Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals

Represented in this work are theoretical basics for description of fields created by scattered coherent radiation with using the new correlation parameter – degree of local depolarization (DLD). The authors have adduced data of measurements of coordinate distributions for DLD in laser images of heal...

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Дата:	2011
Автори:	Ushenko, Yu.A., Angelsky, A.P.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2011
Назва видання:	Semiconductor Physics Quantum Electronics & Optoelectronics
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Цитувати:	Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals / Yu.A. Ushenko, A.P. Angelsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 41-50. — Бібліогр.: 23 назв. — англ.

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spelling	irk-123456789-1176072017-05-26T03:03:47Z Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals Ushenko, Yu.A. Angelsky, A.P. Represented in this work are theoretical basics for description of fields created by scattered coherent radiation with using the new correlation parameter – degree of local depolarization (DLD). The authors have adduced data of measurements of coordinate distributions for DLD in laser images of healthy and pathologically changed skin of a rat. Investigated are the values and ranges of changes in statistical (moments of the first to fourth orders), correlation (correlation area) and fractal (slopes and dispersion of extremes of logarithmic dependences for power spectra) parameters of coordinate distributions for DLD. Defined are objective criteria for diagnostics of oncological changes in the structure of rat skin. 2011 Article Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals / Yu.A. Ushenko, A.P. Angelsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 41-50. — Бібліогр.: 23 назв. — англ. 1560-8034 PACS 78.20.Fm, 87.64.-t http://dspace.nbuv.gov.ua/handle/123456789/117607 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Represented in this work are theoretical basics for description of fields created by scattered coherent radiation with using the new correlation parameter – degree of local depolarization (DLD). The authors have adduced data of measurements of coordinate distributions for DLD in laser images of healthy and pathologically changed skin of a rat. Investigated are the values and ranges of changes in statistical (moments of the first to fourth orders), correlation (correlation area) and fractal (slopes and dispersion of extremes of logarithmic dependences for power spectra) parameters of coordinate distributions for DLD. Defined are objective criteria for diagnostics of oncological changes in the structure of rat skin.
format	Article
author	Ushenko, Yu.A. Angelsky, A.P.
spellingShingle	Ushenko, Yu.A. Angelsky, A.P. Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Ushenko, Yu.A. Angelsky, A.P.
author_sort	Ushenko, Yu.A.
title	Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals
title_short	Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals
title_full	Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals
title_fullStr	Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals
title_full_unstemmed	Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals
title_sort	degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/117607
citation_txt	Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals / Yu.A. Ushenko, A.P. Angelsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 1. — С. 41-50. — Бібліогр.: 23 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv	AT ushenkoyua degreeoflocaldepolarizationdeterminedforfieldsoflaserradiationscatteredbymultilayerbirefringentnetworksofproteincrystals AT angelskyap degreeoflocaldepolarizationdeterminedforfieldsoflaserradiationscatteredbymultilayerbirefringentnetworksofproteincrystals
first_indexed	2025-07-08T12:32:09Z
last_indexed	2025-07-08T12:32:09Z
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. PACS 78.20.Fm, 87.64.-t Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringentnetworks of protein crystals Yu.A. Ushenko1, A.P. Angelsky2 1Chernivtsi National University, Department for Correlation Optics, 2, vul. Kotsyubins’kogo, 58012 Chernivtsi, Ukraine; yuriyu@gmail.com 2Chernivtsi National University, Department for Optics and Spectroscopy, 2, vul. Kotsyubins’kogo, 58012 Chernivtsi, Ukraine Abstract. Represented in this work are theoretical basics for description of fields created by scattered coherent radiation with using the new correlation parameter – degree of local depolarization (DLD). The authors have adduced data of measurements of coordinate distributions for DLD in laser images of healthy and pathologically changed skin of a rat. Investigated are the values and ranges of changes in statistical (moments of the first to fourth orders), correlation (correlation area) and fractal (slopes and dispersion of extremes of logarithmic dependences for power spectra) parameters of coordinate distributions for DLD. Defined are objective criteria for diagnostics of oncological changes in the structure of rat skin. Keywords: laser, polarization, complex degree of coherency, birefringence, correlation, statistical moments, fractal. Manuscript received 10.09.10; accepted for publication 02.12.10; published online 28.02.11. 