Optical properties of a dielectric–metallic superlattice:
the complex photonic bands
Optical properties of a dielectric–metallic superlattice:
the complex photonic bands
Para realizar la Fig. 1 de este paper, primeramente graficamos la funcion dielectrica con el siguiente programa
import numpy as np import matplotlib.pyplot as plt import math import cmath from scipy import linalg c = 3.0e8 wp = 10 # eV h_bar = 6.58e-16 im = complex(0,1) wp = wp/h_bar gamma = 0.0*wp Nw = 100 wi = 0.0*wp wf = 2.0*wp dw = (wf-wi)/Nw w = np.zeros(Nw) epsi = np.zeros(Nw,dtype=complex) for iw in range(1,Nw+1): w[iw-1] = wi+iw*dw epsi[iw-1] = 1.0-(wp*wp)/(w[iw-1]*(w[iw-1]+im*gamma))
Luego graficamos con el codigo
plt.plot(w/wp,epsi.real,'-k',w/wp,epsi.imag,'.k') plt.xlabel(r'$\omega/\omega_p$',fontsize=20) plt.ylabel(r'$\varepsilon(\omega$)',fontsize=20) #plt.xlim(0,0.5) plt.ylim(-1,1) plt.grid() plt.show()
n=np.sqrt(epsi) plt.plot(w/wp,n.real,'-k',w/wp,n.imag,'.k') plt.xlabel(r'$\omega/\omega_p$',fontsize=20) plt.ylabel(r'$n(\omega$)',fontsize=20) plt.xlim(0,1.5) plt.ylim(0,5) plt.grid() plt.show()
Para hacer el panel b de la figura 1 es necesario usar gamma=0.1wp, para tener
import numpy as np
import matplotlib.pyplot as plt
import math
import cmath
from scipy import linalg
c = 3.0e8
wp = 10 # eV
h_bar = 6.58e-16
im = complex(0,1)
d = 100e-9
a = 0.1*d
b = d-a
wp = wp/h_bar
gamma = 0.00001*wp
Nw = 100
wi = 0.0*wp
wf = 2.0*wp
dw = (wf-wi)/Nw
w = np.zeros(Nw)
epsi1 = np.zeros(Nw,dtype=complex)
n1 = np.zeros(Nw,dtype=complex)
aux10 = np.zeros(Nw,dtype=complex)
for iw in range(1,Nw+1):
w[iw-1] = wi+iw*dw
epsi1[iw-1] = 1.0-(wp*wp)/(w[iw-1]*(w[iw-1]+im*gamma))
n1[iw-1] = cmath.sqrt(epsi1[iw-1])
n2 = 1.0
aux1 = cmath.cos(n1[iw-1]*(w[iw-1]/c)*a)*math.cos(n2*(w[iw-1]/c)*b)
aux2 = 0.5*(n1[iw-1]/n2+n2/n1[iw-1])
aux3 = cmath.sin(n1[iw-1]*(w[iw-1]/c)*a)*math.sin(n2*(w[iw-1]/c)*b)
aux10[iw-1] = aux1-aux2*aux3
# coseno[iw-1] = cmath.acos(aux4)/(2.0*math.pi)
plt.plot(w/wp,aux10.real,'-k')
plt.xlabel(r'$\omega/\omega_p$',fontsize=20)
plt.ylabel(r'$f_1$ or $f_2$',fontsize=20)
plt.xlim(0,1.5)
plt.ylim(-1.5,2.5)
plt.grid()
plt.show()
import numpy as np
import matplotlib.pyplot as plt
import math
import cmath
from scipy import linalg
c = 3.0e8
wp = 10 # eV
h_bar = 6.58e-16
im = complex(0,1)
d = 100e-9
a = 0.1*d
b = d-a
wp = wp/h_bar
gamma = 0.01*wp
Nw = 100
wi = 0.0*wp
wf = 2.0*wp
dw = (wf-wi)/Nw
w = np.zeros(Nw)
epsi1 = np.zeros(Nw,dtype=complex)
n1 = np.zeros(Nw,dtype=complex)
