Interferometer techniques for gravitationalwave detection
Abstract
Several kmscale gravitationalwave detectors have been constructed worldwide. These instruments combine a number of advanced technologies to push the limits of precision length measurement. The core devices are laser interferometers of a new kind; developed from the classical Michelson topology these interferometers integrate additional optical elements, which significantly change the properties of the optical system. Much of the design and analysis of these laser interferometers can be performed using wellknown classical optical techniques; however, the complex optical layouts provide a new challenge. In this review, we give a textbookstyle introduction to the optical science required for the understanding of modern gravitational wave detectors, as well as other highprecision laser interferometers. In addition, we provide a number of examples for a freely available interferometer simulation software and encourage the reader to use these examples to gain handson experience with the discussed optical methods.
Keywords
Gravitational waves Gravitationalwave detectors Laser interferometry Optics Simulations Finesse1 Introduction
1.1 The scope and style of the review
The historical development of laser interferometers for application as gravitationalwave detectors (Pitkin et al. 2011) has involved the combination of relatively simple optical subsystems into more and more complex assemblies. The individual elements that compose the interferometers, including mirrors, beam splitters, lasers, modulators, various polarising optics, photo detectors and so forth, are individually well described by relatively simple, mostlyclassical physics. Complexity arises from the combination of multiple mirrors, beam splitters etc. into optical cavity systems that have narrow resonant features, and the consequent requirement to stabilise relative separations of the various components to subwavelength accuracy, and indeed in many cases to very small fractions of a wavelength.
Thus, classical physics describes the interferometer techniques and the operation of current gravitationalwave detectors. However, we note that at signal frequencies above a couple of hundreds of Hertz, the sensitivity of current detectors is limited by the photon counting noise at the interferometer readout, also called shotnoise. The next generation systems such as Advanced LIGO (Fritschel 2003; Aasi 2015), Advanced Virgo (Acernese 2015) and KAGRA (Aso et al. 2013) are expected to operate in a regime where the quantum physics of both light and mirror motion couple to each other. Then, a rigorous quantummechanical description is certainly required. Sensitivity improvements beyond these ‘Advanced’ detectors necessitate the development of nonclassical techniques; a comprehensive discussion of such techniques is provided in Danilishin and Khalili (2012). This review provides a brief introduction to quantum noise in Sect. 6 but otherwise focusses on the nonquantum aspects of interferometry that play an important role in overcoming other limits to current detectors, due to, for example, thermal effects and feedback control systems. At the same time these classical techniques will provide the means for implementing new, nonclassical schemes and just remain as important as ever.
The optical components employed tend to behave in a linear fashion with respect to the optical field, i.e., nonlinear optical effects need hardly be considered. Indeed, almost all aspects of the design of laser interferometers are dealt with in the linear regime. Therefore the underlying mathematics is relatively simple and many standard techniques are available, including those that naturally allow numerical solution by computer models. Such computer models are in fact necessary as the exact solutions can become quite complicated even for systems of a few components. In practice, workers in the field rarely calculate the behaviour of the optical systems from first principles, but instead rely on various wellestablished numerical modelling techniques. An example of software that enables modelling of interferometers and their component systems is Finesse (Freise et al. 2004; Freise 2015). This was developed by some of us (AF, DB), has been validated in a wide range of situations, and was used to prepare the examples included in the present review.
The target readership we have in mind is the student or researcher who desires to get to grips with practical issues in the design of interferometers or component parts thereof. For that reason, this review consists of sections covering the basic physics and approaches to simulation, intermixed with some practical examples. To make this as useful as possible, the examples are intended to be realistic with sensible parameters reflecting typical application in gravitational wave detectors. The examples, prepared using Finesse, are designed to illustrate the methods typically applied in designing gravitational wave detectors. We encourage the reader to obtain Finesse and to follow the examples (see “Appendix A”).
1.2 Overview of the goals of interferometer design for gravitationalwave detection
Quantum noise is just one example of the challenges that need to be overcome to reach the desired sensitivity. Many new technologies and concepts have been—and are still being—invented, tested and refined to further develop these laserinterferometric gravitational wave detectors. It was this endeavour that finally resulted in the spectacular first detections of gravitational waves in 2015 (Abbott et al. 2016a, b). In this review we focus on those ideas that affect the optical layout and that use new interferometer configurations.
The evolution of gravitationalwave detectors can be seen by following their development from prototypes and early observing systems towards the socalled ‘Advanced detectors’, which are currently under construction, or in the case of Advanced LIGO, in the first phase of scientific observing (as of late 2015). Starting from the simplest Michelson interferometer (Forward 1978), then by the application of techniques to increase the number of photons stored in the arms: delay lines (Herriott et al. 1964), Fabry–Perot arm cavities (Fabry and Perot 1899; Fattaccioli et al. 1986) and power recycling (Billing et al. 1983; Drever et al. 1983). The final step in the development of classical interferometry was the inclusion of signal recycling (Meers 1988; Heinzel et al. 1998), which, among other effects, allows the signal from a gravitationalwave signal of approximatelyknown spectrum to be enhanced above the noise.
Reading out a signal from even the most basic interferometer requires minimising the coupling of local environmental effects to the detected output. Thus, the relative positions of all the components must be stabilised. This is commonly achieved by suspending the mirrors etc. as pendulums, often multistage pendulums in series, and then applying closedloop control to maintain the desired operating condition. The careful engineering required to provide lownoise suspensions with the correct vibration isolation and lownoise actuation is described in many works, for example, Braccini et al. (1996), Plissi et al. (2000), Barriga et al. (2009) and Aston et al. (2012).
As the interferometer optics become more complicated the resonance conditions become more narrowly defined, i.e., the allowed combinations of intercomponent path lengths required to allow the photon number in the interferometer arms to reach a maximum. It is likewise necessary to maintain angular alignment of all components so that beams required to interfere are correctly coaligned. Typically the beams need to be aligned within a small fraction, and sometimes a very small fraction, of the farfield diffraction angle: the requirement can be in the low nanoradian range for kmscale detectors (Morrison et al. 1994; Freise et al. 2007). Therefore, for each optical component there is typically one longitudinal, i.e., along the direction of light propagation, plus two angular degrees of freedom: pitch and yaw about the longitudinal axis. A complex interferometer consists of up to around seven highly sensitive components and so there can be of order 20 degrees of freedom to be measured and controlled (Acernese 2006; Winkler et al. 2007).
Although the light fields are linear in their behaviour the coupling between the position of a mirror and the complex amplitude of the detected light field typically shows strongly nonlinear dependence on mirror positions due to the sharp resonance features exhibited by cavity systems. The fields do vary linearly, or at least they vary smoothly close to the desired operating point. So, while wellunderstood linear control theory suffices to design the control system needed to maintain the optical configuration at its operating point, the act of bringing the system to that operating condition is often a separate and more challenging nonlinear problem. In the current version of this work we consider only the linear aspects of sensing and control.
Control systems require actuators, and those employed are typically electricalforce transducers that act on the suspended optical components, either directly or—to provide enhanced noise rejection—at upper stages of multistage suspensions. The transducers are normally coilmagnet actuators, with the magnets on the moving part, or, less frequently, electrostatic actuators of varying design. The actuators are frequently regarded as part of the mirror suspension subsystem and are not discussed in the current work.
To give order to our review we consider the main physics describing the operation of the basic optical components: mirrors, beam splitters, modulators, etc., required to construct interferometers. Although all of the relevant physics is generally well known and not new, we take it as a starting point that permits the introduction of notation and conventions. It is also true that the interferometry employed for gravitationalwave detection has a different emphasis than other interferometer applications. As a consequence, descriptions or examples of a number of crucial optical properties for gravitational wave detectors cannot be found in the literature.
The purpose of this review is especially to provide a coherent theoretical framework for describing such effects. With the basics established, it can be seen that the interferometer configurations that have been employed in gravitationalwave detection may be built up and simulated in a relatively straightforward manner.
1.3 Planewave analysis
The main optical systems of interferometric gravitationalwave detectors are designed such that all system parameters are well known and stable over time. The stability is achieved through a mixture of passive isolation systems and active feedback control. In particular, the light sources are some of the most stable, lownoise continuouswave laser systems so that electromagnetic fields can be assumed to be essentially monochromatic. Additional frequency components can be modelled as small modulations in amplitude or phase. The laser beams are well collimated, propagate along a welldefined optical axis and remain always very much smaller than the optical elements they interact with. Therefore, these beams can be described as paraxial and the wellknown paraxial approximations can be applied.
with \(E_0\) as the (constant) field amplitude in V/m, \( \mathbf {e}_p\) the unit vector in the direction of polarisation, such as, for example, \(\mathbf {e}_y\) for \(\mathscr {S}\)polarised light, \(\omega \) the angular oscillation frequency of the wave, and \(\mathbf {k}=\mathbf {e}_k \omega /c\) the wave vector pointing in the direction of propagation. The absolute phase \(\varphi \) only becomes meaningful when the field is superposed with other light fields.
1.4 Frequency domain analysis
2 Optical components: coupling of field amplitudes
2.1 Mirrors and spaces: reflection, transmission and propagation
The core optical systems of current interferometric gravitational interferometers are composed of two building blocks: a) resonant optical cavities, such as Fabry–Perot resonators, and b) beam splitters, as in a Michelson interferometer. In other words, the laser beam is either propagated through a vacuum system or interacts with a partiallyreflecting optical surface.
The term optical surface generally refers to a boundary between two media with possibly different indices of refraction n, for example, the boundary between air and glass or between two types of glass. A real fused silica mirror in an interferometer features two surfaces, which interact with a reflected or transmitted laser beam. However, in some cases, one of these surfaces has been treated with an antireflection (AR) coating to minimise the effect on the transmitted beam.
The \(\pi /2\) phase shift upon transmission (here given by the factor \(\mathrm {i}\,\)) refers to a phase convention explained in Sect. 2.4.
The free propagation of a distance D through a medium with index of refraction n can be described with the following set of equations (Fig. 6):
In the following we use \(n=1\) for simplicity.
Note that we use above relations to demonstrate various mathematical methods for the analysis of optical systems. However, refined versions of the coupling equations for optical components, including those for spaces and mirrors, are also required, see, for example, Sect. 2.6.
2.2 The twomirror resonator
2.3 Coupling matrices
In these examples the matrix simply transforms the impinging amplitudes into the outgoing amplitudes.
Coupling matrices for numerical computations
Coupling matrices for a compact system descriptions
The advantage of this matrix method is that it allows compact storage of any series of mirrors and propagations, and potentially other optical elements, in a single \(2 \times 2\) matrix. The disadvantage inherent in this scheme is the lack of information about the field amplitudes inside the group of optical elements.
2.4 Phase relation at a mirror or beam splitter
Composite optical surfaces
Please note that we only have the freedom to chose convenient phase factors when we do not know or do not care about the details of the coating, which performs the beam splitting. If instead the details are important, for example, when computing the properties of a thin coating layer, such as antireflex coatings, the proper phase factors for the respective interfaces must be computed and used. Similarly, for a simple glass plate this convention cannot be used.
2.5 Lengths and tunings: numerical accuracy of distances
A simple and elegant solution to this problem is to split a distance D between two optical components into two parameters (Heinzel 1999): one is the macroscopic ‘length’ L, defined as the multiple of a constant wavelength \(\lambda _0\) yielding the smallest difference to D. The second parameter is the microscopic tuning T that is defined as the remaining difference between L and D, i.e., \(D=L+T\). Typically, \(\lambda _0\) can be understood as the wavelength of the laser in vacuum, however, if the laser frequency changes during the experiment or multiple light fields with different frequencies are used simultaneously, a default constant wavelength must be chosen arbitrarily. Please note that usually the term \(\lambda \) in any equation refers to the actual wavelength at the respective location as \(\lambda =\lambda _0/n\) with n the index of refraction at the local medium.
2.6 Revised coupling matrices for space and mirrors
2.7 Finesse examples
2.7.1 Mirror reflectivity and transmittance
2.7.2 Length and tunings
3 Light with multiple frequency components
3.1 Modulation of light fields
Laser interferometers typically use three different types of light fields: the laser with a frequency of, for example, \(f\approx 2.8\cdot 10^{14}\mathrm {\ Hz}\), radio frequency (RF) sidebands used for interferometer control with frequencies (offset to the laser frequency) of \(f\approx 1\cdot 10^{6}\) to \(150\cdot 10^{6}\mathrm {\ Hz}\), and the signal sidebands at frequencies of 1–10 000 Hz.^{4} As these modulations usually have as their origin a change in optical path length, they are often phase modulations of the laser frequency, the RF sidebands are utilised for optical readout purposes, while the signal sidebands carry the signal to be measured (the gravitationalwave signal plus noise created in the interferometer).
3.2 Phase modulation
3.3 Frequency modulation
3.4 Amplitude modulation
In contrast to phase modulation, (sinusoidal) amplitude modulation always generates exactly two sidebands. Furthermore, a natural maximum modulation index exists: the modulation index is defined to be one (\(m=1\)) when the amplitude is modulated between zero and the amplitude of the unmodulated field.
3.5 Sidebands as phasors in a rotating frame
A common method of visualising the behaviour of sideband fields in interferometers is to use phase diagrams in which each field amplitude is represented by an arrow in the complex plane.
Phasor diagrams can be especially useful to see how frequency coupling of light field amplitudes can change the type of modulation, for example, to turn phase modulation into amplitude modulation. An extensive introduction to this type of phasor diagram can be found in Malec (2006).
3.6 Phase modulation through a moving mirror
3.7 Coupling matrices for beams with multiple frequency components
3.8 Finesse examples
3.8.1 Modulation index
This file demonstrates the use of a modulator. Phase modulation (with up to five higher harmonics is applied to a laser beam and amplitude detectors are used to measure the field at the first three harmonics. Compare this to Fig. 18 as well (Fig. 22).
3.8.2 Mirror modulation
4 Optical readout
In previous sections we have dealt with the amplitude of light fields directly and also used the amplitude detector in the Finesse examples. This is the advantage of a mathematical analysis versus experimental tests, in which only light intensity or light power can be measured directly. This section gives the mathematical details for modelling photo detectors.
4.1 Detection of optical beats
4.2 Signal demodulation
A typical application of light modulation, is its use in a modulationdemodulation scheme, which applies an electronic demodulation to a photodiode signal. A ‘demodulation’ of a photodiode signal at a userdefined frequency \(\omega _{x}\), performed by an electronic mixer and a lowpass filter, produces a signal, which is proportional to the amplitude of the photo current at DC and at the frequency \(\omega _0\pm \omega _x\). Interestingly, by using two mixers with different phase offsets one can also reconstruct the phase of the signal, or to be precise the phase difference of the light at \(\omega _0 \pm \omega _x\) with respect to the carrier light. This feature can be very powerful for generating interferometer control signals.
4.3 Finesse examples
4.3.1 Optical beat
In this example two laser beams are superimposed at a 50:50 beam splitter. The beams have a slightly different frequency: the second beam has a 10 kHz offset with respect to the first (and to the default laser frequency). The plot illustrates the output of four different detectors in one of the beam splitter output ports, while the phase of the second beam is tuned from 0° to 180°. The photodiode ‘pd1’ shows the total power remaining constant at a value of 1. The amplitude detectors ‘ad1’ and ‘ad10k’ detect the laser light at 0 Hz (default frequency) and 10 kHz respectively. Both show a constant absolute of \(\sqrt{1/2}\) and the detector ‘ad10k’ tracks the tuning of the phase of the second laser beam. Finally, the detector ‘pd10k’ resembles a photodiode with demodulation at 10 kHz. In fact, this represents a photodiode and two mixers used to reconstruct a complex number as shown in Eq. (4.15). One can see that the phase of the resulting electronic signal also directly follows the phase difference between the two laser beams (Fig. 25).
5 Basic interferometers
The large interferometric gravitationalwave detectors currently in operation are based on two fundamental interferometer topologies: the Fabry–Perot interferometer and the Michelson interferometer. The main instrument is very similar to the original interferometer concept used in the famous experiment by Michelson and Morley (1887). The main difference is that modern instruments use laser light to illuminate the interferometer to achieve much higher accuracy. Already an early prototype in 1971 has thus achieved a sensitivity a million times better than Michelson’s original instrument (Moss et al. 1971). In addition, the Michelson interferometer used in current gravitationalwave detectors has been enhanced by resonant cavities, which in turn have been derived from the original idea for a spectroscopy standard published by Fabry and Perot (1899). The following section will describe the fundamental properties of the Fabry–Perot interferometer and the Michelson interferometer. A thorough understanding of these basic instruments is essential for the study of the highprecision interferometers used for gravitationalwave detection.
5.1 The twomirror cavity: a Fabry–Perot interferometer

