Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties
Performed in this work is a complex statistical, correlation and fractal analysis of coordinate distributions for Jones-matrix elements describing birefringent networks observed in the main types of human amino acids. Determined are the values and ranges for changing the statistical, correlation...
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Ushenko, Yu. O. Balanetsk, V. O. Angelsky, O. P. 2017-05-26T16:30:33Z 2017-05-26T16:30:33Z 2011 Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties / Yu. O. Ushenko, V. O. Balanetska, O. P. Angelsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 3. — С. 365-374. — Бібліогр.: 31 назв. — англ. 1560-8034 PACS 33.50.-j, 34.35.+a, 73.20.Mf, 78.30.-j https://nasplib.isofts.kiev.ua/handle/123456789/117766 Performed in this work is a complex statistical, correlation and fractal analysis of coordinate distributions for Jones-matrix elements describing birefringent networks observed in the main types of human amino acids. Determined are the values and ranges for changing the statistical, correlation and spectral moments of the 1-st to 4-th orders that characterize Jones-matrix images of these biological polycrystalline protein structures. We have ascertained objective criteria for classification and differentiation of optical properties inherent to polycrystalline networks of amino acids with different types of spatial symmetry – dendrite, spherolite and cluster ones. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties Article published earlier |
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Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties |
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Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties Ushenko, Yu. O. Balanetsk, V. O. Angelsky, O. P. |
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Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties |
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Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties |
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Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties |
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jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties |
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Ushenko, Yu. O. Balanetsk, V. O. Angelsky, O. P. |
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Ushenko, Yu. O. Balanetsk, V. O. Angelsky, O. P. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Performed in this work is a complex statistical, correlation and fractal analysis
of coordinate distributions for Jones-matrix elements describing birefringent networks
observed in the main types of human amino acids. Determined are the values and ranges
for changing the statistical, correlation and spectral moments of the 1-st to 4-th orders
that characterize Jones-matrix images of these biological polycrystalline protein
structures. We have ascertained objective criteria for classification and differentiation of
optical properties inherent to polycrystalline networks of amino acids with different types
of spatial symmetry – dendrite, spherolite and cluster ones.
|
| issn |
1560-8034 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117766 |
| citation_txt |
Jones-matrix images corresponding to networks of biological crystals for diagnostics and classification of their optical properties / Yu. O. Ushenko, V. O. Balanetska, O. P. Angelsky // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 3. — С. 365-374. — Бібліогр.: 31 назв. — англ. |
| work_keys_str_mv |
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| first_indexed |
2025-11-24T18:37:47Z |
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2025-11-24T18:37:47Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
PACS 33.50.-j, 34.35.+a, 73.20.Mf, 78.30.-j
Jones-matrix images corresponding to networks of biological crystals
for diagnostics and classification of their optical properties
Yu. O. Ushenko, V. O. Balanetska, O. P. Angelsky
1Chernivtsi National University
2, Kotsyubinsky vul., 58012 Chernivtsi, Ukraine
Abstract. Performed in this work is a complex statistical, correlation and fractal analysis
of coordinate distributions for Jones-matrix elements describing birefringent networks
observed in the main types of human amino acids. Determined are the values and ranges
for changing the statistical, correlation and spectral moments of the 1-st to 4-th orders
that characterize Jones-matrix images of these biological polycrystalline protein
structures. We have ascertained objective criteria for classification and differentiation of
optical properties inherent to polycrystalline networks of amino acids with different types
of spatial symmetry – dendrite, spherolite and cluster ones.
Keywords: laser, polarization, birefringence, Jones matrix, amino acid, statistical
moment, autocorrelation, power spectrum.
Manuscript received 30.08.11; accepted for publication 14.09.11; published online 21.09.11.
