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PDH_doppler_mach-zender/PDH_doppler_mach-zender_eom_amont.py
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#!/usr/bin/python
# -*- coding: utf-8 -*-
'''Pound-Drever-Hall setup
Mach-Zender with one photodiode
----- --- ---
---------|EOM|-|\|------------|/|-----------( )
w_laser ----- --- --- cavity
| |
| |
| ----- ---
\-|AOM|-------|\|
----- ---
^ |
| U
| |
| V
|-------->(X)
| | -----
(W_aom) |---|LPF|--> eps_dop
V -----
(W_pdh)---------->(X)
|
|
V eps_pdh
'''
from time import time
tic = time()
from sympy import *
from sympy.simplify.fu import *
init_printing()
#constants
E0, J0, J1, w_laser, w_pdh, w_aom, t = symbols('E0, J0, J1, omega_laser, Omega_pdh, Omega_aom, t', imaginary=False, real=True)
v_x, d_x = symbols('v_x, delta_x', imaginary=False, real=True)
c = symbols('c', imaginary=False, real=True)
#laser and phase-mod laser
E_laser = E0*exp(I*(w_laser*t))
E_aom_eom = \
E0*( \
J0*exp(I*( ( w_laser+w_aom )*t) ) \
+ J1*exp(I*( ( w_laser+w_aom+w_pdh )*t) ) \
- J1*exp(I*( ( w_laser+w_aom-w_pdh )*t) ) )
E_eom = \
E0*( \
J0*exp(I*( ( w_laser )*t ) ) \
+ J1*exp(I*( ( w_laser+w_pdh )*t ) ) \
- J1*exp(I*( ( w_laser-w_pdh )*t ) ) )
#approximation of F(w) near a resonance
dnu, w_cav = symbols('delta_nu, omega_cav', imaginary=False, real=True)
def F(phi):
dw = phi.diff(t) - w_cav
return -(I/pi)*(dw/dnu)
#reflected phase-mod laser and dephased by doppler effect and by actuator
dx = v_x*t + d_x
E_ref = \
E0*( \
F( w_laser*t - w_laser*(dx)/(2*pi*c) ) \
*J0*exp( I*( (w_laser )*t - 2*w_laser*dx/c ) ) \
+ -1*J1*exp( I*( (w_laser+w_pdh)*t - 2*w_laser*dx/c ) ) \
- -1*J1*exp( I*( (w_laser-w_pdh)*t - 2*w_laser*dx/c ) ) \
)
#optical mixer
E_mz = sqrt(2)/2 * E_aom_eom + sqrt(2)/2 * E_ref
#intensity of mixed wave
I_mz = abs(E_mz)**2
I_mz = expand(TR8(expand(expand_complex(I_mz))))
#Q demodulation of I_mz at Omega_aom for doppler error signal obtention
kphidop = symbols('k_phi_doppler', real=True)
eps_dop = 2 * kphidop * I_mz * cos(w_aom*t-pi/2)
eps_dop = expand(TR8(TR7(expand(eps_dop))))
#Q demodulation of I_mich at Omega_pdh for doppler error signal obtention
kphipdh = symbols('k_phi_pdh', real=True)
eps_pdh = 2 * kphipdh * eps_dop * cos(w_pdh*t)
eps_pdh = expand(TR8(TR7(expand(eps_pdh))))
toc = time()
print('Elapsed time : %fs'%(toc-tic))
'''results
### EOM sur un bras
eps_pdh =
0
eps_dop =
2 ⎛ ω_laser⋅vₓ⎞ ⎛2⋅δₓ⋅ω_laser 2⋅ω_laser⋅t⋅vₓ⎞
E₀ ⋅J₀ ⋅k_φ_doppler⋅⎜ω_laser - ω_cav - ──────────⎟⋅cos⎜──────────── + ──────────────⎟
⎝ c ⎠ ⎝ c c ⎠
- ─────────────────────────────────────────────────────────────────────────────────────
π⋅δ_ν
2 2 ⎛ δₓ⋅ω_laser ω_laser⋅t⋅vₓ⎞
+ 2⋅E₀ ⋅J₁ ⋅k_φ_doppler⋅sin⎜2⋅────────── + 2⋅────────────⎟
⎝ c c ⎠
'''
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