Relacion de dispersion

Analisis del medio de los medios dermis y stratum

La funcion dielectrica es

import numpy as np
import cmath
import math 
import matplotlib.pyplot as plt

im    =    complex(0.0,1.0)  

# Stratum
ka = 0.235
ca = 3600
ra = 1500
ta = 1

# Dermis

kb = 0.445
cb = 3300
rb = 1116
tb = 20

ea  = np.zeros(Nf,dtype=complex)


Nf = 60
fi = 0.01 # Hz
ff = 1.0 # Hz
df = (ff-fi)/Nf
f  = np.zeros(Nf)

ea  = np.zeros(Nf,dtype=complex)
eb  = np.zeros(Nf,dtype=complex)

for ic in range(Nf):
    f[ic]=fi+ic*df
    w    = 2.0*math.pi*f[ic]
    ea[ic] = 1.0 + im/(w*ta)
    eb[ic] = 1.0 + im/(w*tb)
    

plt.plot(f,ea.real,'-b',label=r'$\Re(\varepsilon_a)$')
plt.plot(f,ea.imag,'--b',label=r'$\Im(\varepsilon_a)$')

plt.plot(f,eb.real,'.r',label=r'$\Re(\varepsilon_b)$')
plt.plot(f,eb.imag,'--r',label=r'$\Im(\varepsilon_b)$')

plt.ylim([0,2])
plt.xlim([0,1])

plt.xlabel('f')
plt.ylabel(r'$\varepsilon(f)$')

plt.legend()
plt.show()

Ahora graficamos el indice

import numpy as np
import cmath
import math 
import matplotlib.pyplot as plt

im    =    complex(0.0,1.0)  

# Stratum
ka = 0.235
ca = 3600
ra = 1500
ta = 1
Cca = math.sqrt(ka/(ca*ra*ta))

# Dermis

kb = 0.445
cb = 3300
rb = 1116
tb = 20
Ccb = math.sqrt(kb/(cb*rb*tb))


Nf = 60
fi = 0.01 # Hz
ff = 1.0 # Hz
df = (ff-fi)/Nf
f  = np.zeros(Nf)

ea  = np.zeros(Nf,dtype=complex)
eb  = np.zeros(Nf,dtype=complex)

na  = np.zeros(Nf,dtype=complex)
nb  = np.zeros(Nf,dtype=complex)

gamma_a  = np.zeros(Nf,dtype=complex)
gamma_b  = np.zeros(Nf,dtype=complex)

etaa  = np.zeros(Nf,dtype=complex)
etab  = np.zeros(Nf,dtype=complex)

for ic in range(Nf):
    f[ic]=fi+ic*df
    w    = 2.0*math.pi*f[ic]
    ea[ic] = 1.0 + im/(w*ta)
    eb[ic] = 1.0 + im/(w*tb)
    na[ic] = cmath.sqrt(ea[ic])
    nb[ic] = cmath.sqrt(eb[ic])
    gamma_a[ic] = (w/Cca)*na[ic]
    gamma_b[ic] = (w/Ccb)*nb[ic]
    etaa[ic] = -ka/(1-im*w*ta)
    etab[ic] = -kb/(1-im*w*tb)

####################################
plt.subplot(2, 2, 1)

plt.plot(f,ea.real,'-b',label=r' $\Re(\varepsilon_a)$')
plt.plot(f,ea.imag,'--b',label=r'$\Im(\varepsilon_a)$')

plt.plot(f,eb.real,'-r',label=r'$\Re(\varepsilon_b)$')
plt.plot(f,eb.imag,'--r',label=r'$\Im(\varepsilon_b)$')

plt.xlim([0,1])
plt.ylim([0,3])

plt.ylabel(r'$\varepsilon(f)$')
plt.xlabel(r'Frequency $(f)$')

plt.grid()
plt.legend()

####################################

plt.subplot(2, 2, 2)

plt.plot(f,na.real,'-b',label=r' $\Re(n_a)$')
plt.plot(f,na.imag,'--b',label=r'$\Im(n_a)$')

plt.plot(f,nb.real,'-r',label=r'$\Re(n_b)$')
plt.plot(f,nb.imag,'--r',label=r'$\Im(n_b)$')

plt.xlim([0,1])
plt.ylim([0,3])

plt.ylabel(r'$n(f)$')
plt.xlabel(r'Frequency $(f)$')

plt.grid()
plt.legend()

