Optical Sensor System for 3D Jones Matrix Reconstruction of Optical Anisotropy Maps of Self-Assembled Polycrystalline Soft Matter Films

Our work uses a polarization matrix formalism to analyze and algorithmically represent optical anisotropy by open dehydration of blood plasma films. Analytical relations for Jones matrix reconstruction of optical birefringence maps of protein crystal networks of dehydrated biofluid films are found. A technique for 3D step-by-step measurement of the distributions of the elements of the Jones matrix or Jones matrix images (JMI) of the optically birefringent structure of blood plasma films (BPF) has been created. Correlation between JMI maps and corresponding birefringence images of dehydrated BPF and saliva films (SF) obtained from donors and prostate cancer patients was determined. Within the framework of statistical analysis of layer-by-layer optical birefringence maps, the parameters most sensitive to pathological changes in the structure of dehydrated films were found to be the central statistical moments of the 1st to 4th orders. We physically substantiated and experimentally determined the sensitivity of the method of 3D polarization scanning technique of BPF and SF preparations in the diagnosis of endometriosis of uterine tissue.

The most accurate (gold standard) and mandatory stage of diagnosis is histological and immunohistochemical examination of tumor biopsy, which allows us to make a final conclusion, determine the histological picture of the pathological condition with an accuracy that is close to the maximum (~100%) [41].At the same time, these studies are not rapid, require the use of expensive reagents, are also largely subjective, and do not provide the possibility of quantitative analysis of the parameters of microscopic images of internal organ preparations.
Currently, instrumental high-precision (~95%) polarization interference methods for differential diagnosis of histological sections of prostate tumor biopsy have been developed, which are devoid of the above disadvantages of traditional histological examination [42].
At the same time, obtaining a biopsy is a traumatic and sometimes dangerous operation for patients' health.Therefore, it is relevant to develop new rapid instrumental biophysical methods for differential diagnosis of prostate tumors using a minimally invasive procedure, an optical study of the polycrystalline structure of dehydrated films of biological fluids of human organs.
Polarimetric studies of the polarization manifestations of BF anisotropy make it possible to detect pathological changes in the quaternary and tertiary polycrystalline structure of basic proteins (albumin and globulin) in blood plasma and other biological fluids, which are inaccessible in the most common biochemical methods.In a series of works [43][44][45][46][47][48][49][50][51], the sensitivity of polarimetric methods to inflammatory, pathological and necrotic changes in the polycrystalline structure of films of blood plasma, urine, synovial fluid, and cerebrospinal fluid was demonstrated.
Despite such promising results in diagnosing the structure of biofluid films, there are still a number of unresolved problems.Among them are the following: There is no unified (unified) analytical description of polarization manifestations of optical properties of complex phase and amplitude anisotropic structure of networks of biological crystals in the volume of dehydrated BF films.

2.
Algorithms for polarization reconstruction of histograms of polycrystalline networks of biological crystal maps of various optical anisotropy mechanisms, birefringence and dichroism, have not been developed.

3.
Dehydrated BF films exhibit a complex spatially inhomogeneous volumetric structure of networks of biological crystals.On the other hand, polarization methods provide integrally averaged information over the whole volume of the biological layer in the form of 2D matrix element distributions.As a consequence, the sensitivity of existing methods of matrix polarimetry does not provide the capability to detect spatially localized changes of optical anisotropy of networks of biological crystals.
There is no unified medical-physical approach to the objective evaluation of diagnostic efficiency of matrix polarimetry methods.
Thus, a new and urgent issue of matrix polarimetry is the development of the Jones matrix model of analytical description of optical anisotropy of various types of dehydrated films of biological fluids followed by determination of algorithmic relationships between parameters of optical anisotropy of dehydrated BF films and values of partial matrix elements.
This polarimetric approach can be developed and generalized into a synthesis of polarization interference mapping and digital holographic reconstruction of object fields of complex amplitudes for layer-by-layer reconstruction of optical anisotropy maps of dehydrated BF films.Therefore, to improve the sensitivity and accuracy of polarimetric matrix diagnostics, it is urgent to create a new method for the layer-by-layer study of Sensors 2024, 24, 1589 3 of 14 polycrystalline films of biological fluids by synthesizing polarization (Jones matrix) and interference methods [43][44][45][46][47][48][49][50].
In addition, the potential of the Jones matrix tomography technique can be expanded to other equally relevant areas, such as instrumental medical diagnostics COVID-19 [51][52][53] Our work aims to create and validate a new experimental method of polarization interference multilayer Jones matrix scanning of the polycrystalline structure of blood plasma (BPF) and saliva (SF) for diagnosis and differentiation of prostate cancer stages.
where n 0 , n 90 , n 45 , n 135 and χ 0 , χ 90 , χ 45 , χ 135 are refractive indices-absorption for orthogonal amplitude components 0 • -90 • and 45 Given the insignificant absorption of dehydrated BPF and SF in our work, we limit ourselves to taking into account the effects of birefringence.
We describe the polarization manifestations of each optical anisotropy mechanism by the Jones partial matrix operators of the formula below [48]:

J(LB
An analytical form of the generalized Jones matrix of a polycrystalline medium with complex anisotropy was found in [48] {J} = j 11 j 12 Sensors 2024, 24, 1589 4 of 14 Here V-generalized anisotropy vector where    L 0;90 = LB 0;90 ; L 45;135 = LB 45;135 ; From ( 11) and ( 12) we obtain an explicit form of the vector modulus On the basis of ( 7)-( 13), we obtain theoretical interrelations between the parameters of linear and circular birefringence dichroism and elements of the Jones matrix {J} of an optically anisotropic layer The algorithm for calculating the elements of the Jones matrix {J} (Expression (1)) includes the irradiation of a polycrystalline film sample by linearly polarised beams with azimuths 0 • (Jones vector 1 0 ) and 90 • (Jones vector 0 1 ) U 90 y = 0 0 0 1 As a result, we obtain the following expressions for the Jones matrix tomography of the polycrystalline structure of a biological fluid film |V| = 2arccos 0.5 U 0 x exp iϕ

Optical Scheme of 3D Jones Matrix Scanning of BPF and SF
Figure 1 shows a diagram of the polarization interferometry of Jones layer-by-layer distributions of dehydrated films JMI [46,47,49].
x FOR PEER REVIEW 5 of 14 In further experimental studies, we will use the generally accepted parameters for the different types of birefringence
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles  = 0°;  = 90°.
3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform (, ) [49]: where  , 0,90 , orthogonal components of complex amplitude for different orientations,  = 0°;  = 90° ; * denotes the complex conjugation operation; (, ) are the spatial frequencies and ( = 1120,  = 960) are the number of pixels of the CCD camera.4. The results of the digital Fourier transform (Relations (25), (26)) are used to obtain distributions of complex amplitudes according to the following algorithms:

The Method of 3D Jones Matrix Scanning of BPF and SF
1. Two states of linear polarisation are sequentially formed in the "irradiating" (Ir) and "reference" (Re) parallel laser beams-Ir(0 2. For each of the polarization states (p and r), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles ensors 2023, 21, x FOR PEER REVIEW 5 of 14 In further experimental studies, we will use the generally accepted parameters for the different types of birefringence
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where 4. The results of the digital Fourier transform (Relations ( 25), ( 26)) are used to obtain distributions of complex amplitudes according to the following algorithms:
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations ( 25), ( 26)) are used to obtain distributions of complex amplitudes according to the following algorithms: = 90 • .3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform F(υ, ν) [49]:   4. The results of the digital Fourier transform (Relations ( 25), ( 26)) are used to obtain distributions of complex amplitudes according to the following algorithms: where In further experimental studies, we will use the generally accepted parameters for the different types of birefringence

Optical Scheme of 3D Jones Matrix Scanning of BPF and SF
Figure 1 shows a diagram of the polarization interferometry of Jones layer-by-layer distributions of dehydrated films JMI [46,47,49].
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations ( 25), (26)) are used to obtain distributions of complex amplitudes according to the following algorithms: In further experimental studies, we will use the generally accepted parameters for the different types of birefringence

Optical Scheme of 3D Jones Matrix Scanning of BPF and SF
Figure 1 shows a diagram of the polarization interferometry of Jones layer-by-layer distributions of dehydrated films JMI [46,47,49].
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations (25), (26)) are used to obtain distributions of complex amplitudes according to the following algorithms:
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations (25), (26)) are used to obtain distributions of complex amplitudes according to the following algorithms:
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations (25), ( 26)) are used to obtain distributions of complex amplitudes according to the following algorithms: Sensors 2023, 21, x FOR PEER REVIEW 5 of 14 In further experimental studies, we will use the generally accepted parameters for the different types of birefringence

Optical Scheme of 3D Jones Matrix Scanning of BPF and SF
Figure 1 shows a diagram of the polarization interferometry of Jones layer-by-layer distributions of dehydrated films JMI [46,47,49].
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations (25), ( 26)) are used to obtain distributions of complex amplitudes according to the following algorithms: (24