1. Introduction
Among the methods for optical diagnostics of biological
layers, the most widely spread are those of laser
polarimetric diagnostics aimed at optical-anisotropical
structure inherent to human tissues [1 - 31]. The main
“information product” of these methods is obtaining the
coordinate distributions for elements of Mueller and
Jones matrixes corresponding to biological tissues (BT)
[1 - 5] with the following statistical (statistical moments
of the first to fourth orders [5, 6, 10, 14, 19, 25, 26, 30]),
correlation (auto- and mutual-correlation functions [12,
17, 18, 21, 26]), fractal (fractal dimensionalities [5, 6,
25]), singular (distributions of amounts of linear and
circularly polarized states), wavelet (sets of wavelet
coefficients for various scales of biological crystals [22,
28]) analyses. As a result, one can determine
interrelations between a set of these parameters and
distributions of optical axis directions as well as the
birefringence value inherent to networks of optically
single-axis protein (myosin, collagen, elastin, etc.) fibrils
in optically-anisotropic component of BT layer. Being
based on this approach, a large amount of methods for
diagnostics and differentiation of pathological changes
in BT structure that are related with their degenerative-
dystrophic as well as oncological changes [4 - 6, 12, 19,
20-22, 27, 29, 31]. The following progress in
diagnostical possibilities of laser polarimetry can be
related with account of anisotropy typical not only to
fibrillar networks but to more deep level of their
organization – structure of polycrystalline amino acids.
Our work is aimed at searching the possibilities for
diagnostics and classification of optical properties
inherent to polycrystalline networks of the main types of
human amino acids via determination of coordinate
distributions corresponding to Jones-matrix elements
with the following statistical, correlation and fractal
analyses of these distributions.
2. Main analytical relations
As a base for modeling the optical properties of
polycrystalline networks corresponding to main types of
amino acids in biological liquids, we took the following
conceptions developed for optically-anisotropic protein
fibrils [1-4, 7, 9, 14, 16, 23-27, 30]:
• separate (partial) amino acid crystals are
optically single-axis and birefringent;
• optical properties of a partial crystal is
exhaustively full described with the Jones operator [5]
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
365
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Here, is the direction of the optical axis; ρ
ndΔλ
π=δ 2 – phase shift between orthogonal
components and of the amplitude of illuminating
laser wave with the wavelength ; - birefringence
index for the crystal with geometric dimension .
xE yE
λ nΔ
d
The Jones matrix of a flat layer corresponding
to polycrystalline network can be defined as a sum of
operators for
{ }ℑ
{ }lJ K separate amino acid crystals [5,
11, 30]
{ } { }∑
=
=ℑ
K
l
lJ
1
, (2)
It can be shown that for Q sequentially placed flat
layers the net Jones matrix is defined with the expression
{ } { } { } { } { } { }121
1
... ℑ×ℑ××ℑ×ℑ=ℑ=ℵ −
=
∏ QQ
Q
q
q . (3)
In an expanded version, the matrix elements ikℵ
possess a cumbersome analytical look. Therefore, to
consider the structure of generalized Jones matrix }{ℵ
in more convenient manner, we limit ourselves (without
losses in fullness of analysis) by the case of bilayer
( ) polycrystalline network. With account of this
approximation as well as … look of dependences
(relation (1)) for matrix elements , it is possible
to show that the elements of generalized Jones matrix
can be expressed with the following dependences
2=Q
( δρ,ikJ )
( ) ( ) ( ) ( ) ( )δ−ρ+δ−ρ+ρ=ℵ iUiTR ikikikik 2expexp . (4)
Here, are the coefficients
expressed via quasi-harmonic functionals
( ) from coordinate changes in
orientations of optical axis
ikikik UTR ;;
sin"cos";"sin";"cos~" 22 −
( )yx,ρ
{ } ( ) ( )[ ]
( )[ ] ( );expcossin;exp1sincos
;exp1sincos;expsincos
22
22
2221
1211
δ−ρ+ρδ−−ρρ
δ−−ρρδ−ρ+ρ
==
ii
ii
JJ
JJ
J . (1)
Our comparative analysis of relations (1), (2) and
(3) – (7) has shown that the coordinate distributions of
ikℵ elements in Jones matrixes describing multi-layer
polycrystalline networks of optically birefringent
crystals simultaneously possess various properties:
• complex multi-parametric dependences of ( )δρℵ ,ik
values on specificity of distributions inherent to
orientation ( )ρf and phase parameters; ( )δg
• ikℵ distributions are superposition of various
harmonic components ( ); sin"cos";"sin";"cos~" 22 −
• dependences of matrix elements are scale
repeated.
ikℵ
Thus, an objective analysis of coordinate
distributions for elements of the Jones matrixes
corresponding to polycrystalline networks needs the
complex (statistical, correlation and fractal) analytical
approach.
