Method for measuring dielectric tensor of material

Abstract

The disclosure relates to a method for measuring a dielectric tensor of a material. Firstly, a partial conversion matrix T.sub.p and a transmission matrix T.sub.t are determined by a predetermined initial value ε(E) of the dielectric tensor of the material to be measured, thereby obtaining a transfer matrix of an electromagnetic wave on a surface of the material to be measured by the partial conversion matrix T.sub.p, the transmission matrix T.sub.t, and an incident matrix T.sub.i, a theoretical Mueller matrix spectrum MM.sub.Cal(E) of the material to be measured is determined by the transfer matrix T.sub.m. A fitting analysis is performed on the theoretical Mueller matrix spectrum MM.sub.Cal(E) and a measured Mueller matrix spectrum MM.sub.Exp(E) of the material to be measured to obtain the dielectric tensor of the material to be measured.

Claims

1. A method for measuring a dielectric tensor of a material, comprising: S1: determining, by a processor, a partial conversion matrix T.sub.p and a transmission matrix T.sub.t by a predetermined initial value ε(E) of the dielectric tensor of the material to be measured, thereby obtaining a transfer matrix T.sub.m=T.sub.i.sup.−1T.sub.pT.sub.t of an electromagnetic wave on a surface of the material to be measured by the partial conversion matrix T.sub.p, the transmission matrix T.sub.t, and an incident matrix T.sub.i; and S2: determining, by the processor, a theoretical Mueller matrix spectrum MM.sub.Cal(E) of the material to be measured by the transfer matrix T.sub.m, S3: measuring a measured Mueller matrix spectrum MM.sub.Exp (E) through a Mueller matrix ellipsometer, S4: performing, by the processor, a fitting analysis on the theoretical Mueller matrix spectrum MM.sub.Cal(E) and the measured Mueller matrix spectrum MM.sub.Exp (E) of the material to be measured to obtain the dielectric tensor of the material to be measured, wherein a Trust-Region-Reflective algorithm is adopted in the fitting analysis, wherein the Mueller matrix ellipsometer obtains a plurality of sets of the measured Mueller matrix spectra MM.sub.Exp (E) of the material to be measured from a combination of three incident angles and three azimuth angles of three planes to be measured, and fits the plurality of sets of the measured Mueller matrix spectra MM.sub.Exp (E) with the theoretical Mueller matrix spectrum MM.sub.Cal(E) at the same time, wherein the dielectric tensor of the material to be measured is a dielectric tensor at a highest coincidence degree of fitting, wherein the dielectric tensor of the material is used to improve properties of devices using the material, wherein the material includes: quantum dots, nanowires (nanotubes), and two-dimensional materials.

2. The method for measuring the dielectric tensor of the material according to claim 1, wherein the incident matrix T.sub.i is calculated by a following equation:
T.sub.i[E.sub.is E.sub.rs E.sub.ip E.sub.rp].sup.T=[(E.sub.ip−E.sub.rp)cos θ.sub.i E.sub.is−E.sub.rs(B.sub.rs−B.sub.is)cos θ.sub.i B.sub.ip+B.sub.rp].sup.T wherein, E.sub.is is an electric field of the incident s-polarized light and E.sub.rs is an electric field of a reflected s-polarized light, E.sub.ip is an electric field of an incident p-polarized light and E.sub.rp is an electric field of a reflected p-polarized light, B.sub.is is a magnetic field of the incident s-polarized light and B.sub.rs is a magnetic field of the reflected s-polarized light, B.sub.ip is a magnetic field of the incident p-polarized light and B.sub.rp is a reflected magnetic field of the reflected p-polarized light, and θ.sub.i is an incident angle of the electromagnetic wave.