aux10 = np.zeros(Nw,dtype=complex)
for iw in range(1,Nw+1):
w[iw-1] = wi+iw*dw
epsi1[iw-1] = 1.0-(wp*wp)/(w[iw-1]*(w[iw-1]+im*gamma))
n1[iw-1] = cmath.sqrt(epsi1[iw-1])
n2 = 1.0
aux1 = cmath.cos(n1[iw-1]*(w[iw-1]/c)*a)*math.cos(n2*(w[iw-1]/c)*b)
aux2 = 0.5*(n1[iw-1]/n2+n2/n1[iw-1])
aux3 = cmath.sin(n1[iw-1]*(w[iw-1]/c)*a)*math.sin(n2*(w[iw-1]/c)*b)
aux10[iw-1] = aux1-aux2*aux3
aux10[iw-1] = cmath.acos(aux10[iw-1])/(2.0*math.pi)
plt.plot(aux10.real,w/wp,aux10.imag,w/wp)
#plt.xlim(0,1.5)
plt.ylim(0,1.5)
plt.grid()
plt.show()
Comparacion con el metodo de estructura de bandas complejas
import numpy as np
import matplotlib.pyplot as plt
import math
import cmath
from scipy import linalg
c = 3.0e8
wp = 10 # eV
h_bar = 6.58e-16
im = complex(0,1)
d = 100e-9
a = 0.1*d
b = d-a
f = b/d
wp = wp/h_bar
gamma = 0.01*wp
Nw = 100
wi = 0.0*wp
wf = 2.0*wp
dw = (wf-wi)/Nw
w = np.zeros(Nw)
epsi1 = np.zeros(Nw,dtype=complex)
n1 = np.zeros(Nw,dtype=complex)
aux10 = np.zeros(Nw,dtype=complex)
q0 = np.zeros(Nw,dtype=complex)
q1 = np.zeros(Nw,dtype=complex)
q2 = np.zeros(Nw,dtype=complex)
q3 = np.zeros(Nw,dtype=complex)
for iw in range(1,Nw+1):
w[iw-1] = wi+iw*dw
epsi1[iw-1] = 1.0-(wp*wp)/(w[iw-1]*(w[iw-1]+im*gamma))
n1[iw-1] = cmath.sqrt(epsi1[iw-1])
epsi2 = 1.0
n2 = math.sqrt(epsi2)
aux1 = cmath.cos(n1[iw-1]*(w[iw-1]/c)*a)*math.cos(n2*(w[iw-1]/c)*b)
aux2 = 0.5*(n1[iw-1]/n2+n2/n1[iw-1])
aux3 = cmath.sin(n1[iw-1]*(w[iw-1]/c)*a)*math.sin(n2*(w[iw-1]/c)*b)
aux10[iw-1] = aux1-aux2*aux3
aux10[iw-1] = cmath.acos(aux10[iw-1])/(2.0*math.pi)
e_0 = epsi1[iw-1]+f*(epsi2-epsi1[iw-1])
e_p1 = f*(epsi2-epsi1[iw-1])*(math.sin(math.pi*f)/(math.pi*f))
e_m1 = f*(epsi2-epsi1[iw-1])*(math.sin(-math.pi*f)/(-math.pi*f))
Om = (w[iw-1]*d)/(2*math.pi*c)
AA = [[1,0],[0,1]]
BB = [[-2,0],[0,0]]
CC = [[1-Om*Om*e_0, -Om*Om*e_m1],
[ -Om*Om*e_p1,-Om*Om*e_0]]
ZZ = [[0,0],[0,0]]
II = [[1,0],[0,1]]
AAA = [[CC[0][0],CC[0][1],BB[0][0],BB[0][1]],
[CC[1][0],CC[1][1],BB[1][0],BB[1][1]],
[ZZ[0][0],ZZ[0][1],II[0][0],II[0][1]],
[ZZ[1][0],ZZ[1][1],II[1][0],II[1][1]]]
BBB = [[ZZ[0][0],ZZ[0][1],-AA[0][0],-AA[0][1]],
[ZZ[1][0],ZZ[1][1],-AA[1][0],-AA[1][1]],
[II[0][0],II[0][1], ZZ[0][0], ZZ[0][1]],
[II[1][0],II[1][1], ZZ[1][0], ZZ[1][1]]]
eigen,X = linalg.eig(AAA,BBB)
eigen.sort()
q0[iw-1] = eigen[0]
q1[iw-1] = eigen[1]
q2[iw-1] = eigen[2]
q3[iw-1] = eigen[3]
plt.plot(aux10.real,w/wp,'-b',aux10.imag,w/wp,'-r')
plt.plot(q1.real,w/wp,'.b',q1.imag,w/wp,'.r')
plt.xlim(-0.2,0.5)
plt.ylim(0,1.5)
plt.grid()
plt.show()