when \(T_1<T_2\) the cavity is undercoupled

when \(T_1=T_2\) the cavity is impedance matched

when \(T_1>T_2\) the cavity is overcoupled
5.2 Michelson interferometer
We came across the Michelson interferometer in Sect. 2.4 when we discussed the phase relation at a beam splitter. The typical optical layout of the Michelson interferometer is shown again in Fig. 30, a laser beam is split by a beam splitter and sent along two perpendicular interferometer arms. The four directions seen from the beam splitter are often labelled North, East, West and South. Another common naming scheme, also shown in Fig. 30 refers to the interferometer arms as X and Y; the two outputs are labelled as the symmetric port (towards the laser input) and antisymmetric port respectively. Both conventions are common in the literature and we will make use of both in this article.
The above seems to indicate that the macroscopic armlength difference plays no role in the Michelson output signal. However, this is only correct for a monochromatic laser beam with infinite coherence length. In real interferometers care must be taken that the armlength difference is well below the coherence length of the light source. In gravitationalwave detectors the macroscopic armlength difference is an important design feature; it is kept very small in order to reduce coupling of laser noise into the output but needs to retain a finite size to allow the transfer of phase modulation sidebands from the input to the output port; this is illustrated in the Finesse example below and will be covered in detail in Sect. 8.11.
5.3 Michelson interferometer and the sideband picture
In the context of gravitational wave detection the Michelson interferometer is used for measuring a very small differential change in the length of one arm versus the other. The very small amplitude of gravitational waves, or the equivalent small differential change of the arm lengths, requires additional optical techniques to increase the sensitivity of the interferometer. In this section we briefly introduce the interferometer configurations and review their effect on the detector sensitivity.
Consider a Michelson interferometer which is to be used to measure a differential arm length change. As an example for a signal to noise comparison we consider the phase noise of the injected laser light. For this example the noise can be represented by a sinusoidal modulation with a small amplitude at a single frequency, say 100 Hz. Therefore we can describe the phase noise of the laser by a pair of sidebands superimposed on the main carrier light field entering the Michelson interferometer. Equally the change of an interferometer arm represents a phase modulation of the light reflected back from the end mirrors and the generated optical signal can be represented by a pair of phase modulation sidebands, see Sect. 5.5.
In order to get an estimation of the signal to noise ratio we can trace the individual sidebands through the interferometer and compute their amplitude in the output port. Figure 32 shows the setup of a basic Michelson interferometer, indicating the insertion of the noise and signal sidebands. It also provides a plot of the sideband amplitude in the South output port as a function of the differential arm length of the Michelson interferometer. We can see that a tuning of 90\(^{\circ }\) corresponds to the dark fringe, the state of the interferometer in which the injected light (the carrier and laser noise) is reflected back towards the laser and is not transmitted into the South port. The plot reveals two advantages of the dark fringe as an operating point: first of all the transmission of the signal sidebands to the photo detector is maximised while the laser phase noise is minimised. More generally at the dark fringe, all common mode effects, such as laser noise, or common length changes of the arms, produce a minimal optical signal at the output port, whereas differential effects in the arms are maximised. Furthermore at the dark fringe the least amount of carrier light is transmitted to the photo detector. This is an advantage because it is technically often easier to make an accurate light power measurement when the total detected power is low.

Apply an RF modulation to the laser beam, either before injecting it into the interferometer or inside the interferometer. A small macroscopic length asymmetry between the two arms (Schnupp asymmetry, see Sect. 8.13) allows a significant amount of the RF sidebands to reach the South port when the interferometer is operating on the dark fringe for the carrier. The RF sideband fields can be used as a local oscillator.

Set the Michelson such that it is close to, but not exactly on, the dark fringe. The carrier leaking into the South port can thus be used as a local oscillator. This scheme preserves the advantages of the dark fringe but relies on very good power stability of the carrier light.

Superimpose an auxiliary beam onto the output before the photodetector. For example, a pickoff beam from the main laser can be used for this. The main disadvantage of this concept is that it requires a very stable auxiliary beam (in phase as well as position) thus creating new control problems.
5.4 Michelson interferometer signal readout with DC offset, or RF modulation
Another option for providing a local oscillator is by phase modulating the input laser light, which is typically done at radiofrequencies (RF). This method of readout is also referred to as a heterodyne readout scheme. When the Michelson interferometer is set up with a small, macroscopic arm length difference (Schnupp asymmetry) the RF sidebands will have a different interference condition at the beam splitter compared to the carrier, and the inteferometer can be setup so that the RF sidebands are present at the output port, to be used as a local oscillator, whilst the carrier field is at a dark fringe.
See Sect. 8.13 for an more detailed comparison of the DC and RF techniques to produce control signals and Sect. 8.16 for detailed arguments for the advantages and disadvantages of both techniques.
5.5 Response of the Michelson interferometer to a gravitational waves signal
5.6 Finesse examples
5.6.1 Cavity power
5.6.2 Michelson power
The power in the South port of a Michelson detector varies as the cosine squared of the microscopic arm length difference. The maximum output can be equal to the input power, but only if the Michelson interferometer is symmetric and lossless. The tuning for which the South port power is zero is referred to as the dark fringe (Fig. 35).
5.6.3 Michelson gravitational wave response
This is a simple Finesse example showing how the arm spaces can be modulated to produce the effect a gravitational wave would have on it. It outputs the amplitude and phase of the upper sideband that reaches the output port. The dips in amplitude occur when the travel time of the photons along the interferometer arms equals one gravitational wave period and hence the signal accumulated in the first and second half of the travel time cancel each other (the plot above does not have enough resolution to show that the dips indicate zero signal, the nonzero amplitudes are an artefact of the numerical plotting routine). In this example the frequencies of the dips are given as \(f=N\,c/1200\,\mathrm{m} = N\,250\,\mathrm{kHz}\), with N a positive integer (Fig. 36).
6 Radiation pressure and quantum fluctuations of light
Once classical noise sources are sufficiently reduced, the quantum fluctuations of light become one of the limiting noise sources for interferometric gravitationalwave detectors (Braginskii and Vorontsov 1975; Jaekel and Reynaud 1990; Meers and Strain 1991; Niebauer et al. 1991). To reduce this quantum noise the basic Michelson interferometer has been significantly altered over time, as we discuss in Sect. 7. This section aims to outline what quantum noise is and how its effects can be calculated.
The coupling of the quantum fluctuations of light into the output signal of the detector has traditionally been described as two separate effects: shot noise in the output current of the photodiodes and radiation pressure effects due to the use of suspended optics. Caves has shown that both noise components can be understood as originating from vacuum fluctuations coupling into the dark port of the Michelson interferometer (Caves 1981) and the twophoton formalism suggested by Caves and Schumaker (1985) has led to a large body of work towards understanding and reducing quantum noise in gravitational wave interferometers (Miao et al. 2014; McClelland et al. 2011; Chen et al. 2010; MüllerEbhard et al. 2009; Corbitt et al. 2005; Buonanno et al. 2003).
In the following we outline a method to compute quantum noise in interferometer output ports using sidebands and the classical framework presented in Sects. 2, 3 and 4. We apply this method to investigate the quantum noise limits of several interferometer readout schemes and finally discuss how suspended optics effect the quantum noise.
The interested reader can explore this topic further with a modern and comprehensive treatment of quantum noise in the review provided in Danilishin and Khalili (2012) and the following references: the standard quantum limit (Caves 1981; Jaekel and Reynaud 1990) squeezing (Loudon and Knight 1987; Vahlbruch et al. 2007) and quantum nondemolition interferometry (Braginsky et al. 2000; Giovannetti et al. 2004).
6.1 Quantum noise sidebands
Noise power spectral densities
The description of quantum noise with semiclassical sidebands has the advantage that the propagation of a stochastic signal through a linear system is described by the same transfer functions as for a deterministic signal. Therefor we can use the classical model of the optical system to compute the propagation of the quantum noise as well as any signal.
6.2 Vacuum noise and gravitationalwave detector readout schemes
When no nonlinear optical effects (effects proportional to the beam’s power) are present in an interferometer and the only quantum noise present is uncorrelated vacuum noise, there will always be the same amount of vacuum noise incident on any photodiode. This is irrespective of the topology of the interferometer or components used because noise can never be effectively lost from the system; an equivalent amount of uncorrelated noise is always injected back in. In such cases propagation of the noise sidebands through the interferometer do not need to be computed. Instead, when computing \(S_I\) at any of the photodiodes shown in Fig. 39 we only need to consider pure vacuum noise sidebands and the local oscillator field, \(E_{LO}\); the source of location of the vacuum noise sources is not of importance. This is why for early generation gravitationalwave detectors, which had negligible nonlinear optical effects, the semiclassical Schottky expression could be used to estimate the quantum noise correctly.
The detailed computation of quantum noise limited sensitivity of a detector depends on the readout scheme used. Early generations of gravitational wave detectors such as LIGO, Virgo, GEO 600 and TAMA300 used heterodyne readout schemes, where RF modulation sidebands applied to the input field are used as local oscillators at the output (see Sect. 5.4). However, such schemes included some technical challenges, the oscillator noise of the RF modulator being one of them, and also increase the shotnoise level when demodulating the photocurrent (Meers and Strain 1991; Niebauer et al. 1991; Rakhmanov 2001; Buonanno et al. 2003). Thus the next generation of detectors opted for a DC readout scheme (Fricke et al. 2012; Hild et al. 2009), see Sect. 8.16. Both schemes depicted in Fig. 39 use a form of DC readout, which we will analyse in more detail in the following sections. We do not cover the computation of quantum noise with RF modulation readout schemes, the interested reader should see Harms et al. (2007), Rakhmanov (2001) and Buonanno et al. (2003).
Noisetosignal ratio for DC offset
Noisetosignal ratio for balanced homodyne
No current generation gravitationalwave detector uses this form of homodyne readout for extracting gravitational wave signals. This has been due to the additional technical challenges which are not present when using DC readout. It is however used extensively for quantum noise measurements when nonvacuum states are injected into interferometers (Stefszky et al. 2012; Chua et al. 2014) and offers potential benefits over DC readout if the technical challenges can be overcome, as we show later in this section. Although not currently used, such a readout scheme is a current topic of investigation for future generations of detectors for extracting gravitationalwave signals (Fritschel et al. 2014).
There are two possible sources for the local oscillator field when using balanced homodyne detection: a separate laser system or a pickoff of the same carrier field used in the interferometer. The former is technically challenging as the separate system must be locked to the input laser to ensure temporal coherence when beating with the signal sidebands. The latter option of using a pickoff beam does not have this issue as it is from the same laser. Other technical challenges that exist for both options are (McKenzie et al. 2007) that the beam splitter is exactly 50:50; that the signal sidebands and local oscillator fields have a particularly good spatial overlap, also referred to as modematching and that the local oscillator does not backscatter into the output port of the interferometer.
6.3 Quantum noise with nonlinear optical effects or squeezed states
Up to this point we have only considered pure vacuum noise and linear optical effects, both of which are valid approximations for previous generations of gravitational wave detectors. As the effective laser power in the interferometer is increased to reduce shot noise, the radiation pressure exerted on suspended optics by the circulating laser beams will increase another noise significantly, the radiation pressure noise. As the suspended mirrors are free to move under the influence of radiation pressure, any fluctuation in the laser’s power will couple back into itself as phase modulation. This is a nonlinear process as the amplitude of the motion is proportional to the power in the beam, which leads to the upper and lower sidebands becoming correlated with one another. As explained in Sect. 6.4, such noise is prominent at low frequencies.
Then there is also the possibility of squeezing the vacuum noise, whereby still satisfying the relationship 6.6, the uncertainty in either the phase or amplitude is increased whilst being and decreased in the other. This squeezed noise can be represented by correlated noise sidebands (Caves and Schumaker 1985; Danilishin and Khalili 2012), Fig. 42 shows qualitatively the effect of squeezed vacuum noise on a coherent field. Injecting squeezed noise into the output port of the interferometer can thus reduce the dominant vacuum noise. Upon returning to the output port the noise should still be squeezed, but to a slightly lesser degree due to various optical losses which degrade the amount of squeezing. If squeezing is implemented effectively, the noise can be reduced below the typical shotnoise level of \(2P_0\hbar \omega _0\), thus providing a broad improvement in shotnoise limited regions of the detectors sensitivity (Caves 1981). Although we will not cover squeezing in detail in this article, squeezed light injection has been used routinely by the GEO 600 since 2010 (LIGO Scientific Collaboration 2011), and further upgrades to advanced gravitationalwave detectors using squeezed light sources are actively being developed (Oelker et al. 2014).
One important aspect to note here is that when either nonlinear optical effects or nonvacuum states of light are significant, the correlations introduced in the propagation of light fields through the interferometer have to be considered. This is due to the fact that the noise sidebands will be altered in amplitude and phase and the correlation between sidebands introduced as they propagate becomes an important feature. Such a calculation involves constructing the full interferometer matrix, see Sect. 2.3, for the noise sideband frequencies and including, if so required, the radiation pressure coupling at suspended mirrors as discussed in the next section.
6.4 Radiation pressure coupling at a suspended mirror
Mechanical transfer functions

the motion of any optic is small, \(\delta z \ll \lambda \), when the interferometer is controlled and wellbehaved, and we can linearise equations in \(\delta z\),

any highfrequency fluctuations in the beam are negligible due to \(H(\varOmega ) \propto 1/\varOmega ^2\) and we ignore the effects of RF sidebands on the optics,