3. The method for measuring the dielectric tensor of the material according to claim 1, wherein the step of determining the partial conversion matrix T.sub.p by the predetermined initial value ε(E) of the dielectric tensor of the material to be measured specifically comprises: (1) determining a matrix Δ.sub.B by the predetermined initial value ε(E) of the dielectric tensor of the material to be measured: ${\Delta }_B=\left[\begin{array}{cccc}-n_i\sin {\theta }_i\left({\epsilon }_{31}/{\epsilon }_{33}\right)& -n_i\sin {\theta }_i\left({\epsilon }_{32}/{\epsilon }_{33}\right)& 0& 1-{\left(n_i\sin {\theta }_i\right)}^2/{\epsilon }_{33}\\ 0& 0& -1& 0\\ {\epsilon }_{23}\left({\epsilon }_{31}/{\epsilon }_{33}\right)-{\epsilon }_{21}& {\left(n_i\sin {\theta }_i\right)}^2-{\epsilon }_{22}+{\epsilon }_{23}\left({\epsilon }_{32}/{\epsilon }_{33}\right)& 0& n_i\sin {\theta }_i\left({\epsilon }_{23}/{\epsilon }_{33}\right)\\ {\epsilon }_{11}-{\epsilon }_{13}\left({\epsilon }_{31}/{\epsilon }_{33}\right)& {\epsilon }_{12}-{\epsilon }_{13}\left({\epsilon }_{32}/{\epsilon }_{33}\right)& 0& -n_i\sin {\theta }_i\left({\epsilon }_{13}/{\epsilon }_{33}\right)\end{array}\right]$ wherein, θ.sub.i is the incident angle of the electromagnetic wave, n.sub.i is a refractive index of an external medium, and the predetermined initial value of the dielectric tensor is $\epsilon \left(E\right)=\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}\end{array}\right];$ and (2) determining the partial conversion matrix T.sub.p=exp[i(ω/c)Δ.sub.B(−d)] by the matrix Δ.sub.B, where ω is an angular frequency of the electromagnetic wave, c is a speed of light in vacuum, d is a thickness of the material to be measured, and i is a unit imaginary number.

4. The method for measuring the dielectric tensor of the material according to claim 3, wherein the step of determining the transmission matrix T.sub.t by the predetermined initial value ε(E) of the dielectric tensor of the material to be measured specifically comprises: (1) calculating the transmission matrix T.sub.t by a following equation when the material to be measured is an optical homogeneous substance:
T.sub.t[E.sub.ts 0 E.sub.tp 0].sup.T=[E.sub.tp cos θ.sub.t E.sub.ts−B.sub.ts cos θ.sub.t B.sub.tp].sup.T wherein, θ.sub.t is a transmission angle of the electromagnetic wave, E.sub.ts is an electric field of a transmitted s-polarized light and B.sub.ts is a magnetic field of the transmitted s-polarized light, and E.sub.tp is an electric field of a transmitted p-polarized light and B.sub.tp is a magnetic field of the transmitted p-polarized light; and (2) performing a characteristic analysis on the matrix Δ.sub.B determined by the predetermined initial value ε(E) of the dielectric tensor when the material to be measured is a non-optical homogeneous substance, thereby obtaining the transmission matrix T.sub.t.

5. The method for measuring the dielectric tensor of the material according to claim 1, wherein the step of determining the theoretical Muller matrix spectrum MM.sub.Cal(E) of the material to be measured by the transfer matrix T.sub.m specifically comprises: (1) determining a theoretical Jones matrix J.sub.Cal by the transfer matrix T.sub.m: $J_{\mathrm{Cal}}=\frac{1}{t_{11}{t}_{33}-t_{13}{t}_{31}}\left[\begin{array}{cc}\left(t_{11}{t}_{43}-t_{13}{t}_{41}\right)& \left(t_{33}{t}_{41}-t_{31}{t}_{43}\right)\\ \left(t_{11}{t}_{23}-t_{13}{t}_{21}\right)& \left(t_{21}{t}_{33}-t_{23}{t}_{31}\right)\end{array}\right]$ wherein, the transfer matrix $T_m=\left[\begin{array}{cccc}{t}_{11}& t_{12}& t_{13}& t_{14}\\ t_{21}& t_{22}& t_{23}& t_{24}\\ t_{31}& t_{32}& t_{33}& t_{34}\\ t_{41}& t_{42}& t_{43}& t_{44}\end{array}\right];$ and (2) determining the theoretical Muller matrix spectrum MM.sub.Cal(E) of the material to be measured by the theoretical Jones matrix J.sub.Cal: ${\mathrm{MM}}_{\mathrm{Cal}}\left(E\right)=A\left(J_{Cal}.Math.J_{Cal}^{*}\right)A^{-1},\mathrm{where}A=\left[\begin{array}{cccc}1& 0& 0& 1\\ 1& 0& 0& -1\\ 0& 1& 1& 0\\ 0& i& -i& 0\end{array}\right].$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a flowchart of a method for measuring a dielectric tensor of a material according to an embodiment of the disclosure.