1. Introduction By tradition, light scattering processes in phase- inhomogeneous biological objects are considered in a statistical approach (theory of radiation transfer [1], Monte-Carlo modeling [2]). Use of modern laser technique in investigations of biological tissues (BT) stimulates development of new approaches to an analysis and description of fields inherent to coherent radiation scattered by them. In recent 10 to 15 years, in optics of scattering media there arose a separate direction – laser polarimetry [3 – 31]. Being based on it and using the approach of single light scattering, there determined are interrelations between the set of statistical moments of the first to fourth orders [5 - 7, 11, 15, 20, 26, 27, 31], correlation functions [13, 18, 19, 22, 27], fractal dimensionalities [6, 7, 26], networks of polarization singularities [23, 29] that characterize the distribution of polarization states for laser fields, and parameters of optical anisotropy inherent to optically thin BT layers. When multiple scattering takes place, polarization information upon the BT structure becomes ambiguous and, as a consequence, loses its diagnostic sense [9, 10, 12, 16, 19]. This phenomenon got the name of depolarization of optical radiation due to statistical averaging the polarization states [12, 16, 19]. On the other hand, when scattered is coherent laser radiation, the speckle field with 100 % degree of local polarization is formed [8]. Thereof, it seems topical to develop new correlation approaches [3, 4] to descriptions of mechanisms providing depolarization of laser radiation multiply scattered in layers of optically thick BT. Our work is aimed at studying the possibilities for diagnostics of optically anisotropic BT components (epithelial and dermal skin layers) with application of statistical, correlation and fractal analyses for coordinate distributions inherent to local depolarization degree in the case of multiply scattered laser radiation. 2. Optical modeling for processes providing conversion of laser radiation parameters by a skin layer As a base to analyze the structure of laser radiation field converted by a skin layer, we use the following model [5-7, 14-17, 19, 30]: © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine • we consider this layer as a two-component system consisting of an optically anisotropic derma and rough surface optically isotropic epithelium (Fig. 1); 41 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. Fig. 1. On the analysis of optical modeling aimed at polarization-phase properties of the skin layer. ρi is the direction of fibril optical axis with the birefringence index Δn; γk - slope angle of the epithelium plate, which is formed by directions of micro- normal mj and macro-normal N; (E0), (EA) and (E*) - Maxwell’s vectors of the illuminating laser beam scattered by the collagen network and epithelium layer. • optical properties of the network inherent to birefringent collagen fibrils in the derma layer are characterized with the Jones matrix [27] { } 2221 1211 aa aa A = , (1) where ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )⎪ ⎩ ⎪ ⎨ ⎧ δ−ρ+ρ= δ−−ρρ== δ−ρ+ρ= =δρ .expcossin ;exp1sincos ;expsincos ,, 22 22 2112 22 11 rirra rirraa rirra raik (2) Here, is the direction of optical axis; ρ ndΔλ π=δ 2 - phase shift between amplitude orthogonal components; - wavelength; - geometric distance; - index of birefringence. λ d nΔ • optical properties of surface epithelium plane plates are characterized with the Jones operator of the following look [16, 19] { } ; 10 0 y x p p P = (3) • net Jones matrix for the skin layer { }D is determined with the product of partial operators (1) and (3) { } { }{ } ( ) ( ( ) ( © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine ) ) . 