PDH_doppler_michelson_with_pzt.py
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#!/usr/bin/python
# -*- coding: utf-8 -*-
'''Pound-Drever-Hall setup
Michelson with piezo actuator in cavity arm and AOM in reference arm
-----
(2*W_aom)-->(X)---|LPF|--->
| ----- eps_dop
O
|
----- --- ---
------------|EOM|---|\|---|/|-----------( )
w_laser ----- --- --- cavity
^ | |
| ----- U
| |AOM| |
| ----- |
| _|_ |
| //// |
| 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, w_aom, t, c = symbols('E0, J0, J1, omega_laser, Omega_pdh, Omega_aom, t, c', imaginary=False, real=True)
'''#laser and phase-mod laser
E_laser = E0*exp(I*(w_laser*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)) \
)'''
#aom double shifted phase-mod wave in reference arm
E_aom = \
E0*( \
J0*exp(I*((w_laser +2*w_aom)*t)) \
+ J1*exp(I*((w_laser+w_pdh+2*w_aom)*t)) \
- J1*exp(I*((w_laser-w_pdh+2*w_aom)*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
v_x, d_x, v_u, d_u = symbols('v_x, delta_x, v_u, delta_u', imaginary=False, real=True)
dx = v_x*t + d_x
du = v_u*t + d_u
E_ref = \
E0*( \
F( w_laser *t - (dx+du)*(w_laser)/c ) \
*J0*exp( I*( w_laser *t - 2*(dx+du)*(w_laser)/c ) ) \
+ -1*J1*exp( I*( (w_laser+w_pdh)*t - 2*(dx+du)*(w_laser)/c ) ) \
- -1*J1*exp( I*( (w_laser-w_pdh)*t - 2*(dx+du)*(w_laser)/c ) ) \
)
#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_aom + 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 = expand(TR8(TR7(expand(eps_dop))))
toc = time()
print('Elapsed time : %fs'%(toc-tic))
'''results
eps_pdh =
2 ⎛ ω_laser ⎞
4⋅E₀ ⋅J₀⋅J₁⋅k_φ_pdh⋅⎜ω_laser - ω_cav - ───────⋅(vₓ+vᵤ)⎟
⎝ c ⎠
+ ───────────────────────────────────────────────────────
π⋅δ_ν
eps_dop =
2 2 ⎛ ω_laser ⎞ ⎛2⋅[(δₓ+t⋅vₓ)+(δᵤ+t⋅vᵤ)]⋅ω_laser⎞
E₀ ⋅J₀ ⋅k_φ_doppler⋅⎜ω_laser - ω_cav - ───────⋅(vₓ+vᵤ)⎟⋅cos⎜───────────────────────────────⎟
⎝ c ⎠ ⎝ c ⎠
- ────────────────────────────────────────────────────────────────────────────────────────────
π⋅δ_ν
2 2 ⎛2⋅[(δₓ+t⋅vₓ)+(δᵤ+t⋅vᵤ)]⋅ω_laser⎞
+ 2⋅E₀ ⋅J₁ ⋅k_φ_doppler⋅sin⎜───────────────────────────────⎟
⎝ c ⎠
'''