(2) FIG. 2 are schematic diagrams of crystal structures and ellipsometry of BaGa.sub.4Se.sub.7 according to an embodiment of the disclosure.

(3) FIG. 3 is a test and a theoretical Muller matrix spectrum of a BaGa.sub.4Se.sub.7 crystal in the energy range of 0.73-6.42 eV according to an embodiment of the disclosure.

(4) FIGS. 4A to 4D are schematic diagrams of a dielectric tensor of the BaGa.sub.4Se.sub.7 crystal according to an embodiment of the disclosure, where FIG. 4A is element ε.sub.11, FIG. 4B is element ε.sub.12, FIG. 4C is element ε.sub.22, and FIG. 4D is element ε.sub.33.

DETAILED DESCRIPTION OF DISCLOSED EMBODIMENTS

(5) For the objective, technical solutions, and advantages of the disclosure to be clearer, the disclosure is further described in detail below with reference to the accompanying drawings and the embodiments. It should be understood that the specific embodiments described here are only used to explain the disclosure, but not to limit the disclosure. In addition, the technical features involved in the various embodiments of the disclosure described below may be combined with each other as long as there is no conflict therebetween.

(6) A method for measuring a dielectric tensor of a material provided by an embodiment of the disclosure, as shown in FIG. 1, includes the following steps.

(7) S1: a measured Mueller matrix spectrum MM.sub.Exp(E) within a specified energy range of a material to be measured is determined, where E represents the energy of the electromagnetic wave, which one-to-one corresponds to the frequency. The measured Muller matrix spectrum MM.sub.Exp(E) is obtained through experimental measurement, literature review, etc., and is preferably measured through a Muller matrix ellipsometer. Further, during measurement, the Mueller matrix ellipsometer obtains a plurality of sets of the measured Mueller matrix spectra MM.sub.Exp(E) of the material to be measured from a plurality of incident angles and azimuth angles.

(8) S2: the form of the dielectric tensor of the material to be measured is determined. The general form of the dielectric tensor is as shown in Equation (1). Equation (1) is preferably simplified through a diagonalization operation, thereby obtaining the most simplified form of a dielectric tensor ε(E);

(9) $\begin{array}{cc}\epsilon \left(E\right)=\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}\end{array}\right]& \left(1\right)\end{array}$
wherein, ε.sub.ij (i, j=1, 2, 3) are the elements of the dielectric tensor, and each element is a complex number and is a function of energy (frequency). Specifically, for crystal materials, the number of independent elements and the simplest form of the ε(E) in a certain energy range are determined according to the symmetry thereof through checking the crystal structure manual. Generally speaking, the lower-symmetry of the crystal, the more number of independent elements in the ε(E), and it is more difficult to determine. For example, a triclinic crystal has 6 independent elements. For composite materials, the number of independent elements and the simplest distribution form of the ε(E) in a certain energy range are effectively given through the molecular structure and mesoscopic orientation.

(10) Then, by using experimental measurement and data analysis, literature survey, and theoretical simulation, the predetermined initial value of the dielectric tensor ε(E) are calculated. Specifically, for materials with low-symmetry, the principle of selecting the initial values of the elements of the dielectric tensor is: the main diagonal elements are similar and the non-diagonal elements are close to 0.

(11) S3: a transfer matrix T.sub.m of the material is determined.

