## Sinusoidal Phase Modulation

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* = *λ _{0} *+

*δλ*· sin(

*ωt*)

where *λ _{0} *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*sin

_{2ω}·*(2ωt) + S*cos

_{3ω}·*(3ωt) + ···,*

with

*S _{ω}(t) =* J

_{1}[

*z(t)*]

*·*sin[

*α(t)*]

*S _{2ω}(t) = J_{2}*[

*z(t)*]

*·*cos[

*α(t)*]

*S _{ω}(t) = J_{3}*[

*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*, are 90° out of phase, or in other words, are in quadrature.

_{ω}(t);S_{2ω}(t)]## Working Distance - Bessel Functions

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.

As explained earlier, the Bessel functions are dependent on the modulation amplitude, which means that varying δλ shrinks or stretches the Bessel envelope, and thus, the achievable working range. When larger working distances are desired, the period length of these functions increases. This allows for larger working ranges without encountering points where signal quality drops below the threshold.

Here It Is Important to distinguish between working range and working distance: The former refers to the range over which a measurement can be carried out, whereas the latter refers to the distance between sensor and target. The working range is therefore the difference between the lowest and highest working distance encountered during a measurement.