any lowfrequency fluctuations are very small, such that the magnitude of any sidebands is much less than the magnitude of its carrier field, which allows us to identify a well defined carrier field in our calculations.
6.5 Semiclassical Schottky shotnoise formula
6.6 Optical springs
Optical springs are a result of an optomechanical feedback process that couples the intensity fluctuations in an optical field and the motion of a suspended mirror being restored by gravity. In the following we show how this feedback process introduces a force analogous to that of a damped spring with resonance frequency and damping coefficient defined by the optical properties of the interferometer and mechanical properties of the suspended mirrors, using the properties of the optomechanical coupling, introduced in the previous section.
At high circulating powers optical springs can significantly alter the behaviour of a suspended interferometer. Layouts such as optical bars (Braginsky et al. 1997) and techniques such as detuned signalrecycling (Buonanno and Chen 2002) use optical springs to improve the sensitivity of detectors (Rehbein et al. 2008). Due to their potential impact on current and future generations of gravitationalwave detectors, there have been several efforts to experimentally characterise their behaviour (Virgilio et al. 2006; Sheard et al. 2004; Corbitt et al. 2006). In this article only the longitudinal motion of a mirror along the axis of the optical beam axis is considered. However, rotational optical springs from torques (Sidles and Sigg 2006; Hirose et al. 2010; Dooley et al. 2013) or couplings to higherorder elastic vibrational modes of the mirror, known as parametric instabilities (Braginsky et al. 2001; Evans et al. 2015; Brown 2016), also exist and can pose significant challenges for controlling the interferometer at high laser powers.
Adiabatic optical spring
Full steady state optical spring
To compute the full response of a suspended mirror we have to consider the propagation and transformation of the sidebands generated by a moving mirror through the rest of the interferometer and finally back to the mirror in question. This process of scattering and feedback is represented by the block diagram in Fig. 44. A sinusoidal force \(F(\varOmega )\) is acting on a mirror with mechanical susceptibility \(H(\varOmega )\); the motion \(\delta z(\varOmega )\) combined with the incident carrier field \({a}_c\) scatters light into the upper, \({a}_s^+\), and lower, \({a}_s^\), sidebands leaving the mirror. The IFO plant, \( G(\pm \varOmega ) \equiv G^\pm \), is the optical transfer functions for either the upper or lower sidebands leaving the mirror to those returning to it. Lastly the incident upper and lower sidebands, \({a}'^\pm _s = G^\pm {a}_s^\pm \), are combined again with the carrier field to compute the radiation pressure force \(F_\mathrm {rp}(\varOmega )\) along with an external excitation \(\varDelta F(\varOmega )\) to feedback into the mirror.
The above analysis is applicable in the case of a single optical field. If there are multiple optical fields of comparable amplitude, the sum of the multiple radiation pressure forces must be considered to compute the overall value of \(\kappa \). The result (6.51) is also only applicable for optical fields with dominating radiation pressure on one single side of a near perfectly reflective mirror. The analytical calculation of other cases, such as, suspended beam splitters, multiple suspended optics, or multiple carrier frequencies with higher order spatial modes, can become very complicated. Tools such as Finesse take all these effects into account to ease studying such systems.
Optical spring in a cavity
6.7 Finesse examples
6.7.1 Optical spring
6.7.2 Homodyne detector and squeezed light
A laser and a squeezed light source are mixed with a beam splitter and then detected with a homodyne detector. The nominal quantum noise of an unsqueezed light field in the units of the blue trace are \(2 \hbar f\). The squeezing level of the squeezed light source is 10 dB, which means that the noise in one quadrature is 10 times lower than this whereas the other quadrature should be 10 times higher. With the phase of the local oscillator the homodyne detector can be tuned to measure the different quadratures. The green trace shows a computation of an effective squeezing level from the detected quantum noise using the Schottky equation (Fig. 47).
6.7.3 Quantumnoise limited interferometer sensitivity
This example shows the quantumnoise limited sensitivity of an advanced detectors. See “Appendix B” for the optical layout of the detector and Sect. 8.12 for more details about the interferometer operation. The model is loosely based on the Advanced LIGO design file and thus we expect to see the peak sensitivity around 100 Hz at a sensitivity of about \(10^{23}/\sqrt{\mathrm{Hz}}\). We can see the both the ‘qnoised’ and ‘qshot’ detectors agree at high frequencies, where the sensitivity is purely limited by shot noise. At low frequencies the two traces differ because only ‘qnoised’ takes into account the radiation pressure effects (Fig. 48).
The Finesse input file for this example is more complex than for other examples because it contains a more complex interferometer setup and uses relatively advanced concepts such as setting mechanical transfer function. See “Appendix A” for more information on Finesse and where to find the documentation, such as the syntax reference, required to follow this example.
7 Advancing the interferometer layout
The first generation of interferometric gravitationalwave detectors was limited in the upperfrequency band by shot noise, one manifestation of the quantum noise of the laser light, see Sect. 6. We can improve the ratio between gravitationalwave signal and shot noise in several ways, for example, by increasing the arms’ length or by increasing the injected laser power. The lengths of the arms is typically limited by the associated costs of the building the infrastructure. For example, due to the curvature of the Earth’s surface, a 40 km long interferometer arm would require a trench or tunnel approximately 30 m below the surface in the middle of the arm.
Highpower lasers are used; however do not come near the power levels required for the anticipated sensitivity. For example, the design sensitivity of Advanced LIGO requires a light power or several hundred kilowatts in the interferometer arms. The Advanced LIGO laser can provide up to 200 W of power, and represents a state of the art system (for a CW laser with the required stability in frequency, amplitude and beam profile) (Kwee et al. 2012).
In order to increase the laser power inside the arms further we can utilise the concept of resonant light enhancement in the Fabry–Perot cavity: socalled advanced interferometer topologies are created by introducing optical cavities to the Michelson interferometer. In the following we will briefly introduce the most common concepts, which are used by modern gravitationalwave detectors today.
We have shown in Sect. 5.2 how the dark fringe operating point allows to maximise the throughput of differential signals (with respect to common mode noise), using the sideband picture. Similarly we can compute the transfer functions of the signal sidebands to illustrate the concepts behind the advanced interferometer layout. The motivation for all the advanced concepts shown below is the improvement of the ratio between signal and shot noise. However, we will ignore here the detailed computation of the shot noise and quantum noise discussed in Sect. 6. Instead we will compute only the transfer functions of the signal to the photo detector using the sideband picture. We will ignore radiation pressure noise and shotnoise contributions from any light field but the local oscillator. Thus the amplitude of the signal sidebands in the detection port give a good figure of merit for the shotnoise limited sensitivity of the detector.
7.1 Michelson interferometers with power recycling
7.2 Michelson interferometers with arm cavities
Another way to employ cavities to enhance the light power circulating in the interferometer arms is to place optical cavities into these arms, as socalled arm cavities, as shown in Fig. 51. This optical configuration sometimes referred to as Fabry–Perot–Michelson interferometer. Similar to powerrecycling the finesse of the cavity determines the enhancement of the light power.
The arm cavities have another effect on the detector sensitivity: they affect not only the power of the circulating carrier field, but also that sidebands generated by a length change. This results in a further increase of the sensitivity for signals with a frequency within the linewidth of the arm cavities but to a decrease in sensitivity regarding signals with frequencies that fall outside the linewidth of the cavities. This can be shown again very clearly with the sideband amplitudes detected at the interferometer output as shown in Fig. 52. We can compare this results to the powerrecycling case (Fig. 50): when the reflectivity of the PRM and ITMs is set to \(R=0.99\), the expected gain for the carrier field inside the cavities must be the same and equal to 400, assuming an overcoupled case. At low frequencies the signal sidebands will experience the same enhancement, namely by a factor of 400 in power. Thus the total enhancement for the signal sidebands in the Michelson with arm cavities is 16 000, which gives the amplitude of 400 shown for sideband amplitude in Fig. 52. Therefore the arm cavities also change the detector response function in a way that limits the possible sensitivity increase.
7.3 Signal recycling, dual recycling and resonant sideband extraction
Soon after the development of power recycling in which an additional mirror is used to ‘recycle’ the laser light leaving the Michelson interferometer through the symmetric port, Brian Meers recognised that it would be of interest to employ a similar technique in the antisymmetric port. In the ideal Michelson interferometer on the dark fringe, the carrier light and the signal sidebands become separated at the central beam splitter and leave the interferometer though different ports. Meers (1988) suggested the addition of a signalrecycling mirror at the antisymmetric port, to form a signalrecycling cavity with the Michelson interferometer. In a similar manner to the powerrecycling cavity the signalrecycling cavity could resonantly enhance the light circulating within, i.e., the signal sidebands. The optical layout of a signalrecycled Michelson interferometer is shown in Fig. 53.
When both recycling techniques are used together, power recycling for enhancing the carrier power and signal recycling for increasing the signal interaction time, the combination of the two methods is called dual recycling. It was actually the concept of dual recycling which Meers (1988) proposed, and this was demonstrated first as a tabletop experiment by the Glasgow group in 1991 (Strain and Meers 1991).
The combination of arm cavities and a signalrecycling mirror is sometimes also called resonant sideband extraction (Mizuno et al. 1993). The difference between signalrecycling and resonant sideband extraction is that in the latter case the arm cavities have a very high finesse and the signalrecycling mirror is tuned to or near the antiresonant operating point, thus effectively increasing the bandwidth of the detector for the signal sidebands. An analysis of the different techniques can be found in the thesis of Mizuno (1995). It is interesting to note that for all variants of the signal recycling the total integrated gain remains constant. For example, the areas under curves for the different detunings shown in Fig. 54 are constant.^{10} This means that signalrecycling is used to shape the response function of the detector with respect to the signaltoshotnoise ratio.
7.4 Sagnac interferometer
7.5 Finesse examples
7.5.1 Michelson interferometer with arm cavities
This example shows how to setup a Michelson interferometer, tune it to the dark fringe and compute a transfer function from the differential length change to the output signal, using the sideband amplitude for simplicity (Fig. 57).
7.5.2 Michelson interferometer with signal recycling
This example recreates the plot shown in Fig. 54, the four traces show the transfer function for a Michelson interferometer with different signal recycling tunings (Fig. 58).
7.5.3 Sagnac interferometer
This example demonstrates how compute the frequency response of a simple Sagnac interferometer (Fig. 59).
8 Interferometric length sensing and control
In this section we introduce interferometers as length sensing devices. In particular, we explain how the Fabry–Perot interferometer and the Michelson interferometer can be used for highprecision measurements and that both require a careful control of the base length (which is to be measured) in order to yield their large sensitivity. In addition, we briefly introduce the general concepts of error signals, transfer functions and relevant elements of control theory, which are used to describe most essential features of length sensing and control.
In addition to sensing and controlling the distances between the components of an interferometer, alignment sensing and control is required for correct operation. While we do not deal with this aspect in detail, all of the ideas we develop for length sensing and control can be applied. The essential differences are that split photodetectors are required to sense the relative angles of optical wavefronts, and control is be means of actuators that are able to adjust the angles of optical components. For an introduction to the essential ideas see Sect. 9.2 for an introduction to the relevant theory and Morrison et al. (1994), for details of a practical implementation.
8.1 An overview of the control problem
A complete interferometer can have a large number of control loops for the various mirrors and beam splitters, their suspension systems and many other components, such as the laser, active vibrationisolation systems etc. For practical purposes these are usually divided into two broad classes that are often considered separately in the design process. These divisions reflect a degree of independence of the various categories of control and simplify the design process by allowing the problem to be split into a number of more easily tractable design elements.
The set of control loops that obtain signals from the detection of interference conditions or other properties of the light within the interferometer, and act on the major optical components of the interferometer to control those properties, is usually called global control. As an example, a description of the global control system of Virgo can be found in Arnaud et al. (2005). On the other hand, loops that sense properties associated with a single component, and act on that component are called local loops. A good example of local control is the system employed to damp the rigidbody modes of a mirror suspension, for an example from Advanced LIGO see Aston et al. (2012). By ‘cooling’ or quieting the motion of individual mirrors, the task faced by the global control system can be simplified. Further division of global interferometer control is frequently made between systems that control longitudinal degrees of freedom, i.e., relative positions of the mirrors and beam splitters along the direction of propagation of the light, and angular (alignment) control systems that are designed to stabilise the pointing of components.
Due to the presence of strong nonlinearity throughout much of the phasespace volume, there has been no attempt thus far to solve the multidimensional control problem as a whole. At least up to the present, the problem has been divided into several smaller parts, with methods developed to deal with the particular details of each facet of the system, and each stage of operation from completely uncontrolled to held at the operating point—a condition that is called ‘locked’.
This leads to yet another division: it is normal to separate the startup phase i.e., the process called acquisition of lock from the stable running condition (‘in lock’). This split is motivated, at least in part, by the consideration that signal sizes can differ greatly between the two stages. During acquisition electronic signals tend to be large—corresponding to adjusting mirror positions by of order wavelengths, or more. By contrast, in operation the signals representing residual motion in the sensitive frequency band may be 12 or more orders of magnitude smaller than the wavelength of the light, to deliver the required measurement sensitivity.^{11}
The jump in signal size between these two states is often dealt with by switching gain levels or even substituting large parts of the control system: starting with largerange but noisy methods for acquisition and switching to lownoise, but smallrange controls for operation. A good example of a more substantial transition is the arm length stabilization (ALS) scheme in Advanced LIGO, which employs additional lasers, mirror coatings and interferometric methods to provide wavelengthrange sensing during the acquisition phase (Evans et al. 2009). When the long cavities (see “Appendix B”) are locked, the control systems are switched over to the highsensitivity, lownoise signals derived from the main interferometer systems.
During acquisition of lock, the instantaneous operating point frequently lies in a nonlinear region of the control space. Several methods have been developed to cope with this problem.
The simplest approach, employed in the early interferometer prototypes, was to wait for a random coincidence of suitable values to occur then to catch the system quickly enough to hold it in the desired state. As the complexity of the interferometer topologies increased, and with that the number of degrees of freedom, the probability of the desired state occurring in a conveniently short time became rare. This led to the development of more sophisticated techniques for the first longbaseline interferometers.
As a first step it was realised that digital logic, implemented directly in electronics or as software, could be employed to identify when one or more degrees of freedom happened to fall close to the desired operating point, and to activate the relevant control loop. This prevents false signals, frequently present in regions of phase space close to the desired operating point, being fed back to the actuators and perturbing the system. In Pound–Drever–Hall locking, for example, when the phase modulation sidebands pass through resonance in the cavity, the error signal has the opposite sign from that produced by a carrier resonance. The acquisition process can be improved by enabling the control system only when the circulating power within the cavity has exceeded the maximum possible power that a sideband can produce. By this means the control system is only activated close to the desired operating point, improving the chances of a successful lock.
A second way to improve matters is to linearise the behaviour, and so to increase the capture range: i.e., the volume of phase space within which the various control signals are valid. This improvement can be accomplished by normalising the relevant error signal according to some estimate of its slope, as measured by another signal such as the circulating optical power. As an example, the linear range in the Pound–Drever–Hall signal for locking a cavity may be extended by normalising with respect to the power within the cavity, as measured by probing the light transmitted by the cavity.
Another approach, is to arrange for the first locking to be in a region of phase space that is relatively smooth, compared to the region in phase space surrounding the final operating point. This was an enabling technique for GEO 600 when it was first operated in dualrecycled mode. It was found that by locking with signal recycling detuned by a few kHz, an initial lock was possible. The tuning was then stepped towards the target value, in steps chosen to be small enough to avoid perturbing the lock. By this means it was possible to reach a location in phase space which would essentially never have occurred by chance (Grote 2003).
After lock has been achieved by one or more of the above means, the control task is generally managed by linear control systems that may be analysed using standard linear time invariant (LTI) control theory. Two generic approaches have been employed with success. In one approach there is a set of separate singleinput single output (SISO) controllers, one for each degree of freedom. The alternative is to deal with several degrees of freedom in a single multiinput multioutput (MIMO) controller. Recently, since the advent of computerbased, digital control systems, the MIMO approach has become much more practical than it would be if implemented in analogue electronics. An important difference between the SISO and MIMO approaches concerns how crosscoupling between the degrees of freedoms can dealt with.
Crosscoupling is commonly seen in both sensing and actuation, and considerable effort is needed to develop control systems that operate correctly in the presence of undesired mixing of signals. The main approaches to solving these problems with MIMO controllers is described in Sect. 8.7. Otherwise we discuss SISO controllers to provide illustrative examples of control in idealised interferometers where there is no mixing of degrees of freedom at the point of sensing or actuation.
We introduce standard terminology from control theory. In each control loop a point of reference is taken, called the error point at which we measure how the gain of the loop acts to suppress deviations from the desired operating point. Since a loop has no end, the selection of this point is somewhat arbitrary, but it is usually convenient to take the output from a photodetector or its associated demodulator.
Likewise, we choose an actuation or feedback point at the interface between the control electronics and the interferometer—again the precise division is somewhat arbitrary, but the electronic signal input to an actuator is frequently employed as the point of reference.
With these points defined, the part of the loop from errorpoint to feedbackpoint is called the controller or just the feedback, and the rest of the loop from feedback point to the error point, in the causal direction, is called the plant.
Before a loop is activated, the signal that would be measured at the error point is called the error signal. In interferometry this is usually derived as an output from the optical system and its photodetectors, as explained in Sect. 8.5.
8.2 Linear timeinvariant control theory: introductory concepts
A full description of linear timeinvariant (LTI) theory is beyond the scope of this article, therefore we restrict our description to a short summary of the essential concepts, with some relevant examples presented in the following sections.
In LTI models the superposition principle applies, frequencies do not mix and it is possible to represent any physical timedomain signals in the frequency domain through their Fourier transforms. The timeinvariance means that the response of a system to an input does not depend on the time at which that input is applied. This implies that the differential equations describing the system are linear and homogeneous with coefficients that are constant in time. In this case it is common to solve these equations by employing methods based on the Laplace transform. The response of the system is represented by its transfer function which is the Laplace transform of its impulse response—i.e., the output produced in the time domain when the input is a Diracdelta function. We look at transfer functions in mode detail in Sect. 8.5.
By implication, if the input to the system is a sinusoid (a single Fourier component) the output is also a sinusoid with, generally, different amplitude and phase as described by the transfer function. If the system consists of a series of optical, electronic and mechanical stages or subsystems, the overall transfer function is the product of the individual transfer functions. If there is feedback, then loopalgebra can be applied, and this represents summation of the signal from one point in a loop into an earlier point (feedback), or in the case of feedforward to a later point. This is also a linear operation, and so the system with feedback or feedforward retains its LTI property. The most complicated case that need be considered is when there are two or more nested feedback loops (or equivalently two or more actuators that adjust one degree of freedom (Shapiro et al. 2015)), in this case there are loopalgebraic techniques that allow the system to be reduced to a single equivalent loop, such that nested loops of any practical topology may be considered step by step.
