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PDH_doppler_michelson/PDH_doppler_michelson_without_pzt.py
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#!/usr/bin/python
# -*- coding: utf-8 -*-
'''Pound-Drever-Hall setup
Michelson with piezo actuator and AOM in cavity arm
-----
(2*W_aom)-->(X)---|LPF|--->
| ----- eps_dop
O
|
----- --- --- -----
------------|EOM|---|\|---|/|---|AOM|---( )
w_laser ----- --- --- ----- cavity
^ | |
| | U
| | |
| | |
| _|_ |
| //// |
| v
|---------->(X)
(W_pdh) |
-----
|LPF|
-----
|
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, t = symbols('E0, J0, J1, omega_laser, Omega_pdh, t', imaginary=False, real=True)
v_x, d_x, c = symbols('v_x, delta_x, c', imaginary=False, real=True)
w_aom, dw, dphi = symbols('Omega_aom, delta_Omega, delta_phi', imaginary=False, real=True)
phi_u = dw*t #+ dphi
phi = w_aom*t + phi_u
'''#
#laser
E_laser = E0*exp(I*(w_laser*t))
'''
#phase-mod laser
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 with double aom shift
dx = v_x*t + d_x
E_ref = \
E0*( \
F( (w_laser )*t - dx*(w_laser+phi.diff(t))/c + phi ) \
* J0*exp( I*( (w_laser )*t - 2*dx*(w_laser+phi.diff(t))/c + 2*phi ) ) \
+ -1*J1*exp( I*( (w_laser+w_pdh)*t - 2*dx*(w_laser+phi.diff(t))/c + 2*phi ) ) \
- -1*J1*exp( I*( (w_laser-w_pdh)*t - 2*dx*(w_laser+phi.diff(t))/c + 2*phi ) ) \
)
#intensity of reflected wave
I_pdh = abs(E_ref)**2
I_pdh = expand(TR8(expand(expand_complex(I_pdh))))
#optical mixer
E_mich = sqrt(2)/2 * E_eom + sqrt(2)/2 * E_ref
#intensity of mixed wave
I_mich = abs(E_mich)**2
I_mich = expand(TR8(expand(expand_complex(I_mich))))
#Q demodulation of I_pdh at Omega_pdh for PDH error signal obtention
kphipdh = symbols('k_phi_pdh', real=True)
eps_pdh = 2 * kphipdh * I_pdh * cos(w_pdh*t-pi/2)
eps_pdh = expand(TR8(TR7(expand(eps_pdh))))
#Q demodulation of I_mich at 2*Omega_aom for doppler error signal obtention
kphidop = symbols('k_phi_doppler', real=True)
eps_dop = 2 * kphidop * I_mich * cos(2*w_aom*t-pi/2)
#eps_dop = 2 * kphidop * I_mich * cos(2*phi.diff(t)*t-pi/2)
eps_dop = expand(TR8(TR7(expand(eps_dop))))
toc = time()
print('Elapsed time : %fs'%(toc-tic))
'''results
eps_pdh =
2 ⎛ ω_laser + Ω_aom + δ_Ω ⎞
4⋅E₀ ⋅J₀⋅J₁⋅k_φ_pdh⋅⎜ω_laser + Ω_aom - ω_cav + δ_Ω - ─────────────────────⋅vₓ⎟
⎝ c ⎠
+ ──────────────────────────────────────────────────────────────────────────────
π⋅δ_ν
eps_dop =
2 2 ⎛ ω_laser + Ω_aom + δ_Ω ⎞ ⎛ 2⋅(ω_laser + Ω_aom + δ_Ω)⋅(vₓ⋅t+δₓ)⎞
E₀ ⋅J₀ ⋅k_φ_dop⋅⎜ω_laser + Ω_aom - ω_cav + δ_Ω - ─────────────────────⋅vₓ⎟⋅cos⎜2⋅δ_Ω⋅t - ───────────────────────────────────⎟
⎝ c ⎠ ⎝ c ⎠
─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
π⋅δ_ν
2 2 ⎛ 2⋅(ω_laser + Ω_aom + δ_Ω)⋅(vₓ⋅t+δₓ)⎞
+ 2⋅E₀ ⋅J₁ ⋅k_φ_dop⋅sin⎜2⋅δ_Ω⋅t - ───────────────────────────────────⎟
⎝ c ⎠
'''
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