð; nÞ are the Fresnel amplitude coe±cients of re°ection for orthogonal (x; yÞ components of laser wave amplitude; is the tilt angle of micro-area with respect to macro-surface; n is the epithelium refractive index. 4. Optically thin (attenuation coe±cient 0:1) single scattered biological PhIL with anisotropic component can be described by the Mueller matrices for circular (optical activity) fV g and linear (linear birefringence) fW g phase anisotropygroup 2: Here, is the direction (orientation azimuth) of fast optical axis; ¼ 2=Ánd is the value of linear phase anisotropy, phase shift between orthogonally linearly polarized components of laser amplitude ( 2 ½0; Þ; is the wavelength; d is the geometric path; Án is the index of birefringence; is the value of circular phase anisotropy (rotation angle of polarization plane of laser radiation), 2 ½0; . 5. For optically thick (attenuation coe±cient > 0:1) multiple scattered biological PhILgroup 3the main mechanism of local polarization state formation is interferometric superposition of amplitudes of coherent waves with di®erent polarization.The result of such superposition reveals the formation of elliptical polarization described by the following equation where U x ; U y are the orthogonal components of amplitude U, ' is the phase shift between them.Let us consider, in the framework of vectorparametric approach, the process of polarizationsingular states formation (S 4 ¼ 0; AE1).In the common case, the polarization transformation processes can be described using the following vectorparametric equation where S 0 ; S Ã are the Stokes vectors of illuminated and are transformed by the objects beam correspondingly; fMg is the Mueller matrix of biological layer.

Circular phase anisotropy
There are only L-states (S 4 ¼ 0) forming in this case irrespective of the value of phase shift between the circularly polarized components of amplitude of laser radiation transformed by optically active structures (3) of biological layer. .Linear phase anisotropy There are L-states (S 4 ¼ 0) forming on the condition of coincidence of probing beam polarization plane 0 and direction of optical axis of birefringent ¯bril There are AEC-states (S 4 ¼ AE1) forming on the condition Polarization-singular states in the ¯eld of multiple scattered laser radiation (5) are determined by the conditions Thus, we have demonstrated the basic interconnections between the optical-geometric structure of di®erent types of biological layers and mechanisms of polarization-singular states formation.

Experimental Setup for Polarimetric Investigation
Our study of polarization-inhomogeneous laser images inherent to PhIL was performed using the optical scheme of a laser polarimeter (see Fig. 1). 27he object of investigation was illuminated by collimated (1 ¼ 10 4 m) HeÀNe laser beam ( ¼ 0:6328 m) with the power of 50 W. Polarization light source (quarter-wave plates 3, 5 and polarizer 4) formed the ensemble of Stokes vectors of the illuminating beam fS 0 j¼1;2;3;4 g.By means of micro-objective 7 (focal distance -1.5 cm, aperture -0.2, magni¯cation -4x), the polarization images of biological tissue were projected into the plane of light-sensitive area of CCD camera (total amount of pixels -800 Â 600, light-sensitive area size -4000 Â 3000 m, deviation of photosensitive characteristic from linear oneno more than 15%), which provided the range of measuring the structural elements of biological tissue with the resolution 2À2000 m.Maximal resolution veri¯cation (2 m) was performed using the stage micrometer (linear scale), the image of which was projected into the light-sensitive area of CCD camera with the help of micro-objective 7. Minimal resolution (2000 m) corresponds to the situation, when the light-sensitive area of CCD camera is entirely ¯lled by two equal-sized structural elements (light and dark) of stage micrometer.The conditions of the experiment were chosen in such a way that it enabled to reduce the space-angular aperture ¯ltering while forming the biological tissue images.This was ensured by conformance of angular characteristics of indicatrices of light scattering by the biological tissue samples ( % 16 ) and angular aperture of micro-objective (Á! ¼ 20 ).Here, is the solid angle within which 98% of all the energy of light-scattered radiation is concentrated.
The technique of measuring the number of polarization-singular states in the plane of image of biological layers consists of the following steps: . Turning the transmission axis of analyzer 9 by the angles AE45 relative to the direction of the highest velocity axis for the quarter-wave plate 8, we determined the intensities of right (I ) and left (I È ) circularly polarized components for each separated pixel (r ik ) of CCD camera 10. .It served as a base to calculate the coordinate distributions of the fourth parameter in the Stokes vector V 4 ðm Â nÞ describing the laser image of PhIL, using the relation . The two-dimensional array (11) was scanned along the horizontal direction x 1; . . .; m with the step Áx ¼ 1 pix. .Within the limits of each local sample ð1 pix Â n pix Þ ðk¼1;2;...;mÞ , we calculated the amount (N) of characteristic values V AEC ; . . .; N ðmÞ AEC Þ in this work we have used the statistic approach. 28Distributions N L;AEC ðxÞ for the set of polarization-singular states in laser images of PhIL are characterized by the ensemble of statistic moments of the 1stÀ4th orders Z j¼1;2;3;4 calculated using the following relations where M ¼ 800 Â 600 is the amount of pixels in CCD camera 10 (see Fig. 1).