In modelling LTI systems it is common to pick a representation from one of a mathematicallyequivalent set of options. The alternatives are described in, e.g., Franklin et al. (1998). To give some examples, individual loops or blocks with one input and one output (called SISO—single input, single output systems), are commonly represented by their transfer functions.
For LTI systems, the transfer functions can be written as rational polynomials in the complex frequency variable s. The polynomials on both the numerator and denominator of the transfer function can be factorised. This leads to an equivalent mathematical representation of the transfer function as a list of zeros (zeros of the numerator) and poles (zeros of the denominator). In addition an overall gain factor, independent of frequency, is usually required. The poles and zeros may be represented by their coordinates in the complex (s)plane, or more phenomenologically by their resonant frequencies and damping factors. This leads to the socalled (z, p, k)notation, where z represents one or more zeros, p the poles, and k represents an overall frequencyindependent gain factor.
For MIMO systems arrays of transfer functions can be employed to describe all possible input–output relations, but it is more common to use the statespace representation. Here a single set of matrices encapsulates the behaviour of the entire system. A description of this important method is beyond the scope, but full detail is given in Franklin et al. (1998).
In briefest summary, the statespace method involves writing the set of N second order differential equations representing the internal dynamics of a system which has N degrees of freedom. These are reduced to 2N firstorder equations by introducing the time derivatives, i.e., generalised velocities, of the displacementlike coordinates. The solution of the resulting set of equations is then usually carried out numerically, using matrix methods.
8.3 Digital signal processing for control
In the past two decades digital control systems have been introduced into the control of interferometers, in modern instruments the majority of control systems contain digital processing elements, although the interfaces with the interferometer remain analogue in almost all cases. The essential principles remain the same as in the continuoustime systems, and a common approach to the design of digital control systems starts by designing and simulating a continuoustime analogue model. When this model operates as required in simulation, the result is transformed to the discretetime mathematics of digital control. The resulting filters are then implemented in a combination of software and hardware. In discretetime models only a finite set of frequencies exist, limited at high frequencies by the Nyquist frequency, i.e., half of the sampling rate in the digital system.
Digital models also have finite amplitude resolution, with the practical resolution limits occurring at the analoguedigital and digitalanalogue interfaces (ADC, DAC respectively), rather than in the digital signal processing. These limitations are generally handled by whitening the signal at the input and dewhitening at the output. For example, input signals with predominant lowfrequency content may be highpass filtered to render their spectral content relatively uniform, i.e., white, before sampling to make best use of the available resolution. The converse process can be applied at the output, with sufficient lowpass filtering applied to ensure that white noise resulting from the output conversion (DAC) is suppressed within the gravitational wave band.
In the description of digital controllers, the discrete mathematics of the zplane replaces the continuous nature of the splane (Franklin et al. 1998). Transforming from one space to the other is something of an art, mainly due to the consequences of finite precision in the associated calculations. A bilinear transformation is commonly employed. To avoid problems of numerical accuracy in the associated calculations, complicated systems are broken down in to a series of secondorder sections. These subsystems have up to two poles and two zeros, i.e., the transfer functions have no higher order than quadratic numerator and denominator. Such subsystems can be transformed more reliably.
One further consequence of discrete time is that there is a finite delay associated with the analogue to digital, signal processing and digital to analogue steps. This must be considered in the development of feedback loops based on digital controls. In practice, even more severe limits to highfrequency performance often arise from antialiasing or antiimaging filters that may be required on the analogue input and outputs, respectively.
Aliasing occurs when the ingoing signal contains significant amplitude components at Fourier frequencies above the Nyquist limit. If these are not filtered out, they are incorrectly recorded their beat frequencies with the nearest harmonic of the sampling frequency. At the output, the digital signal has discrete steps from one sample to the next. To properly reconstruct the required analogue signal these steps require to be removed by the lowpass action of an antiimaging (or reconstruction) filter. Further detail of the sampling and reconstruction processes is found in Franklin et al. (1998).
8.4 Degrees of freedom and operating points
We consider the optical components to be rigid bodies, each with six degrees of freedom. With practical, highquality spherical surfaces, only three degrees of freedom per component are important: position along the direction of propagation of the light, referred to as the longitudinal coordinate, the yaw angle with respect to that direction, i.e., in the horizontal plane, and the pitch angle in the vertical plane. The other three degrees of freedom (vertical, horizontal normal to the beam and roll around the axis of the beam) may be important with respect to noise coupling into the length measurement in the case of imperfect mirrors. As discussed in Sect. 11.5, the mirrors typically have only small deviations from ideal spheres, so the coupling factors are small and do not significantly affect the control of the interferometer.
In the interferometer as a whole, one component, for example, a mirror or beam splitter, may be chosen as the origin for the coordinate system. This allows one position and a pair of angles to be predefined. The positions and angles of the other components may then be described with respect to this origin. Note, however, that the longitudinal degrees of freedom are measured with an optical ‘ruler’ that is based on the wavelength of the light, and so the wavelength should be counted as one longitudinal degree of freedom in the system as a whole (in the sense that the light has the same frequency everywhere which is usually true to a good approximation in the ultrastable environment of a gravitationalwave detector). Similarly the direction of the light beam entering the interferometer defines two angles.
To take an example, a simple cavity that is to be held on resonance with ingoing light has two meaningful longitudinal degrees of freedom. For a cavity in isolation it would be usual to consider the position of one mirror relative to the other and the frequency, or wavelength, of the light as the important parameters. Mathematically there are other equivalent choices, but in the control and operation of interferometers the point is to find a convenient set of control variables.
Similarly, a simple Michelson interferometer has three components and three longitudinal degrees of freedom. Again it would be usual to consider one component as a reference. If the beam splitter is fixed, the three degrees of freedom are the two armlengths and the optical wavelength, or frequency.
If a pair of cavities were to be placed, one each, into the arms of the simple Michelson, the single degree of freedom of each mirror is replaced with the two of the cavity, for a total of five: once again, the same as the number of components. Fixing the beam splitter, these are the laser wavelength, the two distances from the beam splitter to the near mirrors of the cavities and the two lengths of the cavities.
An example of the degrees of freedom relevant to longitudinal control of a more complex system is shown in Sect. 8.12.
The choice between employing absolute or relative coordinates for the positions (and angles) of interferometer components is reflected in differences of approach in the available modelling software. In a Finesse model of a twomirror cavity, for example, the longitudinal positions of the two mirrors are specified, and adjusting either of them changes the resonant condition of the cavity (see, for example, Sect. 5.6.1). Likewise, adjusting the position of the input mirror changes the phase of both the light in the cavity, and the light reflected from the cavity. See Sect. 2.1 for a discussion of this point.
An operating interferometer requires various interference conditions to be maintained, e.g., cavities should be kept on resonance, the darkfringe condition in a Michelson interferometer must be met, and so forth. For each degree of freedom this implies that there is an optimum value for best sensitivity or an operating point in the multidimensional space representing the degrees of freedom. This point is not usually unique: for example, signals repeat modulo one roundtrip wavelength, see below.
As most or all degrees of freedom are subject to movement or drift, they must be controlled, generally by designing and implementing a separate control loop for each one. These loops must be designed to hold the value of the degree of freedom close to such an operating point, where ‘close’ is determined by tolerance bounds that must be determined by calculation.
In most cases it is possible to evaluate a tolerance interval around the operating point. The limits usually arise in the consideration of the coupling of some kind of noise into the sensitive measurement (frequency noise, power noise, beam direction noise, etc.). For example, in the case of the dark fringe, sensitivity to laser power noise is at a minimum at the perfectly dark condition, and the tolerable increase in this coupling may be used to set bounds on deviations from the operating point.
Bounds may also be set by considering the required linearity of signals. Nonlinearity can lead to beating, which mixes noise into the measurement band. For example if there is a narrow spectral feature or ‘line’, such as a calibration line that may be applied to monitor instrumental sensitivity, or a suspension violin mode^{12} in the measurement band, beating this with low frequency motion of suspended components will produce sidebands on either side of the narrow feature, and these may be of higher amplitude than the noise background at the frequencies of interest near the line. Nonlinear operation may also cause problems for control systems, as its presence implies that the gain of control loops will depend on the magnitude of deviations from the nominal operating point. In the following (Sect. 8.5) it will be seen that the normal process of sensing the length of a cavity is reasonably linear only within a very narrow range, in comparison to the wavelength of light, around the operating point, at least for a cavity of high finesse—a range of distance of order \(\lambda /{ F }\) (see Sect. 5.1), or smaller.
The diminutive scale of the useful volume in phase space can be illustrated by means of a simplified example. Here we consider the case of two degrees of freedom in a power recycled Michelson interferometer. Even fixing the location of the beam splitter, there are three degrees of freedom (i.e., common arm length, differential arm length and longitudinal position of the power recycling mirror. However, we choose to produce a contour plot showing signal sizes as a function of just two degrees of freedom, Fig. 62. Here we vary the difference in the lengths of the two arms while keeping the average (or common mode) arm length fixed, and also to vary the position of the power recycling mirror. In a practical interferometer there would be several other degrees of freedom associated with, for example, arm cavities, signal recycling and control of the commonmode arm length (or laser frequency), and in most cases the cavities would be of higher finesse producing even narrower features—see, for example, the parameters for Advanced LIGO in “Appendix B”.
The complexity of sensing and control becomes apparent when one considers that, in the common case of the freelysuspended optical components in a groundbased interferometric gravitational wave detector, the initial condition, at the point of ‘switching on’ the controls can be any random point within the space, with—in addition—a wide range of initial velocities associated with each degree of freedom: up to perhaps of order one wavelength per second, in a typical groundbased instrument. How this is dealt with is summarised in Sect. 8.1.
8.5 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 userdefined value, the operating point, \(p_0\). 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, e.g., as Bode plots (see the following section). 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 modelled similarly: applying a sinusoidal signal \(~\sin (\omega _s t)\) to the interferometer, e.g., as a position modulation of a cavity mirror, will create phase modulation sidebands with a frequency offset of \(\pm \omega _s\) 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 \(\omega _s\), which can be extracted, for example, by means of demodulation (see Sect. 4.2).
8.6 Bode plots: traditional control theory for SISO loops
An essential feature of a control system is stability, i.e., for a finite input the output should always be bounded. This is equivalent to requiring all of the transfer function poles to correspond to decaying exponentials, so their real parts must be strictly negative.
Prior to the routine application of computers, a number of tools (plots) were developed to facilitate control system design. Although the rootlocus, Nyquist and Bode plots continue to be applied, computer models remove the practical (calculational) advantages of one over another. All of these methods present essentially equivalent information, and the choice of one over another is a matter of convenience or familiarity. Since Bode plots provide a complete description of minimumphase, singleinput singleoutput (SISO) LTI control loops we choose to describe that approach as an example.
A Bode plot of a system shows its transfer function in the form of logmagnitude and linearphase graphs against a logarithmic frequency axis—conventionally in verticallystacked plots with matching, aligned frequency axes, see Fig. 64 for a simple example. In the context of the design of complete negative feedback loops, it is common, though not universal, to add \(\pi \) to the phase to represent the overall negative sign—this convention is assumed here. The standard procedure starts with consideration of the openloop Bode plot. In this, the loop is broken (in the model) at a convenient point, and the transfer function from there back to just before the break is calculated and plotted. Remember that the total transfer function is computed as the product of individual transfer functions of parts of the loop that are connected in series. Particular attention is paid to the regions close to points where the transfer function magnitude crosses unity (i.e., zero on the log scale), called the unity gain point(s), and where the phase crosses \(\pi \) in absolute terms, not modulo \(2\pi \). The transfer function is then characterised by the phase margin and the gain margin. The phase margin is the phase of the transfer function plus \(\pi \) at the frequency where the gain is unity, and the gain margin is the inverse of the gain where the phase is \(\pi \) (or the negative of the log gain). If there are multiple unity gain points the smallest phase margin, and the smallest gain margin, dominate. If the smallest gain and phase margins are both positive, the system is stable. Note that if there are multiple paths or ‘loops’, these are dealt with by applying loop algebra to reduce the system to a single feedback loop without subsidiary loops.
Traditionally these methods were extended to reveal properties of the closedloop system, i.e., of the original model without any break. This was done because, for transfer functions of low order (one, two or three poles), there are simple expressions that relate the phase margin to the ringing, or equivalently damping, of the closedloop response. When computer models are employed for systems of greater complexity there is no need for these rules or guidelines and it is common to transform back to the time domain, calculate the impulse response of the closed loop system, and characterise its resonant frequencies and damping without reliance on rules.
The construction and utility of the Bode plot originates in part from the properties of a common subset of transfer functions that represents stable, causal systems. Such systems are called minimum phase as a consequence of the locations of their zeros in the splane. In a causal system the output lags the input. Stable, causal LTI systems are also invertible, i.e., the transfer function numerator and denominator can be swapped, or equivalently all the poles and zeros may be exchanged resulting in another stable, causal system. For this to work the zeros of the system must have negative real parts, so that when they become poles in the inverse system they are damped. It can be shown that in such a system there is a strict relationship between the phase and the slope of the logmagnitude, as shown on a Bode plot—one is a Hilbert transform of the other. In practice this is equivalent to writing that the when the magnitude graph has a slope of \(f^{n}\), where f is the frequency, the phase approaches \(n\pi /2\). This method allows the loop to be designed to meet various goals that are usually expressed in terms of gain (or attenuation) that must be achieved in one or more range of frequencies, with stability checked by reading off the phase and gain margins.
In interferometry the optical transfer function is usually a significant aspect of control loops. Such transfer functions may be measured or found by calculation (e.g., with Finesse). The corresponding transfer functions can be found by applying the techniques described in Sect. 8.5.
8.7 Separating mixtures of the degrees of freedom: control matrices
In practice, each error signal intended to represent a particular degree of freedom of the optical arrangement also contains some information about other degrees of freedom. To give a simple example of the mixing that may occur, any motion that leads to a change in the circulating light power in a cavity is likely to couple, at some level, to every signal that depends on the intracavity light, unless the signal is precisely zero.
In most cases such mixing is undesirable as it is easier to design control systems to deal with one degree of freedom in isolation. In the worst case, if the mixing, or crosscoupling is strong, it can lead to the formation of unintended feedback paths. If the transfer function of such loops has a magnitude exceeding unity, there is a chance that the loop may be unstable. A common cause of such instability is a resonance in the unintended or ‘parasitic’ loop. At such a resonance high gain is typically accompanied by a phase lag of \(\pi \) which will tend to be unstable unless some compensation is included, e.g., in the form of a notch filter to cancel the resonance.
Unwanted mixing of signals can also occur at the point of actuation. For example, a mirror may be common to two degrees of freedom of an interferometer. In an interferometer with arm cavities, the cavity mirrors closest to the beam splitter behave in this way. Moving such a mirror must then affect at least two length degrees of freedom. This can be seen in Fig. 51 where motion of either of the two mirrors labelled ITMX and ITMY affects the phase of the light in the respective arm cavity and also the interference condition of the Michelson interferometer. In contrast, the end mirrors (ETMX, ETMY) each affect only one longitudinal degree of freedom.
A further possible source of mixing between degrees of freedom arises at the point of actuation. Feedback to control a mirror is often carried out in practice using an array of actuators, such as coilmagnet pairs, that push on the mirror at various points on its surface. For example, it is common to employ a squarearray of four magnets attached to the rear surface of the mirror, as these allow longitudinal, pitch and yaw adjustment. If they are mounted close to the perimeter of the rear surface they may be out of the way of a transmitted light beam. With such an arrangement, each individual actuator causes changes to a mixture of angular and longitudinal degrees of freedom. If the actuators are not of precisely uniform strength and alignment, this leads to unintended components in the resultant force produced by the array. An actuation matrix, with frequencydependent elements where necessary, can be employed to orthogonalise the response of the system to commands from the controller, at least to some degree of precision.
The elements of actuation and sensing matrices are typically determined as a result of simulation and measurement. Modelling may yield a set of starting values that suffice to allow the interferometer to operate. When operational residual mixing is normally determined by carrying out all possible transfer function measurements. The measurements allow coupling matrices to be determined, and inverting the coupling matrix provides the appropriate matrix necessary to remove unwanted mixing. This process is somewhat involved and benefits from automation.
8.8 Modern control methods in gravitationalwave detectors
During the past few decades new methods of designing sophisticated controllers based on digital signal processing have emerged. A major benefit of the resulting ‘digital controls’ is that the response of a control filter can be adjusted by changing filter coefficients, this can even be achieved while the controller is operating, if that is required.
Digital control facilitates the application of socalled modern control methods in which optimisation methods are employed. As an indication of the possible advantages that may arise from this, we briefly mention two approaches to modern control of application in interferometry. For a relevant description of these see, e.g., Franklin et al. (1998).
In the first approach, we consider the generation of an optimal filter with fixed coefficients (gain, poles and zeros). In such a case, the plant to be controlled is characterised by some means, and the results are used in the design of an optimal filter. For example, if it can be assumed that a measurement produces an estimate of the system contaminated by noise, and a model of the system with the correct number of degrees of freedom exists, a Wiener filter may be formed as a result of leastsquares fitting the model to the data. If the result is to be inverted to provide a compensating filter in a control system, then the fit must be constrained produce a causal filter (with all poles and zeros having negative real parts, in an analogue model). The are standard methods by which this may be accomplished.
The next step up in sophistication is to find a controller that remains optimal even if the underlying plant changes (or if its parameters cannot be measured accurately before the controller is put into operation). Such an adaptive controller, employs a Kalman filter—also called a Linear Quadratic Estimator. This is implemented as an algorithm that operates on a series of measurements taken over time. These measurements are assumed to be contaminated with noise. The algorithm operates recursively to produce an optimal estimate of the state of the physical system. During this process a model of the system, i.e., a representation of the equations of motion, with relevant coefficients available to be adjusted, is iteratively updated. The model is assumed to have errors either as a result of poor starting estimates or due to drifting of parameters over time. A weighting function, also called a cost function, is applied to the measured data to allow less noisy or otherwise more important aspects of the data to have a stronger influence on the outcome. At each iteration the model is employed to predict the current state, this is then compared with the actual state and the results of the comparison are used to refine and update the model. When this method is made to operate, the model of the underlying system converges to an optimal solution for the given weighting function.
8.9 Fabry–Perot length sensing