2222122121221121 2212121121121111 2221 1211 2221 1211 apapapap apapapap aa aa pp pp APD ++ ++ = =×== (4) • conversion processes for the amplitude ( )yx EE , and phase δ of laser radiation in the skin layer are described with the following matrix equation ( ) { } ( )( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ↔= 0 0 2221 12110 y x y x E E dd dd E E EDE , (5) where ( )( )0E and ( )E are the Jones vectors for illuminating and converted laser beams; ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) .; ; exp ; exp 000 00 0 0 0 yxxyyxxy xyy x y x xyy x y x iU U E E iU U E E δ−δ=δδ−δ=δ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ δ− =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ δ− =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ (6) Here, is the phase shift between real parts , of orthogonal components , of laser wave; xyδ xU yU xE yE • polarization-phase properties of every field point in the case of laser radiation scattered in skin are described with the coherence matrix { [18, 19, 25] }J { } ( ) ( ) ( ) ( ) . exp exp 2 2 yxyyx xyyxx yyxx xxxx yyyx xyxx UiUU iUUU EEEE EEEE JJ JJ J δ δ− = === ∗∗ ∗∗ (7) Using the operator ( ){ }rJ , let us characterize the polarization degree ( )rT of the scattered radiation field with the classic expression {4, 18] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] . ,,,, ,,,,,,,,4 1 2 ττ+ττ ⎥⎦ ⎤ ⎢⎣ ⎡ ττττ−ττττ −= = ∗∗ ∗∗∗∗ rErErErE rErErErErErErErE rT yyxx xyyxyyxx (8) Here denotes the operation of averaging by time. In our case of the field inherent to scattered coherent radiation ( EE = ) and ( xyxy δδ = ), the expression (8) is transformed to some constant ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] .0.1 4 1 2 = + − −= ∗∗ ∗∗∗∗ rErErErE rErErErErErErErE rT yyxx xyyxyyxx (9) In other words, all the points of polarization- inhomogeneous field of scattered laser radiation are 100 % polarized, ( ) 0.1=rT . 3. Correlation nature of the depolarization degree for the field of laser radiation In [4], to describe depolarization of these fields, the authors offered another, “two-point” correlation approach. The main its idea consists in determining the degree of correlation similarity between orthogonal components of laser wave amplitudes in the points and within the field of scattered coherent radiation [22, 27]. To qualitatively estimate this yx EE , 1r 2r 42 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. similarity, they used the parameter of a complex degree of coherency (CDC) ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⋅ =μ ◊ ),(),( ),(),(),( 2211 2121 21 rrTrWrrWTr rrWrrWTrrr . (10) Here, is the transversally spectral density matrix )r,( 21rW ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ∗∗ ∗∗ )()()()( )()()()( ),( 2121 2121 21 rErErErE rErErErE rrW yyxy yxxx , (11) where is the Hermitian conjugated matrix to the matrix ; ),( 21 rrW ◊ ),( 21 rrW Tr - matrix spur. Let us consider the possibility to use the above correlation approach in description of a polarization- inhomogeneous field formed by the layer of optically anisotropic derma (relations (1), (2)) with a rough surface (relation (3)). With this purpose, let us rewrite the expression (11) in the following manner )()()()( rDrWrDrW inout ⋅⋅= ◊ . (12) Here, is the Jones matrix (4) for the skin layer in the point )(rD r ; - Hermitian conjugated Jones’ matrix; transversally spectral density matrix for the probing beam )(rD◊ )(rWin ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ∗∗ ∗∗ )()()()( )()()()( )( 0000 0000 rErErErE rErErErE rW yyxy yxxx in . (13) Our analysis of the complex expression (12) with account of (1) – (4) and (12), (13) enabled us to reveal that the CDC module ( )rμ in every point r of the scattered coherent radiation field is a constant value coinciding with the polarization degree T(r) ( ) ( ) 0.1==μ rTr . (14) On the other hand, some new additional information possibility to describe scattered laser radiation is brought by the phase ( )( ) ( )rr Φ≡μΦ of CDC that is determined as ( ) ( ) ( ) ( ) ( ) ( )rUrU rrUrU r yx xyyx 22 cos2 + δ =Φ . (15) An explicit form of the value for the phase shift between amplitude orthogonal components for laser wave in the point r can be determined from the matrix equation (5): ( )rxyδ ( ) ( )rUrU yx , ( ) ( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( )[ Let us rewrite the expression (16) (without reduction in analysis completeness) for the case when the skin layer is illuminated with a laser wave linearly polarized with the azimuth 00 ( ) ( ) ( ) 0;0.1 00 =≡ yx EE ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( ) . cos1 sin cos1sin1 sinsin exp1sincosRe exp1sincosIm expsincosRe expsincosIm Re Im Re Im 2 2 22 22 21 21 1111 1111 ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ δ+ δ − ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ δ−ρ− δρ = ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ δ−−ρρ δ−−ρρ − − ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ δ−ρ−ρ δ−ρ−ρ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ =δ y y x x y y x x xy arctgarctg i i arctg i iarctg a aarctg ap aparctgr (17) As seen from the analysis of (17), the dominant role in formation of phase modulation inherent to the field of scattered laser radiation is played by the subsurface birefringent layer of collagen fibrils. Moreover, the range of changes in phase shifts ( )rxyδ is maximum high π≤δ≤ xy0 . (18) The condition (18) corresponds to the following range of changes in CDC phase: ( ) 0.10 ≤Φ≤ r . (19) Thus, to