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模擬架構如上圖左，觀測左旋圓偏振、右旋圓偏振、S波、P波入射膽固醇液晶層的穿透與反射
其中介電常數表示式如圖所示，液晶層pitch number 設定為 8 , 厚度為150 nm
我們對照的理論解析解程式為
https://github.com/Berreman4x4/Berreman4x4
此理論解析解方程為python語言所寫，請安裝python來執行。理論解析解程式碼請看最下方。
設定參數
設定參數
1. 選取模擬的維度
2. 定義總x,y,z長度相等於模型檔案，然後調整適當的各軸解析度
3. 按【建立網格 Cad to Grids (Create)】按鈕建立模擬網格
勾選 Anisotropic ☑ er。
內建結構的寫法則如下
gdx是(grid) dx的大小, gdy = dy. gdz = dz;
ib= X grids, jb= Y grids, kb= Z grids;
icenter= ib/2; jcenter= jb/2; kcenter=kb/2;
%========膽固醇液晶 Cholesteric Liquid-Crystal=====
pitch=300e-9; %(thickness=pitch/2)
Number=8
kstart=25*gdz;
kend=kstart+Number*pitch/2;
ne=1.7;
no=1.5;
dn=0.5*(ne^2-no^2);
epsavg=(ne^2+no^2)/2;
beta=1; %rotate
Aniso_CholestericLiquidCrystal(kstart,kend,pitch,epsavg,beta,dn,no);
%========膽固醇液晶 Cholesteric Liquid-Crystal=====
figure(1);
hold on;
plot((dz:dz:kb*gdz)-kstart,squeeze(sqrt(epsrxx(1,1,:))),'k');
plot((dz:dz:kb*gdz)-kstart,squeeze(sqrt(epsryy(1,1,:))),'r');
plot((dz:dz:kb*gdz)-kstart,squeeze(sqrt(epsrzz(1,1,:))),'b');
legend('esprxx','epsryy','epsrzz')
hold off;
xlabel('wavelength')
ylabel('refractive index n')
函數 Aniso_CholestericLiquidCrystal 寫法如下
function Aniso_CholestericLiquidCrystal(kstart,kend,pitch,epsavg,beta,dn,no)
choice='E_AnIso';
gridtype=0
Globalsetup
for k=1:kb
for j=1:jb
for i=1:ib
ii=i*gdx;
jj=j*gdy;
kk=k*gdz;
if(kk>=kstart & kk <=kend)
phi=(4*pi/(pitch))*(kk-kstart);
if(i~=1);
epsryx(i-1 ,j ,k) =beta*dn*sin(phi);
epsryy(i-1 ,j ,k) =epsavg-dn*cos(phi);
epsryz(i-1 ,j ,k) =0;
epsrzx(i-1 ,j ,k) =0;
epsrzy(i-1 ,j ,k) =0;
epsrzz(i-1 ,j ,k) =no^2;
geoy(i-1 ,j ,k) =5.1;
geoz(i-1 ,j ,k) =5.1;
end
if(j~=1);
epsrxx(i ,j-1 ,k) =epsavg+dn*cos(phi);
epsrxy(i ,j-1 ,k) =beta*dn*sin(phi);
epsrxz(i ,j-1 ,k) =0;
epsrzx(i ,j-1 ,k) =0;
epsrzy(i ,j-1 ,k) =0;
epsrzz(i ,j-1 ,k) =no^2;
geox(i ,j-1 ,k) =5.1;
geoz(i ,j-1 ,k) =5.1;
end
if(k~=1);
epsrxx(i ,j ,k-1)=epsavg+dn*cos(phi);
epsrxy(i ,j ,k-1)=beta*dn*sin(phi);
epsrxz(i ,j ,k-1)=0;
epsryx(i ,j ,k-1)=beta*dn*sin(phi);
epsryy(i ,j ,k-1)=epsavg-dn*cos(phi);
epsryz(i ,j ,k-1)=0;
geox(i ,j ,k-1)=5.1;
geoy(i ,j ,k-1)=5.1;
end
epsrxx(i ,j ,k) =epsavg+dn*cos(phi);
epsrxy(i ,j ,k) =beta*dn*sin(phi);
epsrxz(i ,j ,k) =0;
epsryx(i ,j ,k) =beta*dn*sin(phi);
epsryy(i ,j ,k) =epsavg-dn*cos(phi);
epsryz(i ,j ,k) =0;
epsrzx(i ,j ,k) =0;
epsrzy(i ,j ,k) =0;
epsrzz(i ,j ,k) =no^2;
geox(i ,j ,k) =5.1;
geoy(i ,j ,k) =5.1;
geoz(i ,j ,k) =5.1;
end
end;
end;
end;
確定無誤後，按下【輸出結構】即可輸出記憶體內描述結構的矩陣，好讓執行檔讀取
這時會產生Data_Materials.csv（判斷哪個地方有結構及是什麼材料）
Data_Epsr_Anisotropic.csv (紀錄epsrxx...epsrzz的值)
Python 執行後的理論解析解
這個是p波與s波 及 unpolarized 波，下圖是折射率n在z軸上的變化情形
下圖是左旋圓偏振及右旋圓偏振的穿透率及反射率頻譜
再來是FonSinEM計算結果的比對，實線為Berreman4x4的理論解析解值，點符號為FonSinEM(SFM一樣)所計算出的穿透率值。這裡無偏振波是兩垂直線偏振的平均值，也就是(p1+p2)/2;下圖中還是能看到理論值與FDTD計算出的值有些許的差異。非等向性材料時由於會使用epsrxx,epsrxy ..epsrzz 三個方向不同的相對電導率，其所產生的誤差會較其他材料來的明顯一點。
這個是左旋圓偏振與右旋圓偏振理論值與FDTD計算的比較圖。實線為理論值、點符號為FDTD計算結果。
理論解析解 Berreman4x4
https://github.com/Berreman4x4/Berreman4x4
請先在上面連結下載解析解程式，此理論解析解方程為python語言所寫，請使用python執行。對照用理論解析解程式碼如下
其中會改到的我用紅色標示起來
#!/usr/bin/python
# encoding: utf-8
# Berreman4x4 example
# Author: O. Castany, C. Molinaro
# Example of a cholesteric liquid crystal
import numpy, Berreman4x4
from numpy import sin, sqrt, abs
from Berreman4x4 import c, pi, e_y
import matplotlib.pyplot as pyplot
# Materials
glass = Berreman4x4.IsotropicNonDispersiveMaterial(1.)