The cavity must be held as near as possible to the resonance for maximum sensitivity. This is the reason that active servo control systems play an important role in modern laser interferometers.

If we want to use the power directly as an error signal for the length, we cannot use the cavity directly on resonance because there the optical gain is zero. A suitable error signal (i.e., a bipolar signal) can be constructed by adding an offset to the light power signal. A control system utilising this method is often called DClock or offsetlock. However, we show below that more elegant alternative methods for generating error signals exist.

The differentiation of the cavity power looks like a perfect error signal for holding the cavity on resonance. A signal proportional to such differentiation can be achieved with a modulationdemodulation technique.
8.10 The Pound–Drever–Hall length sensing scheme
8.11 Michelson length sensing
However, the main difference is that the measurement is made differentially by comparing two lengths. This allows one to separate a larger number of possible noise contributions, for example noise in the laser light source, such as amplitude or frequency noise. This is why the main instrument for gravitationalwave measurements is a Michelson interferometer. However, the resonant enhancement of light power can be added to the Michelson, for example, by using Fabry–Perot cavities within the Michelson as introduced in Sect. 7.2. This construction of new topologies by combining Michelson and Fabry–Perot interferometers has culminated in the dualrecycled Fabry–Perot–Michelson configuration that is the subject of the following section.
8.12 Advanced LIGO: an example of a complex interferometer
In this section we present a simplified overview of the dualrecycled Fabry–Perot–Michelson interferometer (DRFPMI) topology, as exemplified by the Advanced LIGO detectors (Harry and the LIGO Scientific Collaboration 2010). At this level of detail, the description applies equally to Advanced Virgo (Acernese 2015).
The final cavity is formed by placing a partially transmitting mirror between the output or ‘dark’ port of the Michelson and the detection optics (consisting of a photodetector, and perhaps some other components). This mirror recycles light that carries signal information to the photodetector, and is called the signal recycling mirror—see Sect. 7.3 for an introduction to this aspect of the interferometer configuration.
The idea of a bright fringe or port and dark fringe or port can be extended to form one of the central concepts in the control of complex interferometers. In the condition described, with the input or power recycling port maintained in the bright state, and the output or signal recycling port held in the dark state, there is a separation of lightfield components to one or other port according to their relative state in the interferometer arms. Here ‘component’ means light at a single frequency, i.e., a carrier or a sideband, and in a single optical mode (for a discussion of spatial modes, see Sect. 9). Such lightfield components, which have spatial and temporal coherence, can interfere. If they have the same phase in the two arms they interfere constructively at the bright port. If they have the opposite phase in the two arms they interfere constructively at the dark port. Note that this arises because of the choice of interference of the carrier light to create the bright and dark ports.
In the same way that the carrier light which has a common phase in the two arms appears at the bright port, any perturbation of the interferometer that is common to the two arms generates higher order modes and/or sidebands that have the same phase in the two arms and thus causes an effect on the optical field at the bright port. Examples of this would be inphase armlength changes, or the addition of the same amount of optical loss in the two arms. On the other hand, perturbations that are exactly outofphase between the two arms have an effect on the light field at the dark port. An example would be that gravitational waves produce differential phase modulation sidebands that have opposite phase in the two arms, and these interfere constructively at the output port.
The distinction between effects that are either inphase or have opposing phases is frequently important in the control of interferometers. As noted in the previous section for the case of the simple Michelson, it has become standard to consider the two physical degrees of freedom associated with the arms of an interferometer in logicalcombination as the common mode and the differential mode. For the same reasons, the bright port is also called the symmetric port and the dark port is called the antisymmetric port.
The advantage of the choice of common and differential modes may be seen in consideration of control loops to deal with laserfrequency fluctuations and to keep the interferometer locked at the dark fringe, to give but two examples. In a nearlysymmetrical interferometer a fluctuation of the frequency of the ingoing light will lead to a primarily commonmode effect, and it makes sense to stabilise the laser frequency with respect to the common mode of the two arms. Similarly the gravitational wave signal may be readout as part of the error signal of a control loop for the dark fringe. Such a loop should act on the differential mode, rather than on the length of one arm cavity, or the other.

CARM Commonmode arm length, CARM \(=L_x+L_y\). This corresponds to the average length of the arm cavities and is adjusted to keep both armcavities on resonance.

DARM Differential arm length, DARM \(=L_xL_y\). This corresponds to the difference in length of the two arm cavities and is used to maximise the constructive interference, at the output port, of sidebands resulting from differential armlength changes (this degree of freedom is therefore the source of the gravitational wave channel).

MICH Michelson arm length difference, MICH \(=l_xl_y\). MICH corresponds to the difference in length of the short arms of the Michelson, between the ITMs and the beam splitter, and determines the state of interference at the output port. In Advanced LIGO, the Michelson is operated close to the dark fringe.

PRCL Power recycling cavity length, PRCL \(=L_p +\frac{l_x+l_y}{2}\). The power recycling cavity is operated on resonance to maximise the power coupled into the central interferometer.

SRCL Signal recycling cavity length, SRCL \(=L_s+\frac{l_x+l_y}{2}\). This corresponds to the resonance condition of the signal recycling cavity. The operating point of SRCL depends on the mode of operation of the interferometer. It can be tuned for a particular frequency of gravitational wave or for broadband operation.
In the following sections we discuss, in general terms, how the error signals can be extracted from the optical system, combined and processed to provide signals representing the degrees of freedom to be controlled, and how the resulting signals can be fedback to force the optical system into the desired condition.
8.13 The Schnupp modulation scheme
In this and the following Sects. 8.14, 8.15 and 8.16, we introduce techniques for reading out signals from interferometers. These approaches complement and extend the Pound–Drever–Hall method for readout from Fabry–Perot cavities presented in Sect. 8.10.
Similar to the Fabry–Perot cavity, the Michelson interferometer is also often used to set an operating point where the optical gain of a direct light power detection is zero. This operating point, given by \(\varDelta L/\lambda =(2N+1)\cdot 0.25\) with N a nonnegative integer, is called dark fringe. This operating point has several advantages, the most important being the low (ideally zero) light power on the diode. Highly efficient and lownoise photodiodes usually use a small detector area and thus are typically not able to detect large power levels. By using the dark fringe operating point, the Michelson interferometer can be used as a null instrument or null measurement, which generally is a good method to reduce systematic errors (Saulson 1994).
Another option is to employ phase modulated light, similar to the Pound–Drever–Hall scheme described in Sect. 8.10. The optical layout of such a scheme is depicted in Fig. 70: an electrooptical modulator is used to apply a phase modulation at a fixed frequency, usually in the RF range, to the monochromatic laser light before it enters the interferometer. The photodiode signal from the interferometer output is then demodulated at the same frequency. This scheme allows one to operate the interferometer precisely on the dark fringe. The method originally proposed by Lise Schnupp is also sometimes referred to as frontal modulation.
The optical gain of a Michelson interferometer with Schnupp modulation is shown in Fig. 75 in Sect. 8.17.
8.14 Extending the Pound–Drever–Hall technique to more complicated optical systems
To recap Sect. 8.10, in the Pound–Drever–Hall (PDH) or RFreflection locking technique sinusoidal radiofrequency phase modulation is applied to the light to produce phase modulation sidebands. With phase modulation, higher order sidebands are imposed on the light, though the beats due to these are generally not employed in the normal implementation of the Pound–Drever–Hall technique. The light is then incident on the cavity that is to be controlled. The signal is obtained by detecting the reflected light on a photodetector which has a squarelaw response to the light amplitude, and analysing the resulting beats. The important beats are between the carrier and the first order RF sidebands. The electronic signal from the photodiode is filtered to pass the beats in a frequency range around the modulation frequency, and multiplied or ‘mixed’ with an electronic signal at the modulation frequency: an electronic local oscillator. The output from the mixer is then lowpass filtered to remove oscillations at harmonics of the modulation frequency. The useful signal is in one quadrature of the output from the photodetector at the modulation frequency. The phase of the local oscillator is chosen to select the required quadrature.
During the 1980s and 1990s, the question arose of how to obtain control signals for systems of coupled cavities and systems with combination of cavities in a Michelson interferometer. A good example is the powerrecycled Fabry–Perot–Michelson interferometer configuration as employed in initial LIGO and Virgo. In such a system, one possibility is to add pickoffs (lowreflectivity beam splitters) to remove some of the light reflected from each arm cavity for detection. This approach introduces a conflict between efficient power recycling that requires low loss, and the generation of a lownoise control signal, which argues for more highly reflecting beam splitters. It is of interest to identify other approaches that do not require additional detection ports. With this restriction, the problem becomes one of sensing all internal degrees of freedom by analysing light fields reflected from or transmitted by the entire interferometer. This has been accomplished for the dualrecycled Michelson topology of GEO 600 (Grote 2003), for the dualrecycled Fabry–Perot–Michelson configuration, e.g., Advanced LIGO—see “Appendix B” for a short description of the sensing scheme.
The scope of this section permits discussion only of design principles. It is worth noting, however, some practical matters that constrain the acceptable solutions. For example, the choice of modulation frequencies is usually restricted. One limit is the speed of photodetectors, and in particular quadrant photodetectors, for alignment sensing. This restriction makes the use of modulation frequencies above of order 100 MHz highly challenging. Another limitation results from the presence of modecleaning cavities in the path from the laser to the interferometer. Due to practical difficulties in the design of invacuum modulators, modulation is usually applied prior to the light passing the modecleaner. In this case the only available modulation frequencies are wholemultiples of the freespectralrange of the modecleaner. Note however that invacuum modulation is possible, and has been applied in GEO 600 allowing a relatively free choice of modulation frequency which is important for the method of locking the dual recycled system (Grote 2003).
The essence of the Pound–Drever–Hall method is that the light field is divided, according to frequency, into a component that suffers a phase change in response to variation of the target degree of freedom for measurement, and a component that does not. Therefore, a starting point in the discovery of alternatives is to create circumstances in which different light components, distinguished by frequency, resonate in different locations. Secondly, to produce a useful error signal, the output from the detection process should contain a dominant linear component in terms of its magnitude as a function of the target degree of freedom. Although it is desirable that the signal crosses zero at the operating point, it may be necessary and acceptable to subtract a (hopefully steady) offset to obtain the required result. These aspects are dealt with in turn.
First we consider how zerocrossing signals may be obtained from beats. The desired zerocrossing linear slope is achieved most directly if the components of the light are in quadrature, as is the case in Pound–Drever–Hall sensing: see Sects. 3.2 and 8.10. This ensures that the measurement depends on the relative phase of the optical field components, rather than their amplitudes. As an example of an alternative, quadrature is also achieved in the case of beating amplitude modulation sidebands against phase modulation sidebands.
In cases like this, where beats are obtained between various sidebands, rather than by beating with the carrier, the demodulated signal may either be obtained directly by mixing the electronic signal with a local oscillator at the beat frequency, or by employing double demodulation. A description of this process is shown in Sect. 4.2.
The condition for quadrature requires pairs of sidebands to be symmetrical so that they represent either pure phase modulation or pure amplitude modulation. In either of these cases, their resultant sum maintains a constant phase over time. If there is an imbalance of the amplitude of the lower and upper sidebands, the phase of the resultant must oscillate. This is equivalent to saying that the sidebands represent a mixture of amplitude and phase modulation, or equivalently, that there is an unbalanced singlesideband component. Extraction of useful error signals is still possible, but it is to be expected that there will be an offset in the demodulated signal, rather than a zerocrossing at the desired resonance condition.
Such sideband imbalance arises naturally in interferometers with detuned signal recycling, see Sect. 7.3. In these interferometers, the resonance of the signal recycling cavity is not centered on the carrier and so the response to upper and lower modulation sidebands can be expected to be asymmetrical. The beats produced on detection of the unbalanced sidebands may still produce a useful linear component, corresponding to the part of the amplitude that is in the appropriate quadrature.
As an example of obtaining signals from beats between sidebands, we cite the important method called thirdharmonic demodulation, introduced and explained in detail in Arai et al. (2000). In brief summary, this technique exploits the natural presence of higher harmonics in phase modulation for moderate to large modulation indices, e.g., 0.8 rad in the cited work. As noted above, such harmonics are passed by a modecleaner that is resonant at the first harmonic, and depending on the design of the interferometer, at least some can be expected to be resonant in the power recycling cavity (the odd members of the series in the scheme described by Arai et al. (2000). By combining this method of demodulation with the introduction of asymmetry in the geometry of the interferometer, as described in the following section, it is possible to construct a sensing system that provides well separated readout of the various degrees of freedom. In the cited scheme, neither the first or third order sidebands are strongly affected by the phase of the arm cavities (when the carrier is on resonance), and the method allows relatively independent control of the other degrees of freedom.
The third harmonic demodulation approach has been extended, with results proven in a series of investigations on prototype interferometers, including a 4 m interferometer with resonant sideband enhancement (Kawazoe et al. 2006), and experiments on the CalTech 40 m apparatus (Miyakawa et al. 2006) as part of the development of control systems for Advanced LIGO, in which thirdharmonic demodulation is employed—see “Appendix B”.
Next we return to the question of how sideband fields may be separated by breaking the symmetry of the interferometer. To reduce noise couplings, interferometers are usually designed and built to be as symmetrical as possible. For instance, an interferometer with perfectly matched arms is insensitive to the frequency of the light. In the design process it is usually assumed that the long arms of the interferometer must be kept as symmetrical as can be arranged in practice, but that controlled amounts of asymmetry can be introduced in the paths from the beam splitter to the arm cavities or recycling mirrors as appropriate to facilitate the design of sensing schemes.
The methods discussed in this section stem from the Schnupp modulation technique described in Sect. 8.13. In the unmodified Michelson interferometer, shown in Fig. 70, the asymmetry required to maximise the strength of the sidebands at the output, with modulation frequencies in the usual range (typically 10–100 MHz) is one quarter of the RF wavelength. The addition of powerrecycling lowers the required asymmetry because in this case optimum transfer of sideband power occurs when the asymmetry leads to an outcoupling of equal strength to the transmission of the power recycling mirror. This is in direct analogy with the transmission of light through an equalmirror Fabry–Perot cavity.
An example of this ‘classical’ application of Schnupp modulation is found in GEO 600. Here the approximately 1200 m (optical path) arms are adjusted to differ in length by about 10 cm, and this provides efficient transfer of \({\approx } 15\) MHz sidebands to the output port. The approach is described in Grote (2003).
The idea of Schnupp modulation influenced the development of Advanced LIGO see, for example, Strain et al. (2003). It had been decided that phase modulation would be applied prior to the invacuum modecleaner, thus constraining the modulation sidebands to fall in a harmonic series. A detailed description of these methods is beyond the scope of this review, but some important features are described below.
The objective is always to cause distinct modulation sidebands to resonate in different physical regions within the interferometer. In a dualrecycling Fabry–Perot–Michelson configuration, it is necessary to control the (inner) Michelson, the power recycling cavity and the signal recycling cavity. Controlling the arm cavities may be achieved by beating the carrier with suitable sidebands, the hard part of the problem is to remove the influence of arm cavities on signals for the other degrees of freedom. For control of the signal recycling cavity, for example, at least one sideband must be directed towards the signal recycling mirror. This can be accomplished by choosing a difference in the lengths of the two arms of the Michelson to arrange that one sideband is on a bright fringe, and therefore strongly directed towards the signal recycling mirror. For further detail of this aspect of interferometer sensing, see Strain et al. (2003) and “Appendix B”.
One last design ingredient is that, in a ‘closed’ configuration like the dualrecycling Fabry–Perot–Michelson, light travelling back from one of the arms ‘sees’ another (effective) Michelson interferometer formed by the beam splitter and the two recycling mirrors. A variation of the Schnupp technique can also be applied in that case, by adjusting the optical paths from the beam splitter to the recycling mirrors to be unequal. This provides further control over sideband resonance conditions in the various parts of the interferometer.
It can be appreciated that the design problem rapidly becomes too complex for a full description in this review, but all of the main principles are included, and numerical calculation allows these principles to be developed into a complete sensing scheme.
8.15 Complementary techniques: internal modulation, external modulation and dithering
In the technique of external modulation (Man et al. 1990), a phase modulated field is derived from the common mode light within the interferometer as shown in Fig. 72. Light pickedoff from a convenient location, usually close to the beam splitter or even at its imperfectly antireflection coated rear surface, is phase modulated and recombined with the main output field, by means of a second beam splitter. This Mach–Zehnder interferometer geometry is distinguished from general heterodyne methods in that, when power recycling is present, the modulated field is obtained from within the power recycling cavity, where the light field may be more stable than the ingoing light, due to the passive filtering provided by the power recycling cavity. External modulation adds significant complexity to the output optics of an interferometer, and is disfavoured in advanced interferometers where the application of squeezed light is considered.
Another approach to the generation of suitable signals is dithering, this is, effectively, the application of phase modulation sidebands by modulating parameters of the system, usually the positions or angles of mirrors, rather than modulating the ingoing light. In principle, dithering could be applied at distinct frequencies to as many components of the system as there are degrees of freedom requiring to be controlled.
There are practical limitations that restrict the application of dithering, and it is normally applied to lock auxiliary degrees of freedom where the signal to noise requirements are less severe. The limitations arise because dithering is commonly applied by mechanical means, resulting in restricted actuation force (to avoid either causing damage or adding noise due from powerful actuators). This imposes a limit to the product of imposed displacement and (dither) frequencysquared, resulting in typical dither frequencies that do not exceed a few kHz. Dithering is, therefore, typically employed to monitor and control slowly varying aspects of the interferometer. A relatively recent application of dithering is in locking an output modecleaner for use with DC readout. This is discussed in the following section and in Ward et al. (2008).
8.16 Circumstances in which offset locking is favoured over modulationbased techniques
As mentioned in Sects. 5.4 and 8.13, the idea of offsetlocking of Michelson interferometers to produce a zerocrossing error signal for the differential displacement arises naturally. There are, however, disadvantages associated with this method of readout, and it has only become favoured over heterodyne methods due to particular circumstances that associated with recently developed interferometer designs, as explained below.
In a simple Michelson interferometer, the steepest gradient in the length to intensity transfer function occurs halfwayup the fringe. However, operating in this condition has two disadvantages: half of the light is directed back towards the laser and sensitivity to laser power fluctuations is maximised. The latter problem can be ameliorated by symmetrising the readout through the addition a photodetector for the reflected light. On subtracting the signals from the detectors at the two ports of the interferometer, the displacement signals add while laser power fluctuations cancel, to the extent that balance is achieved. In this case, however, all the light is detected and there is no possibility to take advantage of lowloss optics by adding power recycling.
A further problem when a simple Michelson is offsetlocked is that the optical local oscillator for the measurement is a relatively noisy component of the light field. Indeed this last concern led to the choice of radio frequency modulation in the Pound–Drever–Hall and other techniques described above. In those techniques modulation frequencies are chosen to fall at Fourier frequencies where technical laser noise is less than shot noise in the detected light power. This is typically true above about 10 MHz for detection of the tens of mW of light from the argonion or Nd:YAG lasers typically employed.
During the design of Enhanced and Advanced LIGO, Advanced Virgo and GEOHF, three motivations emerged to prompt reconsideration of offsetlocking methods. As noted in Sect. 3.1 it had been shown that modulation generally worsens shotnoise limited performance, and these arguments were extended to show that it is impractical to benefit from squeezed light in modulation based readout (Buonanno et al. 2003). Secondly, it was realised that, for the interferometer to achieve the planned sensitivity, the light within the power recycling cavity in a system such as Advanced LIGO, must be more stable than the best available RF oscillators, at Fourier frequencies of interest, and so the arguments against employing this light for signal readout scheme become moot. Finally, whether the detected light amplitude is shot noise limited or not depends on the power that is detected, because the shot noise in the detection of small light power can make technical noise unimportant. It was realised that, by adding a modecleaner on the output of the interferometer, to pass the signal, which would predominantly be in the \(\mathrm {TEM}_{00}\) mode of the arms, but exclude other light resulting from imperfect interference, mainly in other modes, it would suffice to detect relatively low light power, at which level the measurement should be shot noise limited. See Sect. 10 for a description of modes resulting from imperfect interference.
In modern detectors this scheme, where signals are read out directly in the baseband i.e., near zero frequency or ‘DC’, is often called DC readout. As an example of its application, the details of the DC readout scheme developed for Advanced LIGO are described in Ward et al. (2008). The technique has also been tested on GEO 600, where the method has been shown to be compatible with squeezing (LIGO Scientific Collaboration 2011).
It should be noted that offset locking applies to the control of one length degree of freedom per interferometer, and the remaining degrees of freedom are typically sensed using the modulation methods described above.
8.17 Finesse examples
8.17.1 Michelson modulation
This example demonstrates how a macroscopic arm length difference can cause different ‘dark fringe’ tuning for injected fields with different frequencies. In this case, some of the 10 MHz modulation sidebands are transmitted when the interferometer is tuned to a dark fringe for the carrier light. This effect can be used to separate light fields of different frequencies. It is also the cause for transmission of laser noise (especially frequency noise) into the Michelson output port when the interferometer is not perfectly symmetric (Fig. 73).
8.17.2 Cavity power and slope
Figure 74 (same as Fig. 65) shows a plot of the analytical functions describing the power inside a cavity and its differentiation by the cavity tuning. This example recreates the plot using a numerical model in Finesse.
8.17.3 Michelson with Schnupp modulation
Figure 75 shows the demodulated photodiode signal of a Michelson interferometer with Schnupp modulation, as well as its differentiation, the latter being the optical gain of the system. Comparing this figure to Fig. 68, it can be seen that with Schnupp modulation, the optical gain at the dark fringe operating points is maximised and a suitable error signal for these points is obtained.
9 Beam shapes: beyond the plane wave approximation
In previous sections we have introduced a notation for describing the onaxis properties of electric fields. Specifically, we have described the electric fields along an optical axis as functions of frequency (or time) and the location z. Models of optical systems may often use this approach for a basic analysis even though the respective experiments will always include fields with distinct offaxis beam shapes. A more detailed description of such optical systems needs to take the geometrical shape of the light field into account. One method of treating the transverse beam geometry is to describe the spatial properties as a sum of ‘spatial components’ or ‘spatial modes’ so that the electric field can be written as a sum of the different frequency components and of the different spatial modes. Of course, the concept of modes is directly related to the use of a sort of oscillator, in this case the optical cavity. Most of the work presented here is based on the research on laser resonators reviewed originally by Kogelnik and Li (1966). Siegman has written a very interesting historic review of the development of Gaussian optics (Siegman 2000a, b) and we use whenever possible the same notation as used in his textbook ‘Lasers’ (Siegman 1986).
9.1 A typical laser beam: the fundamental Gaussian mode
The beam produced from a real laser is not a plane wave, but has some intensity distribution. This is typically a roughly circular beam with a peak brightness near the centre. The intensity pattern of a beam generated by an ideal laser based on a stable optical cavity with spherical mirrors would resemble a Gaussian beam. Figure 76 shows the intensity and amplitude distribution of a typical Gaussian beam, often characterised by the beam spot size, w, the radius within which \({\sim }86\,\%\) (\(\frac{1}{e^2}\)) of the light power is contained. As the beam propagates the beam spot size changes slowly, which produces a narrow beam of light with a small diffraction angle.
9.2 Describing beam distortions with higherorder modes
In an ideal interferometer the laser beam would be a perfect Gaussian beam, with wavefronts exactly matched to the shape of the mirrors. However, in a real interferometer mismatches between the beam and mirror curvatures, misalignments from the optical axis and deviations of the mirror surfaces from a perfect sphere all contribute to distort the beam from the ideal Gaussian beam.
Small distortions of the fundamental beam can be described by the addition of higherorder modes. Higherorder modes have the same basic properties of the fundamental Gaussian beam, with two exceptions: higherorder modes have different intensity patterns from the simple spot of the fundamental mode and modes of different order pick up an extra phase upon propagation (the Gouy phase, see Sect. 9.10).
One simple example is a misaligned beam, whose centre has been shifted from the optical axis. This can be described by the addition of an order ‘1’ Hermite–Gauss mode, HG\(_{10}\) (Sect. 9.7), as illustrated in the left panel of Fig. 78. Such a distortion is a first order effect and, as long as the misalignment is small, can be described with just this one additional mode. In a similar way the second order effect such as a mismatch in beam size can be described by the addition of a single order ‘2’ mode, in this case the Laguerre–Gauss mode LG\(_{10}\) (Sect. 9.11). A mismatch in beam size is illustrated in the right panel of Fig. 78.
9.3 The paraxial approximation
9.4 Transverse electromagnetic modes
9.5 Properties of Gaussian beams
9.6 Astigmatic beams: the tangential and sagittal plane
If the interferometer is confined to a plane (here the x–z plane), it is convenient to use projections of the threedimensional description into two planes (Rigrod 1965): the tangential plane, defined as the x–z plane and the sagittal plane as given by y and z.
Remember that these Hermite–Gauss modes form a base system. This means one can use the separation into sagittal and tangential planes even if the actual optical system does not show this special type of symmetry. This separation is very useful in simplifying the mathematics. In the following, the term beam parameter generally refers to a simple case where \(w_{0,x}=w_{0,y}\) and \(z_{0,x}=z_{0,y}\) but all the results can also be applied directly to a pair of parameters.
9.7 Higherorder Hermite–Gauss modes
9.8 The Gaussian beam parameter
9.9 Properties of higherorder Hermite–Gauss modes

