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PDH/PDH.py
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#!/usr/bin/python
# -*- coding: utf-8 -*-
'''Pound-Drever-Hall setup
----- ---
------------|EOM|------|/|-----( )
w_laser ----- --- cavity
^ |
| U
| |
| v
|------->(X) π/2
(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, c = symbols('E0, J0, J1, omega_laser, Omega_pdh, t, c', imaginary=False, real=True)
'''laser and phase-mod laser
E_laser = E0*exp(I*(w_laser*t))
E_inc = \
E0*( \
J0*exp(I*((w_laser )*t)) \
+ J1*exp(I*((w_laser+w_pdh)*t)) \
- J1*exp(I*((w_laser-w_pdh)*t)) \
)'''
#reflection near 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
v_x, d_x = symbols('v_x, delta_x', imaginary=False, real=True)
dx = v_x*t + d_x
E_ref = \
E0*( \
F( (w_laser )*t - w_laser*dx/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 ) ) \
)
#intensity of reflected phase-mod laser
I_pdh = abs(E_ref)**2
I_pdh = expand(TR8(expand(expand_complex(I_pdh))))
#demodulation of I_pdh at Omega_pdh
kphipdh = symbols('k_phi_pdh', imaginary=False, real=True)
eps_pdh = 2 * kphipdh * I_pdh * cos(w_pdh*t-pi/2)
eps_pdh = expand(TR8(TR7(expand(eps_pdh))))
toc = time()
print('Elapsed time : %fs'%(toc-tic))
'''low-passed demodulated I_pdh signal
eps_pdh =
2 ⎛ ω_laser ⎞
4⋅E₀ ⋅J₀⋅J₁⋅k_φ_pdh⋅⎜ω_laser - ω_cav - ───────⋅vₓ⎟
⎝ c ⎠
+ ──────────────────────────────────────────────────
π⋅δ_ν
'''
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