Sinusoidal Phase Modulation Interferometry

As mentioned in the chapter about Michelson interferometry, we at SmarAct apply sinusoidal phase modulation to our laser source to generate an interference signal containing harmonics of the modulation frequency. This is generated by alternating the injection current of the laser diode, which results in a modulation of the emitted wavelength λ that can be expressed as:

λ = λ₀ + δλ · sin(ωt)

where λ₀ is the carrier wavelength, δλ is the modulation amplitude and ω is the angular modulation frequency. When applied, the interference signal can be expressed by:

I(t) = S_ω · cos(ωt) + S_2ω · sin(2ωt) + S_3ω · cos(3ωt) + ···,

with

S_ω(t) = J₁[z(t)] · sin[α(t)]

S_2ω(t) = J₂[z(t)] · cos[α(t)]

S _ω(t) = J₃[z(t)] · sin[α(t)]

Where z(t) is the modulation depth dependent on the modulation amplitude δλ, and J_n are the n-th Bessel function of the first kind. Using standard demodulation at the frequencies n · ω , it is possible to extract the components S_{nε{1,2,3,...} · ω}. Successive pairs, like [S_ω(t);S_2ω(t)], are 90° out of phase, or in other words, are in quadrature.

The quadrature signal (S_ω;S_2ω) can be expressed in terms of sine and cosine functions, as well as Bessel functions of the first kind. A reliable measurement is only possible when the amplitude of both signal components is above the detection limit of the system.