The size of the intensity profile of any sum of Hermite–Gauss modes depends on z while its shape remains constant over propagation along the optical axis.

The phase distribution of Hermite–Gauss modes shows the curvature (or radius of curvature) of the beam. The curvature depends on z but is equal for all higherorder modes.
9.10 Gouy phase
9.11 Laguerre–Gauss modes
9.12 Tracing a Gaussian beam through an optical system
Whenever Gauss modes are used to analyse an optical system, the Gaussian beam parameters (or equivalent waist sizes and locations) must be defined for each location at which field amplitudes are to be computed (or at which coupling equations are to be defined). In our experience the quality of a computation or simulation and the correctness of the results depend critically on the choice of these beam parameters. One might argue that the choice of a basis should not alter the result. This is correct, but there is a practical limitation: the number of modes having nonnegligible power might become very large if the beam parameters are not optimised, so that in practice a good set of beam parameters is usually required.
In general, the Gaussian beam parameter of a mode is changed at every optical surface in a welldefined way (see Sect. 9.13). Thus, a possible method of finding reasonable beam parameters for every location in the interferometer is to first set only some specific beam parameters at selected locations and then to derive the remaining beam parameters from these initial ones: usually it is sensible to assume that the beam at the laser source can be properly described by the (hopefully known) beam parameter of the laser’s output mode. In addition, in most stable cavities the light fields should be described by using the respective cavity eigenmodes. Then, the remaining beam parameters can be computed by tracing the beam through the optical system. ‘Trace’ in this context means that a beam starting at a location with an alreadyknown beam parameter is propagated mathematically through the optical system. At every optical element along the path the beam parameter is transformed according to the ABCD matrix of the element (see below).
9.13 ABCD matrices
Transmission through a mirror
A mirror in this context is a single, partlyreflecting surface with an angle of incidence of 90°. The transmission is described by (Fig. 87)
with \(R_{\mathrm {C}}\) being the radius of curvature of the spherical surface. The sign of the radius is defined such that \(R_{\mathrm {C}}\) is negative if the centre of the sphere is located in the direction of propagation. The curvature shown above (in Fig. 87), for example, is described by a positive radius. The matrix for the transmission in the opposite direction of propagation is identical.
Reflection at a mirror
The reflection at the back surface can be described by the same type of matrix by setting \(C=2n_2/R_{\mathrm {C}}\).
Transmission through a beam splitter
where f is the focal length. The matrix for the opposite direction of propagation is identical. Here it is assumed that the thin lens is surrounded by ‘spaces’ with index of refraction \(n=1\).
Transmission through a free space
The matrix for the opposite direction of propagation is identical.
9.14 Computing a cavity eigenmode and stability
A cavity eigenmode is defined as the optical field whose spatial properties are such that the field after one roundtrip through the cavity will be exactly the same as the injected field. In the case of resonators with spherical mirrors, the eigenmode will be a Gaussian mode, defined by the Gaussian beam parameter \(q_\mathrm{{cav}}\). For a generic cavity (an arbitrary number of spherical mirrors or lenses) a roundtrip ABCD matrix \(M_\mathrm{{rt}}\) can be defined and used to compute the cavity’s eigenmode. Chapter 21 of Siegman (1986) provides a comprehensive description of different optical resonators including a derivation and discussion of stability criteria. Here we provide a brief introduction focussing on the specific case of closed and stable resonators with spherical mirrors.
9.15 Roundtrip Gouy phase and higherordermode separation
As discussed in Sect. 9.10, as a higher order optical mode propagates it accumulates an additional phase, the Gouy phase, proportional to its mode order. To determine how such a mode resonantes within an optical cavity the accumulated Gouy phase on one roundtrip through the cavity must be included. The roundtrip Gouy phase will determine which order of optical modes are resonant within a cavity. As the resonance condition of a mode is dependent on its order, this allows an optical setup to select particular orders of optical modes from an incident field. This behaviour is the basis of modecleaner cavities; such as those used for the input and output light of gravitationalwave detectors.
9.16 Coupling of higherordermodes
Now that we are able to compute the eigenmode of a particular cavity, what happens if a beam with a slightly different eigenmode is injected into it? The aim of this section is to outline the problem. In reality producing a perfect Gaussian laser which matches exactly the eigenmode of a cavity is essentially impossible, there will always be a minor difference. However we are still able to inject lasers into a cavity and produce a resonance. This is because as long as the eigenmode of the incoming laser is nearly the same, the majority of the laser light will ‘fit’ into the cavity and resonate, the rest will be reflected from it.