####################################

plt.subplot(2, 2, 3)

plt.plot(f,etaa.real,'-b',label=r' $\Re(\eta_a)$')
plt.plot(f,etaa.imag,'--b',label=r'$\Im(\eta_a)$')

plt.plot(f,etab.real,'-r',label=r'$\Re(\eta_b)$')
plt.plot(f,etab.imag,'--r',label=r'$\Im(\eta_b)$')

plt.xlim([0,1])
#plt.ylim([0,3])

plt.ylabel(r'$\eta(f)$')
plt.xlabel(r'Frequency $(f)$')

plt.grid()
plt.legend()

####################################

plt.subplot(2, 2, 4)

plt.plot(gamma_a.real,f,'-b',label=r' $\Re(\gamma_a)$')
plt.plot(gamma_a.imag,f,'--b',label=r'$\Im(\gamma_a)$')

plt.plot(gamma_b.real,f,'-r',label=r'$\Re(\gamma_b)$')
plt.plot(gamma_b.imag,f,'--r',label=r'$\Im(\gamma_b)$')

#plt.xlim([0,1])
#plt.ylim([0,3])

plt.xlabel(r'$\gamma(f)$')
plt.ylabel(r'Frequency $(f)$')

plt.grid()
plt.legend()
plt.show()

import numpy as np
import cmath
import math 
import matplotlib.pyplot as plt
im    =    complex(0.0,1.0)  
# Stratum
ka = 0.235
ca = 3600
ra = 1500
ta = 1
Cca = math.sqrt(ka/(ca*ra*ta))
# Dermis
kb = 0.445
cb = 3300
rb = 1116
tb = 20
Ccb = math.sqrt(kb/(cb*rb*tb))
Nf = 60
fi = 0.01 # Hz
ff = 1.0 # Hz
df = (ff-fi)/Nf
f  = np.zeros(Nf)
ea  = np.zeros(Nf,dtype=complex)
eb  = np.zeros(Nf,dtype=complex)
na  = np.zeros(Nf,dtype=complex)
nb  = np.zeros(Nf,dtype=complex)
gamma_a  = np.zeros(Nf,dtype=complex)
gamma_b  = np.zeros(Nf,dtype=complex)
etaa  = np.zeros(Nf,dtype=complex)
etab  = np.zeros(Nf,dtype=complex)

f=4.0
w    = 2.0*math.pi*f
ea = 1.0 + im/(w*ta)
eb = 1.0 + im/(w*tb)
na = cmath.sqrt(ea)
nb = cmath.sqrt(eb)
gamma_a = (w/Cca)*na
gamma_b = (w/Ccb)*nb


Nx = 1000
xi = 0.00
xf =  0.001
dx = (xf-xi)/Nx
xp   = np.zeros(Nx)
xm   = np.zeros(Nx)
Tap = np.zeros(Nx,dtype=complex)
Tam = np.zeros(Nx,dtype=complex)
Tbp = np.zeros(Nx,dtype=complex)
Tbm = np.zeros(Nx,dtype=complex)

for ix in range(Nx):
    xp[ix]   = xi + ix*dx
    Tap[ix] = cmath.exp(+im*gamma_a*xp[ix])
    Tbp[ix] = cmath.exp(+im*gamma_b*xp[ix])

xi = -0.001
xf =  0.000
dx = (xf-xi)/Nx

for ix in range(Nx):
    xm[ix]   = xi + ix*dx
    Tam[ix] = cmath.exp(-im*gamma_a*xm[ix])
    Tbm[ix] = cmath.exp(-im*gamma_b*xm[ix])

plt.subplot(2,1,1)
plt.grid()
plt.title('f=4 Hz')
plt.plot(xp,Tap.real,'-b',label=r'$T_a^+$')
plt.plot(xp,Tbp.real,'-r',label=r'$T_b^+$')
plt.legend()

plt.subplot(2,1,2)
plt.grid()
plt.plot(xm,Tam.real,'-b',label=r'$T_a^-$')
plt.plot(xm,Tbm.real,'-r',label=r'$T_b^-$')
plt.legend()


plt.show()