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#!/usr/bin/env mpython_q
def ntest(lda,T,pol=0,MgO=0,MgO_th=5.0,E = 0): return 1.5
def nLNO1(lda,T,pol,MgO=0,MgO_th=5.0,E = 0):
"""
This is Sellmeyer eq. for Lithium Niobate from "Influence of the defect structure on the refractive indices
of undoped and Mg-doped lithium niobate", U.Schlarb and K.Betzler, Phys. Rev. B 50, 751 (1994).
Temperature in Centigrades, wavelength in microns. pol=0 "ordinary", pol=1 "extraordinary" MgO
=0 MgO concentration in % MgO_th the threshold concentration. External electric field E in V/cm
"""
import math
if pol: # extraordinary
w0 = 218.203
mu = 6.4047E-6
A0 = 3.9466E-5
A_NbLi = 23.727E-8
A_Mg = 7.6243E-8
A_IR = 3.0998E-8
A_UV = 2.6613
rr = 33E-10 # r_33 cm/V
else: # ordinary
w0 = 223.219
mu = 1.1082E-6
A0 = 4.5312E-5
A_NbLi = -1.4464E-8
A_Mg = -7.3548E-8
A_IR = 3.6340E-8
A_UV = 2.6613
rr = 11E-10 # r_13 cm/V
B = A0+A_Mg*MgO
if MgO < MgO_th: B+=(MgO_th-MgO)*A_NbLi
def f(T):
return (T + 273)**2 + 4.0238E5*(1/math.tanh(261.6/(T + 273)) - 1)
def n(lda,T):
return math.sqrt(B/((w0+(f(T)-f(24.5))*mu)**(-2)-lda**(-2))-A_IR*lda**2+A_UV)
return n(lda,T)+0.5*rr*E*n(lda,T)**3
def nLNO1_disp(lda,T,pol,MgO=0,MgO_th=5.0):
"""
dispersion dn/dlda (in nm) for Lithium Niobate dispersion nLNO1
"""
import math
if pol: # extraordinary
w0 = 218.203
mu = 6.4047E-6
A0 = 3.9466E-5
A_NbLi = 23.727E-8
A_Mg = 7.6243E-8
A_IR = 3.0998E-8
A_UV = 2.6613
else: # ordinary
w0 = 223.219
mu = 1.1082E-6
A0 = 4.5312E-5
A_NbLi = -1.4464E-8
A_Mg = -7.3548E-8
A_IR = 3.6340E-8
A_UV = 2.6613
B = A0+A_Mg*MgO
if MgO < MgO_th: B+=(MgO_th-MgO)*A_NbLi
def f(T):
return (T + 273)**2 + 4.0238E5*(1/math.tanh(261.6/(T + 273)) - 1)
def n(lda,T):
return math.sqrt(B/((w0+(f(T)-f(24.5))*mu)**(-2)-lda**(-2))-A_IR*lda**2+A_UV)
return -(B*lda**(-3)*((w0+(f(T)-f(24.5))*mu)**(-2)-lda**(-2))**(-2)+A_IR*lda)/n(lda,T)
def nBBO(lda,T,pol=0,MgO=0,MgO_th=5.0,E=0):
"""
This is Sellmeyer eq. for BBO from http://refractiveindex.info/
(Handbook of Optics, 3rd edition, Vol. 4. McGraw-Hill 2009);
T is the angle (in degrees) between the optical axis and polarization:
T = 0 extraordinary, T = 90 ordinary.
"""
import math
a = 2.7405,2.3730 # ordinary, extraordinary
b = 0.0184,0.0128
c = 0.0179,0.0156
d = 0.0155,0.0044
lda = lda/1000.0 # nm -> microns
no = math.sqrt(a[0]-d[0]*lda*lda+b[0]/(lda*lda-c[0]))
ne = math.sqrt(a[1]-d[1]*lda*lda+b[1]/(lda*lda-c[1]))
o = math.sin(math.pi*T/180)/no
e = math.cos(math.pi*T/180)/ne
return (o*o+e*e)**(-0.5)
def nBBO1(lda,T,pol=0,MgO=0,MgO_th=5.0,E=0):
"""
This is Sellmeyer eq. for BBO from Optics Communications 184 (2000) 485-491;
T is the angle (in degrees) between the optical axis and polarization:
T = 0 extraordinary, T = 90 ordinary.