use the parameter is efficient when determining the measure of correlation between orthogonal components of the complex amplitude in different points ( )rΦ r within the field of scattered laser radiation. In this sense, this parameter will be named as the degree of local depolarization (DLD) for points of polarization-inhomogeneous coherent field. ( )rΦ 4. Optical setup and method for measuring thr degree of local depolarization in the field of scattered laser radiation Shown in Fig. 2 is the optical setup for measuring coordinate distributions of DLD in the field of laser light transformed by a skin layer [5]. As objects for investigation, we used optically thick (geometric thickness d = 75 µm, extinction coefficient 75.0≤τ ) histological sections of rat skin prepared using the standard technique with a freezing microtome [4]. ] ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] . Re Im Re Im 0 22 0 21 0 22 0 21 0 12 0 11 0 12 0 11 ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ + + − − ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ + + =δ yx yx yx yx xy ErdErd ErdErd arctg ErdErd ErdErd arctgr (16) Illumination of the rat skin histological sections was made by a parallel beam (Ø = µm) of the Hе-Nе laser (λ = 0.6328 µm) 1. Using the polarizer 4 and quarter-wave plate 5, we could form arbitrary states of polarization for the probing laser beam. The image of the skin samples 6 was projected with the micro-objective 7 onto the plane of light-sensitive area ( 410 pixpixnmr 600800 ×=×≡ ) of CCD camera 10. © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 43 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. The method of measuring coordinate distributions of DLD consists of the following stages: ( )rΦ © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine Fig. 2. Optical setup for measurements: 1 – He-Ne laser; 2 – collimator; 3, 5, 8 – quarter-wave plates; 4, 9 – polarizer and analyzer; 6 – object; 7 – micro-objective (x4); 10 – CCD camera; 11 – personal computer. ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ nmn rr rr I ... ....... ... 1 111 min rotation angles ⎞ ⎜ ⎜ ⎜ ⎛ ≡ ⎟ ⎟ ⎟ ⎞ ⎜ ⎜ ⎜ ⎛ ⎟ ⎟ ⎟ ⎞ ⎜ ⎜ ⎝ Θ min 1111 ......... ... ......... ... I rr I r r m n m . for the azimuth and ellipticity of laser field polarization 1. The transmission plane of the polarizer- analyzer 9 is sequentially oriented under the angles , and measured are the respective discrete ( ) sets of values for intensities , . 00=Θ 090=Θ nm× ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ nmn m rr rr I ... ......... ... 1 111 0 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ nmn m rr rr I ... ......... ... 1 111 90 2. Using rotation of the analyzer transmission plane within the limits , one determines a set of values for maximum and minimum intensity levels , ⎟ ⎟m .. for every separate pixel of the CCD camera as well as respective ⎜ ⎛ 11r 00 1800 ÷=Θ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ nmn m rr rr I ... ......... ... 1 111 max ⎟ ⎟ ⎟ ⎟ ⎠⎝ ⎠⎝⎠ 11 ...... rrr nmnnm 3. Then, calculated are the coordinate distributions ( )( ) ( ) ( ) . ... max 1 rnm ⎟ ⎠ ......... ... ; 2 ... ......... ... min 111 min 1 111 rI rI arctg r rr rI rr rr n m nmn m =⎟ ⎟ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ β π −Θ= ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ α (20) 4. Using (20), one calculates the coordinate distributions for phase shifts ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ δ nmn m rr rr ... ......... ... 1 111 ( ) ( ) ( ) .2 ⎥ ⎦ ⎢ ⎣ α rtg ⎜ ⎝ nmn rr ...1 ( ) ( ) ( ) ( ( ) ( ) ) rIrI rrIrI arctgr 900 5,0 90 5,0 0 cos + δ =Φ . (22) 5. Algorithms for statistical, correlation and fractal analyses of coordinate distributions for the degree of local polarization To get an objective estimation of the coordinate distributions ( )nymx ÷=−÷=Φ 1;11 inherent to laser images of histological skin sections, we used the complex statistical, correlation and fractal analysis. The set of statistical moments of the first to fourth orders for DLD distributions was calculated using the following relations [6, 7, 26, 31] Φ = 4,3,2,1jZ ∑ = Φ Φ= N i iN Z 1 1 1 , ∑ = Φ Φ= N i iN Z 1 2 2 1 , ( ) ∑ = Φ Φ Φ= N i iNZ Z 1 3 3 2 3 11 , ( ) ∑ = Φ Φ Φ= N i iNZ Z 1 4 4 2 4 11 . (23) Here, nmN ×= is the number of pixels in the CCD camera. As a base for the correlation analysis of DLD distributions we took the method of autocorrelation with using the function [19, 22, 27] ( ) ( )[ ] ( )[ ]∫ Δ−ΦΦ=Δ → Φ ÷= m ii m ni dmmmm m mK 1 0 1 1lim . (24) Here, ( )pixm 1=Δ is the step for changing the coordinates mx ÷=1 of the DLD distribution for the separate i -th line of pixels in the digital camera. ⎤⎡ β =δ rtgarctgr (21) 5. The coordinate distribution of DLD ⎜ ⎜ ⎛ Φ mrr ......... ... 