3. Criteria for estimation of Jones-matrix images
corresponding to polycrystalline networks
Distributions of values inherent to elements ikℵ of the
Jones matrix can be characterized with the set of
statistical moments of the 1-st to 4-th orders M; σ; A; E
calculated using the following relations [5, 6, 25, 30]:
( ) ( )
( ) ( ) .11,11
,1,1
1
4
4
1
3
3
1
2
1
∑∑
∑∑
==
==
ℵ
σ
=ℵ
σ
=
ℵ=σℵ=
N
j
jik
N
j
jik
N
j
jik
N
j
jik
N
E
N
A
NN
M
(8)
where is the amount of local values ( within
the limits of coordinate distribution corresponding to the
Jones-matrix image.
N ) jikℵ
( ) ( ) ( )
( ) ( ) ( )[ ( )
( ) ( )[ ] ( )⎪
⎪
⎩
⎪⎪
⎨
⎧
δ−ρΔ+ρρ+ρΔ+ρρ=
δ−ρΔ+ρρ−ρΔ+ρρ+ρΔ+ρρ=
ρΔ+ρρ+ρΔ+ρρ=
.2exp2sin2sin2sinsin
;exp2sin2sin2cossinsincos
;2sin2sin5,0coscos
22
11
2222
11
422
11
iU
iT
R
] (5)
( ) ( )[ ]
( ) ( )[ ] ( )
( )[ ] ( )⎪
⎪
⎩
⎪⎪
⎨
⎧
δ−ρρ+ρΔ+ρρ=
δ−ρΔ+ρρ+ρΔ+ρρ=
ρΔ+ρρ+ρΔ+ρρ=
.2expcos2sin2sinsin25,0
;exp2sinsincos2sin25,0
;cos2sin2sincos25,0
22
21;12
22
21;12
22
21;12
iU
iT
R
(6)
( ) ( ) ( )
( ) ( ) ( )[ ] ( )
( ) ( )[ ] ( )⎪
⎩
⎪
⎨
⎧
−Δ++Δ+=
−Δ+−Δ++Δ+=
Δ++Δ+=
.2exp2sin2sin2coscos
;exp2sin2sin2sincoscossin
;2sin2sin5,0sinsin
22
22
2222
22
422
22
δρρρρρρ
δρρρρρρρρρ
ρρρρρρ
iU
iT
R
(7)
366
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
As a base for analyzing the coordinate structure of
distributions, we took the autocorrelation method
with using the function [12, 21, 26]
( ) jikℵ
( ) ( )[ ] ( )[ ]∫ Δ−ℵℵ=Δ
0
10
1
X
ikik dxxxx
X
xG . (9)
Here, is the “step” of changing the coordinates
.
xΔ
01 Xx ÷=
As parameters that characterize the dependences
, we chose the set of correlation moments of the
1-st to 4-th orders that are defined like to
relations (8).
( )xG Δ
4;3;2;1=lK
Estimating the degree of self-similarity or
repeatability for different geometric ( ) scales of the
structure inherent to coordinate distributions of
elements corresponding to the Jones matrix of
polycrystalline networks was performed by calculation
of logarithmic dependences for power spectra
that were approximated using the
least-squares method to the curves . For the
straight parts of the curves
d
ikℵ
( ) )log(log 1−−ℵ dJ ik
( )ηΦ
( )ηΦ , determined are the
slope angles and calculated are the values of fractal
dimensionalities for the sets of values by using the
relations [5, 6, 11, 25]
iη
ikℵ
ii tggD η−= 3)( . (10)
Classification of coordinate distributions for matrix
elements was carried out in accord with the
criteria offered in [5].
( yxik ,ℵ
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
)
)
)
If the value of the slope angle in the
dependences for 2 or 3 decades of changing the
sizes , then the distributions are fractal.
Under condition that several constant slope angles are
available in the curve , the sets are multi-
fractal. When no stable slope angles are available over
the whole interval of changing the sizes , the
sets are considered as random.
const=η
( )ηΦ
d ( yxik ,ℵ
( )ηΦ ( yxik ,ℵ
d
( yxik ,ℵ
To make this comparative analysis of
dependences more objective, let
us introduce the conception of spectral moments from
the 1-st to 4-th orders - the relation (8).
