6.1 Error signals and transfer functions

In general, we will call an error signal any measured signal suitable for stabilising a certain experimental parameter p with a servo loop. The aim is to maintain the variable p at a user-defined value, the operating point, p0. Therefore, the error signal must be a function of the parameter p. In most cases it is preferable to have a bipolar signal with a zero crossing at the operating point. The slope of the error signal at the operating point is a measure of the ‘gain’ of the sensor (which in the case of interferometers is a combination of optics and electronics).

Transfer functions describe the propagation of a periodic signal through a plant and are usually given as plots of amplitude and phase over frequency. By definition a transfer function describes only 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.

Experimentally, network analysers are commonly used to measure a transfer function: one connects a periodic signal (the source) to an actuator of the plant (which is to be analysed) and to an input of the analyser. A signal from a sensor that monitors a certain parameter of the plant is connected to the second analyser input. By mixing the source with the sensor signal the analyser can determine the amplitude and phase of the input signal with respect to the source (amplitude equals one and the phase equals zero when both signals are identical).

Mathematically, transfer functions can be modeled similarly: applying a sinusoidal signal sin(ω t) s to the interferometer, e.g., as a position modulation of a cavity mirror, will create phase modulation sidebands with a frequency offset of ± ωs to the carrier light. If such light is detected in the right way by a photodiode, it will include a signal at the frequency component ωs, which can be extracted, for example, by means of demodulation (see Section 4.2).

Transfer functions are of particular interest in relation to error signals. Typically a transfer function of the error signal is required for the design of the respective electronic servo. A ‘transfer function of the error signal’ usually refers to a very specific setup: the system is held at its operating point, such that, on average, ¯p = p0. A signal is applied to the system in the form of a very small sinusoidal disturbance of p. The transfer function is then constructed by computing for each signal frequency the ratio of the error signal and the injected signal. Figure 32View Image shows an example of an error signal and its corresponding transfer function. The operating point shall be at

xd = 0 and xEP(xd = 0) = 0 (100 )
The optical transfer function Topt,xd with respect to this error signal is defined by
^xEP (f) = Topt,x Tdet^xd (f ), (101 ) d
with Tdet as the transfer function of the sensor. In the following, Tdet is assumed to be unity. At the zero crossing the slope of the error signal represents the magnitude of the transfer function for low frequencies:
| | |dxEP| | ||dx--|| | = |Topt,xd||f→0 (102 ) d |xd=0
The quantity above will be called the error-signal slope in the following text. It is proportional to the optical gain |Topt,xd|, which describes the amplification of the gravitational-wave signal by the optical instrument.
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