111 can be obtained using the following relation ⎟ ⎟ ⎟ ⎠ ⎞ The net expression for the autocorrelation function was obtained by averaging the partial functions (24) over all the lines i n÷1 = ( ) ( ) n mK mK n i i∑ = Φ Φ Δ =Δ 1 . (25) To qualitatively characterize the autocorrelation de- pendences ( )mK ΔΦ , we chose the “correlation area” ΦS ( ) ; 1 ∫ Δ= ΦΦ m dmmKS (26) 44 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine The fractal analysis of )( nm×Φ distributions was based on calculatio f the logarithmic depend pectra where are the spatial frequencies that determined by geometrical dimensions of struc elements in the skin layer. e de t d us to deter ns o ences ( ) 1loglog −−Φ dJ for power s ( ) ∫ +∞ νπνΦ=Φ dJ 2cos , (27) ( )ΦJ ∞− 1−=ν d are tural d Th pendences ( ) 1loglog −−Φ dJ were approximated using the least squares method to the curves ( )ηV , straight par s of which enable mine the slope angles η and corresponding fractal dimensionalities ΦF [6, 7] η−=Φ tgF 3 . (28) Classification he c )( nm×Φ was performed in of t oordinate distributions accord with the following criteria [23]: angle • )( nm×Φ is fractal or self-similar when the slope onstant consis c t=η within the limits of 2 or 3 decades in changi nsions ; ng the dime d • are multi-fractal when several slope angles of ( ) )( nm×Φ ηV are ; • )( nm×Φ is random when no available stable slope angles of ( )ηV are available over the whole range of changing dimension s All racte d . the distributions ( ) 1loglog −−Φ dJ were cha rized with the dispersion ( )[ ]∑ = Φ iD 21 . 6. Statistical, correlation and frac of coordinate distributions for the degree depolarization of laser field light scattered by skin ns lly d for crossed and app ental v , there th types enabled us to reveal their phase- inho −−Φ= i dJ N 1 loglog1 (31) N tal parameters of local layers in different physiological states Shown in Fig. 3 are laser images of histological sectio inherent to healthy (fragment (a)) and oncologica changed (fragment (b)) rat skin obtaine transmission planes ( 090=Θ ) of the polarizer 4 and analyzer 9 (Fig. 2). Our choice of the studied objects was stipulated both with fundamental lied reasons. From the fundam iewpoint, the sample of skin with rough surface is a classical example of multilayer biological tissue providing simultaneous manifestation of various mechanisms (both bulk (1), (2) and surface (3)) in the scattering of coherent radiation, and one can observe polarization-inhomogeneous field. From the applied viewpoint, the samples of rat skin allows to experimentally create oncological states and fore, can be efficiently used in optimization of correlation-phase laser diagnostics of pathological changes. Our analysis of laser images corresponding to the skin of bo mogeneous structure (relations (6), (8) – (16)) that can be visualized with crossed polarizer and analyzer. In the case of pathologically changed skin (Fig. 3, fragment (b)), there observed is a higher level of blooming. This fact can be explained by growth of birefringence in newly created collagen fibrils [5, 8, 10, 14, 17, 24, 25, 31] as well as increasing phase modulation ( ( ) ( )maxmin xyxy δ↔δ ) in the points of scattered radiation field. Quantitatively, these mechanisms of coherent radiation conversion are characterized by the value and range of DLD changes (relation (19)) in laser images of rat skin in different physiological states (Fig. 4). From the obtained laser images of histological sections prepared from the skin of both types, it is seen that these are characterized with the widest ( 10 ≤Φ≤ ) range of DLD changes. At the same time, the coordinate structure of Φ(m×n) distributions is rather com d individual both for healthy (fragment (a)) and oncologically changed (fragment (b)) skin. plex an Fig. 3. Laser images of histological sections inherent to healthy (a) and oncologically changed (carcinoma) (b) rat skin. Fig. 4. Coordinate distributions Φ(m×n) of laser images for healthy (fragment (a)) and oncologically changed (fragment (b)) skin. 45 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. Fig. 5. Coordinate (fragments (a), (b)), correlation (fragments (c), (d)) and fractal (fragments (e), (f)) structures of the sampling Φ(m×n) = 1 of two-dimensional DLD distributions in laser images of histological sections of healthy rat (fragments (a), (c), (e)) and that sick of cancer (fragments (b), (d), (f)). See explanations in the text. To determine statistical (relation (23)), correlation (relations (24)) and fractal (relations (26) – (28)) parameters of coordinate DLD distributions sensitive to changes in the structure of laser images, we used the following method of information selection. From the whole set, we chose characteristic or extreme values: ( nm×Φ © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine ) • “minimal” ( ) that correspond to absence of phase modulation (δ 0min =Φ xy = 0, relation (15)) of orthogonal components inherent to the laser wave amplitude. From the physical viewpoint, this sampling