CN110850523A - Full-wave mixed spectrum element method-based liquid crystal filled photonic crystal fiber analysis method - Google Patents

Full-wave mixed spectrum element method-based liquid crystal filled photonic crystal fiber analysis method Download PDF

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CN110850523A
CN110850523A CN201911259471.3A CN201911259471A CN110850523A CN 110850523 A CN110850523 A CN 110850523A CN 201911259471 A CN201911259471 A CN 201911259471A CN 110850523 A CN110850523 A CN 110850523A
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刘娜
吴依娜
蔡国雄
柳清伙
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Abstract

A liquid crystal filled photonic crystal fiber analysis method based on a full-wave mixed spectrum element method relates to photonic crystal fibers. And selecting a calculation region which is not repeatedly calculated according to the solid model, and carrying out mesh subdivision on a limited number of sub-domains on the calculation region, wherein each sub-domain unit is subdivided into a quadrilateral structure. The method utilizes a mode of combining full-wave Helmholtz equation and Gauss's law, constructs a basis function by a Gauss-Legendre-Lobarton polynomial, and obtains the energy band characteristic of the photonic crystal with the complex medium in a full-wave calculation mode of mutual coupling of TM mode and TE mode, and can inhibit the generation of zero spurious mode. The photonic crystal filled with the liquid crystal is calculated by a full-wave mixed spectrum element method, so that different photonic band gaps under different parameters can be obtained, the flexible control of the photonic crystal fiber on transmission spectrum is realized, and the problem that the transmission characteristic of the traditional photonic crystal fiber is not changed once the photonic crystal fiber is drawn is solved.

Description

Full-wave mixed spectrum element method-based liquid crystal filled photonic crystal fiber analysis method
Technical Field
The invention relates to a photonic crystal fiber, in particular to a liquid crystal filled photonic crystal fiber analysis method based on a full-wave mixed spectrum element method.
Background
Photonic crystal fibers are typically constructed of a single material (e.g., undoped quartz or polymer) with air holes in the order of microns periodically distributed in two dimensions across the fiber cross-section. The photonic crystal fiber has the characteristics of no-cutoff single mode, low loss, capability of carrying out microstructure design transformation and the like, and is concerned in recent years (the influence of Du-Muqing, Zhang Linli, Liu Yong Jun. liquid crystal and different filling structures thereof on the transmission characteristics of the photonic crystal fiber [ J ] liquid crystal and display, 2018,33(02):116- & 122.). The transmission characteristics of photonic crystal fibers do not change as the photonic crystal fibers are drawn, which greatly limits their range of applications. Therefore, the liquid crystal is filled into the air holes of the photonic crystal fiber cladding, and the parameters of the liquid crystal can be controlled by changing the external environment, so that the transmission spectrum of the photonic crystal fiber can be flexibly controlled. Therefore, when studying and designing photonic crystal fibers, it is important to fully understand the propagation characteristics of electromagnetic waves in the photonic crystal structure.
Meanwhile, the basis for researching the photonic crystal is to accurately calculate the energy band characteristics of the photonic crystal. Currently, commonly used calculation methods include a plane wave expansion method (PWE), a Transmission Matrix Method (TMM), a finite difference time domain method (FDTD), a Finite Element Method (FEM), and the like. If the material properties of the two-dimensional photonic crystal are isotropic, the numerical model described above is sufficient for band structure analysis. However, anisotropic materials are sometimes intentionally introduced into a two-dimensional photonic crystal structure to alter the mode of the band structure and thereby control the behavior of the bandgap. For two-dimensional photonic crystals made of diagonally anisotropic materials, these numerical models may also suffice, as long as the completeness of the irreducible brillouin zone that constructs the band structure is carefully modified, yet a complete band structure can still be obtained (Hsu SM, Chang H c. full-vertical finite element method for the analysis of 2D photonic crystals with the array 3D analytical copy J. Optics Express,2007,15(24):15797 15811). However, the photonic crystal is not in a diagonal form in the case of permittivity and permeability tensor anisotropy, and the above numerical method cannot obtain a correct band structure by separating the TE and TM modes. If the numerical pseudo-model and the physical model are mixed together, the numerical values cannot be resolved and a correct solution is screened out.
