**6.6 Diagnostic feasibilities of polarization mapping of optically thin layers of BTs**

To look for the feasibilities for differentiation of geometrical/optical structure of architectonics of BTs, comparative study of statistical and fractal structures of BT maps has been carried out [103, 119 - 121]. There are BTs of interest: (i) physiologically normal and dystrophically changed muscular tissue (MT), i.e. structured BT with ordered architectonics; (ii) physiologically normal and septically inflamed pulmonary tissue (PT), i.e. BT with island architectonics. The set of figures (Figs. 23 and 24) illustrates polarization maps of the mentioned objects.

Fig. 23. Polarization maps of physiologically normal (left part) and dystrophycally changed (right part) MT: (a), (b) – 2D distributions (in degrees) of the azimuth of polarization and ellipticity; (c), (d) – histograms of polarization parameters.

Potentialities for statistical differentiation of polarization maps are illustrated by comparison of magnitudes of the statistical moments of the 1st to the 4th orders for the azimuth of polarization and ellipticity at images of physiologically normal and pathologically changed BTs, represented in Tables 2 and 3.


Table 2. Statistical moments of the 1st to the 4th orders for the coordinate distributions of the azimuth of polarization and ellipticity at images of histological tomes of normal and dystrophically changed MT

Fig. 24. Polarization maps of physiologically normal (left part) and septically inflamed (right part) PT: (a), (b) – 2D distributions (in degrees) of the azimuth of polarization and ellipticity; (c), (d) – histograms of polarizarion parameters.

One can see from the represented data that magnitudes of asymmetry and excess of the coordinate distributions of the azimuth of polarization and ellipticity at BT images with ordered architectonics (Table 2) exceed by 3 to 5 times the magnitudes of the corresponding parameters characterizing statistics of the 3rd and the 4th orders of inhomogeneous in polarization BT images with island architectonics (Table 3).

Potentialities for statistical differentiation of polarization maps are illustrated by comparison of magnitudes of the statistical moments of the 1st to the 4th orders for the azimuth of polarization and ellipticity at images of physiologically normal and pathologically changed

(c) (d)

(a) (b)

*Norm (21 samples) Pathology (22 samples)* 

azimuth of polarization and ellipticity at images of histological tomes of normal and

Fig. 24. Polarization maps of physiologically normal (left part) and septically inflamed (right part) PT: (a), (b) – 2D distributions (in degrees) of the azimuth of polarization and ellipticity;

One can see from the represented data that magnitudes of asymmetry and excess of the coordinate distributions of the azimuth of polarization and ellipticity at BT images with ordered architectonics (Table 2) exceed by 3 to 5 times the magnitudes of the corresponding parameters characterizing statistics of the 3rd and the 4th orders of inhomogeneous in

*M*<sup>1</sup> *0,26 0,013 0,12 0,01 M*<sup>1</sup> *0,21 0,001 0,18 0,011 M*<sup>2</sup> *0,12 0,01 0,08 0,004 M*<sup>1</sup> *0,19 0,013 0,12 0,006 M*<sup>3</sup> *6,7 0,469 4,9 0,294 M*<sup>1</sup> *9,4 0,846 6,2 0,434 M*<sup>4</sup> *17,9 1,61 14,5 1,595 M*<sup>1</sup> *25,4 2,54 21,6 2,592*  Table 2. Statistical moments of the 1st to the 4th orders for the coordinate distributions of the

*M*<sup>1</sup>

BTs, represented in Tables 2 and 3.

(c) (d)

(c), (d) – histograms of polarizarion parameters.

polarization BT images with island architectonics (Table 3).

(a) (b)

dystrophically changed MT

*M*<sup>1</sup>


Table 3. Statistical moments of the 1st to the 4th orders for the coordinate distributions of the azimuth of polarization and ellipticity at images of histological tomes of normal and septically inflamed PT

On the other hand, structured BTs possess hierarchical, self-similar structure of architectonics [97, 113, 120]. That is why one must determine optical manifestations of such geometry of anisotropic component of structured BTs. The study of self-similarity of polarization maps of BTs has been carried out for skin derma.

Figure 25 shows the set of log-log dependences, log *P d* ( ) lo g(1 / ) , log *P d* ( ) lo g(1 / ) , for power spectra of the distributions of polarization parameters at images of physiologically normal and pathologically changed skin derma.

Fig. 25. log-log dependences of power spectra of the azimuth of polarization and ellipticity at images of physiologically normal (a), (b) and pathologically changed (c), (d) skin derma.

The dependence log *P d* ( ) lo g(1 / ) manifests two slopes within three decades of scales of architectonics of skin derma's elements; the dependence log *P d* ( ) lo g(1 / ) consists of the part with constant slope and statistical part, as it is seen from Fig. 1.6 (a) and (b). Pathological state of such BT manifests itself in randomization of distributions of polarization parameters at images of the corresponding histological tomes, i.e. for approximating curves (Fig. 25 (c) and (d)) steady slope is absent.