corresponds to mechanisms of radiation transformation (extinction) by the optically isotropic skin component; • “mean” ( 5.0~ =Φ ) that correspond to the phase shifts ( 3 ~ π=δ ) formed by the network of collagen fibrils with the averaged statistical dimensions md μ≈ 75 , and the birefringence index . This sampling of DLD values is characteristic both to the healthy and pathologically changed optically anisotropic component of skin derma [14, 17, 30]; 3105.1 −×=Δn • “maximal” ( 1max =Φ ) that correspond to extreme phase shifts ( 2 π=δxy ). In our opinion, this level of phase modulation is more probable for some sub-ensemble of “newly-created” skin derma fibrils in the process of carcinoma formation. Within the limits of every column ( )( )mk pixpix n ...,,2,11 =× of two-dimensional array ( )nm×Φ by scanning along the horizontal direction mx ...,,1≡ with the step pixx 1=Δ , we calculated the amount ( ) of values N minΦ , – ( ); , – (( )kNmin Φ ~ ( )kN~ ) and maxΦ , – ( ( )kNmax ). Thus, we determined the dependences ( ) )...,,,( )( min )2( min )1( minmin mNNNxN ≡ ; ( ) )~...,,~,~(~ )()2()1( mNNNxN ≡ and 46 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine ( ) )...,,,( )( max )2( max )1( maxmax mNNNxN ≡ of the amount of extreme DLD values within the limits of a laser image for a skin histological section. Shown in Fig.5 to Fig. 7 are ( )nm×Φmin samplings (Fig. 5, fragments (a), (b)); ( nm×Φ )~ (Fig. 6, fragments (a), (b)); ( nm )×Φmax (Fig. 7, fragments (a), (b)); autocorrelation functions ( ),min xK Δ ( )xK Δ ~ , (fragments (c), (d)) as well as logarithmic dependences ( )xK Δmax 1 maxmin log,~,log −− dNNN for power spectra of distributions , )(min xN )(~ xN , (fragments (e), (f)), which characterize the coordinate DLD structure of laser images inherent to skin histological sections of healthy rat (left columns) and that sick of cancer (right columns). )(max xN Our comparative analysis of the set of statistical, correlation and fractal parameters characterizing the distributions , ( )xNmin ( )xN~ , enabled us to reveal: ( )xNmax • tendency to growth of the amount of extreme values minΦ in the coordinate DLD distribution of laser images corresponding to the layer of skin with carcinoma (Figs 5 to 7, fragments (a) and (b)); Fig. 6. Coordinate (fragments (a), (b)), correlation (fragments (c), (d)) and fractal (fragments (e), (f)) structures of the sampling Φ(m×n) = 0.5 for two-dimensional DLD distributions corresponding to laser images of histological sections taken from healthy rat (fragments (a), (c), (e)) and that sick with cancer (fragments (b), (d), (f)). See explanations in the text. • sequential ( ( ) ( ) ( )xNxNxN minmax ~ →→ ) and more fast decrease in the correlation area of autocorrelation functions characterizing the coordinate structure of extreme sampling for DLD of laser images corresponding to the histological section of healthy skin as compared with the respective parameter obtained for the samples of pathologically changed skin (Figs 5 to 7, fragments (c) and (d)); ( )xK Δ • transformation of fractal distributions ( )xNmax , ( )xN~ (Figs 5 and 6, fragments (e), (f)) to the statistical one (Fig. 7, fragment (e)) obtained for the sampling ( ) 0=×Φ nm . The obtained results can be related with growth of the concentration inherent to collagen proteins that form the fibrilar network (Fig. 3, fragment (b)) in the rat skin with carcinoma. This biochemical process results in 47 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. Fig. 7. Coordinate (fragments (a), (b)), correlation (fragments (c), (d)) and fractal (fragments (e), (f)) structures of the sampling Φ(m×n) = 0 for two-dimensional DLD distributions corresponding to laser images of histological sections taken from healthy rat (fragments (a), (c), (e)) and that sick with cancer (fragments (b), (d), (f)). See explanations in the text. increasing optical anisotropy of the subsurface derma layer as well as in respective growth of the probability for extreme values (relations (15) - (19)) in the coordinate DLD distribution of laser images corresponding to samples of pathologically changed skin. ( ) 0=×Φ nm In the case of healthy skin samples, the probability and, consequently, amount of these DLD values are significantly less. Using this basis, one can explain the lower values of correlation area (Fig. 7, fragments (c) and (d)) as well as absence of any stable slope of the approximating curve to the logarithmic dependence (Fig. 7, fragments (e) and (f)). ΦS ( ) 1 min loglog −− dNJ © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine From the quantitative viewpoint, the dependences , , characterize statistical , correlation and fractal ( )xNmin ( )xN~ ( )xNmax Φ −= 41iZ ΦS ΦF , parameters determined within the limits of two groups for skin samples (Table 1). ΦD The comparative analysis of the obtained data has shown that diagnostically sensitive to the oncological state are: • statistical moments of ( )min41 NZi Φ −= ( )xNmin distributions for the amount of extreme values ( ) 1=×Φ nm , differences between them reach 2 to 3 times; • correlation area for autocorrelation functions ΦS ( )xK Δ~ and ( )xK Δmin , differences between them reach 2 and 5 times; • ( )xNmin distribution for laser images corresponding to histological sections of a healthy rat and that sick of cancer are, respectively, random and self-similar; • dispersion of the logarithmic dependences ΦD ( ) 1 min loglog −− dNJ in the case of pathological changes is 2-fold decreased. 