However, for the mixed spectral element method, based on the vector helmholtz equation and gauss's law, when calculating photonic crystals of any three-dimensional anisotropic and/or inhomogeneous medium, the band structure without the zero spurious mode can be correctly obtained, compared to the conventional method.
Disclosure of Invention
The invention aims to provide a full-wave mixed spectrum element method for analyzing a liquid crystal filled photonic crystal fiber, which is used for calculating the band gap of a two-dimensional liquid crystal photonic crystal by using the advantage of no zero pseudo mode in a characteristic value, and obtaining the energy band characteristic of the photonic crystal by using a full-wave calculation mode of mutual coupling of a TM mode and a TE mode when the two-dimensional liquid crystal photonic crystal of a complex medium is analyzed, thereby being beneficial to the analysis and design of the photonic crystal fiber.
The invention comprises the following steps:
1) establishing a model, selecting a lattice structure and material parameters of a two-dimensional liquid crystal filled photonic crystal, simplifying a single cell structure, and selecting a calculation area which is not repeatedly calculated;
2) carrying out mesh subdivision on a limited number of sub-fields in the calculation region, subdividing each sub-field unit into a quadrilateral structure, and recording the information of each unit in the calculation region;
3) and establishing a mixed spectrum element method implementation and a matrix form intrinsic equation of the two-dimensional liquid crystal filled photonic crystal to finally obtain the energy band structure of the two-dimensional liquid crystal filled photonic crystal structure.
4) If the photonic band gap size obtained in the step 3) does not meet the design of the photonic crystal fiber, the parameters of the liquid crystal can be changed by changing the external environment conditions, and the processes of the steps 1) to 3) are repeated by using the updated material information until the photonic band gap size meets the conditions, the iteration is finished, and the energy band structure diagram is output.
In the step 1), the material parameters of the two-dimensional liquid crystal filled photonic crystal comprise refractive indexes, dielectric constants, magnetic conductivities, sizes and the like of various materials; the lattice structure can be one of a square lattice, a rectangular lattice, a triangular lattice, a hexagonal lattice and the like; the material of the two-dimensional liquid crystal filled photonic crystal can be isotropic, anisotropic or dispersive;
if the two-dimensional liquid crystal filled photonic crystal is isotropic, dividing the energy band structure according to symmetry, including rotational symmetry and reflection symmetry, and selecting a calculation area which is not repeatedly calculated; if the two-dimensional liquid crystal-filled photonic crystal is an anisotropic material, the electromagnetic wave at this time is highly dependent on the direction, and the anisotropy breaks the symmetry of the structure, so that it is necessary to construct a band structure having more wavevector k directions in the first brillouin zone and select it as a calculation region.
In step 2), the information of each unit in the record calculation area comprises vertex coordinates, material properties, boundary characteristics and the like.
In step 3), the specific steps of establishing the mixed spectrum element method implementation and the matrix form eigenequation of the two-dimensional liquid crystal filled photonic crystal may be:
① obtaining variation forms (2.a) and (2.b) by vector wave equation and Gauss's law, namely equations (1.a) and (1.b) by applying variation principle;
the equations (1.a), (1.b), (2.a), (2.b) are as follows:
Figure BDA0002311215720000021
Figure BDA0002311215720000031
Figure BDA0002311215720000032
Figure BDA0002311215720000033
wherein U is the electric field, k0Is the wave number in free space, [ mu ]r]Is the permeability tensor, [ epsilon ]r]Is the dielectric constant tensor, defines
Figure BDA0002311215720000034
V, q is a test function;
② the divergence term of the above equation (2.b) is introduced as Lagrangian multiplier into the above equation (2.a) to obtain the following variational forms (3.a), (3. b):
Figure BDA0002311215720000035
Figure BDA0002311215720000036
③ converts the integral operation of the physical unit coordinates (x, y) into a calculation in the reference unit coordinates (ξ), and performs a corresponding covariate mapping on the basis functions, the mapping relation being as follows:
wherein the jacobian matrix J is:
Figure BDA0002311215720000038
④ field distribution U and of in-plane propagating wave modes of two-dimensional liquid crystal photonic crystalThe operator is decomposed into a tangential component and a longitudinal component, respectively, denoted as (6) and (7), respectively. Both equations (6) and (7) are taken into the equations (3.a), (3.b) above, and again using the Gauss-Legendre-Lobaroto polynomialConstructing a basis function, and taking the basis function as a test function to obtain a discretized matrix form eigen equation (8);
Figure BDA00023112157200000311
Figure BDA0002311215720000041
wherein, UtAnd UzRespectively the transverse component and the longitudinal component of the electric field U,
Figure BDA0002311215720000042
and
Figure BDA0002311215720000043
are respectively
Figure BDA0002311215720000044
Transverse component and longitudinal component of operator, matrix S is rigidity matrix, matrix M is quality matrix, matrix K is discretized matrix of Gaussian law, and definition
Figure BDA0002311215720000045
e is the column vector of the electric field along the edge and y is the column vector of the electric field on all nodes.