"""
import math
a = 2.7359,2.3753 # ordinary, extraordinary
b = 0.01878,0.01224
c = 0.01822,0.01667
d = 0.01471,0.01627
x = 0.0006081,0.0005716
y = 0.00006740,0.00006305
lda = lda/1000.0 # nm -> microns
no = math.sqrt(a[0]-d[0]*lda*lda+b[0]/(lda*lda-c[0])+x[0]*lda**4-y[0]*lda**6)
ne = math.sqrt(a[1]-d[1]*lda*lda+b[1]/(lda*lda-c[1])+x[1]*lda**4-y[1]*lda**6)
o = math.sin(math.pi*T/180)/no
e = math.cos(math.pi*T/180)/ne
return (o*o+e*e)**(-0.5)
def FSR_simple(R,lda,T,pol,L,q = 1,MgO=0,MgO_th=5.0,E = 0, n = nLNO1):
import math
#from WGM_lib import nLNO1
from scipy.special import ai_zeros
q -= 1
AiRoots = -ai_zeros(40)[0]
a = 29.9792/(2*math.pi*R*n(lda,T,pol,MgO,MgO_th,E))
b = 1+0.5*AiRoots[q]*((L/2.0+0.5)**(1.0/3.0)-(L/2.0-0.5)**(1.0/3.0))
return a*b
def Get_frac_L(R,r,lda,T,pol,q = 1,p = 0,MgO=0,MgO_th=5.0,E = 0, n = nLNO1):
"""
This iteratively solves the WGM dispersion equation for a spheroid at the target wavelength lda
and returs the fractional orbital number L
pol=0 "ordinary", pol=1 "extraordinary" MgO =0 MgO concentration in %
MgO_th the threshold concentration. External electric field E in V/cm
R, r are big and small radia in mm
"""
import math
#from WGM_lib import nLNO1
from scipy.special import ai_zeros
q -= 1
r = math.sqrt(r*R) # redefine the rim radius to spheroid semi-axis
AiRoots = -ai_zeros(40)[0]
freq_term = 2*math.pi*R*n(lda,T,pol,MgO,MgO_th,E)*1E7/lda
geom_term = (2*p*(R-r)+R)/2.0/r
pol_term = n(lda,T,pol,MgO,MgO_th,E)**(1-2*pol)/math.sqrt(n(lda,T,pol,MgO,MgO_th,E)**2-1)
L = freq_term
dL = 1
while math.fabs(dL) >= 1E-9:
L1 = freq_term - AiRoots[q]*(L/2.0)**(1.0/3.0) - geom_term + pol_term\
- 3*AiRoots[q]**2*(L/2.0)**(-1.0/3.0)/20
dL = L1-L
L = L1
return L
def WGM_freq(R,r,L,T,pol,q = 1,p = 0,MgO=0,MgO_th=5.0,E = 0, n = nLNO1):
"""
This iteratively solves the WGM dispersion equation for a spheroid at the target
orbital number L and returs the wavelength (nm) and frequency (GHz)
pol=0 "ordinary", pol=1 "extraordinary" MgO =0 MgO concentration in %
MgO_th the threshold concentration. External electric field E in V/cm
R, r are big and small radia in mm
"""
import math
#from WGM_lib import nLNO1
from scipy.special import ai_zeros
q -= 1