(12) As shown in FIG. 2, an incident surface of an electromagnetic wave of the material to be measured is taken as an xoy surface, and a coordinate system is established with the thickness direction of the material as the z-axis direction. Electromagnetic waves passing through upper and lower surfaces of the material may be connected by the 4×4 transfer matrix T.sub.m, which is specifically expressed as:
[E.sub.x E.sub.y H.sub.x H.sub.y].sub.z=0.sup.T=T.sub.m[E.sub.x E.sub.y H.sub.x H.sub.y].sub.z=d.sup.T=T.sub.i.sup.−1T.sub.pT.sub.t[E.sub.x E.sub.y H.sub.x H.sub.y].sub.z=d.sup.T  (2)
wherein, E.sub.x is an electric field in the x-direction and E.sub.y is the electric field in the y-direction, and H.sub.x is a magnetic field in the x-direction and H.sub.y is a magnetic field in the y-direction; and z=0 represents the incident surface (upper surface) of the electromagnetic wave of the material to be measured, z=d represents another surface (lower surface) of the material to be measured, and d is the thickness of the material to be measured.

(13) It can be known from Equation (2) that: the transfer matrix T.sub.m=T.sub.i.sup.−1T.sub.pT.sub.t; wherein, T.sub.i is the incident matrix, which projects the electromagnetic wave obliquely incident on the upper surface of the material to the upper surface of the material along the positive direction of the z-axis, and the inverse matrix T.sub.i.sup.−1 converts the electromagnetic wave on the upper surface of the material into an obliquely reflected electromagnetic wave along the negative direction of the z-axis; T.sub.t is the transmission matrix, which projects the electromagnetic wave transmitted from the lower surface of the material to the lower surface of the material along the negative direction of the z-axis; and T.sub.p is the partial conversion matrix, which links electromagnetic fields on the upper and lower surfaces of the material.

(14) Preferably, when the material to be measured is a bulk crystal, the transfer matrix of electromagnetic waves on the upper and lower surfaces thereof is T.sub.m=T.sub.i.sup.−1T.sub.t.

(15) Specifically, the calculation method of each matrix is as follows.

(16) (1) The incidence matrix T.sub.i is calculated according to the in-plane form of the electromagnetic field at z=0 of the material, which is then inverted to obtain T.sub.i.sup.−1. The incidence matrix T.sub.i is specifically calculated by the following equation:
T.sub.i[E.sub.is E.sub.rs E.sub.ip E.sub.rp].sup.T=[(E.sub.ip−E.sub.rp)cos θ.sub.i E.sub.is−E.sub.rs(B.sub.rs−B.sub.is)cos θ.sub.i B.sub.ip+B.sub.rp].sup.T   (3)
wherein, E.sub.is is an electric field of the incident s-polarized light and E.sub.rs is an electric field of a reflected s-polarized light, E.sub.ip is an electric field of an incident p-polarized light and E.sub.rp is an electric field of a reflected p-polarized light, B.sub.is is a magnetic field of the incident s-polarized light and B.sub.rs is a magnetic field of the reflected s-polarized light, and B.sub.ip is a magnetic field of the incident p-polarized light and B.sub.rp is a magnetic field of the reflected p-polarized light; and θ.sub.i is the incident angle of the electromagnetic wave.

(17) (2) The partial conversion matrix is solved according to the Berreman equation, wherein, n.sub.i is the refractive index of an external medium, i is the unit imaginary number, ω is the angular frequency of the electromagnetic wave, c is the speed of light in vacuum, d is the thickness of the material to be measured, and Δ.sub.B is the fourth-rank matrix in the Berreman equation:

(18) $\begin{array}{cc}{\Delta }_B=\left[\begin{array}{cccc}-n_i\sin {\theta }_i\left({\epsilon }_{31}/{\epsilon }_{33}\right)& -n_i\sin {\theta }_i\left({\epsilon }_{32}/{\epsilon }_{33}\right)& 0& 1-{\left(n_i\sin {\theta }_i\right)}^2/{\epsilon }_{33}\\ 0& 0& -1& 0\\ {\epsilon }_{23}\left({\epsilon }_{31}/{\epsilon }_{33}\right)-{\epsilon }_{21}& {\left(n_i\sin {\theta }_i\right)}^2-{\epsilon }_{22}+{\epsilon }_{23}\left({\epsilon }_{32}/{\epsilon }_{33}\right)& 0& n_i\sin {\theta }_i\left({\epsilon }_{23}/{\epsilon }_{33}\right)\\ {\epsilon }_{11}-{\epsilon }_{13}\left({\epsilon }_{31}/{\epsilon }_{33}\right)& {\epsilon }_{12}-{\epsilon }_{13}\left({\epsilon }_{32}/{\epsilon }_{33}\right)& 0& -n_i\sin {\theta }_i\left({\epsilon }_{13}/{\epsilon }_{33}\right)\end{array}\right]& \left(4\right)\end{array}$

(19) (3) The transmission matrix T.sub.t method is calculated and determined according to the physical properties of the material to be measured, which specifically includes the following steps.