The Investigation Objects Characteristics
The following types of histological sections of skin dermis samples taken from the human stomach region were used as the objects of investigation: . super¯cial optically thin (the attenuation coe±cient ¼ 0:085) epidermis layer, which is formed by the set of keratinized isotropic epithelium platesgroup 1; .optically thin ( ¼ 0:095) dermis layer, where the anisotropic component is formed by a network of collagen birefringent ¯brilsgroup 2; .optically thick ( ¼ 0:45) dermis layergroup 3.
The histological sections were prepared with the standard technique by freezing microtome.They have the following characteristics: the absorption coe±cient a ¼ 2:2 cm À1 , the scattering coe±cient s ¼ 185 cm À1 , the anisotropy parameter g ¼ 0:82, value of birefringence ÁnðSDÞ % 1:45 Â 10 À3 , the geometric thickness d ¼ 15 m ( ¼ 0:095) d ¼ 40 m ( ¼ 0:45Þ. Polarization manifestations of peculiarities of morphological structure are presented by coordinate (100 Â 50 pix) distributions of the fourth parameter for the Stokes vector V 4 ðm Â nÞ inherent to laser images of PhIL in all the groups (see Fig. 2).
Our qualitative analysis of coordinate distributions V 4 ðm Â nÞ for laser images of PhIL (see Fig. 2) enabled to reveal that: . practically all the images of the rough surface of skin [see Fig. 2(a)] are a linearly polarized ¯eld V 4 ðm Â nÞ ¼ 0. This fact is in good agreement with the performed modeling (relations ( 2) and ( 6)).Availability of a small amount of the parts V 4 ðm Â nÞ 6 ¼ 0 polarized otherwise can be related to the interference e®ects of multiple-scattered coherent waves with adjacent micro-roughnesses; .the image of the optically thin layer of the dermis [see Fig. 2 9), (10).
The second one is the interference of di®erently polarized partial fronts and the formation of polarization-inhomogeneous speckle ¯eld.

L-states of laser images
The series of coordinate (V 4 ðm Â nÞ ¼ 0) and quantitative (N L ðxÞ) distributions for polarizationsingular L-states in laser images of PhIL is summarized in Fig. 3.In the case of surface-scattering epidermis layer, the main mechanisms of transformation of polarization state are the local re°ection (refraction) acts by keratinized, optically isotropic epithelium plates.Such interaction acts lead to the formation of polarization plane rotation, the value of which is determined by the tilt angle of microasperity and refraction index of matter.Therefore, the amount of L-states of polarization in the corresponding laser image appears to be dominant [see Figs.3(a)  and 3(d)].
In the case of optically thin dermis layer, the simultaneous realization of two interaction mechanisms occurs.This results in the formation of both L-states (circular phase anisotropy (7) and C-states (linear phase anisotropy ¼ ð2q þ 1Þ 2 , ( 8)).Due to this fact, the amount of L-states in laser image of such a layer decreases [see Figs.As for multiple-scattered skin dermis layer, the phase modulation, which arises due to linear phase anisotropy with subsequent interference of partial coherent waves, appears to be the prevailing mechanism of formation of local polarization states.Such a mechanism optically reveals in the essential increase of probability of elliptically polarized states formation (including C-states) under the simultaneous decrease of the total amount of L-states [see Figs.3(c) and 3(f)].Probably, the processes of multiple scattering of polarized laser radiation by the optically anisotropic collagen network reveal in subsequent decrease (increase) of the value of the ¯rst (second) statistic moments, which characterize the N L ðxÞ distribution for the group 3 objects.
Quantitatively speci¯ed regularities of L-states formation in laser images of biological layers of all groups reveal the following facts.
The values of high-order statistic moments, which characterize the distributions of linearly polarized states N L ðxÞ, are considerably small in comparison with the statistic moments of the 1st-2nd orders -Z L j¼3;4 ( Z L j¼1;2 .The distinctions between the distributions of L-states in laser images of various PhIL, which are observed as variations of the 1st and 2nd statistic moments -Z L 1 ¼ 0:62; Z L 2 ¼ 0:105 (group 1); Z L 1 ¼ 0:39; Z L 2 ¼ 0:19 (group 2) and Z L 1 ¼ 0:19; Z L 2 ¼ 0:28 (group 3) can be used as classi¯cation and diagnostic parameters of object optical properties.Apparently, for PhIL of the 1st, 2nd and 3rd groups, the mean value Z L 1 is by 1.5 and 3.3 times decreased.In addition, vice versa, the dispersion Z L 2 is by 1.5 and 2.5 times increased.

