Transfer Functions and Error Signals

Two common tasks for interferometer analysis are the computations of error signals and optical transfer functions. In an interferometer, several degrees of freedom for the optical com- ponents exist (e.g. positions, alignment angles) and active stabilisation is necessary to enhance the sensitivity.

An error signal is the output of a sensor (or in general a measurable signal) as a function of one degree of freedom (of the interferometer). The transfer function now gives the frequency-dependent coupling of a signal that is present in that particular ‘degree of freedom’ into the error signal. Transfer functions can be used to compute the coupling of noise in the interferometer and thus to estimate the sensitivity.

Error signals

In general, an error signal is an output of any detector that is suitable for stabilising a certain parameter \(p\) with a servo loop. The error signal must be a function of that parameter. In most cases it is preferable to have a bipolar signal with a zero crossing at the operating point \(p_0\). The slope at the operating point is a measure of the sensor gain.

Transfer Functions

A transfer function describes how a system responds to a small periodic disturbance. If we “wiggle” some parameter of the system sinusoidally at a frequency \(f\), the transfer function tells us:

  • how large the resulting output oscillation is (amplitude response), and

  • how delayed it is relative to the excitation (phase response).

Transfer functions describe the propagation of a periodic signal through a plant and are usually given as frequency plots of amplitude and phase. A transfer function describes the linear coupling of signals inside a system. This means a transfer function is independent of the actual signal size.

For small signals or small deviations, most systems can be linearised and correctly described by transfer functions. Physically, this means we approximate the system as responding proportionally to tiny perturbations around its operating point.

Experimentally, network analysers are commonly used to measure a transfer function: one connects a periodic signal (the source) to an actuator of the plant and compares it to a measured sensor signal. By mixing the source with the sensor signal the analyser determines the amplitude and phase of the response relative to the excitation.

Mathematically, applying a sinusoidal signal \(\sin(\omega_s t)\) to the interferometer, for example as a position modulation of a cavity mirror, creates phase-modulation sidebands offset by \(\pm\omega_s\) from the carrier light. If detected appropriately, the photodiode output contains a component at \(\omega_s\) which can be extracted by demodulation.

Modulation-demodulation methods

Several standard techniques exist to generate error signals for controlling an interferometer. Many of them use modulation-demodulation schemes in which a light field is modulated (in phase or amplitude) at a fixed frequency and the detector output is demodulated at that same frequency.

This technique shifts low-frequency information to higher frequencies, where technical noise is often lower, thereby improving the signal-to-noise ratio.

Modelling Transfer Functions

To model a transfer function in Finesse there is a specfic recipe you must follow. The steps of this recipe are:

  1. Initiate a signal simulation: Add an fsig(1) command to tell Finesse to run a ‘signal simulation’. This creates a signal called fsig with an initial frequency of 1 Hz.

  2. Apply an input signal: Add a signal generator sgen to the injection node, e.g., sgen sig laser.amp, which injects the signal into the laser amplitude.

  3. Measure the output signal: Add a detector to demodulate this signal, e.g. pd1 tf s1.p1.i f=fsig.f. This measures the signal we added before.

  4. Sweep the frequency of the input signal: To scan the signal frequency we can use xaxis(fsig.f, log, 1e4, 1e9, 1000). This allows us to see how our detector responds to different frequencies.

In Finesse 3 you can use multiple sgen components at once in a simulation. This is useful when you have two effects happening at once. For example, a differential motion of the mirrors in the arms of a detector are being moved at the same time but out of phase with each other, as one moves in one direction, the other mirror moves in the opposite. This signal can be described with the following two sgen commands:

"""
sgen sig1 MX.z amplitude=1 phase=0
sgen sig2 MY.z amplitude=1 phase=180
"""

'\nsgen sig1 MX.z amplitude=1 phase=0\nsgen sig2 MY.z amplitude=1 phase=180\n'

This would shake mirrors (i.e., modulate their position along the optical axis) MX and MY but out of phase by 180 degrees.

See more about transfer functions in Modelling Transer Functions and Calculating General Transfer Functions