(20) (3.1) If the material to be measured is an optical homogeneous substance, the transmission matrix T.sub.t is calculated according to the in-plane form of the electromagnetic field at z=d of the material. The transmission matrix T.sub.t is specifically calculated by the following equation:
T.sub.t[E.sub.ts 0 E.sub.tp 0].sup.T=[E.sub.tp cos θ.sub.t E.sub.ts−B.sub.ts cos θ.sub.tB.sub.tp].sup.T  (5)
wherein, θ.sub.t is the transmission angle of the electromagnetic wave, E.sub.ts is an electric field of a transmitted s-polarized light and B.sub.ts is a magnetic field of the transmitted s-polarized light, and E.sub.tp is an electric field of a transmitted p-polarized light and B.sub.tp is a magnetic field of the transmitted p-polarized light.

(21) (3.2) If the material to be measured is a non-optical homogeneous substance, the transmission matrix T.sub.t is determined through a characteristic analysis of the matrix Δ.sub.B. For materials with no or negligible back reflection, the eigenvectors corresponding to the non-negative eigenvalues of the Δ.sub.B are written to the first and third columns of the fourth-rank empty matrix to obtain the transmission matrix T.sub.t.

(22) S4: a theoretical Muller matrix spectrum MM.sub.Cal(E) of the material to be measured is determined by the transfer matrix T.sub.m, which specifically includes the following steps.

(23) (1) A theoretical Jones matrix J.sub.Cal is determined by the transfer matrix T.sub.m:

(24) $\begin{array}{cc}{J}_{\mathrm{Cal}}=\frac{1}{t_{11}{t}_{33}-t_{13}{t}_{31}}\left[\begin{array}{ccc}(t_{11}{t}_{43}& -t_{13}{t}_{41})& \left(t_{33}{t}_{41}-t_{31}{t}_{43}\right)\\ (t_{11}{t}_{23}& -t_{13}{t}_{21})& \left(t_{21}{t}_{33}-t_{23}{t}_{31}\right)\end{array}\right]& \left(6\right)\end{array}$

(25) (2) The theoretical Muller matrix spectrum MM.sub.Cal(E) of the material to be measured is determined by the theoretical Jones matrix J.sub.Cal:
MM.sub.Cal(E)=A(J.sub.Cal.Math.J.sub.Cal*)A.sup.−1  (7)
wherein, t.sub.ij (i, j=1, 2, 3, 4) are the elements in the transfer matrix T.sub.m; and J.sub.Cal* is the complex conjugate of the J.sub.cal, and

(26) $A=\left[\begin{array}{cccc}1& 0& 0& 1\\ 1& 0& 0& -1\\ 0& 1& 1& 0\\ 0& i& -i& 0\end{array}\right].$

(27) S5: the Levenberg-Marquardt or Trust-Region-Reflective algorithm is adopted to perform fitting and matching analyses at each energy frequency on multiple sets of the measured Mueller matrix spectra MM.sub.Exp(E) and the theoretical Mueller matrix spectrum MM.sub.Cal(E) The dielectric tensor at the highest coincidence degree of element matching is selected as the most reliable dielectric tensor of the material to be measured at the frequency, thereby obtaining the complete dielectric tensor of the material to be measured.

(28) The following are specific embodiments.

Embodiment 1

(29) The dielectric tensor of a BaGa.sub.4Se.sub.7 crystal in an energy E range of 0.73-6.42 eV is calculated, which specifically includes the following steps.