#### **6.7 Degree of mutual polarization of laser images of BTs**

For laser light fields scattered by BTs, coordinate changes of polarization characteristics can be characterized by the complex degree of mutual polarization, CDMP, at two points *r r* 1 2 ; [6]:

$$\mathcal{W}(r\_1, r\_2) = \frac{\left(\mathcal{U}\_x(r\_1)\mathcal{U}\_x(r\_2) - \mathcal{U}\_y(r\_1)\mathcal{U}\_y(r\_2)\exp\left(i\left(\mathcal{S}\_2(r\_2) - \mathcal{S}\_1(r\_1)\right)\right)\right)^2}{\left(\mathcal{U}\_x^2(r\_1) + \mathcal{U}\_y^2(r\_1)\right)\left(\mathcal{U}\_x^2(r\_2) + \mathcal{U}\_y^2(r\_2)\right)}.\tag{27}$$

The following interconnections between the real part of the CDMP, Re , *W Wrr* 1 2 , and the magnitudes at the azimuth of polarization, *r* , ellipticity, *r* , at points *r* of BT's image, and orthogonal components , *U IU I x x y y* of the complex amplitude with phase difference between them *r* have been established in study [100]:

$$\overline{\mathcal{W}}\left(r\_2, r\_1\right) = \frac{\left(\left(I\_x\left(r\_2\right)I\_x\left(r\_1\right)\right)^{\frac{1}{2}} - \left(I\_y\left(r\_2\right)I\_y\left(r\_1\right)\right)^{\frac{1}{2}}\cos\left(\delta\_2\left(r\_2\right) - \delta\_1\left(r\_1\right)\right)\right)^2}{I\left(r\_2\right)I\left(r\_1\right)}\,\tag{28}$$

where

$$\mathcal{S}\left(r\_1\right) = \tan^{-1}\left[\frac{\tan\left[2\,\mathcal{J}\left(r\_1\right)\right]}{\tan\left[\,\alpha\left(r\_1\right)\right]}\right]; \quad \mathcal{S}\left(r\_2\right) = \tan^{-1}\left[\frac{\tan\left[2\,\mathcal{J}\left(r\_2\right)\right]}{\tan\left[\,\alpha\left(r\_2\right)\right]}\right].\tag{29}$$

#### **6.8 Statistical approach to analysis of coordinate distributions of the CDMP at BT images**

Normal and dystrophycally changed tissues of skeleton muscle has been investigated in paper [100]. Coordinate distributions of the CDMP *W x*(,) *y* at images of histological tomes of such samples are shown in Figure 26.

Statistical analysis of the coordinate distributions of magnitudes *W x*(,) *y* shows for the map of tissue of skeleton muscle differences between the statistical moments of the 1st to the 3rd orders (within 30%-50%), while magnitudes of excess (the 4th order statistical moment) differ by 2 to 2.5 times.

part with constant slope and statistical part, as it is seen from Fig. 1.6 (a) and (b). Pathological state of such BT manifests itself in randomization of distributions of polarization parameters at images of the corresponding histological tomes, i.e. for

For laser light fields scattered by BTs, coordinate changes of polarization characteristics can be characterized by the complex degree of mutual polarization, CDMP, at two points *r r* 1 2 ;

The following interconnections between the real part of the CDMP, Re , *W Wrr* 1 2 ,

*r* of BT's image, and orthogonal components , *U IU I x x y y* of the complex

1 2 2 22 2

exp , *xx yy*

*x yx y*

*Ur Ur Ur Ur*

*U rU r U rU r i r r*

112 2

 

*Ir Ir*

1 1 1 2

tan 2 tan 2 tan ; tan . tan tan

 

Normal and dystrophycally changed tissues of skeleton muscle has been investigated in paper [100]. Coordinate distributions of the CDMP *W x*(,) *y* at images of histological tomes

Statistical analysis of the coordinate distributions of magnitudes *W x*(,) *y* shows for the map of tissue of skeleton muscle differences between the statistical moments of the 1st to the 3rd orders (within 30%-50%), while magnitudes of excess (the 4th order statistical moment) differ

**6.8 Statistical approach to analysis of coordinate distributions of the CDMP at BT** 

*xx yy IrIr IrIr r r*

2 1

(29)

*r r*

1 2

*r r*

<sup>2</sup> <sup>2</sup> 2 1 2 1 22 11

. (27)

<sup>2</sup> <sup>1</sup> <sup>1</sup>

cos

, (28)

 

*r* , ellipticity,

1 2 1 2 22 11

architectonics of skin derma's elements; the dependence log *P d* ( ) lo

approximating curves (Fig. 25 (c) and (d)) steady slope is absent.

**6.7 Degree of mutual polarization of laser images of BTs** 

and the magnitudes at the azimuth of polarization,

1 2

*r r*

amplitude with phase difference between them

g(1 / ) manifests two slopes within three decades of scales of

g(1 / ) consists of the

2

*r* have been established in study

 

 

  *r* , at points

The dependence log *P d* ( ) lo

*Wrr*

*Wr r*

of such samples are shown in Figure 26.

,

2 1

[6]:

[100]:

where

**images** 

by 2 to 2.5 times.

Fig. 26. Coordinate distributions of the CDMP at images of histological tomes of normal (a) and dystrophycally changed (b) tissue of skeleton muscle.


