Method of conical anisotropic rigorous coupled wave analysis for grating and computing device
A method of conical anisotropic rigorous coupled wave analysis for grating and a computing device are disclosed. The method includes: obtaining a target geometric phase δ′ g for the anisotropic-material-based grating; obtaining a slow axis azimuth angle ϕ c (x) of the anisotropic-material-based grating according to the target geometric phase δ′ g ; obtaining a permittivity tensor of the anisotropic-material-based grating, wherein the anisotropic-material-based grating has an ordinary index n o and an extraordinary index n e , the anisotropic-material-based grating has a slow axis polar angle θ c and slow axis azimuth angle ϕ c (x), and the permittivity tensor is based on n o , n e , θ c and ϕ c (x); applying the permittivity tensor into Maxwell equations; obtaining electromagnetic field for the anisotropic-material-based grating by using boundary conditions of at least two layers or sublayers of the anisotropic-material-based grating to obtain a diffraction efficiency for the anisotropic-material-based grating.
1. A method of obtaining a diffraction efficiency of an anisotropic-material-based grating having at least two layers or sublayers in augmented reality (AR)/virtual reality (VR) devices, comprising:
obtaining, by a processor, a target geometric phase δ′ g for the anisotropic-material-based grating;
obtaining, by the processor, a slow axis azimuth angle ϕ c (x) of the anisotropic-material-based grating according to the target geometric phase δ′ g ;
obtaining, by the processor, a permittivity tensor of the anisotropic-material-based grating, wherein the anisotropic-material-based grating has an ordinary index n, and an extraordinary index ne, the anisotropic-material-based grating has a slow axis polar angle θ c and slow axis azimuth angle ϕ c (x), and the permittivity tensor is based on n o , n e , θ c and ϕ c (x);
applying, by the processor, the permittivity tensor into Maxwell equations;
obtaining, by the processor, electromagnetic field for the anisotropic-material-based grating by using boundary conditions of the at least two layers or sublayers of the anisotropic-material-based grating and Maxwell equations for each layer or sublayer, to obtain the diffraction efficiency for the anisotropic-material-based grating,
determining if the at least two layers or sublayers are suitable to be integrated into one stack for an output coupler grating or an input coupler grating of a waveguide in the augmented reality/virtual reality devices based on the diffraction efficiency; and
analysing the diffraction efficiency by a color dispersion and light control within the augmented reality/virtual reality devices.
2. The method according to claim 1 , wherein obtaining electromagnetic field for the anisotropic-material-based grating by using boundary conditions of the at least two layers or sublayers of the anisotropic-material-based grating and Maxwell equations for each layer or sublayer further includes:
obtaining a matrix F 1 for a first layer or sublayer of the anisotropic-material-based grating by using a first boundary condition between the first layer or sublayer and a region 1 through a first equation as below:
[
sin
ψ
δ
i
0
j
sin
ψ
n
1
cos
θ
δ
i
0
-
j
cos
ψ
n
1
δ
i
0
cos
ψ
cos
θ
δ
i
0
]
+
[
I
0
-
j
Y
I
0
0
I
0
-
j
Z
I
]
[
R
s
R
p
]
=
F
1
·
C
obtaining a matrix F L for a final layer or sublayer of the anisotropic-material-based grating by using a last boundary condition between the final layer or sublayer and a region 3 through a second equation as below:
F
L
·
C
=
[
I
0
-
j
Y
I
I
0
0
I
0
-
jZ
II
]
[
T
s
T
p
]
;
obtaining respective matrix F l of lth layer or sublayer by using respective boundary condition between adjacent layers or sublayers according to an iteration equation as below:
[
sin
ψ
δ
i
0
j
sin
ψ
n
1
cos
θ
δ
i
0
-
j
cos
ψ
n
1
δ
i
0
cos
ψcosθδ
i
0
]
+
[
I
0
-
j
Y
I
0
0
I
0
-
j
Z
I
]
[
R
s
R
p
]
=
∏
l
=
1
L
F
l
·
(
F
l
′
)
-
1
[
I
0
-
j
Y
I
I
0
0
I
0
-
jZ
II
]
[
T
s
T
p
]
;
and
obtaining electromagnetic field for the anisotropic-material-based grating by using boundary conditions obtained through the first, second equations,
wherein region 1 is an input region for the anisotropic-material-based grating and region 3 is an output region for the anisotropic-material-based grating,
wherein a total layer or sublayer number of the anisotropic-material-based grating is L,
wherein matrix F′ l is obtained through the boundary condition between layer/sublayer (l−1) and layer/sublayer l,
wherein matrix F l represents a matrix function of lth layer or sublayer of the anisotropic-material-based grating, where l=1 . . . L, which can include or is derived from part or all of the following vectors: diagonal matrix K x , convolution permittivity matrix E, eigenvalue matrix Q l , and eigen matrix W l ,
wherein the diagonal matrix K x has i, i element being equal to k xi /k 0 , k 0 is the incident wavevector k 0 ={circumflex over (x)}k x0 +ŷk y0 +{circumflex over (z)}k z0 , and k xi is the ith order wavevector in x axis in wavevector k i ={circumflex over (x)}k xi +ŷk yi +{circumflex over (z)}k zi ,
wherein the convolution permittivity matrix E is formed by permittivity harmonic components, with i, p element being equals to a permittivity ∈ (i-p) , i represents ith diffraction order, and p represents pth layer/sublayer,
wherein the eigenvalue matrix Q l is formed by eigenvalues q l,i,m , wherein q l,i,m represents an mth eigenvalue of ith diffraction order in lth layer/sublayer,
wherein the eigen matrix W l is formed by eigenvector ω l,i,m , wherein w l,i,m represents an mth eigenvector of ith diffraction order in lth layer/sublayer,
wherein matrix I represents an identity matrix,
wherein matrix C represents constant vector used to reconstruct the linear combination of the eigenvectors,
wherein matrices Y I , Y II , Z I , and Z II represent diagonal matrices with diagonal element (k 1,zi /k 0 ), (k 2,zi /k 0 ), (k 1,zi /k 0 n 1 2 ), and (k 2,zi /k 0 n 1 2 ), respectively,
wherein ψ is an angle between an electric field vector and a plane of incidence in region 1, δ i0 is a geometric phase in region 1, n 1 is an index of the region 1, and θ is a polar angle of the electric field vector in region 1.
3. The method according to claim 2 , wherein the anisotropic-material-based grating is a liquid-crystal-based grating, and the sublayers of the liquid-crystal-based grating is formed through hologram in each layer of the liquid-crystal-based grating.
4. The method according to claim 2 , wherein the ordinary index n, and the extraordinary index n e vary with a wave length of an incident light.
5. The method according to claim 1 , wherein the anisotropic-material-based grating is a transmissive and/or reflective grating.
6. A method of obtaining parameters of an anisotropic-material-based grating, comprising:
obtaining diffraction efficiencies of different anisotropic-material-based gratings by using the method according to claim 1 ;
obtaining a merit function involving a diffraction efficiency for an anisotropic-material-based grating;
obtaining a desired anisotropic-material-based grating based on the merit function using the diffraction efficiencies of different anisotropic-material-based gratings; and
obtaining parameters of the desired anisotropic-material-based grating.
7. A computing device, comprising a non-transitory machine-readable memory and a processing unit, wherein the memory includes executable instructions, when the executable instructions are executed by the processing unit, the processing unit is made to implement the processes of the method according to claim 1 .