Reversible polar decomposition method of backscattering mueller matrix

Abstract

The present invention discloses a reversible polar decomposition method of a backscattering Mueller matrix, which comprises the following steps: Step 1, the reversible polar decomposition of the backscattering Mueller matrix is ​​based on the relationship between the forward and backward path dichroic matrices and the phase delay matrix, and the backscattering Mueller matrix is ​​converted into a symmetric matrix QM; Step 2, the dichroic matrix M is obtained through the QMG matrix. D1 ; Step 3, obtain the eigenvalues ​​and corresponding eigenvectors through orthogonal decomposition; Step 4, sort the eigenvectors to obtain the debiased matrix M Δd and the phase delay matrix M R1 ; Step 5, the depolarization matrix M is obtained Δd and the phase delay matrix M R1 The polarization parameters are obtained. This method gives a systematic solution for backscattering Mueller matrix decomposition for the first time, and obtains polarization parameters (such as azimuth angle, phase delay and depolarization coefficient, etc.) that characterize the microstructure of the medium.

Description

TECHNICAL FIELD

[0001] The present application particularly relates to a reversible polar decomposition method of backscattering Mueller matrix. BACKGROUND

[0002] Polarization technology is a non-labeled and non-invasive detection technology based on the interaction between polarized light and medium. Polarization measurement can obtain microstructure information of medium or biological tissue. Due to the existence of scattering and polarization effect, 16 elements of Mueller matrix have different responses to polarization state changes, and there is a complex mutual relationship between the elements. Therefore, it is necessary to decompose and transform the Mueller matrix to obtain parameters representing various polarization characteristics of the medium. The decomposed sub-matrix and various coefficients can independently reflect a polarization property of the medium, and further represent optical properties and microstructure of the medium, and also provide feasibility for imaging and detection. In biomedicine, Mueller matrix decomposition is used to judge the state of tissue, monitor optical processes of tissue, distinguish cancerous tissue from normal tissue, and analyze the source of tissue lesions. In chemistry and material science, it can be used for dynamic monitoring of structural changes and material growth in materials.

[0003] At present, the most common decomposition is Lu-chipman polar decomposition method, which is proposed by Lu S Y and Chipman R A in Interpretation of Mueller matrices based on polar decomposition published in J.opt.soc.am.a in 1996. The method is widely used in analysis of interaction between polarized light and medium. However, the decomposition is only suitable for forward scattering and cannot be applied to backscattering. Backscattering is more common than forward scattering, has simpler experimental conditions and higher practical value. However, backscattering Mueller matrix is more complex. Therefore, how to decompose backscattering Mueller matrix is crucial for backscattering polarization technology. SUMMARY

[0004] In view of the deficiencies in the prior art, the present application aims to provide a reversible polar decomposition method of backscattering Mueller matrix.

[0005] To achieve the above object, the present application provides the following technical scheme.

[0006] A reversible polar decomposition method of backscattering Mueller matrix comprises the following steps:

[0007] Step one, the reversible polar decomposition of backscattering Mueller matrix According to the relationship between the forward and backward path dichroism matrix and phase retardation matrix, the backscattering Mueller matrix is transformed into a symmetric matrix QM;

[0008] Step two, the dichroism matrix M is obtained by QMG matrix D1 ;

[0009] Step three, the eigenvalue and corresponding eigenvector are obtained by orthogonal decomposition;

[0010] Step four, the eigenvector is sorted again, so as to obtain the depolarization matrix M Δd and the phase retardation matrix M R1 ;

[0011] Step five, the polarization parameters are obtained from the obtained depolarization matrix M Δd and the phase retardation matrix M R1 .

[0012] In step one, the backscattering Mueller matrix is defined as M=M D2 M R2 M Δd M R1 M D1 , wherein M R1 is the phase retardation matrix of the forward path, M D1 is the dichroism matrix of the forward path, M Δd is the depolarization matrix, M R2 is the phase retardation matrix of the backward path, M D2 is the dichroism matrix of the backward path;

[0013] According to the principle of light path reversibility in backscattering, the dichroism matrix and the phase retardation matrix of the backward path are expressed by the dichroism matrix and the phase retardation matrix of the forward path

[0014]

[0015]

[0016] Wherein Q=diag(1,1,-1,1);

[0017] The backscattering Mueller matrix is transformed into a symmetric matrix by , wherein M' Δd = QM Δd = diag(d0,d1,-d2,d3), I is a 3x3 unit matrix, δ ijis Kronecker function, ε ijk is Levi-civita permutation symbol, R is phase retardation, is normalized Stokes vector of fast axis.

[0018] By using ensure the symmetry of matrix QM.

[0019] In step two, by using and get and get dichroic vector from it, and get dichroic matrix M D1 and M', wherein G = diag(1,-1,-1,-1).