If the optical axes of the beam and the cavity do not overlap perfectly, the setup is called misaligned,

If the beam size or shape at cavity input does not match the beam shape and size of the (resonant) fundamental eigenmode (\(q_1(z_{\mathrm {cav}})\ne q_2(z_{\mathrm {cav}})\)), the beam is then not modematched to the second cavity, i.e., there is a mode mismatch.
To compute the amount of coupling the beam must be projected into the base system of the cavity or beam segment it is being injected into. This is always possible, provided that the paraxial approximation holds, because each set of Hermite–Gauss modes, defined by the beam parameter at a position z, forms a complete set. Such a change of the basis system results in a different distribution of light power in the new Hermite–Gauss modes and can be expressed by coupling coefficients that yield the change in the light amplitude and phase with respect to mode number.
9.17 Finesse examples
9.17.1 Beam parameter tracing
This example illustrates a possible use of the beam parameter detector ‘bp’: the beam radius of the laser beam is plotted as a function of distance to the laser. For this simulation, the interferometer matrix does not need to be solved. ‘bp’ merely returns the results from the beam tracing algorithm of Finesse (Fig. 95).
9.17.2 Telescope and Gouy phase
This example shows the fine tuning of a telescope. The optical setup is similar to the optical layout on the Virgo Northend detection bench, resembling the telescope for the beam transmitted by the end mirror of one arm. The telescope consists of four sequential lenses with the purpose of reducing the beam size and provide a user defined Gouy phase for a split photo detector which is used for the alignment sensing system (Fig. 96).
9.17.3 LG33 mode
Finesse uses the Hermite–Gauss modes as a base system for describing the spatial properties of laser beams. However, Laguerre–Gauss modes can be created using the coefficients given in Eq. (9.43). This example demonstrates this and the use of a ‘beam’ detector to plot amplitude and phase of a beam cross section (Fig. 97).
10 Imperfect interferometers
Imperfections in a Michelson interferometer can refer to any of the differences between a real interferometer and the perfect design. These include, but are not limited to: deviations of the optical properties of the mirrors from the design; the limits of longitudinal and alignment control of the mirrors; additional noise sources not included in our models (i.e., electronic noise); and effects which distort the shape of the beam. To estimate the impact of such imperfections on the Michelson’s performance is complicated and requires substantial modelling. The greatest impact on the sensitivity arises from asymmetries between the two arms. For accurate differential measurements, such as those made in gravitational wave interferometers, the mirrors are very carefully manufactured to make the arms as similar as possible. Differences between the two arms, for example, imbalances in the finesse of the two arm cavities, will couple extra light into the antisymmetric port of the interferometer where it adds additional noise to the detection photodiode.
It is important to understand how imperfections in an interferometer affect the resonating beams and impact the sensitivity of the instrument. For this we need accurate models which can simulate complex interferometers in the presence of such imperfections. This is crucial for the design of interferometers, such as gravitationalwave detectors, and the commissioning process, in which deviations of the interferometer behaviour from the expected design must be diagnosed. In this review we will consider imperfections in the form of distortions of the beam and we discuss these effects for gravitational wave interferometers; firstly in terms of the behaviour of distorted optics and how this effects the performance of different optical configurations; and secondly in terms of solutions to these distortion problems and implications for the design process.
10.1 Spatial modes in optical cavities
In the previous chapter the idea of representing distortions of a beam as higherorder Gaussian modes was introduced. Here we use this description to investigate the behaviour of interferometers with distorted beams.
This property of an optical cavity to act as a filter of spatial modes is utilised in gravitationalwave detectors. Firstly, the input laser beam is ‘cleaned’ of spatial modes by passing through an input mode cleaner, an optical cavity carefully designed to transmit the fundamental mode and filter out most higherorder modes before the beam enters the main interferometer. Within the multiple cavities of the central interferometer careful design can take advantage of these resonant properties to suppress distortions of the beam. Finally, the output beam containing the gravitational wave signal is cleaned of spatial modes and control sidebands using an output mode cleaner. These design features are discussed in grater detail in Sect. 10.7.
10.2 Cavity alignment in the mode picture
Summary of the parameters defining the Gaussian eigenmode of an Advanced LIGO arm cavity
\(R_{C,1}\) (m)  \(R_{C,2}\) (m)  \(w_0\) (cm)  \(w_1\) (cm)  \(w_2\) (cm)  \(z_1\) (m)  \(z_2\)  \(z_R\) (m)  \(\frac{\varPsi }{2}\) (\(^{\circ }\)) 

1934  2245  1.2  5.3  6.2  \(\)1834  2160  425  24.3 
The consequence of a misalignment of an optical cavity is the creation of first order modes. If we chose to describe the problem using the Gaussian beam parameters and axis of the incoming beam as our basis, then the incoming beam is a pure fundamental beam and the first order modes are created when the light enters (and leaves) the cavity. Alternatively we can use the cavity eigenmode and cavity axis as our basis. In this case the higherorder modes are already present in the incoming beam. Either of these approaches is valid for such a simple distortion.
In more realistic cases the circulating field in a cavity is not completely described by a fundamental Gaussian beam, due to deviations of real mirrors from an ideal sphere. This can be modelled using the closest Gaussian eigenmode (from now on referred to as the eigenmode of the cavity) superimposed with higherorder modes. One can say that the higherorder modes are created when the fundamental mode interacts with the distorted mirrors.
In Fig. 100 the effects of misalignment on intracavity power is illustrated. In this example an Advanced LIGO cavity has been modelled with a misaligned end mirror. The circulating power exhibits several peaks, corresponding to the resonances of the different higherorder modes created due to the misalignment. Most of the power remains in the fundamental mode (peak at \({\sim } 0^{\circ }\)). The misalignment of the cavity has induced higherorder modes, mostly the first order HG\(_{10}\) mode, whose resonance is observed at \({\sim } 24^{\circ }\). In this case the extent of the misalignment also results in the creation of the order 2 mode HG\(_{20}\), resonant at \({\sim } 50^{\circ }\). During operation, where the cavity is on resonance for the HG\(_{00}\) mode, the relative power in higherorder modes is suppressed. There will still be some higherorder modes in the beam at this tuning, which degrade the purity of the beam transmitted and reflected from the cavity.
10.3 Mode mismatch
10.4 Spatial defects
Misalignment and mode mismatch are the lowest order distortions of the beam and are well described analytically. These low order distortions are carefully controlled in an interferometer, using alignment control schemes and using lenses and curved optics to mode match beams between different cavities. Higherorder distortions produced from more complex processes, i.e., interaction with distorted mirror surfaces or finite sized optics, cannot currently be controlled. There are many different spatial defects which are likely to be present in real interferometers.
For the design and commission of real detectors we want to represent these more arbitrary defects, in particular the deviation of the mirror surfaces from a perfect sphere. In the case of interferometer design this will help set requirements on the polishing and coating of the mirrors. For the commissioning process this will aide in identifying the output beam shape and other effects associated with distortions of the beam. In this article we will focus on mirror surface errors and thermal effects. The detailed mathematics of these higherorder effects are discussed in Sect. 11. For now we just consider that higherorder modes are created when beams are distorted. The advantage of describing distortions of the beam as higherorder modes is that these spatial modes are easy to trace through the interferometer, to predict the behaviour of a distorted interferometer.
10.5 Operating cavities at high power
 1.
A thermal lens forms within the mirror due to the temperature dependent nature of the index of refraction of the substrate material (fused silica). This distortion can be described mostly as a spherical lens, with some higherorder components (Hello and Vinet 1990; Vinet and the Virgo Collaboration 2001).
 2.
The mirror expands thermally, with the expansion greatest where the mirror is hottest, giving a nonuniform expansion over the mirror surface and effectively distorting the surface from the cold case. This thermal distortion is primarily a change in the radius of curvature of the mirror (Vinet and the Virgo Collaboration 2001; Hello and Vinet 1990).
The design geometric and optical parameters of an Advanced LIGO arm cavity
\(R_C\) (m)  Transmittance  Loss (ppm)  

ITM  1934  1.4 %  37.5 
ETM  2245  5 ppm  37.5 
\(f_\mathrm{{ITM}}\) (km)  \(\delta R_{C,\mathrm {ITM}}\) (km)  \(\delta R_{C,\mathrm {ETM}}\) (km)  

Cold case (0 W)  \(\infty \)  \(\infty \)  \(\infty \) 
Low power (12.5 W)  50  1100  1600 
High power (125 W)  5  110  160 
Beam parameter, q, beam size, w, wavefront curvature, \(R_C\) and distance from the wasit, z of 3 different Gaussian beams, the eigenmode of an advanced LIGO cavity during cold operation (0 W input power) and during hot operation (125 W input power) and the input beam during hot operation
q (m)  w (cm)  \(R_C\) (m)  z (m)  

Cold eigenmode  \(1834.2 + 427.8\mathrm {i}\,\)  5.30  \(1934\)  \(1834.2\) 
Hot eigenmode  \(1832.7 + 499.0\mathrm {i}\,\)  4.95  \(1968.6\)  \(1832.7\) 
Hot input beam  \(1356.2 + 228.1\mathrm {i}\,\)  5.30  \(1394.6\)  \(1356.2\) 
In reality the hot eigenmode and input beam will develop over time as the mirrors heat up and the aberrations evolve. This takes us from the cold case, where the incoming beam is well matched to the cavity, to the hot case where there is a strong mode mismatch. This will have a strong impact on the power injected into the cavity. During operation the arm cavities are ‘locked’ to the resonance of the fundamental cavity mode. In this state the components of the injected beam which do not overlap with the cavity eigenmode will be reflected. As was discussed in Sect. 10.3 these will be primarily order 2 modes.
10.6 The Michelson: differential imperfections

Differences in the loss and finesse of each arm result in different carrier amplitudes in each arm, degrading the interference of the two beams at the dark fringe and leading to additional carrier light at the output port.

Different resonant or interference conditions for the carrier and control sidebands. Advanced gravitationalwave detectors such as Advanced LIGO employ a Schnupp modulation scheme, see Sect. 8.13, to control the interferometer, where by an asymmetric length applied to the short Michelson arms ensures that the dark fringe of the carrier is not equivalent to the dark fringe of the control sidebands, resulting in a proportion of these radio frequency sideband fields at the dark port.

Spatial differences between the beams coming from each arm. An imperfect overlap of the spatial distribution of these beams will degrade their interference and cause light in higherorder modes to leak into the dark port.
An example of the impact of higher order modes on contrast defect in a dual recycled Michelson have been observed at GEO 600, the German–British gravitationalwave detector in Hannover (Danzmann et al. 1994; Abbott 2004). Unlike other gravitationalwave detectors GEO 600 does not include arm cavities, but instead has folded arms to increase the effective arm length of the Michelson. As described in Lück et al. (2004), during the operation of GEO 600 it was discovered that a difference in the radii of curvature of the folding mirrors in the x and y arms (687 m in the x arm, 666 m in the y arm) was causing a significant difference in the wavefront curvatures of the beams returning from each arm. This mismatch between the two beams resulted in a significant amount of power at the interferometer dark fringe: the degrade in overlap between the two beams reduced the effective destructive interference and consequently the output beam on the dark fringe was dominated by the order 2 mode typical of a mode mismatch, LG\(_{10}\). The resulting loss of power into the anti symmetric port increased the effective loss in the power recycling cavity, limiting the power build up to \({\sim } 200\) W/W, a significant reduction from the 300 W/W predicted for this configuration. The mismatch of the two arms also had a negative impact on the longitudinal error signal of the Michelson, reducing the magnitude of the error signal and increasing the susceptibility to misalignments.
To reduce this mismatch and recover the power recycling gain the curvature of one of the folding mirrors required correcting, to match that of the other arm. In this case the thermal properties of the mirrors were exploited, namely the dependence of the radius of curvature of the mirrors to a temperature gradient, as was discussed in Sect. 10.5. In advanced detectors the temperature gradient which develops from high powered beams incident on the mirrors is an unwanted effect which results in the distortions of the mirror surfaces (primarily a change in curvature) and thermal lensing. In GEO 600 this thermal behaviour was manipulated to alter the curvature of the folding mirror in the East arm (x arm) using a ring heater placed behind the mirror substrate to produce an appropriate temperature gradient in the East mirror (Lück et al. 2004). The extent of the change in mirror curvature is dependent on the ring heater power, which can be gradually altered to find the power which corresponds to the optimum curvature (i.e., \({\sim } 666\) m to match the North mirror).
To fully diagnose and understand the nature of this problem these measurements were compared with Finesse simulations of GEO 600. In Fig. 106 the power circulating in the power recycling cavity is plotted against the power at the dark fringe, showing both simulation and experimental results. Two different experimental results are shown. The first result has the optimum curvature compensation applied with the powers measured as the interferometer passes through the dark fringe (solid red trace). The second result is the case where the interferometer is locked to the dark fringe and the curvature compensation is varied (black markers). The experimental and simulation results are sufficiently similar to suggest that our understanding of this problem is correct and that the low intra cavity power/high contrast defect is dominated by a differential mode mismatch. The slight differences between the experimental results and the simulation observed at high intra cavity power can be explained by the limits of the model: no astigmatic or higherorder spatial effects were included in this model and hence the model represents a more simplified system than reality. The experiment and simulation are well matched for low intracavity power where the effects of the spherical mode mismatch dominate.
This experience at GEO 600 illustrates the need to have well matched arms. This can be in terms of mode matching, as shown here, or in terms of mirror surface distortions and other defects. While loworder aberrations such as misalignment and mode mismatch can be corrected during operation by means of additional control systems, higherorder effects are typically not actively controlled. It is crucial that the impact of higherorder modes is considered during the design of an interferometer to avoid a large buildup of unwanted modes in the detector.
10.7 Advanced LIGO: implications for design and commissioning

Misalignment Any tilt or lateral shift between the beam axis and a cavity axis, or between the axes of the multiple interdependent cavities in advanced interferometers, will produce higherorder modes, for small misalignments these are dominated by firstorder modes. In modern gravitationalwave detectors these effects are carefully controlled using alignment systems to maintain consistent optical axes within the interferometer and avoid a large amount of power in first order modes on the detection photodiode.

Mismatch Secondorder modes arise from a mismatch in beam size or wavefront curvature between the cavity eigenmode and incoming beam, or the multiple cavity eigenmodes in complex interferometers. In gravitational wave detectors mismatches are the result of secondorder mirror aberrations from the manufacturing process or environmental processes such as thermal lensing. In Advanced interferometers thermal compensation systems will be in place to correct the curvature of the arm cavity mirrors, to compensate any thermal lensing and to avoid large mode mismatches.

Surface distortions Higherorder distortions of the beam are generally the result of higherorder mirror distortions on the highly reflective mirror surfaces. These defects can arise during the manufacturing process (socalled mirror figure error) or through environmental processes like the thermal distortion of the mirror surfaces. Whilst first and second order distortions of the beam can be corrected it is more difficult to actively correct modes of a higher order. A crucial part of the design process is to determine the tolerances and requirements for the polishing and coating of the interferometer mirrors, to ensure a low higherorder modes content. This is discussed in more detail in Sect. 11.

Apertures Higherorder modes are also generated when the circulating beams encounter the effective aperture caused by the finite size of optical components. The ‘clipping’ of the beam results in a sharp cutoff, equivalent to the addition of high order modes. The design of a well behaved optical setup will ensure the size of the optics, compared to the beam, is sufficiently large such that these higherorder effects are small and we can simple consider the effect of the aperture as a small power loss.
The input mode cleaner
Summary of key design parameters of an Advanced LIGO input mode cleaner (Martin et al. 2013)
Parameter  Value 

Length  16.473 m 
Free spectral range  9, 099, 471 Hz 
Input/end mirror \(R_C\)  \({>}10{,}000\) m 
Input/end mirror T  0.6 % 
Input/end mirror R  99.4 % 
Input/end mirror \(\alpha \)  \(44.59^{\circ }\) 
Curved mirror \(R_C\)  \(27.24 \pm 0.14\) m 
Curved mirror R  \({>}0.9999\) 
Curved mirror \(\alpha \)  \(0.82^{\circ }\) 
Finesse  522 
Recycling cavities
For the design stage we first assume perfect matching of the arm cavities. We can then consider each recycling cavity acting with the arms as a simple coupled cavity. Two examples of a possible coupled cavity setup are shown in Fig. 108. The eigenmode of the arm cavities is selected to produce large beams at the ITM (5.3 cm) and ETM (6.2 cm) to reduce thermal noise, with slightly smaller beams at the ITM as the thermal noise is lower here (fewer coating layers) and to prevent scattering into the recycling cavities. The curvatures are also carefully selected for a specific Gouy phase to avoid higherorder modes easily ringing up in the arms: \(R_C=1934\) m (ITM) and \(R_C=2245\) m (ETM). The beam parameter of the arms is therefore a fixed parameter, and the properties of the recycling cavities should be chosen to mode match the recycling cavity to the arms.
The simplest design for the recycling cavities uses a single mirror coupled with the arm cavities, as shown in the left diagram of Fig. 108, where the curvature of the recycling mirror is matched to the wavefront curvature of the arm cavity eigenmode. This was the layout chosen for power recycling in Initial LIGO. In this layout the eigenmode of the arm and recycling cavity can be matched. However, there is another consideration for the design of the recycling cavities: the separation of higherorder mode resonances. This is determined by the Gouy phase accumulated in the recycling cavity (between RM and ITM). In Fig. 109, the Gouy phase of the eigenmode for the Advanced LIGO arms is shown at different positions along the optical axis. The ITM and ETM are both far from the waist but the difference in Gouy phase (155.7\(^{\circ }\), equivalent to \(24.3^{\circ }\)) is far outside the linewidth of the cavity. With a single recycling mirror the only possible positions do not allow for a large change in Gouy phase, as the ITM is already in the far field. In reality there are additional limitations on the position of the recycling mirror, such as the physical location of the vacuum chambers.
In Advanced LIGO, the issue of unstable recycling cavities becomes more complex due to larger beam sizes, large thermal lensing effects and the addition of signal recycling. Unlike the power recycling cavity the signal recycling cavity coupled with the arm cavities will operate on antiresonance for the carrier, for resonant sideband extraction. Any HOMs will be nearly resonant in an SRC designed with a single recycling mirror. To avoid these problems in Advanced LIGO an alternative recycling geometry was designed. This is shown in the right diagram of Fig. 108, adding 2 folding mirrors to the recycling cavities to alter the beam parameter and gain significant Gouy phase between the ITM and recycling mirror. The curvatures of these mirrors are carefully chosen to gain this required Gouy phase, whilst maintaining a mode matched system. The design parameters for the power and signal recycling cavities for Advanced LIGO are summarised in “Appendix B”.
Thermal distortions
The plots shown in Fig. 111 show that the mode matching between the recycling cavities and arm cavities is relatively independent of the expected thermal lensing. Whilst the eigenmode of the arm cavity is fixed, the recycling cavity eigenmode is affected by the thermal lens. The recycling eigenmode curvature is fixed at the reflective ITM surface, and the beam size at this point only varies a small amount, maintaining the mode matching between the arm and recycling cavity. However, the effect of the lens on the mode parameters is exaggerated during the large divergence between the recycling mirrors R2 and R3 (see Fig. 108) and this has a large impact on the beam size at the recycling mirror, and hence the mode matching between the input beam and recycling cavity is significantly degraded. As we saw previously for a single cavity, during high power operation the power coupled into the interferometer will be significantly reduced.
In Advanced LIGO thermal compensation systems (TCS) will be employed at high power, not only to ensure a large power buildup within the interferometer but to balance the lensing and eigenmodes of the two arms to prevent a high contrast defect (Willems 2009). The first is a ringheater positioned near the antireflective surface of each test mass (Arain et al. 2012). These are used to heat the outer edge of the mirror to produce a curvature in the opposite direction to that from heating by the beam. The ring heater also corrects some of the thermal lens in the ITM substrate. An additional system is required to complete the correction of the thermal lens. This involves a compensation plate, placed in front of the ITMs, made of the same material (fused silica). A heating pattern is projected onto this plate via a CO\(_2\) laser. This pattern is designed to heat the compensation plate in such a way as to correct any thermal lensing in the ITM (Brooks et al. 2012).
The output mode cleaner
Even with state of the art optics, alignment systems to correct any misalignments and thermal compensation systems to correct for differential mismatches some light at the Michelson antisymmetric port will be in higherorder modes. There will also be some power in the control sidebands exiting the interferometer, as the dark fringe for the carrier is not the dark fringe for the sidebands due to the applied Schnupp asymmetry (see Sect. 8.13 and “Advanced LIGO configuration” section in “Appendix B”). The only fields which should be present on the detection photodiode are the gravitational wave signal sidebands and the local oscillator field, in the case of Advanced LIGO this is the leaked carrier light for DC readout (see Sect. 6.2). If the power in the higherorder modes and control sidebands is sufficiently low they can be effectively stripped from the beam using an output mode cleaner (OMC), an optical cavity between the Michelson interferometer and the main photodiode.
Summary of key design parameters for the Advanced LIGO output mode cleaner (Arai et al. 2013)
Parameter  Value 