AEC-states of laser images
Summarized in Fig. 4 is the series of coordinate (V 4 ðm Â nÞ ¼ 1) and quantitative (N AEC ðxÞ) distributions for polarization-singular AEC-states in laser images of PhIL.The absence of C-points in the image of the object of group 1 is designated as \À absent" in fragments (a, d).
The comparative analysis of data presented in Fig. 4 with the parameters of L-points networks (see Fig. 3) reveals su±cient distinctions between them.
In the laser image of the group 1, AEC-states of polarization are absent (see Fig. Optical manifestations of the anisotropic layer of collagen ¯brils are illustrated by the network of AEC-points in the laser image [see Fig. 4(b)].One can see that for laser images of optically thin skin dermis layer (group 2) there is a considerabe large amount of C-points.This fact is the evidence of su±ciently probable forming of the right-and left-circulated polarization components of laser radiation transformed by the set of birefringent collagen ¯bers [relations (7), (8)].
On the other hand, there exist the pronounced intergroup distinctions between the statistic [see Figs.

Polarization-Singular Classi¯cation and Di®erentiation of Optical Properties Inherent to PhIL
In order to de¯ne the possibilities of classi¯cation and di®erentiation of all types of biological layers, a comparative investigation of distributions of polarization singularities amount of laser images within the reliable amount of histological sections samples was performed.The histological sections of dermis samples were obtained from the autopsied material of 21 corpses.The statistical validity of the sampling is proved by con¯dence interval for each group p 0:01.The statistically averaged (within the limits of groups 1 to 3) values and ranges of changing statistic moments Z L;AEC j¼1;2;3;4 that characterize the N AEC ðxÞ dependences for the amount of singular states in laser images of PhIL have been illustrated in Tables 1 and 2.
The performed analysis of the results presented in Tables 1 and 2 for statistic (Z L;AEC j¼1;2;3;4 Þ parameters shows: . the signature of predominance of surface scattering reveals in minimal values of statistic moments of the third and the fourth orders, which characterize the distributions of L-states in the sensibe absence of AEC-states; .for optically thin anisotropic biological layer and for optically thick biological layers the tendency of formation of statistically similar distributions Land AEC-states is the most typical; .the di®erentiation of such types biological layers is possible using both the 1st and the 2nd statistic moments, which characterize the distributions of Lpolarization states, and the 3rd and the 4th statistic moments, which characterize the distributions of AEC-polarization states; .The di®erence between them reaches 1.5-2.6 (Z 1 ) and 2-3 (Z 2 ) times for L-states and 5 (Z 3 ) -7 (Z 4 Þ times for AEC-states.

Conclusions
From the results of performed investigations, one can conclude: . The main mechanisms of formation of polarizationsingular states of laser images and interconnection with optical-geometric structure of surface, optically thin and optically thick skin derma layers have been determined. .The universal, azimuthally stable technique of de¯nition of polarization-singular structure of laser images of biological layers of di®erent types using the statistic analysis of the distribution of the number of characteristic values of the fourth Stokes vector parameter (S 4 ¼ 0; AE1) has been elaborated. .The values and the ranges of changes of statistic moments of the ¯rst to the fourth orders, which characterize the distributions of the number of characteristic values (S 4 ¼ 0; AE1) and can be used in diagnostics and di®erentiation of optical properties of surface, optically thin and optically thick skin derma layers have been de¯ned.
3(b) and 3(e)].Statistically, such a process is displayed in the decrease of the ¯rst statistic moment of N L ðxÞ distribution and, vice versa, in the increase of the second statistic moment.

Fig. 3 .
Fig. 3. Coordinate V 4 ðm Â nÞ ¼ 0 (fragments a, b, c) and quantitative N L ðxÞ distributions (fragments d, e, f ) of L-states in polarization for laser images of the PhIL.
4, left columnfragments a, d), which once again con¯rms the adequacy of model conceptions of mechanisms providing transformation of laser radiation by the set of chaotically oriented micro-areas of the rough surface [relation (2)].
4(d)À4(f)] structures of N AEC ðxÞ distributions in laser images of all groups [see Figs.4(a)À4(c)].This fact represents the background for not only the classi¯cation, but also di®erentiation of biological PhIL optical properties.

Table 1 .
Statistic parameters for the distribution of the amount of L-states in laser images of PhIL.

Table 2 .
Statistic parameters for the distribution of the amount of AEC-states in laser images of PhIL.