( ) )log(log 1−−ℵ dJ ik
4;3;2;1=jS
4. Computer modeling the Jones-matrix
images corresponding to multi-layer
polycrystalline network
With the aim to obtain objective criteria for diagnostics
of the structure inherent to these multilayer birefringent
networks, we have performed computer modeling the
dependence of coordinate distributions for elements of
the Jones matrix ( )nmik ×ℵ on changes of the
orientation-phase structure of flat layers as well as their
number.
Our computer modeling was based on the
following assumptions:
• multi-layer network is considered as a set of
planar sequentially located networks of birefringent
cylinders with parallel optical axes;
• each planar layer consists of parallel
densely-packed cylinders of the length L = 800 µm and
diameter
20=Q
mμ=⊗ 40 and birefringent index
; 21075.1 −×=Δn
• within the limits of cross-section for a partial
cylinder, the full period ( ) of the phase change π÷ 20
( ) ( )⊗÷=Δλ
π=⊗÷=δ 020 rnlr is realized;
• multi-layer network (the amount of layers is )
is formed via sequential arrangement of flat layers
turned to each other by the angle
q
qqq
π+γ=γ −1 .
To calculate coordinate distributions for Jones-
matrix elements ikℵ , the plane XOY where placed is a
partial flat layer of oriented birefringent cylinders was
considered as a two-dimensional array of discrete square
pixels of the linear size d = 2 µm. In other words, there
formed is a 2D analyzed array
pixpixnm 400400 ×=× in the plane of this partial
layer, within the framework of which one can calculate
the Jones-matrix images ( )nmik ×ℵ .
Depicted in Fig. 1 are the results of our computer
modeling the coordinate structure for the Jones-matrix
image of one of the ( )nm×ℵ22 elements corresponding
to 6-layer birefringent polycrystalline network and its
statistical, correlation and fractal parameters.
The analysis of results obtained using computer
modeling the structure of Jones-matrix images
describing the 6-layer birefringent polycrystalline
network has confirmed adequacy of our analytical
modeling (relations (1) to (7)) and shown that:
• coordinate distribution is
“qausi-regular” with sufficiently equiprobable
distribution
( nmz ×ℵ =6
22 )
( )22ℵH within the limits of values 0 to 1;
• dependence of the autocorrelation function
( )nmG ΔΔ ,22 for coordinate distribution ( )nm×ℵ22
drops monotonically with different-frequency
oscillations of intrinsic values;
• logarithmic dependences
for the power spectra corresponding to the coordinate
distribution of Jones-matrix elements
( ) )log(log 1−−ℵ dJ ik
( )nm×ℵ22
possess a stable slope angle characterizing the
approximating curve ( )ηΦ .
367
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
The obtained results can be related with multiple
contribution of “orientation” (ρ ) and “phase” ( δ )
parameters inherent to networks of partial crystals to
values of Jones-matrix elements describing this
polycrystalline network. So, the following values take
place for these six flat layers: ρ = 0; ρ = 0 + 0,1π;
ρ = 0 + 0,2π; …; ρ = 0 + 0,5π and δ = 0 ÷ 2π; 2δ; 3δ;
…; 5δ. Therefore, the coordinate distribution of the
element in the generalized Jones matrix
operator as well as respective autocorrelation function
is a multi-frequency dependence. Besides,
the coordinate-repeated oscillations of intrinsic values
corresponding to Jones-matrix images are the very cause
for formation of scale-similar clusters, which results in
appearance of the fractal structure in the
distribution.
{ }ikℵ
)(22 nm×ℵ
( nmG ΔΔ ,22 )
)(22 nm×ℵ
From the quantitative viewpoint, transformation of
Jones-matrix images and )(22 nm×ℵ )(21;12 nm×ℵ
describing a polycrystalline network of optically uniaxial
birefringent crystals with different numbers ( 6;3=q )
of flat layers can be illustrated with the set of statistical
( EAM ,,,σ ), correlation ( ) and spectral
( ) moments of the 1-st to 4-th orders, which are
summarized in Table 1.
