From calculation, phase fluctuation of reflected beam from length stabilized arm is not disturbing MI lock.
The phase PD at AS port sense is
phi = phi_x - phi_y = 2*l_MICH*omega/c + (phi_X - phi_Y)
where l_MICH is the Michelson differential length change, omega is laser frequency, phi_X and phi_Y are phase of arm reflected beam. From very complicated calculation,
phi_X ~ F/2 * Phi_X
at near resonance. Where F is arm finesse, Phi_X is the round trip phase change in X arm. So,
phi = 2*l_MICH*omega/c + F/2 * 2*L_DARM*omega/c
Our ALS stabilizes arm length in ~ 70 pm(see elogs #6835, #6858). Finesse for IR is ~450. Considering l_MICH is ~ 1 um, MICH signal at AS port should be larger than stabilized DARM signal by an order of magnitude.
Length sensing matrix of FPMI:
Calculated length sensing matrix of 40m FPMI is below. Here, I'm just considering 11 MHz modulation. I assumed input power to be 1 W, modulation index 0.1i, Schnupp asymmetry 26.6 mm. PRM/SRM transmissivity is not taken into account.
[W/m] DARM CARM MICH
REFL_I 0 1.69e8 0
REFL_Q 7.09e1 0 -3.61e3
AS_I 0 0 0
AS_Q 1.04e6 0 3.61e3
Maybe we should use REFL_Q as MICH signal, but since IQ separation is not perfect, we see too much CARM. I tried to lock MI with REFL11_Q yesterday, but failed.