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 components 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

There are two different ways to compute transfer functions in Finesse which are well described in modelling transfer functions. That section describes a ‘manual’ method and a more general method using the frequency response actions.

The manual method explicitly injects a signal using the sgen element and measure the response with a demodulated power detector pd1. You can see an example of this in the living review optical spring example, which is a direct translation of the Finesse 2 syntax. While this syntax can be illustrative in simple examples, for signal simulations with multiple inputs and outputs it is cumbersome to add multiple signal generators and detectors. Therefore we recommend using the frequency response actions for your simulations.