48 Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. Table 1. Statistical , correlation SΦ −= 41iZ Φ and fractal FΦ, DΦ parameters for the distributions Nmax(x), ( )xN~ , Nmin(x) of the amount of extreme values Φ(m×n) in laser images of skin histological sections N(x) Nmax(x) )(~ xN Nmin(x) Norm (q = 21) Cancer (q = 19) Norm (q = 21) Cancer (q = 19) Norm (q = 21) Cancer (q = 19) Φ 1Z 0.51±0.063 0.47±0.052 0.31±0.034 0.42± 0.045 0.09± 0.001 0.24 0.035 ± Φ 2Z 0.14±0.021 0.16±0.019 0.19±0.023 0.15± 0.018 0.43± 0.031 0.17 0.019 ± Φ 3Z 0.28±0.034 0.33±0.038 0.58±0.067 0.47± 0.065 1.65± 0.17 0.72 0.081 ± Φ 4Z 0.41±0.063 0.54±0.067 0.77±0.081 0.63± 0.072 2.11± 0.32 0.82 0.091 ± ΦS 0.25±0.023 0.21±0.021 0.18±0.013 0.09± 0.007 0.11± 0.013 0.02 0.001 ± ΦD 0.18±0.014 0.17±0.018 0.32±0,034 0.19± 0.025 0.48± 0.047 0.21 0.035 ± ΦF 1.98±0.016 1.93±0.07 2.17±0.05 2.03± 0.021 - 2.23 0.031 ± 7. Conclusions Demonstrated in this work is the possibility to use the parameter of degree of local depolarization for describing the processes of the multiple scattering of laser radiation by optically anisotropic biological tissues. Offered is a model approach to the analysis of correlation-phase structure formation in the field of laser radiation scattered by a birefringent skin layer with rough surface. Developed and tested is the experimental method for measuring the coordinate DLD distributions in laser images of rat skin histological sections. Being based on it, the authors have determined and grounded the set of statistical, correlation and fractal criteria for diagnostics of skin cancer. References 1. W.F. Cheong, S.A. Prahl, A.J. Welch, “A review of the optical properties of biological tissues,” IEEE J. of Quant. Elec. 26, 2166-2185 (1990). 2. S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch. A Monte Carlo model of light propagation in tissue. In G. J. Müller and D. H. Sliney, eds, SPIE Proceedings of Dosimetry of Laser Radiation in Medicine and Biology, volume IS 5, pp. 102-111, (Bellingham, WA, 1989. SPIE Opt. Eng. Press). 3. J.Ellis and A.Dogariu, “Complex degree of mutual polarization,” Opt.Lett. 29, 536-538 (2004). 4. Jani Tervo, Tero Setala, and Ari Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003) © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 5. O. V. Angelsky, A. G. Ushenko, Yu. A. Ushenko, V. P. Pishak, “Statistical and Fractal Structure of Biological Tissue Mueller Matrix Images”, in Optical Correlation Techniques and Applications, Oleg V. Angelsky, ed. (Washington: Society of Photo-Optical Instrumentation Engineers), pp. 213- 266 (2007). 6. O.V. Angelsky, A.G. Ushenko, Yu.A. Ushenko, V.P. Pishak, and A.P. Peresunko, “Statistical, Correlation, and Topological Approaches in Diagnostics of the Structure and Physiological State of Birefringent Biological Tissues”, in Handbook of Photonics for Biomedical Science, Valery V. Tuchin, ed. (USA: CRC Press), pp. 21- 67 (2010). 7. Alexander G. Ushenko and Vasilii P. Pishak, “Laser Polarimetry of Biological Tissue: Principles and Applications”, in Handbook of Coherent-Domain Optical Methods: Biomedical Diagnostics, Environmental and Material Science, Valery V. Tuchin, ed. (Boston: Kluwer Academic Publishers), pp. 93-138 (2004). 8. Alexander G. Ushenko, “Polarization structure of laser scattering fields,” Opt. Eng. 34, 1088-1093 (1995). 9. A.G. Ushenko, “Laser diagnostics of biofractals,” Quantum Electron. 29, 1078–1084, (1999). 10. O.V. Angel'skii, A.G. Ushenko, A.D. Arkhelyuk, S.B. Ermolenko, D.N. Burkovets, “Structure of matrices for the transformation of laser radiation by biofractals,” Quantum Electron. 29, 1074-1077 (1999). 11. O.V. Angel'skii, A.G. Ushenko A.D. Arheluk, S.B. Ermolenko, D. N. Burkovets, “Scattering of Laser Radiation by Multifractal Biological Structures,” Optics and Spectroscopy 88, 444-448 (2000). 12. A.G. Ushenko, “Polarization Structure of Biospeckles and the Depolarization of Laser Radiation,” Optics and Spectroscopy 89, 597-601 (2000). 49 http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=64354 http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=64354 http://omlc.ogi.edu/pubs/abs/prahl89.html http://omlc.ogi.edu/pubs/abs/prahl89.html Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 41-50. 