[0020] In step three, by using and get and get its orthogonal decomposition to get eigenvalues and related eigenvectors.

[0021] Sort the obtained eigenvectors, and get M Δd and M R1 .

[0022] Without known information of medium, under the premise of det(M R ) = 1, adopt the strategy of minimum absolute value of total retardation R, and consider the absolute values of optical rotation ψ and linear phase retardation δ to determine the order, so as to obtain M Δd and M R1 .

[0023] The beneficial effects of the present application: the method not only provides a systematic method for backscattering Mueller matrix decomposition, but also obtains polarization parameters (such as azimuth angle, phase retardation and depolarization coefficient) representing the microstructure of the medium. BRIEF DESCRIPTION OF DRAWINGS

[0024] Figure 1 is the specific step diagram of the present application.

[0025] Figure 2 is the schematic diagram of optical image acquisition device in backscattering polarization device.

[0026] Figure 3 is the birefringence target diagram of experimental sample.

[0027] Figure 4 is the contrast diagram of azimuth angle obtained by forward scattering Lu-chipman polar decomposition, backscattering Lu-chipman polar decomposition and backscattering reversible polar decomposition on different direction birefringence target.

[0028] Figure 5 Figure 6 is a contrast plot of linear retardation for different direction birefringent targets resolved by forward scattering Lu-chipman polar decomposition, backscattering Lu-chipman polar decomposition and backscattering reversible polar decomposition.

[0029] Figure 6 Figure 7 is a depolarization coefficient plot for different direction birefringent targets resolved by forward scattering Lu-chipman polar decomposition, backscattering Lu-chipman polar decomposition and backscattering reversible polar decomposition. DETAILED DESCRIPTION

[0030] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of protection of the present application.

[0031] It should be noted that all directionality indications (such as up, down, left, right, front, back, etc.) in the embodiments of the present application are only used to explain the relative positional relationship, motion condition, etc. between components in a certain specific posture (as shown in the drawings), and if the specific posture changes, the directionality indications also change accordingly.

[0032] In the present application, unless otherwise explicitly specified and limited, the terms "connection", "fixation" and the like should be understood in a broad sense, for example, "fixation" can be fixed connection, or detachable connection, or integral; can be mechanical connection, or connection; can be direct connection, or indirect connection through an intermediate medium; can be internal connection of two elements or interaction relationship between two elements, unless otherwise explicitly limited. For those skilled in the art, the specific meaning of the above terms in the present application can be understood according to the specific circumstances.

[0033] As shown in Figure 1 The present application provides a reversible polar decomposition method of backscattering Mueller matrix, characterized in that it comprises the following steps:

[0034] Step one, reversible polar decomposition of backscattering Mueller matrix According to the relationship between the forward and backward path dichroism matrix and the phase retardation matrix, the backscattering Mueller matrix is converted into a symmetric matrix QM.

[0035] Step two, obtain dichroism matrix M D1 from QMG matrix (QMG matrix is the product of QM matrix and G matrix).

[0036] Step three, get eigenvalues and corresponding eigenvectors by orthogonal decomposition.

[0037] Step four, sort the eigenvectors, and get depolarization matrix M Δd and phase delay matrix M R1 .

[0038] Step five, get polarization parameters from the obtained depolarization matrix M Δd and phase delay matrix M R1 .

[0039] In backscattering, the backscattering Mueller matrix M can be expressed as:

[0040] M = M D2 M R2 M Δd M R1 M D1 (1)

[0041] where M R1 is the phase delay matrix of the forward path, M D1 is the dichroism matrix of the forward path, M Δd is the depolarization matrix, M R2 is the phase delay matrix of the backward path, M D2 is the dichroism matrix of the backward path,

[0042] In backscattering, the optical path is reversible. The dichroism matrix and the phase delay matrix of the backward path can be expressed by the dichroism matrix and the phase delay matrix of the forward path:

[0043]

[0044] where Q = diag(1, 1, -1, 1)

[0045] The backscattering Mueller matrix is converted into a symmetric matrix by using formula (3), and the symmetry of the matrix QM is ensured by using formula (4):

[0046]

[0047]

[0048]

[0049]

[0050] M' Δd = QM Δd = diag(d0, d1, -d2, d3) (7)

[0051] where I is a 3x3 identity matrix, δ ij is the Kronecker function, ε ijk is the Levi-civita permutation symbol, R is the phase retardation, is the normalized Stokes vector of the fast axis.

[0052] Using equation (8), equation (9) and equation (4), equation (10) is obtained, equation (10) has and only has one positive eigenvalue

[0053] The relevant eigenvector can obtain the dichroism vector Thus the dichroism matrix M D1 is obtained.