r = math.sqrt(r*R) # redefine the rim radius to spheroid semi-axis
AiRoots = -ai_zeros(40)[0] # >0
geom_term = (2*p*(R-r)+R)/2.0/r
def nm2GHz(lda): return 2.99792E8/lda
def wl1(lda):
freq_term = 2*math.pi*R*n(lda,T,pol,MgO,MgO_th,E)*1E7
pol_term = n(lda,T,pol,MgO,MgO_th,E)**(1-2*pol)/math.sqrt(n(lda,T,pol,MgO,MgO_th,E)**2-1)
return freq_term/(L+AiRoots[q]*(L/2.0)**(1.0/3.0)\
+geom_term-pol_term+3*AiRoots[q]**2*(L/2.0)**(-1.0/3.0)/20)
lda0 = 2*math.pi*R*n(1000,T,pol,MgO,MgO_th,E)*1E7/L # initial wavelength guess based on n(1000 nm)
lda = wl1(lda0)
#print lda0, lda
df = 1
while math.fabs(df) > 1E-6: #1 kHz accuracy
lda1 = wl1(lda)
df = nm2GHz(lda1)-nm2GHz(lda)
lda = lda1
#print df,lda1,n(lda,T,pol,MgO,MgO_th,E)
return lda, nm2GHz(lda)
def Gaunt(L1,m1,L2,m2,L3,m3):
"""
Gaunt formula for Legendre polynomials overlap. Large factorials are treated in Stirling's approximation.
"""
import math
def fa(n):
if n<=10: fac = float(math.factorial(n))
else: fac = math.sqrt(2*math.pi*n)*(n/math.e)**n
return fac
params = [(L1,m1),(L2,m2),(L3,m3)]
params = sorted(params,key=lambda list: list[1])
l,u = params[-1]
prms = params[:-1]
prms = sorted(prms,key=lambda list: list[0])
n,w = prms[0]
m,v = prms[1]
if not v+w == u: return 0
else:
s = (l+m+n)/2.0
if not s-round(s) == 0 : return 0
elif m+n <l : return 0
elif m-n >l : return 0
else:
p = int(max(0, n-m-u))
q = int(min(n+m-u,l-u,n-w))
a = (-1)**(s-m-w)*math.sqrt((2*l+1)*(2*m+1)*(2*n+1)*fa(l-u)*fa(n-w)/2.0/fa(m-v))/4/math.pi/math.pi
b = a*math.sqrt(math.sqrt(2*math.pi*(m+v)*(n+w)/(l+u))/(s-m)/(s-n))/2.0/math.pi
c = b*math.exp(0.5*(m+n-l+u-v-w))*(s-l)**(l-s)
d = c*((n+w)/float(l+n-m))**(s-m)*((n+w)/2.0)**((w+m-l)/2.0)
t0 = d*2**((w-n)/2.0)*((m+v)/float(l+m-n))**((l+m-n)/2.0)*(m+v)**((v-l+n)/2.0)*(l+m-n)**(l-u)
sum = 0
for t in range (p,q+1):
f1 = (-1)**t*fa(m+n-u-t)/fa(t)/fa(l-u-t)/fa(n-w-t)
f2 = f1*((l+u+t)/float(m-n+u+t))**(0.5+t)
f3 = f2*((l+u+t)/float(l+u))**(0.5*l+0.5*u)
f4 = f3*((l+u+t)/float(l+m+n))**((l+m+n)/2.0)*(l+u+t)**((u-m-n)/2.0)
sum+= f4*((l+m-n)/float(u+m-n+t))**(u+m-n)
return t0*sum
def Gaunt_low(L1,m1,L2,m2,L3,m3):
"""
Gaunt formula for Legendre polynomials overlap computed exactly. Good only for low orders.