4;3;2;1=iK
4;3;2;1=iS
Table 1. Statistical, correlation and spectral moments of the
1-st to 4-th orders for ℵik(m × n) distributions typical for a
multi-layer (q) polycrystalline network
q = 3 q = 6 Para-
meters ℵ22(m × n) ℵ12; 21(m × n) ℵ22(m × n) ℵ12; 21(m × n)
M 0.35 0.28 0.52 0.46
σ 0.15 0.11 0.18 0.22
A 0.61 0.55 0.21 0.13
E 0.34 0.49 0.11 0.125
K1 0.49 0.46 0.42 0.44
K2 0.044 0.016 0.072 0.031
K3 2.15 1.89 0.73 1.08
K4 6.36 5.29 3.23 2.14
S1 0.69 0.86 0.44 0.48
S2 0.28 0.45 0.085 0.11
S3 2.22 4.61 1.37 1.16
S4 1.89 2.51 1.19 135
ℵ22(m × n) H(ℵ22)
G22(Δm × Δn) logJ(ℵ22) – log(d–1)
Fig. 1. Coordinate ℵ22(m × n), probability H(ℵ22), correlation G22(Δm × Δn) and self-similar logJ(ℵ22) – log(d–1) structures of
the Jones-matrix element ℵ22 corresponding to the multi-layer (q = 6) birefringent network.
368
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
As seen, the whole set of statistical moments of
the 1-st to 4-th orders, which characterize coordinate
distributions for Jones-matrix elements, differs from
zero. Second, the values of statistical parameters are
individual in Jones-matrix images of different
elements. In this case, the values of statistical moments
of higher orders ( ) considerably exceeds the
values of mean (
EA,
M ) and dispersion ( ). This fact is
indicative of an essential difference of these coordinate
distributions ℵ
σ
ik(m × n) from the normal or Gaussian
ones […].
The set of correlation moments of the 1-st to 4-th
orders that characterize the autocorrelation functions
is not only different from zero but is
extraordinary “sensitive” to peculiarities of the
coordinate structure. The values of correlation moments
of higher orders ( ) essentially exceed by two
orders of magnitude the values of mean ( ) and
dispersion ( ).
( )xGik Δ
4;3=iK
1K
2K
The set of spectral moments of the 1-st to 4-th
orders that characterize the dependences
is also characterized with
individual sensitivity to coordinate self-similarity of the
structure inherent to Jones-matrix images.
4;3;2;1=iS
( ) )log(log 1−−ℵ dJ ik
Our comparative analysis performed for the set of
statistical ( EAM ,,,σ ), correlation ( ) and
spectral ( ) moments of the 1-st to 4-th orders for
distributions of Jones-matrix elements has shown that
with increasing the number of flat layers in birefringent
networks the following tendencies take place:
4;3;2;1=iK
4;3;2;1=iS
Glycin Methionine
• mean M is 1.5 – 1.7 times increased; dispersion σ
is also increased noticeably (by 1.2 – 2 times); the
most expressed are changes in statistical moments
of higher orders ( ) – their values are 3- to 4-
fold decreased;
EA,
• correlation moment of the 2-nd order ( )
increases by 1.5 – 1.85 times; correlation moments
of higher orders decrease, respectively, from 2
times (K
2K
3) up to 3.5 times (K4);
• spectral moments demonstrate the following
behavior: is 1.5 – 1.8 times decreased; is 3 –
4 times decreased; is 1.75 – 4 times decreased;
is 1.3 – 1.9 times decreased.
1S 2S
3S
4S
5. Jones-matrix images of polycrystalline networks in
optically thin amino acid layers
As investigated objects, we chose optically thin
(extinction coefficient 01.0≤τ ) polycrystalline layers of
amino acids (three types), namely: glycin, methionine
and proline. This choice of objects is caused by the fact
that, on the one hand, these compounds are the main
“building” material for formation of protein structures in
biological tissues. On the other hand, the coordinate
structure of these networks sharply differs in its
geometric shape – dendrite-like networks of glycin,
azimuth-symmetrical spherolite networks of methionine,
island (cluster) networks of proline (Fig. 2).
Proline
Fig. 2. Polycrystalline networks of main human amino acid types.
Fig. 3. Optical scheme of the polarimeter: 1 – He-Ne laser; 2 – collimator; 3 – stationary quarter-wave plate; 5, 8 –
mechanically movable quarter-wave plates; 4, 9 – polarizer and analyzer, respectively; 6 – object of investigation; 7 – micro-
objective; 10 – CCD camera; 11 – personal computer.