Optical Scheme of 3D Jones Matrix Scanning of BPF and SF
Figure 1 shows a diagram of the polarization interferometry of Jones layer-by-laye distributions of dehydrated films JMI [46,47,49].
2. For each of the polarization states ( and ), two partial interference patterns ar recorded through the polarizer-analyzer 14 with the orientation of the transmission plan at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film sam ples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientation = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatia frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations ( 25), ( 26)) are used to obtai distributions of complex amplitudes according to the following algorithms: In further experimental studies, we will use the generally accepted parameters for the different types of birefringence
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.
4. The results of the digital Fourier transform (Relations ( 25), ( 26)) are used to obtain distributions of complex amplitudes according to the following algorithms:
2. For each of the polarization states ( and ), two partial interference patterns are recorded through the polarizer-analyzer 14 with the orientation of the transmission plane at angles = 0°; = 90°.3. Analytical processing of microscopic interference images of dehydrated film samples was carried out using the digital Fourier transform ( , ) [49]: where , orthogonal components of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.

Statistical Analysis of Maps of Linear and Circular Birefringence of Polycrystalline Films of Biological Fluids
The resulting set of optical anisotropy maps f ≡ LB(θ k , m, n); CB(θ, m, n) was analyzed in a statistical approach using the following algorithms to calculate mean (Z 1 ), variance (Z 2 ), skewness (Z 3 ) and kurtosis (Z 4 ) [43-51] where K = M × N-CCD pixels quantity.

Diagnostic Method
The search for the most sensitive phase plane was carried out as follows.
1.At the first stage: • Discrete phase scanning was carried out in "maximum" increments ∆θ max k = 0.25 rad.

•
The differences between the obtained values of the statistical parameters were assessed

•
The phase interval ∆θ * = θ max j+1 ÷ θ max j was determined, within which the monotonic increase in the value ends • The macrointerval phase found ∆θ * was analyzed again in finer increments ∆θ min q = 0.05 rad • The new (most sensitive) phase section θ * was determined, in which ∆Z i θ * = max.
Here, a 12 (a 34 ) and b 12 (b 34 ) are the number of correct and incorrect diagnoses within investigation group 2 (group 4); and c 12 (c 34 ) and d 12 (d 34 ) are the same within control group 1 (group 3).

Biological Samples
Blood and saliva samples were taken for the purpose of determination of the prostate tumor [60][61][62][63].
Experimental blood plasma samples were obtained by centrifugation.Liquid drops with a volume of 10 µL were applied onto optically homogeneous glass heated to the temperature of the human body (36.6 • C).The process of complete drying (dehydration) lasted 45-50 min.
Experimental saliva samples were obtained by taking 2-3 mL of liquid into a test tube and settling it until the mucin proteins precipitated.Then, by centrifugation, the liquid and sedimentary components were separated.A drop of liquid with a volume of 10 µL was applied to optically homogeneous glass heated to the temperature of the human body (36.6 • C).The process of complete drying (dehydration) lasted 40-45 min.
It is known [62] that in the process of blood plasma dehydration, a dry (polycrystalline) drop is formed, which contains three zones:
The composition of saliva and blood plasma is correlated with each other.Saliva contains the same protein fractions as blood plasma-albumin and globulins.At the same time, among them, there are significantly fewer albumins and four times more globulins.
When a saliva drop is dehydrated, the crystallization results differ significantly from the polycrystalline structure of a blood plasma drop.This is due to the lower concentration of albumin and globulin proteins in saliva.Therefore, the marginal protein roll is not formed.
A dry drop of saliva has the form of a film quasi-uniform in thickness ( h ∼ 10 µm ÷ 15 µm), where optically isotropic dendritic networks of cubic crystals of NaCl salt are localized.
Such a film contains, in the form of domains, large (linearly birefringent needlelike albumin crystals) and small (circularly birefringent globulin crystals) optically anisotropic aggregates with a range of sizes h = 10 µm ÷ 500 µm.
Four representative groups of blood plasma and saliva polycrystalline films were formed: • Group 1 consisted of n = 36 samples of BPF from healthy donors;
Comparison of the obtained data revealed a correlation between the model analysis (Ratios (1)-( 25)) and the experimental results (Figures 2 and 3).Namely:

•
The presence of mechanisms of linear LB θ * , m, n and circular CB θ * , m, n birefringence of supramolecular networks of dehydrated films of biological fluids with different morphological architectonics (Figures 2 and 3).