[0054]

[0055]

[0056]

[0057] where G = diag(1,-1,-1,-1)

[0058] Using equation (11) and equation (12), M' is obtained, and then the eigenvalue and the relevant eigenvector are obtained by orthogonal decomposition of equation (13)

[0059]

[0060]

[0061]

[0062] The order and the sign of the eigenvalue and the eigenvector obtained by the above decomposition are undetermined, so the order and the sign of the eigenvalue and the eigenvector need to be determined. If there is known information about the medium depolarization characteristics, then the order and the sign of the eigenvector can be determined. In the case of no known information about the medium, under the premise of det(M R ) = 1, the strategy of the absolute value of the total retardation R is the smallest is adopted, considering that the optical rotation (i.e. circular retardation) of the tissue sample is generally small, and the strategy of considering the absolute values of the optical rotation ψ and the linear phase retardation δ to be small is adopted to determine the order, so as to obtain M Δd and M R1 .

[0063] The linear retardation and optical rotation effects can occur simultaneously or in sequence, when in sequence, the azimuth angle θ, the optical rotation ψ, the linear phase retardation δ and the depolarization coefficient Δ are obtained by using equations (14)-(19). When occurring simultaneously, the total retardation R, the azimuth angle θ, the optical rotation ψ and the linear phase retardation δ are obtained by using equations (20)-(26).

[0064]

[0065]

[0066] M LR1 = M CR1 T M R1 (16)

[0067]

[0068]

[0069]

[0070]

[0071]

[0072]

[0073]

[0074]

[0075]

[0076]

[0077] wherein M CR1 and M LR1 are the circular retardation matrix and the linear retardation matrix of M R1 , respectively.

[0078] The experimental setup is shown in Figure 2 Fig. 4. LED (633 nm) light passes through a polarizer (P, Thorlabs) and a quarter wave plate (QW1, Thorlabs), which is reflected by a thin film beam splitter (BS, Thorlabs) onto the sample. Photons scattered from the sample are reflected by a mirror (M) and pass through a quarter wave plate (QW2, Thorlabs) and are received by a polarization camera (PCCD, FLIR BFS-U3-51 S5P-C). The Mueller matrix of each optical element was measured and stray light interference was eliminated to obtain accurate Mueller matrices.

[0079] To study the feasibility of the reversible polar decomposition of Mueller matrix in backscattering configuration, the sample is chosen as NBS1963A birefringence resolution target (Thorlabs, R2L2S1B), as shown in FIG. 1, which has a clear linear retardation and azimuth angle. The red box region in (a) is selected as shown in (b), and the targets in different directions are placed to obtain the results of azimuth angle, linear phase retardation and depolarization coefficient by forward scattering Lu-chipman polar decomposition, backscattering Lu-chipman polar decomposition and backscattering reversible polar decomposition, respectively. Figure 3

[0080] As shown in FIG. 2, the two-dimensional graphs of azimuth angle obtained by three different methods for three different direction targets are shown, wherein (a), (b), (c) are the azimuth angles of three different methods for the horizontally placed target, (d), (e), (f) are the azimuth angles of three different methods for the target rotated counterclockwise by 13°, (g), (h), (i) are the azimuth angles of three different methods for the target rotated clockwise by 11°, and the specific data are shown in Table 1. Figure 4 As shown in FIG. 3, the two-dimensional graphs of linear phase retardation obtained by three different methods for three different direction targets are shown, wherein (a), (b), (c) are the linear phase retardations of three different methods for the horizontally placed target, (d), (e), (f) are the linear phase retardations of three different methods for the target rotated counterclockwise by 13°, (g), (h), (i) are the linear phase retardations of three different methods for the target rotated clockwise by 11°, and the specific data are shown in Table 2.

[0081] Figure 5 As shown in FIG. 4, the two-dimensional graphs of depolarization coefficient obtained by three different methods for three different direction targets are shown, wherein (a), (b), (c) are the depolarization coefficients of three different methods for the horizontally placed target, (d), (e), (f) are the depolarization coefficients of three different methods for the target rotated counterclockwise by 13°, (g), (h), (i) are the depolarization coefficients of three different methods for the target rotated clockwise by 11°, and the specific data are shown in Table 3.

[0082] Figure 6 As shown in FIG. 4, the two-dimensional graphs of depolarization coefficient obtained by three different methods for three different direction targets are shown, wherein (a), (b), (c) are the depolarization coefficients of three different methods for the horizontally placed target, (d), (e), (f) are the depolarization coefficients of three different methods for the target rotated counterclockwise by 13°, (g), (h), (i) are the depolarization coefficients of three different methods for the target rotated clockwise by 11°, and the specific data are shown in Table 3.