⑤ to reduce the computational size of the matrix, equation (8) is integrated into generalized eigenvalue equation (9):
(S+KHKT)e1＝λMe1(9)
wherein H is selected as α I, definition
Figure BDA0002311215720000046
h is the minimum distance between two inner difference points in the calculation region.
⑥ TE mode and TM mode are calculated using the above generalized eigenvalue equation (9)Wave number k in the case of coupled equation0And finally obtaining the energy band structure of the two-dimensional liquid crystal filling photonic crystal structure.
In step 3) part ②, equation (2.a) is equivalent to equation (3.a) and equation (2.b) is equivalent to equation (3.b) will
Figure BDA0002311215720000047
Taken into equation (3.a) and combined with equation (3.b) to yieldAll solutions to the simultaneous equations representing equation (3.a) and equation (3.b) are solutions to the simultaneous equations of equation (2.a) and equation (2. b); vice versa, when p is 0, all solutions to the simultaneous equations representing equation (2.a) and equation (2.b) are solutions to the simultaneous equations of equation (3.a) and equation (3. b).
In the ⑤ part of step 3), the H is selected by considering the size consistency of two sides of the generalized eigenvalue equation (9) when S follows
Figure BDA0002311215720000049
But increases, however, M and K do not change, so H is chosen to be α I,
Figure BDA00023112157200000410
compared with the prior art, the invention has the following advantages:
the invention provides a full-wave mixed spectrum element method-based liquid crystal filled photonic crystal fiber analysis, which can inhibit the generation of a zero pseudo mode when calculating the band gap of a two-dimensional liquid crystal photonic crystal;
secondly, the liquid crystal is filled into the air holes of the cladding of the photonic crystal fiber, and the parameters of the liquid crystal can be controlled by changing the external environment, so that the transmission spectrum of the photonic crystal fiber can be flexibly controlled;
when the liquid crystal material filled in the air holes is analyzed to be any three-dimensional anisotropic dielectric constant and permeability tensor, the traditional numerical analysis method cannot obtain a correct photonic crystal band gap structure by separating a TE mode from a TM mode, and the full-wave mixed spectrum element method is used as a more universal characteristic value analysis method to analyze the photonic crystal band gap structure, so that the energy band characteristics of the photonic crystal can be obtained in a full-wave calculation mode of mutual coupling of the TM mode and the TE mode.
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FIG. 1 is a flow chart of a full-wave mixed spectrum element method-based liquid crystal filled photonic crystal fiber analysis method provided by an embodiment of the invention;
FIG. 2 is a schematic diagram of a triangular lattice of a two-dimensional anisotropic liquid crystal-filled photonic crystal structure and its computational region according to an embodiment of the present invention;
FIG. 3 is a cross-sectional view of a two-dimensional anisotropic liquid crystal-filled photonic crystal and corresponding cell structure diagram according to an embodiment of the present invention;
FIG. 4 is a schematic illustration of energy bands calculated according to an embodiment of the invention;
FIG. 5 is a block diagram of E for calculating the three modes with M points lowest according to an embodiment of the present inventionZField map normalization scheme.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
Referring to fig. 1, the steps of an embodiment of the present invention are as follows:
s1: and establishing a solid model, selecting a lattice structure and material parameters of the two-dimensional liquid crystal filled photonic crystal, dividing the single cell structure into a plurality of equal parts according to the symmetry of the single cell structure, and taking out only one part of the single cell structure as a calculation area to avoid increasing the calculation amount by repeated calculation.
In the embodiment of the invention, a triangular lattice (as shown in fig. 2) is selected as a lattice structure of the two-dimensional liquid crystal filled photonic crystal, and a region on the right half of the triangular lattice is selected as a calculation region (as shown by a shaded part in fig. 2).