369
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
ℵ22(m × n) ℵ12; 21(m × n)
Fig. 4. Coordinate ℵ22; 12; 21(m × n), correlation G22; 12; 21(Δm × Δn) and self-similar logJ(ℵ22; 12; 21) – log(d–1) structures of
Jones-matrix elements corresponding to polycrystalline network of the glycin layer.
ℵ22(m × n) ℵ12; 21(m × n)
Fig. 5. Structure of Jones-matrix elements corresponding to polycrystalline network of the methionine layer.
370
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
ℵ22(m × n) ℵ12; 21(m × n)
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Fig. 6. Structure of Jones-matrix elements corresponding to
polycrystalline network of the proline layer.
Fig. 3 shows the optical setup for measuring the
coordinate distributions of Jones-matrix elements
corresponding to birefringent layers.
Illumination was provided with a parallel
(∅ = 104 µm) beam of a He-Ne laser (λ = 0.6328 µm,
W = 5.0 mW). The polarization illuminator consists of
the quarter-wave plates 3, 5 and polarizer 4, which
provides formation of a laser beam with an arbitrary
polarization state. Using the micro-objective 7
(magnification 4x), images of biological layers were
projected onto the plane of light-sensitive area (800x600
pixels) of the CCD-camera 10 that provided
measurements of structural elements within the range 2
to 2000 µm. The analysis of laser images was performed
using the polarizer 9 and quarter-wave plate 8.
Depicted in Figs 4 to 6 is the series of
experimentally measured Jones-matrix images
(fragments (a)) of various elements ( nm )×ℵ22 (left
columns) and ( nm )×ℵ 21;12 (right columns);
autocorrelation functions ( ) ( )xGxG ΔΔ 21;1222 ;
(fragments (b)); logarithmic dependences
(fragments (c)) of polycrystalline
glycin (Fig. 4), methionine (Fig. 5) and proline (Fig. 6)
layers.
( ) )log(log 1−−ℵ dJ ik
The results of investigations of ℵ22(m × n) and
ℵ12; 21(m × n) elements in Jones-matrix images
corresponding to polycrystalline layers consisting of
human main type amino acids have shown:
1. The significant influence of features inherent to
orientation-phase structure in the network of amino acid
partial crystals. It is confirmed by the wide range of
changes ( 10 11 ≤ℵΔ≤ ) in intrinsic values of matrix
elements corresponding to crystalline layers of all the
amino acids (Figs 4 to 6, fragments (a)). There, all the
coordinate distributions ( nmik × )ℵ are individual for
polycrystalline networks with varying geometrical
structure (Figs 4 to 6, fragments (b)).
2. Autocorrelation functions ( )xG Δ21;12;11 for
coordinate distributions of the Jones-matrix elements
corresponding to amino acid crystalline layers with
expressed dendrite and spherolite geometry are decaying
dependences with clearly pronounced fluctuations of
intrinsic values (Figs 4 and 5, fragments (b)). The found
regularity is in a good accordance with the data of model
analysis (relations (1) to (7)) and data of computer
modeling (Fig. 1) for the structure of respective Jones-
matrix images ( )nmik ×ℵ . By contrast, the spatially
non-oriented cluster structure of proline partial crystals
does not show any fluctuations of the autocorrelation
functions ( )xG Δ21;12;11 .
3. Sets of values for coordinate distributions
( )nm×ℵ22 are practically fractal, since the respective
approximating curves ( )ηΦ independently of the spatial
symmetry type (dendrite, spherolite, cluster) inherent to
birefringent networks (Fig. 3) of amino acid
polycrystalline layers are characterized with a constant
slope angle over the whole range of changes in
geometric sizes of partial crystals (Figs 4 and 5,
fragments (c)). The Jones-matrix image for the element
( )nm×ℵ 21;12 is multi-fractal, since the logarithmic
dependences ( ) )log(log 1
21;12
−−ℵ dJ possess two slope
angles in the approximating curve (Figs 4 and 5,
fragments (c)). The found self-similarity of Jones-matrix
images is in a good correlation with the model data (Fig.