"""
import math
def fa(n):
if n<=10: fac = float(math.factorial(n))
else: fac = math.sqrt(2*math.pi*n)*(n/math.e)**n
return fac
def fa_ratio(n1, n2): #calculates n1!/n2!
p = 1.0
for j in range (min(n1, n2)+1, max(n1, n2)+1):
p = p*j
if n1 >= n2: return p
else: return 1/p
params = [(L1,m1),(L2,m2),(L3,m3)]
params = sorted(params,key=lambda list: list[1])
l,u = params[-1]
prms = params[:-1]
prms = sorted(prms,key=lambda list: list[0])
n,w = prms[0]
m,v = prms[1]
if not v+w == u: return 0
else:
s = (l+m+n)/2.0
if not s-round(s) == 0 : return 0
elif m+n <l : return 0
elif m-n >l : return 0
else:
s = int(s)
p = int(max(0, n-m-u))
q = int(min(n+m-u,l-u,n-w))
#print l,m,n,s
#print fa(l-u),fa(n-w), fa(m-v),fa(s-l)
a1 = (-1)**(s-m-w)*math.sqrt((2*l+1)*(2*m+1)*(2*n+1)*fa(l-u)*fa(n-w)/fa(m-v)/math.pi)/fa(s-l)/float(2*s+1)/4.0
a2 = math.sqrt(fa(m+v)*fa(n+w)/fa(l+u))*fa(s)/fa(s-m)/fa(s-n)
sum = 0
for t in range (p,q+1):
b1 = (-1)**t*fa(m+n-u-t)/fa(t)/fa(l-u-t)/fa(n-w-t)
b2 = fa_ratio(l+u+t,2*s)*fa_ratio(2*s-2*n,m-n+u+t)
sum+= b1*b2
#print a1, a2, sum
return a1*a2*sum
def RadOvlp(R,L1,q1,L2,q2,L3,q3):
"""
Calculates radial overlap of three Airy functions
"""
import math
from scipy.special import airy, ai_zeros
from scipy.integrate import quad
AiRoots = -ai_zeros(40)[0]
q1, q2, q3 = q1-1, q2-1, q3-1
def nk(L,q): return L*(1+AiRoots[q]*(2*L**2)**(-1.0/3.0))
def AiArg(x,L,q): return -AiRoots[q]*(L-nk(L+0.5,q)*x)/(L-nk(L+0.5,q))
def RadFunc(x,L,q): return airy(AiArg(x,L,q))[0]/math.sqrt(x)
L0 = min(L1,L2,L3)
q0 = max(q1,q2,q3)+1
Rmin = 1-5*math.sqrt(q0)*L0**(-2.0/3.0)
N1 = R**3*quad(lambda x: x*x*RadFunc(x,L1,q1)**2, Rmin, 1)[0]
N2 = R**3*quad(lambda x: x*x*RadFunc(x,L2,q2)**2, Rmin, 1)[0]
N3 = R**3*quad(lambda x: x*x*RadFunc(x,L3,q3)**2, Rmin, 1)[0]
#print N1, N2, N3
def Core(x): return RadFunc(x,L1,q1)*RadFunc(x,L2,q2)*RadFunc(x,L3,q3)*x*x
return R**3*quad(lambda x: Core(x), Rmin, 1)[0]/math.sqrt(N1*N2*N3)
def AngOvlp(L1,p1,L2,p2,L3,p3,limit=3):
"""
Calculates angular overlap of three spherical functions. Default limit is sufficient for up to p = 10
"""
import math
from scipy.special import hermite
from scipy.integrate import quad
#def HG(p,L,x): return (2**L*math.factorial(p))**-0.5*hermite(p)(x*math.sqrt(L))*(L/math.pi)**0.25*math.exp(-L*x**2/2.0)
def HG(p,L,x): return hermite(p)(x*math.sqrt(L))*math.exp(-L*x**2/2.0) # removed a large factor that would drop out in normalization anyway
def HGnorm(p,L,x): return HG(p,L,x)*(4*math.pi*quad(lambda x: HG(p,L,x)**2, 0, limit*((p+1)*math.pi/L/2)**0.5)[0])**(-0.5)
return 2*quad(lambda x: HGnorm(p1,L1,x)*HGnorm(p2,L2,x)*HGnorm(p3,L3,x), 0, limit*((max(p1,p2,p3)+1)*math.pi/min(L1,L2,L3)/2)**0.5)[0]
def T_phasematch(L1,p1,q1,L2,p2,q2,L3,p3,q3,R,r,T0=70,MgO=0,MgO_th=5.0,E = 0,Tstep=0.1, n = nLNO1):
'''
Finds 1/wl2 + 1/wl2 = 1/wl3, L1+L2 -> L3 phase matching temperature down to the accuracy deltaf (GHz)
based on the initial guess T0 with initial search step Tstep
Requires importing Sellmeyer equations. This program is unaware of the orbital selection rules!