[0083] ​​​Thorlabs provides the data of NBS 1963A birefringence resolution target horizontally placed, as shown in Table 4, the azimuth and linear phase retardation obtained by backscattering reversible polar decomposition and forward scattering Lu-chipman polar decomposition are basically consistent with the data provided by Thorlabs, the data of each parameter obtained by backscattering reversible polar decomposition and forward scattering Lu-chipman polar decomposition are basically consistent when the horizontally placed target is rotated counterclockwise by 13° and clockwise by 11°, while the backscattering Lu-chipman polar decomposition result is quite different.

[0084] It can be clearly seen from the experimental results that the backscattering reversible polar decomposition can not only obtain the correct azimuth and linear phase retardation, but also obtain the correct azimuth and retardation when placed in different directions, which proves the feasibility of the Mueller matrix reversible polar decomposition in the backscattering structure and can solve the problem of backscattering Mueller matrix decomposition.

[0085] In each embodiment described above, the Mueller matrix of backscattering is converted into a symmetric matrix by using the reversible polar decomposition of the Mueller matrix of backscattering, the dichroism matrix M D is obtained by using the QMG matrix, and then the eigenvalue and the related eigenvector are obtained by using orthogonal decomposition, the eigenvector is sorted, and then the depolarization matrix M Δ , the phase retardation matrix M R , and the polarization parameters (such as azimuth, phase retardation and depolarization coefficient) with physical meaning can be obtained. This method not only greatly reduces the complexity of the backscattering Mueller matrix decomposition, but also can obtain the polarization parameters (such as azimuth, phase retardation and depolarization coefficient) with physical meaning.

[0086] Table 1. Azimuth data of targets in different directions by different methods. #1: horizontally placed target, #2: horizontally placed target rotated counterclockwise by 13°, #3: horizontally placed target rotated clockwise by 11°, θ L : target line azimuth,

[0087] θ B : target background azimuth.

[0088]

[0089] Table 2. Linear phase retardation data of targets in different directions by different methods. #1: horizontally placed target, #2: horizontally placed target rotated counterclockwise by 13°, #3: horizontally placed target rotated clockwise by 11°, δ L : target line linear phase retardation, δ B : target background linear phase retardation.

[0090]

[0091] Table 3. Depolarization data for different orientations of target and different methods. #1 : target placed horizontally, #2: target placed horizontally rotated 13° counter clockwise, #3: target placed horizontally rotated 11° clockwise, Δ L : target line depolarization, Δ B： target background depolarization.

[0092]

[0093] Table 4. Thorlabs horizontally placed birefringent target data

[0094]

[0095] The examples should not be considered as limiting the application but any improvement based on the spirit of the application should be within the scope of the application.

Claims

1. A reversible polar decomposition method for a backscattered Mueller matrix, characterized by: It includes the following steps: Step 1: Reversible polar decomposition of the backscattered Mueller matrix Based on the relationship between the forward and backward path dichroic matrices and the phase delay matrix, the backscattered Mueller matrix is ​​converted into a matrix QM, ; In step 1, the Mueller matrix of backscattering is defined as ,in M R1 is the phase delay matrix of the forward path, M D1 is the dichroic matrix of the forward path, M ∆d is the depolarization matrix, M R2 is the phase delay matrix of the backward path, M D2 is the dichroic matrix of the backward path; According to the principle of reversibility of light path in backscattering, the dichroic matrix and phase delay matrix of the backward path are expressed as the dichroic matrix and phase delay matrix of the forward path in ; pass Convert the backscattered Mueller matrix into a symmetric matrix, in , , , , is a 3x3 unit matrix, , , is the Kronecker function, is the Levi-civita substitution symbol, is the phase delay, is the normalized Stokes vector of the fast axis; pass( QM+ (QM) T ) / 2 Substitution Matrix QM Ensure symmetric matrix QM Symmetry; Step 2: Pass QMG The matrix gets the dichroic matrix of the forward path M D1 ,in ; Step 3: Pass and get , and perform orthogonal decomposition to obtain eigenvalues ​​and related eigenvectors; Step 4: Sort the obtained eigenvectors. According to the convention of sorting eigenvectors, the eigenvectors can be multiplied by ±1 to obtain the minimum delay, that is, m R Closest to the identity matrix, satisfying det( m R )=1, and get the depolarization matrix M Δd and the phase delay matrix of the forward path M R1 ; Step 5: Obtain the depolarization matrix M ∆d and the phase delay matrix of the forward path M R1 Get the polarization parameters, .

2. The reversible polar decomposition method of the backscattered Mueller matrix according to claim 1, characterized in that: In step 2, use 、 and get , and obtain the dichroic vector , and then obtain the dichroic matrix of the forward path M D1 and ,in .

3. The reversible polar decomposition method of the backscattered Mueller matrix according to claim 1, characterized in that: In the absence of known information about the medium, det(m R ) =1, using phase delay R The strategy of minimizing absolute value while taking into account optical rotation ψ Absolute value and linear phase delay δ The absolute value is smaller to determine the order, so as to obtain M Δd and M R1 , ， 。

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