•
Insignificant differences (the magnitude of the average and the amplitude of fluctuations) between topographic maps of linear birefringence of LB θ * , m, n dendritic components of optically anisotropic architectonics of dehydrated plasma films (Figure 2) and saliva (Figure 3) of healthy donors (Fragments (1)) and prostate cancer patients (Fragments (2)).

•
A significant increase in the circular birefringence of CB θ * , m, n spherulite components of the optically anisotropic architectonics of dehydrated plasma films (Figure 2) and saliva (Figure 3) of healthy donors (Fragments (3)) and prostate cancer patients (Fragments (4)).
Quantitative and objective scenarios of changes in various types of birefringence LB θ * , m, n and CB θ * , m, n .The optically anisotropic architectonics of dehydrated films of biological fluids of both types are illustrated by statistical analysis data (ratios ( 29)), which are presented in Table 2.The maximum intergroup differences were established: • BPF-∆Z 3 (CB) = 1.62; ∆Z 4 (CB) = 1.63; • SF-∆Z 3 (CB) = 1.69; ∆Z 4 (CB) = 1.65.For these statistical parameters, Table 3 shows the values of the sensitivity Se, specificity Sp and balanced accuracy Ac of the method of layer-by-layer Jones matrix mapping with reproduction of maps of linear and circular birefringence of polycrystalline films of blood plasma and saliva.The results shown in Table 3 indicate a different efficiency in diagnosing prostate cancer by Jones matrix layered reproduction of maps of linear LB θ * , m, n and circular CB θ * , m, n birefringence of BPF and SF.
It was found that the layer-by-layer mapping of the linear birefringence distributions LB θ * , m, n of the dendritic component of the optically anisotropic architectonics of dehydrated plasma films is quite sensitive and accurate (87.5%) to the pathological change in the structure of supramolecular networks of needlelike albumin crystals.
The accuracy of a similar diagnosis for the dendritic component of the supramolecular network of the dehydrated saliva film appears to be significantly lower (76.4%)due to a lower level of linear birefringence of needle-shaped albumin crystals and a high of optically isotropic cubic NaCl crystals.The most optimal and practically independent of the type of biological fluid was the statistical analysis of the coordinate distributions of the circular birefringence value CB θ * , m, n associated with the concentration of globulins.In this situation, an excellent level of balanced accuracy was achieved: Ac(Z 3;4 ) = 93.05%÷ 95.8%.

Conclusions
For the first time, the Jones matrix analytical model for the optical anisotropy description of different types of dehydrated films of biological fluids, such as blood plasma and saliva, is proposed.New algorithmic relationships between phase anisotropy parameters (birefringence and optical activity) of dehydrated films and values of partial elements of the Jones matrix were found.Based on the synthesis of polarization interference mapping and digital holographic reconstruction of object fields of complex amplitudes, a new technique of phase Jones matrix scanning with the subsequent layer-by-layer reconstruction of optically mapped optically anisotropic structure of dehydrated biological liquid films was developed and tested experimentally for the first time.An original algorithm for determining the cross section in the volume of dehydrated films of biological fluids is proposed, which is most effective for differential diagnosis of pathological changes in optical birefringence.New objective criteria (markers) for high accuracy (93.05% to 95.8%) diagnosis and differentiation (91.7% to 94.4%) of prostate cancer stages have been defined in the statistical Jones matrix analysis of the reconstructed optical anisotropy maps of dehydrated blood plasma and saliva samples.
The present study is one of the first steps in the fundamental development and experimental testing of the diagnostic potential of the Jones express technique matrix tomography of polycrystalline architectonics of dehydrated human fluid films.
The next steps in the development of this method will be experimental studies planned on the scale of clinical institutions of the countries represented by the members of the author's team for the purpose of expert evaluation and determination of the validity of Jones matrix tomography of pathological conditions of human organs.

3 .
Figure 1 shows a diagram of the polarization interferometry of Jones layer-by-layer distributions of dehydrated films JMI [46,47,49].
of complex amplitude for different orientations, = 0°; = 90° ; * denotes the complex conjugation operation; ( , ) are the spatial frequencies and ( = 1120, = 960) are the number of pixels of the CCD camera.

Sensors 2023 ,
21, x FOR PEER REVIEW 5 of 1In further experimental studies, we will use the generally accepted parameters fo the different types of birefringence

Table 1 .
Statistical parameters of optical anisotropy maps of BPF and SF of healthy donors.

Table 2 .
Statistical parameters of optical anisotropy maps in sections θ * (LB) = 0.35rad and θ * (CB) = 0.25 rad fields of complex amplitudes of BPF and SF.

Table 3 .
Characteristics of the diagnostic power of the 3D matrix scanning method of BPF and SF.