1) concerning the influence of coordinate-ordered
changes in orientation
( )ηΦ
( nm× )ρ of optical axes
corresponding to partial crystals of amino acids with
simultaneous multiple modulation of the phase period
( )nm×δ (relations (3) to (7)).
The results of our quantitative analysis of values
and ranges for changing the statistical ( M , σ , , A E ),
correlation ( ), spectral ( ) moments
of the 1-st to 4-th orders that characterize the coordinate
distributions of Jones-matrix elements
4;3;2;1=iK 4;3;2;1=iS
( )nmik ×ℵ
corresponding to polycrystalline layers consisting of
various amino acids are summarized in Table 2.
The analysis of the obtained experimental data
enabled us to ascertain satisfactory correlation with the
results of computer modeling and showed that all the set
371
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
of statistical, correlation and spectral moments, which
characterizes the coordinate distributions of Jones-
matrix elements, possesses individual set of values
dependent on optical-and-geometrical parameters
inherent to amino acid polycrystalline networks. For
example, the change in spatial symmetry of amino acid
partial crystals in the sequence “dendrite network –
spherolite network – cluster ensemble” is pronounced as:
• growth of mean ( M ) and dispersion (σ )
and, just opposite, decrease in the values of statistical
moments of the 3-rd ( ) and 4-th (A E ) orders for
coordinate distributions ℵ22(m × n) and ℵ12; 21(m × n);
• decay of oscillations inherent to
autocorrelation functions of Jones-matrix images as well
as decrease in respective values of correlation moments
of higher orders and for these dependences at
the background of growth typical for the correlation
moment of the 2-nd order ;
3K 4K
2K
• decrease in all the set of spectral moments
of the 1-st to 4-th orders that characterize the
logarithmic dependences . The
range of differences between the values of
experimentally found statistical, correlation and spectral
moments of the 1-st to 4-th orders (Table 2) is 1.5 – 2
times higher than those of computer modeling (Table 1).
4;3;2;1=iS
( ) )log(log 1−−ℵ dJ ik
Table 2. Statistical (M; σ; A; E), correlation (Ki = 1; 2; 3; 4), spectral (Si = 1; 2; 3; 4) parameters of the Jones-matrix images
ℵik(m × n) corresponding to polycrystalline layers consisting of various amino acids
ℵ22(m × n)
6. Conclusions
1. Offered is the method of superposition for
Jones matrixes to describe properties of multi-layer
polycrystalline networks, and ascertained are
interrelations between the orientation-phase structure of
crystalline networks and the structure of Jones-matrix
images.
2. It is also ascertained that Jones-matrix images
of human amino acid polycrystalline layers with
different types of spatial symmetry possess individual
values of statistical, correlation and spectral moments of
the 1-st to 4-th orders.
3. Found and grounded is the complex of criteria
for Jones-matrix diagnostics and classification of optical
properties typical for birefringent dendrite, spherolite
and cluster polycrystalline networks of human amino
acids.
ℵ12; 21(m × n)
Glycine
M 0.31 K1 0.44 S1 0.56 M 0.24 K1 0.48 S1 0.53
σ 0.11 K2 0.14 S2 0.31 σ 0.09 K2 0/14 S2 0.14
A 0.86 K3 1.31 S3 0.96 A 0.44 K3 0/89 S3 0.43
E 0.63 K4 3.16 S4 1.43 E 0.37 K4 1.19
S4 0.37
Methionine
M 0.46 K1 0.52 S1 0.48 M 0.38 K1 0.51 S1 0.35
σ 0.21 K2 0.21 S2 0.23 σ 0.14 K2 0.43 S2 0.11
A 0.35 K3 0.57 S3 0.47 A 0.18 K3 S3 0.54 0.21
E 0.28 K4 1.71 S4 0.39 E 0.12 K4 0.91 S4 0.17
Proline
M 0.65 K1 0.45 S1 0.256 M 0.54 K1 0.48 S1 0.12
σ 0.32 K2 0.36 S2 0.13 σ 0.32 K2 0.67 S2 0.05
A 0.12 K3 0.24 S3 0.18 A 0.09 K3 S3 0.16 0.11
E 0.09 K S E 4 0.73 4 0.11 0.04 K S4 0.15 4 0.08
372
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 365-374.
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