'''
import math
from WGM_lib import WGM_freq
def Df(T): # freqency detuning in GHz
lda_1, freq_1 = WGM_freq(R,r,L1,T,0,q1,p1,MgO,MgO_th,E,n)
lda_2, freq_2 = WGM_freq(R,r,L2,T,0,q2,p2,MgO,MgO_th,E,n)
lda_3, freq_3 = WGM_freq(R,r,L3,T,1,q3,p3,MgO,MgO_th,E,n)
return freq_3-freq_1-freq_2
T = T0
df = Df(T)
while math.fabs(df)>1E-6: #1 kHz accuracy
#print "T = ",T," df = ", df,
df1 = Df(T+Tstep)
ddfdt = (df1-df)/Tstep
dT = df/ddfdt
T -= dT
Tstep = math.fabs(dT/2.0)
df = Df(T)
#print " dT = ", dT, " T -> ",T, " df -> ", df
if T<-200 or T> 500: break
return T
def T_phasematch_blk(wl1,wl2, T0=70,MgO=0,MgO_th=5.0,E = 0,Tstep=0.1, n = nLNO1):
'''
Finds 1/wl2 + 1/wl2 = 1/wl3 (wl in nanometers), collinear bulk phase matching temperature
based on the initial guess T0 with initial search step Tstep
Requires importing Sellmeyer equations.
'''
import math
wl3 = 1/(1.0/wl1+1.0/wl2)
def Dk(T):
k1 = 2*math.pi*n(wl1,T,0,MgO,MgO_th,E)*1e7/wl1 # cm^-1
k2 = 2*math.pi*n(wl2,T,0,MgO,MgO_th,E)*1e7/wl2
k3 = 2*math.pi*n(wl3,T,1,MgO,MgO_th,E)*1e7/wl3
return k3-k2-k1
T = T0
dk = Dk(T)
while math.fabs(dk)>0.1: #10 cm beat length
dk1 = Dk(T+Tstep)
ddkdt = (dk1-dk)/Tstep
dT = dk/ddkdt
T -= dT
Tstep = math.fabs(dT/2)
dk = Dk(T)
#print " dT = ", dT, " T -> ",T, " dk -> ", dk
if T<-200 or T> 500:
print "No phase matching!"
break
return T
def SH_phasematch_blk(angle, wl0=1000,step=1.01, n = nBBO):
'''
Finds the fundamental wavelength (in nm) for frequency doubling o+o->e at a given angle between the optical axis and the e.
Requires importing Sellmeyer equations.
'''
import math
def Dk(wl):
k1 = 2*math.pi*n(wl,90)*1e7/wl # ordinary cm^-1
k2 = 4*math.pi*n(wl/2.0,angle)*1e7/wl # extraordinary with angle
return k2-2*k1
wl = wl0
dk = Dk(wl)
while math.fabs(dk)>0.1: #10 cm beat length
dk1 = Dk(wl*step)
ddkdt = (dk1-dk)/step
dwl = dk/ddkdt
wl = wl/dwl
#step = min(step,math.fabs(dwl/2) )
dk = Dk(wl)
print " dwl = ", dwl, " wl -> ",wl, " dk -> ", dk
return wl
def Phasematch(wlP,p1,q1,p2,q2,p3,q3,R,r,T,MgO=0,MgO_th=5.0,E = 0, n = nLNO1):
'''
NOT FULLY FUNCTIONAL!!