The cross section and the corresponding unit structure of the two-dimensional anisotropic liquid crystal filled photonic crystal in the embodiment of the invention are shown in fig. 3, the period of the two-dimensional liquid crystal photonic crystal is 500nm, and the ordinary refraction n of the liquid crystalo1.5292, extraordinary refractive index ne＝1.7072. Angle theta between liquid crystal director and z-axisc75 DEG and the angle phi between the liquid crystal director and the x-axiscAt 30 deg. and the rest is Te (ordinary refractive index n)o4.8, extraordinary refractive index ne6.2) medium. The radius of the liquid crystal was 0.45 a.
Dielectric tensor [ epsilon ] of liquid crystalr]Can be expressed as follows:
Figure BDA0002311215720000052
Figure BDA0002311215720000053
Figure BDA0002311215720000061
s2: mesh subdivision of a limited number of sub-domains is carried out on the calculation region, each sub-domain unit is subdivided into quadrilateral units, and the vertex coordinates, material properties and boundary characteristics of each unit in the calculation region are recorded;
s3: the implementation of a two-dimensional photonic crystal mixed spectrum element method and the establishment of a matrix form eigen equation comprise the following specific steps:
① obtaining variation forms (3.a) and (3.b) by applying the variation principle according to the vector wave equation and Gauss's law, namely equations (2.a) and (2. b);
the equations (2.a), (2.b), (3.a), (3.b) are as follows:
Figure BDA0002311215720000062
Figure BDA0002311215720000063
Figure BDA0002311215720000064
Figure BDA0002311215720000065
wherein U is the electric field, k0Is the wave number in free space, [ mu ]r]Is the permeability tensor, [ epsilon ]r]Is the dielectric constant tensor, definesV, q is a test function;
② introduces the divergence term of the above equation (3.b) as a lagrange multiplier into the above equation (3.a), so that the above equation (3.a) is transformed into a variational form (4.a), and the equation (4.a) is transformed into a variational form (4. b):
Figure BDA0002311215720000068
③ converts the integral operation of the physical unit coordinates (x, y) into a calculation in the reference unit coordinates (ξ), and performs a corresponding covariate mapping on the basis functions, the mapping relation being as follows:
wherein the jacobian matrix J is:
Figure BDA0002311215720000071
④ field distribution U and ▽ operators of the wave mode of the in-plane propagation of the two-dimensional liquid crystal photonic crystal are respectively decomposed into tangential components and longitudinal components, which are respectively expressed as (7) and (8). the equations (7) and (8) are respectively brought into the equations (4.a) and (4.b), then, a base function is constructed by using a Gauss-Legendre-Lobarton polynomial, and the base function is used as a test function to obtain a discretized matrix form eigenequation (9);
Figure BDA0002311215720000072
Figure BDA0002311215720000073
wherein, UtAnd UzRespectively the transverse component and the longitudinal component of the electric field U,and
Figure BDA0002311215720000076
are respectively
Figure BDA0002311215720000077
Transverse component and longitudinal component of operator, matrix S is rigidity matrix, matrix M is quality matrix, matrix K is discretized matrix of Gaussian law, and definitione is the column vector of the electric field along the edge and y is the column vector of the electric field on all nodes.
Using lagrange interpolation basis functions as N-order gaussian-legendre-lowbartto basis functions, the expression is as follows:
Figure BDA0002311215720000079
wherein, ξk∈[-1,1]Is a gaussian-legendre-lobatto sampling point, j 0, 1.
In the reference cell [ -1,1] × [ -1,1], the gaussian-legendre-lobatto polynomial vector basis function is:
Figure BDA00023112157200000710
Figure BDA00023112157200000711
the node-based scalar basis functions are:
Figure BDA00023112157200000712
⑤ to reduce the computational size of the matrix, equation (9) is integrated into the form of the following generalized eigenvalue equation (13):
(S+KHKT)e1＝λMe1(13)
wherein H is selected as α I, definitionh is the minimum distance between two inner difference points in the calculation region.
⑥ calculating wave number k under the coupling of TE mode and TM mode by using eigen equation obtained in step ⑤0And finally obtaining the energy band structure of the two-dimensional photonic crystal structure.