front = back = Berreman4x4.IsotropicHalfSpace(glass)
# Liquid crystal oriented along the x direction
(no, ne) = (1.5, 1.7)
Dn = ne**2-no**2
n_med = (ne**2 + no**2)/2
LC = Berreman4x4.UniaxialNonDispersiveMaterial(no, ne) # ne along z
R = Berreman4x4.rotation_v_theta(e_y, pi/2) # rotation of pi/2 along y
LC = LC.rotated(R) # apply rotation from z to x
# Cholesteric pitch:
p = 300e-9
"""
0.65e-6
"""
# One half turn of a right-handed helix:
TN = Berreman4x4.TwistedMaterial(LC, p/2, angle=+pi, div=35)
# Inhomogeneous layer, repeated layer, and structure
IL = Berreman4x4.InhomogeneousLayer(TN)
N = 8 # number half pitch repetitions
h = N * p/2
L = Berreman4x4.RepeatedLayers([IL], N)
s = Berreman4x4.Structure(front, [L], back)
# Normal incidence:
Kx = 0.0
# Calculation parameters
lbda_min, lbda_max = 200e-9, 800e-9 # (m)
lbda_B = p * n_med
lbda_list = numpy.linspace(lbda_min, lbda_max, 500)
k0_list = 2*pi/lbda_list
############################################################################
# Analytical calculation for the maximal reflection
R_th = numpy.tanh(Dn/n_med*pi*h/p)**2
lbda_B1, lbda_B2 = p*no, p*ne
############################################################################
# Calculation with Berreman4x4
J = numpy.array([s.getJones(Kx,k0) for k0 in k0_list])
power = abs(J)**2
T_pp = Berreman4x4.extractCoefficient(power, 't_pp')
T_ps = Berreman4x4.extractCoefficient(power, 't_ps')
T_ss = Berreman4x4.extractCoefficient(power, 't_ss')
T_sp = Berreman4x4.extractCoefficient(power, 't_sp')
# Note: the expression for T is valid if back and front media are identical
# Transmission coefficients for incident unpolarized light:
T_pn = 0.5 * (T_pp + T_ps)
T_sn = 0.5 * (T_sp + T_ss)
T_nn = T_sn + T_pn
# Transmission coefficients for 's' and 'p' polarized light, with
# unpolarized measurement.
T_ns = T_ps + T_ss
T_np = T_pp + T_sp
###########################################################################
# Text output: eigenvalues and eigenvectors of the transmission matrix for
# a wavelength in the middle of the stop-band.
i = numpy.argmin(abs(lbda_list-lbda_B)) # index for stop-band center
T = J[i,1,:,:] # transmission matrix
eigenvalues, eigenvectors = numpy.linalg.eig(T)
numpy.set_printoptions(precision=3)
print("\nTransmission in the middle of the stop-band...\n")
print("Eigenvalues of the Jones transmission matrix:")
print(eigenvalues)
print("Corresponding power transmission:")
print(abs(eigenvalues)**2)
print("Corresponding eigenvectors:")
print(eigenvectors)
# Note: the transformation matrix to the eigenvector basis is
# B = numpy.matrix(eigenvectors), and the matrix B⁻¹ T B is diagonal.
print("Normalization to the 'p' componant:")
print(eigenvectors/eigenvectors[0,:])
print("Ratio 's'/'p':")
print(abs(eigenvectors[1,:]/eigenvectors[0,:]))
print("Complex angle (°) (+90°: L, -90°: R)")
print(180/pi*numpy.angle(eigenvectors[1,:]/eigenvectors[0,:]))
# We observe that the eigenvectors are nearly perfectly polarized circular waves
###########################################################################
# Jones matrices for the circular wave basis
Jc = Berreman4x4.circularJones(J)