13. A.G. Ushenko, “Stokes-correlometry of biotissues,” Laser Phys. 10, 1286-1292 (2000). 14. A.G. Ushenko, “The Vector Structure of Laser Biospeckle Fields and Polarization Diagnostics of Collagen Skin Structures,” Laser Phys. 10, 1143- 1149 (2000). 15. A.G. Ushenko, “Laser polarimetry of polarization- phase statistical moments of the object field of optically anisotropic scattering layers,” Optics and Spectroscopy 91, 313-316 (2001). 16. A.G. Ushenko, “Polarization contrast enhancement of images of biological tissues under the conditions of multiple scattering,” Optics and Spectroscopy 91, 937-940 (2001). 17. A.G. Ushenko, “Laser probing of biological tissues and the polarization selection of their images,” Optics and Spectroscopy 91, 932-936 (2001). 18. A.G. Ushenko, “Correlation processing and wavelet analysis of polarization images of biological tissues,” Optics and Spectroscopy 91, 773-778 (2002). 19. A.G. Ushenko, “Polarization correlometry of angular structure in the microrelief pattern or rough surfaces,” Optics and spectroscopy 92, 227-229 (2002). 20. O.V. Angelsky, A.G. Ushenko, Ye.G. Ushenko, “2- D Stokes Polarimetry of Biospeckle Tissues Images in Pre-Clinic Diagnostics of Their Pre-Cancer States,” J. Holography Speckle 2, 26-33 (2005). 21. Oleg V. Angelsky, Alexander G. Ushenko, and Yevheniya G. Ushenko, “Complex degree of mutual polarization of biological tissue coherent images for the diagnostics of their physiological state,” J. Biomed. Opt. 10, 060502 (2005). 22. O. V. Angelsky, A. G. Ushenko, and Ye. G. Ushenko, “Investigation of the correlation structure of biological tissue polarization images during the diagnostics of their oncological changes,” Phys. Med. Biol. 50, 4811-4822 (2005). 23. Oleg V. Angelsky, Alexander G. Ushenko, Yevheniya G. Ushenko, Yuriy Y. Tomka, “Polarization singularities of biological tissues images,” J. Biomed. Opt. 11, 054030 (2006). 24. O.G. Ushenko, S.G. Guminetsky, A.V. Motrich, “Optical properties of urine, blood plasma and pulmonary condensate of the patients with pulmovnary form of tuberculosis,” Photoelectronics 16, 133-139 (2007). 25. S.H. Guminetskiy, O.G. Ushenko, I.P. Polyanskiy, A.V. Motrych, F.V. Grynchuk, “The optical method for investigation of the peritonitis progressing process,” Proc. SPIE 7008, 700827 (2008). 26. Alexander Ushenko, Sergej Yermolenko, Alexander Prydij, Stepan Guminetsky, Ion Gruia, Ovidiu Toma, Konstantin Vladychenko, “Statistical and fractal approaches in laser polarimetry diagnostics of the cancer prostate tissues,” Proc. SPIE 7008, 70082C (2008). 27. A.G. Ushenko, A.I. Fediv, Yu.F. Marchuk, “Correlation and fractal structure of Jones matrices of human bile secret,” Proc. SPIE 7368, 73681Q (2009). 28. A.G. Ushenko, Yu.Ya. Tomka, V.I. Istratiy, “Polarization selection of two-dimensional phase- inhomogeneous birefringence images of biotissues,” Proc. SPIE 7388, 73881L (2009). 29. A.G. Ushenko, A.I. Fediv, Yu.F. Marchuk, “Singular structure of polarization images of bile secret in diagnostics of human physiological state,” Proc. SPIE 7368, 73681S (2009). 30. S.B. Yermolenko, A.G. Ushenko, P. Ivashko, “Spectropolarimetry of cancer change of biotissues,” Proc. SPIE 7388, 73881D (2009). 31. A.G. Ushenko, I. Z.Misevich, V. Istratiy, I. Bachyns’ka, A. P. Peresunko, Omar Kamal Numan, and T. G. Moiysuk, “Evolution of Statistic Moments of 2D-Distributions of Biological Liquid Crystal Net Mueller Matrix Elements in the Process of Their Birefringent Structure Changes,” Advances in Optical Technologies 2010, 423145 (2010). © 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 50 http://www.springerlink.com/content/119843/?p=9965306b055148b0aa7d0d8c7de0d65f&pi=0 http://www.springerlink.com/content/119843/?p=9965306b055148b0aa7d0d8c7de0d65f&pi=0 http://www.springerlink.com/content/119843/?p=9d37ef7cde024ae089812d29a37ffa85&pi=0 http://www.springerlink.com/content/119843/?p=3a87023873fd46849b78890e125f80cb&pi=0 http://www.springerlink.com/content/119843/?p=3a87023873fd46849b78890e125f80cb&pi=0 http://www.ingentaconnect.com/content/asp/jhs;jsessionid=69dj5lalsprb1.alexandra http://bookstore.spie.org/index.cfm?fuseaction=detailpaper&cachedsearch=1&volume=11&fpage=054030&coden=JBOPFO&producttype=pdf&CFID=3429818&CFTOKEN=56766105 http://bookstore.spie.org/index.cfm?fuseaction=detailpaper&cachedsearch=1&volume=11&fpage=054030&coden=JBOPFO&producttype=pdf&CFID=3429818&CFTOKEN=56766105 http://spie.org/app/profiles/viewer.aspx?profile=KHWTUA http://spie.org/x648.xml?product_id=831559 http://spie.org/x648.xml?product_id=853298 http://spie.org/x648.xml?product_id=853298 1. W.F. Cheong, S.A. Prahl, A.J. Welch, “A review of the optical properties of biological tissues,” IEEE J. of Quant. Elec. 26, 2166-2185 (1990).

Degree of local depolarization determined for fields of laser radiation scattered by multilayer birefringent networks of protein crystals

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