Energy band diagram of an embodiment of the invention is shown in fig. 4.
Table 1 gives an exemplary table of suppressing zero spurious modes at point M calculated according to an embodiment of the present invention.
S4: if the photonic band gap size obtained in the step S3 does not satisfy the design of the photonic crystal fiber, the parameters of the liquid crystal can be changed by changing the external environmental conditions, and the processes from the step S1 to the step S3 are repeated by using the updated material information until the photonic band gap size satisfies the conditions, and the iteration is ended to output the energy band structure diagram.
TABLE 1
Frequency of COMSOL Mixed spectral element method
1 5.3471e-07-1.25856e-09i 0.1391
2 5.3500e-07-4.2354e-11i 0.1663
3 5.6027e-07+4.6041e-09i 0.1929
4 5.7275e-07+1.2502e-10i 0.2711
5 2.8859e-07+1.7096e-08i 0.2981
6 0.1392 0.3207
7 0.1663 0.3433
8 0.1929 0.3908
9 0.2712 0.4184
FIG. 5 shows E for calculating the three modes with M points lowest according to an embodiment of the inventionZField map normalization scheme.
According to the entity model, a calculation region which is not repeatedly calculated is selected, grid subdivision of a limited number of sub-domains is carried out on the calculation region, and each sub-domain unit is subdivided into a quadrilateral structure. The method utilizes a mode of combining full-wave Helmholtz equation and Gauss's law, constructs a basis function by a Gauss-Legendre-Lobarton polynomial, and obtains the energy band characteristic of the photonic crystal with the complex medium in a full-wave calculation mode of mutual coupling of TM mode and TE mode, and can inhibit the generation of zero spurious mode. The photonic crystal filled with the liquid crystal is calculated by a full-wave mixed spectrum element method, so that different photonic band gaps under different parameters can be obtained, the flexible control of the photonic crystal fiber on transmission spectrum is realized, and the problem that the transmission characteristic of the traditional photonic crystal fiber is not changed once the photonic crystal fiber is drawn is solved.

Claims (9)

1. The liquid crystal filled photonic crystal fiber analysis method based on the full-wave mixed spectrum element method is characterized by comprising the following steps of:
1) establishing a model, selecting a lattice structure and material parameters of a two-dimensional liquid crystal filled photonic crystal, simplifying a single cell structure, and selecting a calculation area which is not repeatedly calculated;
2) carrying out mesh subdivision on a limited number of sub-fields in the calculation region, subdividing each sub-field unit into a quadrilateral structure, and recording the information of each unit in the calculation region;
3) establishing a mixed spectral element method implementation and a matrix form intrinsic equation of the two-dimensional liquid crystal filled photonic crystal to finally obtain an energy band structure of the two-dimensional liquid crystal filled photonic crystal structure;
4) if the photonic band gap size obtained in the step 3) does not meet the design of the photonic crystal fiber, the parameters of the liquid crystal can be changed by changing the external environment conditions, and the processes of the steps 1) to 3) are repeated by using the updated material information until the photonic band gap size meets the conditions, the iteration is finished, and the energy band structure diagram is output.
2. The method for analyzing liquid crystal-filled photonic crystal fiber based on full-wave mixed spectrum element method according to claim 1, wherein in step 1), the material parameters of the two-dimensional liquid crystal-filled photonic crystal comprise refractive index, dielectric constant, magnetic permeability and size of various materials.
3. The method for analyzing a liquid crystal-filled photonic crystal fiber according to claim 1, wherein in the step 1), the lattice structure is one of a square lattice, a rectangular lattice, a triangular lattice, and a hexagonal lattice.
4. The method for analyzing liquid crystal-filled photonic crystal fiber based on full-wave mixed spectrum element method according to claim 1, wherein in step 1), the material of the two-dimensional liquid crystal-filled photonic crystal is isotropic, anisotropic or dispersive.
5. The method for analyzing liquid crystal-filled photonic crystal fiber based on full-wave mixed spectrum element method as claimed in claim 1, wherein in step 1), if the two-dimensional liquid crystal-filled photonic crystal is isotropic, the energy band structure is divided according to symmetry, including rotational symmetry and reflection symmetry, and a calculation region with non-repeated calculation is selected; if the two-dimensional liquid crystal-filled photonic crystal is an anisotropic material, the electromagnetic wave at this time is highly dependent on the direction, and the anisotropy breaks the symmetry of the structure, so that it is necessary to construct a band structure having more wavevector k directions in the first brillouin zone and select it as a calculation region.