Finds wl1 and wl2 such that 1/wl2 + 1/wl2 = 1/wlP and L1-p1+L2-p2 = L3-p3 at a given temperature T.
Requires importing Sellmeyer equations. R is evaluated at T. This program is unaware of the other orbital
selection rules! L1, L2, L3 are allowed to be non-integer.
'''
import math
from WGM_lib import Get_frac_L
def Dm(wl): # m3-m1-m2, orbital detuning
L3 = Get_frac_L(R,r,wlP,T,1,q3,p3,MgO,MgO_th)
L1 = Get_frac_L(R,r,wl,T,0,q1,p1,MgO,MgO_th)
wl2 = 1.0/(1.0/wlP-1.0/wl)
L2 = Get_frac_L(R,r,wl2,T,0,q2,p2,MgO,MgO_th)
return L2-L3+p3+L1-p1-p2
wl = 2*wlP #try at degeneracy
dm = Dm(wl)
print "dm = ", dm
ddm = 0
wlstep = 1.0 # chosing an appropriate wavelength step so the m variation is not too large or too small
while math.fabs(ddm)<1:
wlstep = 2*wlstep
ddm = Dm(wl+wlstep)-dm
print "wlstep = ", wlstep, " initial slope = ", ddm/wlstep
while math.fabs(dm)>1E-6: # equiv *FSR accuracy
dm1 = Dm(wl+wlstep)
print "dm1 = ", dm1
slope = (dm1-dm)/wlstep
print "slope = ", slope, "nm^-1"
dwl = dm/slope
wl -= dwl
print "new wl = ", wl
wlstep = min(math.fabs(dwl/2), wlstep)
dm = Dm(wl)
print " d wl = ", dwl, " wl -> ",wl, " dm -> ", dm
if math.fabs(dm)> Get_frac_L(R,r,2*wlP,T,0,q1,p1,MgO,MgO_th): break
L1 = Get_frac_L(R,r,wl,T,0,q1,p1,MgO,MgO_th)
wl2 = 1.0/(1.0/wlP-1.0/wl)
L2 = Get_frac_L(R,r,wl2,T,0,q2,p2,MgO,MgO_th)
return L1, L2, wl, wl2
def n_eff_FSR(wl,R,T,FSR,pol,MgO=0,MgO_th=5.0,E = 0,n = nLNO1,n_disp=nLNO1_disp):
'''
Finds effective refraction index (material+geometrical) based on the FSR measuremert (in GHz).
R in cm, wl in nm. Also tries to guess the appropriate q and L. The first order approximation.
'''
import math
from scipy.special import ai_zeros
import bisect
nr = n(wl,T,pol,MgO,MgO_th,E)
wl_deriv = n_disp(wl,T,pol,MgO,MgO_th)
denom = 2*math.pi*3*R*FSR*(nr-wl_deriv*wl)/29.9792-2
neff = nr/denom
AiRoots = -ai_zeros(40)[0]
L = 2*math.pi*R*neff*1e7/wl
a_q = (nr/neff-1)*(2*L*L)**(1.0/3.0)
j = bisect.bisect(AiRoots, a_q)
#q = 0
#if a_q>AiRoots[0]/2.0: q = 1
if j: q = j+(a_q-AiRoots[j-1])/(AiRoots[j]-AiRoots[j-1])
else: q = 1+(a_q-AiRoots[0])/(AiRoots[1]-AiRoots[0])
return neff,q,L
def n_eff(wl,R,r,T,pol,q,p,MgO=0,MgO_th=5.0,E = 0,n = nLNO1):
'''
Finds effective refraction index (material+geometrical) for a given WGM.
R,r in cm, wl in nm.
'''
import math
from WGM_lib import Get_frac_L
L = Get_frac_L(R,r,wl,T,pol,q,p,MgO,MgO_th,E,n)
neff = L*wl*1e-7/(2*math.pi*R)
return neff,L
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