Table 4. Statistical moments of the 1st to the 4th orders for distributions of magnitudes *W x*(,) *y* at images of histological tomes of normal and dystrophically changed tissues of skeleton muscle.

Figure 27 shows autocorrelation functions of the coordinate distributions of magnitudes of the CDMP at images of histological tomes of samples of normal (a) and pathologically changed (b) muscular tissues.


Table 5. Correlation parameters of the distributions of the CDMP, *W x*(,) *y* , for normal and pathologically changed osseous muscular tissues.

Fig. 27. Autocorrelation functions *Gxy* (,) of 2D distributions of the CDMP, *W x*(,) *y* , at images of histological tomes of muscular tissues.

Correlation analysis of 2D distributions of magnitudes *W x*(,) *y* leads to the following conclusions: (*i*) differences between magnitudes of a half-width *L* of autocorrelation functions for distributions *W x*(,) *y* at polarization/correlation maps of muscular tissue's maps are within the interval 0.65 to 2 times; (*ii*) differences between dispersions of the distributions for extrema of power spectra for *W x*(,) *y* reach one order of magnitude.

Thus, the analysis of approaches and metrological techniques of modern polarimetric diagnostics of birefringent nets of BTs leads to the following conclusions.


#### **7. Conclusions**

Thus, new approaches to metrology of partially coherent and partially polarized light fields are derived proceeding from singular optics concept. The first of them concerns to exploring

Fig. 27. Autocorrelation functions *Gxy* (,) of 2D distributions of the CDMP, *W x*(,) *y* , at

distributions for extrema of power spectra for *W x*(,) *y* reach one order of magnitude.

diagnostics of birefringent nets of BTs leads to the following conclusions.

Correlation analysis of 2D distributions of magnitudes *W x*(,) *y* leads to the following conclusions: (*i*) differences between magnitudes of a half-width *L* of autocorrelation functions for distributions *W x*(,) *y* at polarization/correlation maps of muscular tissue's maps are within the interval 0.65 to 2 times; (*ii*) differences between dispersions of the

Thus, the analysis of approaches and metrological techniques of modern polarimetric

 higher-order statistical moments (asymmetry and excess) characterizing the distributions of polarization/correlation parameters of laser images of BTs are of the highest sensitivity in respect to changing optical properties of nets of biological crystals

 interconnection between the real part of the CDMP and parameters of anisotropy (such as direction of an optical axis and phase difference between orthogonal components of scattered laser radiation) at different points of birefringent matrix provides reliable tool for correlation optical diagnostics of pathological changes of biological tissues and, as a consequence, for early and non-invasive diagnostics of wide-spread diseases, which

Thus, new approaches to metrology of partially coherent and partially polarized light fields are derived proceeding from singular optics concept. The first of them concerns to exploring

images of histological tomes of muscular tissues.

of normal and pathologically changed BTs;

challenge Humanity in the Third Millenium.

**7. Conclusions** 

the spatial modulated time-averaged Poynting vector in completely and partially coherent non-paraxial light fields for control the motion of nano- and microparticles in optical currents. The second approach reveals interconnection between polarization singularities inherent in completely polarized and partially polarized optical beams for the general case of partially mutual coherence of orthogonally polarized components. Further, we have shown the feasibility for experimental analysis of the Poynting vector components by combining interferometric and polarimetric metrological techniques. At last, we have demonstrated potentialities of laser 2D Stokes-polarimetric metrological technique developed and implemented in previous consideration in non-invasive pre-clinic diagnostics of physiological state of biological tissues. The represented approaches show fruitfulness of attracting the concepts and metrological tools of singular optics in formation and investigation of unconventional polarization distributions that can be of usefulness in problems of optical correlation diagnostics and optical telecommunications.

There are the important next steps in developing the direction of Metrology represented in this chapter. The study of optical currents using light-scattering particles of nano-scale will provide vital data both on microstructure of light and on intimate processes of interaction of optical beams with extremely small (in scale of a wave length) particles and ensembles of them. It is in prospect to investigate experimentally in more details influence of varying degree of coherence of optical fields on some phenomena considered in this review and associated with peculiarities (*singularities*) of the Poynting vector in complex inhomogeneous optical fields beyond the paraxial approximation. Also, it is important to obtain experimental verification, *viz.* visualization, of the mechanical action of the spin moment in optical beams. The study of influence of polarization characteristics of light field on motion of nanoparticles will be continued, as well as development of biomedical application of notion and techniques of completely/partially coherent, inhomogeneously polarized fields.

Another branch of the following R&D must be focused, in our opinion, on overcoming some disadvantages intrinsic to digital optical data surveying due to automatic gain control applied in the most of register tools, including CCD cameras. Really, dealing with differentiation metrology (such as 2D Stokes polarimetry) one wants to have comparable data in the set of measurements that is not always achievable with self-controlled in sensitivity digital registering devices. As compromise, one can consider combination of analog data recording and digital processing of these data. Nevertheless, this important point is worthy separate investigation for providing higher accuracy and reliability of modern optical metrology.