6. The method for analyzing liquid crystal-filled photonic crystal fiber based on full-wave mixed spectrum element method according to claim 1, wherein in step 2), the information of each unit in the recorded calculation region comprises vertex coordinates, material properties and boundary characteristics.
7. The method for analyzing liquid crystal-filled photonic crystal fiber based on full-wave mixed spectrum element method as claimed in claim 1, wherein in step 3), the concrete steps of establishing the mixed spectrum element method implementation and matrix form eigen equation of the two-dimensional liquid crystal-filled photonic crystal are as follows:
① obtaining variation forms (2.a) and (2.b) by vector wave equation and Gauss's law, namely equations (1.a) and (1.b) by applying variation principle;
the equations (1.a), (1.b), (2.a), (2.b) are as follows:
Figure FDA0002311215710000021
Figure FDA0002311215710000022
Figure FDA0002311215710000023
wherein U is the electric field, k0Is the wave number in free space, [ mu ]r]Is the permeability tensor, [ epsilon ]r]Is the dielectric constant tensor, defines
Figure FDA00023112157100000210
V, q is a test function;
② the divergence term of the above equation (2.b) is introduced as Lagrangian multiplier into the above equation (2.a) to obtain the following variational forms (3.a), (3. b):
Figure FDA0002311215710000026
③ converts the integral operation of the physical unit coordinates (x, y) into a calculation in the reference unit coordinates (ξ), and performs a corresponding covariate mapping on the basis functions, the mapping relation being as follows:
Figure FDA0002311215710000027
wherein the jacobian matrix J is:
④ field distribution U and of in-plane propagating wave modes of two-dimensional liquid crystal photonic crystal
Figure FDA0002311215710000029
The operator is decomposed into a tangential component and a longitudinal component, respectively, which are respectively represented as (6) and (7); bringing both equations (6) and (7) into the equations (3.a) and (3.b), constructing a basis function by using a Gauss-Legendre-Lobarton polynomial, and taking the basis function as a test function to obtain a discretized matrix form eigen equation (8);
Figure FDA0002311215710000031
Figure FDA0002311215710000032
Figure FDA0002311215710000033
wherein, UtAnd UzRespectively the transverse component and the longitudinal component of the electric field U,andare respectively
Figure FDA0002311215710000036
Transverse component and longitudinal component of operator, matrix S is rigidity matrix, matrix M is quality matrix, matrix K is discretized matrix of Gaussian law, and definition
Figure FDA0002311215710000037
e is the column vector of the electric field along the edge, y is the column vector of the electric field on all nodes;
⑤ to reduce the computational size of the matrix, equation (8) is integrated into generalized eigenvalue equation (9):
(S+KHKT)e1＝λMe1(9)
wherein H is selected as α I, definition
Figure FDA0002311215710000038
h is the minimum distance between two inner difference points in the calculation region;
⑥ wave number k in the case of coupling between TE mode and TM mode is calculated by using the above generalized eigenvalue equation (9)0And finally obtaining the energy band structure of the two-dimensional liquid crystal filling photonic crystal structure.
8. The method for analyzing liquid crystal-filled photonic crystal fiber according to claim 7, wherein in step 3) section ②, the equation (2.a) is equivalent to the equation (3.a), and the equation (2.b) is equivalent to the equation (3.b) that will be described in the following
Figure FDA0002311215710000039
Taken into equation (3.a) and combined with equation (3.b) to yield
Figure FDA00023112157100000310
All solutions to the simultaneous equations representing equation (3.a) and equation (3.b) are solutions to the simultaneous equations of equation (2.a) and equation (2. b); vice versa, when p is 0, all solutions to the simultaneous equations representing equation (2.a) and equation (2.b) are solutions to the simultaneous equations of equation (3.a) and equation (3. b).
9. The method as claimed in claim 7, wherein the H is selected in step 3) ⑤ according to the size consistency of both sides of the generalized eigenvalue equation (9) when | | | S | | follows
Figure FDA00023112157100000311
But increases, however, M and K do not change, so H is chosen to be α I,
Figure FDA00023112157100000312
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