power_c = abs(Jc)**2
# Right-circular wave is reflected in the stop-band.
# R_LR, T_LR close to zero.
R_RR = Berreman4x4.extractCoefficient(power_c, 'r_RR')
T_RR = Berreman4x4.extractCoefficient(power_c, 't_RR')
# Left-circular wave is transmitted in the full spectrum.
# T_RL, R_RL, R_LL close to zero, T_LL close to 1.
T_LL = Berreman4x4.extractCoefficient(power_c, 't_LL')
R_LL = Berreman4x4.extractCoefficient(power_c, 'r_LL')
############################################################################
# Plotting
"""
fig = pyplot.figure(1)
ax = fig.add_subplot("111")
# Draw rectangle for λ ∈ [p·no, p·ne], and T ∈ [0, R_th]
rectangle = pyplot.Rectangle((lbda_B1,0), lbda_B2-lbda_B1, R_th, color='cyan')
ax.add_patch(rectangle)
ax.legend(loc='center right', bbox_to_anchor=(1.00, 0.50))
ax.set_title("Right-handed Cholesteric Liquid Crystal, aligned along \n" +
"the $x$ direction, with {:.1f} helix pitches.".format(N/2.))
ax.set_xlabel(r"Wavelength $\lambda_0$ (m)")
ax.set_ylabel(r"Power transmission $T$ and reflexion $R$")
fmt = ax.xaxis.get_major_formatter()
fmt.set_powerlimits((-3,3))
stack = s.drawStructure()
"""
# Draw rectangle for λ ∈ [p·no, p·ne], and T ∈ [0, R_th]
rectangle = pyplot.Rectangle((lbda_B1,0), lbda_B2-lbda_B1, R_th, color='cyan')
stack = s.drawStructure()
fig = pyplot.figure(8)
ax = fig.add_subplot("111")
ax.plot(lbda_list, R_RR, '--', label='R_rihgt circular')
ax.plot(lbda_list, T_RR, label='T_right circular')
ax.plot(lbda_list, R_LL, '--', label='R_left circular')
ax.plot(lbda_list, T_LL, label='T_left circular')
ax.legend(loc='center right', bbox_to_anchor=(1.00, 0.50))
ax.set_title("Right-handed Cholesteric Liquid Crystal, aligned along \n" +
"the $x$ direction, with {:.1f} helix pitches.".format(N/2.))
ax.set_xlabel(r"Wavelength $\lambda_0$ (m)")
ax.set_ylabel(r"Power transmission $T$ and reflexion $R$")
fmt = ax.xaxis.get_major_formatter()
fmt.set_powerlimits((-3,3))
fig = pyplot.figure(9)
ax = fig.add_subplot("111")
ax.plot(lbda_list, T_nn, label='T_nn un polar')
ax.plot(lbda_list, T_ns, label='T_ns S polar')
ax.plot(lbda_list, T_np, label='T_np P polar')
ax.legend(loc='center right', bbox_to_anchor=(1.00, 0.50))
ax.set_title("Right-handed Cholesteric Liquid Crystal, aligned along \n" +
"the $x$ direction, with {:.1f} helix pitches.".format(N/2.))
ax.set_xlabel(r"Wavelength $\lambda_0$ (m)")
ax.set_ylabel(r"Power transmission $T$ and reflexion $R$")
fmt = ax.xaxis.get_major_formatter()
fmt.set_powerlimits((-3,3))
fig = pyplot.figure(10)
ax = fig.add_subplot("111")
ax.plot(lbda_list, R_RR, '--', label='R_rihgt circular')
ax.plot(lbda_list, T_RR, label='T_right circular')
ax.plot(lbda_list, R_RR+T_RR, '--', label='T+R_right circular')
ax.legend(loc='center right', bbox_to_anchor=(1.00, 0.50))
ax.set_title("Right-handed Cholesteric Liquid Crystal, aligned along \n" +
"the $x$ direction, with {:.1f} helix pitches.".format(N/2.))
ax.set_xlabel(r"Wavelength $\lambda_0$ (m)")
ax.set_ylabel(r"Power transmission $T$ and reflexion $R$")
fmt = ax.xaxis.get_major_formatter()
fmt.set_powerlimits((-3,3))
fig = pyplot.figure(11)
ax = fig.add_subplot("111")
ax.plot(lbda_list, R_LL, '--', label='R_left circular')
ax.plot(lbda_list, T_LL, label='T_left circular')
ax.plot(lbda_list, R_LL+T_LL, '--', label='T+R_left circular')
ax.legend(loc='center right', bbox_to_anchor=(1.00, 0.50))
ax.set_title("Right-handed Cholesteric Liquid Crystal, aligned along \n" +
"the $x$ direction, with {:.1f} helix pitches.".format(N/2.))
ax.set_xlabel(r"Wavelength $\lambda_0$ (m)")
ax.set_ylabel(r"Power transmission $T$ and reflexion $R$")
fmt = ax.xaxis.get_major_formatter()
fmt.set_powerlimits((-3,3))
pyplot.show()
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