(30) S1: a measured Mueller matrix spectrum MM.sub.BaGa.sub.4.sub.Se.sub.7.sup.Exp(E) of the BaGa.sub.4Se.sub.7 crystal in the energy range of 0.73-6.42 eV is determined. As shown in FIG. 2, the Mueller matrix ellipsometer (MME) is adopted for measurements of multiple incident angles (θ.sub.i=60°, 65°, 70°) and multiple azimuth angles (three azimuth angles of 45°, 60°, and 75° are selected in each measurement plane) of three measurement planes (xoy, xoz, zoy) of the monoclinic BaGa.sub.4Se.sub.7 crystal. A total of 27 sets of the measured Mueller matrix spectra are obtained. As shown in FIG. 3, the scatters are the measured Muller matrix spectra at the 45° azimuth angle and the 70° incident angle of the measured xoy plane. In the drawing, m.sub.ij (i,j=1, 2, 3, 4) represents elements in the i-th row and the j-th column in the Muller matrix spectrum. The spectrum has been normalized by an m.sub.11 element in the Muller matrix (m.sub.11=1, not shown).

(31) S2: the simplest form of the dielectric tensor of the BaGa.sub.4Se.sub.7 crystal is determined. Referring to the crystal structure manual, the BaGa.sub.4Se.sub.7 crystal belongs to monoclinic crystal system, a space group Pc, and unit cell parameters: a=7.6252(15) Å, b=6.5114(13) Å, c=14.702(4) Å, β=121.24(2)°, and Z=2. The simplest form of the dielectric tensor of the BaGa.sub.4Se.sub.7 crystal is

(32) 0${\epsilon }_{BaG{a}_{4}S{e}_7}\left(E\right)=\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& \\ {\epsilon }_{12}& {\epsilon }_{22}& \\ & & {\epsilon }_{33}\end{array}\right].$

(33) Referring to the literature “Jiyong Yao et al. BaGa.sub.4Se.sub.7: A New Congruent-Melting IR Nonlinear Optical Material. Inorg. Chem. 2010, 49: 9212-9216”, the initial value of a diagonal independent element of the dielectric tensor at 0.73 eV is initially set as ε.sub.0.sup.jj=7.04+i0.68, j=1, 2, 3, and the initial value of a non-diagonal independent element is ε.sub.0.sup.12=0+i0.02.

(34) S3: a transfer matrix

(35) $T_{m_{\left(\mathrm{BaG}{a}_{4}S{e}_7\right)}}$
of the material is determined.

(36) In the embodiment, the sample atmosphere is air and the refractive index n.sub.i≈1. Since BaGa.sub.4Se.sub.7 is a bulk crystal, there is no need to introduce a partial conversion matrix T.sub.p. An inverse matrix

(37) $T_{i\left(BaG{a}_{4}S{e}_7\right)}^{-1}=\left[\begin{array}{cccc}0& 1& -1/2\cos {\theta }_i& 0\\ 1/2& 1& 1/2\cos {\theta }_i& 0\\ 1/2\cos {\theta }_i& 0& 0& 1/2\\ -1/2\cos {\theta }_i& 0& 0& 1/2\end{array}\right]$
of the incident matrix is obtained by Equation (3).

(38) At the same time, BaGa.sub.4Se.sub.7 is a crystal with low-symmetry, and the number of independent elements in the dielectric tensor thereof is 4, which are all complex numbers, so the values to be determined are 8. Therefore, a transmission matrix

(39) $T_{t_{\left(BaG{a}_{4}S{e}_7\right)}}$
is in the form of a non-optical homogeneous substance, that is, a characteristic analysis needs to be performed on the Δ.sub.B matrix in Equation (4) ΔB. When measuring different sections of BaGa.sub.4Se.sub.7, the simplest dielectric tensor thereof needs to be rotated, that is, modulated using a rotation matrix. The rotation matrix is:

(40) $R=\left[\begin{array}{ccc}\cos \mathrm{\varphi cos}\psi -\sin \mathrm{\varphi cos}\mathrm{\theta sin}\psi & -\cos \mathrm{\varphi cos}\psi -\sin \mathrm{\varphi cos\theta cos\psi }& \sin \mathrm{\varphi sin\theta }\\ \sin \mathrm{\varphi cos}\psi +\cos \mathrm{\varphi cos}\mathrm{\theta sin}\psi & -\sin \mathrm{\varphi sin}\psi +\cos \mathrm{\varphi cos\theta cos\psi }& -\cos \mathrm{\varphi sin}\theta \\ \sin \mathrm{\varphi sin\psi }& \sin \mathrm{\varphi cos\psi }& \cos \theta \end{array}\right]$
wherein, (θ, ϕ, Ψ) are the Euler angles. Therefore, a dielectric tensor ε.sub.BaGa.sub.4.sub.Se.sub.7′(E)=Rε.sub.BaGa.sub.4.sub.Se.sub.7(E)R.sup.T in Equation (4) is finally substituted, thereby obtaining the Δ.sub.B to find the eigenvalues and eigenvectors thereof, so as to obtain the transmission matrix

(41) $T_{t\left(BaG{a}_4{\mathrm{Se}}_7\right)}=\left[\begin{array}{cccc}{X}_{a1}& 0& X_{b1}& 0\\ X_{a2}& 0& X_{b2}& 0\\ X_{a3}& 0& X_{b3}& 0\\ X_{a4}& 0& X_{b4}& 0\end{array}\right],$
wherein, the subscripts a and b refer to two positive real part eigenvalues of the Δ.sub.B matrix, and the vectors [X.sub.a1, X.sub.a2, X.sub.a3, X.sub.a4].sup.T and [X.sub.b1, X.sub.b2, X.sub.b3, X.sub.b4].sup.T are respectively eigenvectors corresponding to the eigenvalues a and b.

(42) Then, the

(43) $T_{i_{\left(BaG{a}_{4}S{e}_7\right)}}^{-1}$
is multiplied by the

(44) $T_{t_{\left(BaG{a}_{4}S{e}_7\right)}}$
to obtain the transfer matrix

(45) $T_{m_{\left(\mathrm{BaG}{a}_{4}S{e}_7\right)}}.$

(46) S4: a theoretical Mueller matrix spectrum MM.sub.BaGa.sub.4.sub.Se.sub.7.sup.Cal(E) of the BaGa4Se7 crystal is calculated from Equation (6) and Equation (7). As shown in FIG. 3, the solid line is the theoretical Mueller matrix spectrum at the 45° azimuth angle and the 70° incident angle of the calculated xoy plane.

(47) S5: a dielectric tensor ε.sub.BaGa.sub.4.sub.Se.sub.7(E) of the BaGa4Se7 crystal is extracted. The Levenberg-Marquardt and trust-region-reflective algorithms are adopted to fit and match 27 sets of the theoretical Mueller matrix spectrum MM.sub.BaGa.sub.4.sub.Se.sub.7.sup.Cal(E) and the measured Mueller matrix spectra MM.sub.BaGa.sub.4.sub.Se.sub.7.sup.Exp(E) at each energy (frequency). The dielectric tensor with the highest degree of matching is considered to be the most reliable dielectric tensor of the material at the frequency. It is found that the dielectric tensors obtained by the two intelligent algorithms are basically consistent. As shown in FIG. 3, the 2 main diagonal 2×2 sub-matrix blocks of MM.sub.BaGa.sub.4.sub.Se.sub.7.sup.Cal(E) and MM.sub.BaGa.sub.4.sub.Se.sub.7.sup.Exp(E) have good goodness of fit, and the basic trend of fitting of the 2 vice-diagonal 2×2 sub-matrix blocks is consistent.

(48) Finally, the spectrum of each element of the dielectric tensor of the BaGa.sub.4Se.sub.7 crystal is shown in FIGS. 4A to 4D. In the drawing, Re-ε.sub.ij represents the real part of the element ε.sub.ij, and Im-ε.sub.ij represents the imaginary part of the element ε.sub.ij. Using such analysis method, not only the main diagonal elements ε.sub.11, ε.sub.22, and ε.sub.33 of the dielectric tensor of the BaGa.sub.4Se.sub.7 crystal are obtained, the non-diagonal element ε.sub.12 is also determined at the same time.

(49) Persons skilled in the art may easily understand that the above descriptions are only the preferred embodiments of the disclosure and are not intended to limit the disclosure. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the disclosure should be included in the protection scope of the disclosure.