Length  1.132 m 
Free spectral range  264.8 MHz 
Input/output mirror \(R_C\)  \(\infty \) 
High reflectors \(R_C\)  2.5 m 
Input/output mirror T  8300 ppm 
High reflectors T  50 ppm 
Finesse  390 
Angle of incidence  4\(^{\circ }\) 
10.8 Commissioning
Commissioning describes the process of tuning and improving a gravitationalwave detector after its subsystems have been installed and before the full system is operational. This process typically takes several years because the interferometer couples all the subsystems in a unique and complex way, which cannot be tested in advance. This is particularly important for advanced detectors which employ many cutting edge technologies which, although having been tested in the laboratory and at prototype facilities, have never been implemented together in interferometers of this scale. The efficiency of the commissioning process is crucial to achieving the expected sensitivity and providing an instrument for scientific data taking in a timely manner.
Through the commissioning process we observe effects never seen before, the interferometer will be operated in a new regime, namely a full scale, high power, dual recycled interferometer with arm cavities. In this extremely sensitive configuration previously negligible effects could have a strong impact on interferometer performance. For example, parametric instabilities, where higherorder mode and radiation pressure effects couple together with the potential of ringing up high order sidebands, will likely be a factor in this high powered regime (Braginsky et al. 2001; Evans et al. 2010; Gras et al. 2010). Subsystems of the interferometer also use cutting edge techniques which have yet to be tested within the full framework of our advanced detectors.
During commissioning the interferometers are assembled in increments, building towards the full dual recycled configuration. As the optics are installed many measurements are taken to test the behaviour of various subsystems and finally to test the response and noise budgets of the full interferometer. During this process it is crucial that we have accurate models of the interferometers. These must include possible defects and higherorder mode effects, typically going beyond the more simplified models used in the design phase. For example, in Advanced LIGO measurements of the surfaces of the mirrors were taken prior to installation. This surface data can be used in simulations to model the expected distortion of the beams within the real interferometers. During the commissioning process these models are used to check against experimental measurements. In the case where a measurement is not as expected models are used to investigate the possible causes, adding in more realistic measurements and tuning parameters to recreate the observed behaviour. From such models we can then suggest solutions in the case of underperformance.

Comparisons of alignment signals calculated for Fabry–Perot cavities using three methods: Finesse, an analytic calculation and the FFT propagation simulation OSCAR (Ballmer et al. 2014).

Comparisons of the control signals and sideband build up in Advanced LIGO, as modelled in Finesse and Optickle (Bond et al. 2014a; Bond 2014).

Investigation into the effect of modemismatch in the control signals of the Advanced LIGO interferometer (Bond et al. 2014b).

A dedicated commissioning investigation into power loss at the central beam splitter in Advanced LIGO using Finesse (Bond et al. 2013).

Finesse simulations of the alignment control signal of the Advanced LIGO input mode cleaner (Kokeyama et al. 2013).
10.9 Finesse examples
10.9.1 Higherorder mode resonances
10.9.2 Mode cleaner
10.9.3 Misaligned cavity
In this example a misaligned cavity is scanned and the circulating power is detected. Additional spikes in the cavity scan indicate the higherorder modes (order one and two are visible) created by the misalignment (Fig. 114).
10.9.4 Impact of thermal aberrations
This example shows the power circulating in an Advanced LIGO style arm cavity versus input laser power when we consider the impact of thermal effects (lensing of the input mirror and change in curvature of the mirror surfaces). The mode mismatches these aberrations cause results in less power coupled into the cavity (Fig. 115).
Finally we scale the power circulating in an individual arm cavity ($Pc) by the gain afforded by the power recycling cavity (45 W/W) and the beam splitter (0.5) to represent the power in an arm of the full power recycled Michelson configuration. We also plot the theoretical linear circulating power, when thermal effects are not considered. Here the 280.7 W is the circulating power in a cavity simulated with no thermal or higher order mode defects.
11 Scattering into higherorder modes
Spatial variations in the optics that compose a laser interferometer, such as distortions of the mirror surfaces, will change the shape of the circulating beams. Methods for quantifying such optical imperfections and their effects are required during the design of an interferometer and for modelling efforts to characterise the instrument during operation. In particularly, this is crucial during the design phase in order to produce, for example, polishing requirements for the mirror surfaces. At first glance it is not obvious how such optical defects should be characterised and we will show that the nature of the problem determines which approach to use.
Previously we introduced the idea that these distortions can be described as higherorder Gaussian modes and considered the impact of such modes on the interferometer performance. In this section we consider the mechanisms and mathematics of this scattering into higherorder modes, with particular emphasis on this process for mirror surface distortions. We will explore how different types of surface distortions impact the beam shape and quantify which mirror shapes produce which higher order modes. Throughout this section we use measured data from the Advanced LIGO mirrors, kindly provided by GariLynn Billingsley of the LIGO Laboratory (Billingsley 2015).
11.1 Light scattering in interferometers
The term ‘scattering’ in interferometers can refer to several different processes. Most commonly it refers to imperfections of high spatial frequency that scatter light at large angles away from the optical axis, effectively scattering light out of the path of the beam. This is a different problem to scattering into higher order modes, which occurs when the light is scattered back into the path of the beam (i.e., small angle scattering). Light scattered at large angles has the potential to be rescattered back into the path of the beam by interactions with, for example, the walls of the beam tube. This will couple new noises into the interferometer, from the beam tubes into the circulating light field. Low angle scattering into higher order modes can introduce noise in other ways, as was discussed in Sect. 10. The effects of scattered light and mitigation solutions are an ongoing research topic in the gravitational wave community (Vinet et al. 1997; Yamamoto 2007; Accadia 2010; VanderHyde et al. 2015).
The different scattering processes require different methods for efficient, accurate modelling. Whereas lowangle scattering can be modelled using a paraxial approach, either via a description of higherorder modes or using a Fourier propagation model, highangle scattering is outside the paraxial approximation and can require computationally heavy numerical algorithms for accurate results.
In this review we focus on low angle scattering which manifests itself as changes in the beam shape. Of course low and high angle scattering are not two separate phenomena, and we see the paraxial method fail at scattering angles greater than \({\sim }20^{\circ }\). This region between high and lowangle scattering can be difficult to model, falling between the two regimes. In addition, the finite size of the mirrors in real interferometers prevents the buildup of very highorder modes as these are wider than the mirrors and experience significant larger losses. In this way the finite size of the cavity mirrors can set a limit for high angle scattering.
 1.
Flatness, describing the overall shape of a mirror and its large scale, low spatial frequency features. These defects impact the shape of the beam within the path, as can be described with higher spatial modes.
 2.
Roughness, the high spatial frequency distortions of the mirror which scatter light out of the path of the beam.
11.2 Mirror surface defects

misalignment and curvature mismatch, i.e., a mismatch between the position, orientation and shape of the optics with respect to the laser beam

nonuniform mirror phase effects, distorted surfaces and substrates will change the phase distribution of a reflected or transmitted beam

nonuniform amplitude effects: dirty or distorted optics can cause nonuniform absorption and reflection

apertures created by the finite size of the optics.
11.3 Coupling between higherorder modes
In interferometer simulations such as Finesse that use modes to describe the beam shape, a maximum order of the modes included \(O_\mathrm{max}\) must be defined for each model. The coupling between all modes with an order less than the maximum order of modes is calculated in reflection and transmission of a distorted optic. This is represented as a coupling coefficient matrix, as described in Sect. 9.16, which computes the transformation of the incident light field as it interacts with the distorted optic. These coupling matrices are inserted into the matrix describing the interferometer behaviour, as described in Sect. 2.3, giving the higherorder mode content at any position within the simulated setup. For well behaved optics, such as those installed in gravitationalwave detectors, we can accurately model realistic distortions of the beam with a finite number of modes, as long as we chose a good Gaussian basis (eigenmode) to work in. The further from the ideal eigenmode the more modes you will require to converge to the correct result. It has been our experience that the best eigenmode is most often that of the optical cavity, as given by the mirror curvatures and positions.
11.4 Simulation methods
 1.
Modal decomposition with light fields expressed as linear combinations of Gaussian modes (solutions to the paraxial wave equation).
 2.
Fast Fourier Transform (FFT) methods where the light fields are represented as finite numerical grids which are propagated through an optical system in the Fourier domain.
11.5 Mirror surface maps
In order to analyse the effects of mirror surface distortions we require numerical descriptions of actual mirrors. In this section we discuss several methods for representing mirror surface data, with some methods more suited to use in numerical simulations, whilst others allow an analytic analysis.
A powerful way to implement mirror surface distortions in modal models is by using a numerical grid representing the surface height of the real mirrors, known as mirror maps. This is how mirror surface effects are implemented in the interferometer simulation Finesse (Freise 2015; Freise et al. 2004). The surface data is given as a function over the x–y surface of the optic and can either be measured data from a real mirror or data generated from mathematical functions, for example, describing the expected thermal distortion of a mirror. Mirror map data can be produced for surface height, reflectivity, transmissivity or absorption over the surface of the optic. Unless otherwise noted we use the generic term mirror map referring to surface height. Figure 117 shows an example of a mirror map depicting the surface of an end test mass produced for Advanced LIGO (shown here with any curvature, tilt and offset removed) to illustrate the kinds of distortions of the mirrors we can expect. Note the nanometer scale of the graph, which is typical for mirrors in such highprecision interferometers. We can also see that the central region of the mirror exhibits less surface height variation. Again this is expected, as the requirements on the polishing of the mirror are much more stringent in the centre of the optic where the beam is most intense.
11.6 Spectrum of surface distortions
Low spatial frequency distortions correspond to the overall mirror shape, higher spatial frequencies refer to the roughness of the mirror. The amplitude of the lower spatial frequencies is significantly higher, as expected. Higher spatial frequencies occur naturally with smaller amplitudes but are also required to be very small in gravitational wave mirrors to reduce wide angle scattering out of the beam path.
11.7 Surface description with Zernike polynomials
Using numerical integration routines real surface data can be represented as a sum of Zernike polynomials. This is illustrated in Fig. 121, where an Advanced LIGO mirror map is recreated using low order Zernike polynomials (\(n\le 20\)). The overall shape of the Zernike surface looks very similar to the original map, but lacks the high spatial frequencies. These are shown in the residual map which also illustrates the high polishing requirements for the central 16 cm region. Although high spatial frequencies can be represented by Zernike polynomials it is often convenient for mirror surface analysis to consider only the low order Zernike polynomials, with the rest of the mirror description contained in spectra of spatial frequencies. In Fig. 122 the spectrum of an Advanced LIGO mirror map is shown, as well as the spectra for Zernike maps recreated using polynomials up to a given order, illustrating how low order polynomials correspond to low spatial frequencies. Including more polynomials in our model tends towards the original map.
11.8 Mode coupling due to mirror surfaces defects
In Sect. 11.3 the method for calculating coupling coefficients numerically, for a generic surface distortion, was discussed. For design of new laser interferometers we want tools to predict which types of distortions will couple light into which higherorder Gaussian modes. Such a tool would allows us to compute specific requirements for the distortions in optics for future detectors. For example, in Bond et al. (2011) the proposal of a new input laser mode, LG\(_{33}\), is analysed in terms of the performance of such a highorder mode with the current mirrors. This involves an analysis of the mirror shapes which will couple between LG\(_{33}\) and other modes of the same order, as these modes have the potential to seriously degrade the performance. In such a case an analytic approach to coupling, where the distortions are described by functions such as the Zernike polynomials, is highly desirable.
Scattering into HOMs
Firstly we consider coupling from specific spatial frequencies within a mirror surface. Such an approach was also considered by Winkler et al. (1994), for an incident HG\(_{00}\) mode. Here we expand on this work to present an analytic approach to scattering of light in the modal picture from an arbitrary incident mode.
Typically individual \(k_{n.n'}\) (where \(n\ne n'\)) are of the order \(10^{3}\) for 1 nm distortions. For coupling back into the same mode \(k_{n,n}\approx 1\) for small distortions. We would therefore expect coupling from HG\(_{n,m}\) to HG\(_{n,m'}\) or HG\(_{n',m}\) would be significantly larger than coupling where both indices change, as these are of the order \(10^{3}\) rather than \(10^{6}\).
The analytical coupling approximation described here is discussed in detail in Bond (2014). This provides a quick analytical tool to predict the modes produced on interaction with distorted mirrors and can be used to provide mirror surface requirements and during the commissioning stage to predict the modal content of the resonating laser beams.
Zernike coupling
Common analysis using spatial frequencies involves taking a statistical approach: performing numerous simulations using randomly generated realisations of mirror surfaces to determine the higher order mode behaviour for a mirror conforming to a particular spectrum of spatial frequencies. Such an approach for an LG\(_{33}\) mode investigation is detailed in Hong et al. (2011). Here we present an analytic approach which aims to identify the exact shapes which couple between different modes.
Azimuthal index (m) of the Zernike shapes required to cause first order coupling from an incident LG\(_{33}\) mode into each of the other order 9 modes
mode (p, l)  4, 1  2, 5  4, \(\)1  1, 7  3, \(\)3  0, 9  2, \(\)5  1, \(\)7  0, \(\)9 

m  2  2  4  4  6  6  8  10  12 
11.9 Efficient coupling matrix computations with multiple distortions
Evaluating coupling coefficients numerically is a computationally expensive task if an analytic solution is not known for a particular distortion to the beam shape. Analytic solutions such as those from BayerHelms (1984) for modemismatches and misalignments (see Sect. 9.16) provide a fast way to compute the matrices for such effects. However, if a surface defect or some other distortion is also applied to a mirror this can require full numerical integration which is very slow, especially if the simulation varies the modemismatch or alignments. Different distortions can mathematically be separated into multiple coupling coefficient matrices, allowing a fast method to solve one which varies often, like modemismatch, and a slow numerical integration which often need only be performed once.
The expansion beam parameter \(q_l\) can in theory be set to any value; however the computational requirements can be reduced if it is chosen sensibly. Remember that a modemismatch is present if \(q_1 \ne q_2\), so if \(q_l\) is chosen to be either \(q_1\) or \(q_2\) the modemismatch is present in only one of the matrices. This is beneficial as coupling coefficient matrices are Hermitian if there is no modemismatch. Thus only one half of the matrix elements need to be computed—when solving via numerical integration this can save a great deal of time. There is also the issue of matrix commutation, A(x, y) and B(x, y) are interchangeable in the derivation thus it appears \([\hat{A}, \hat{B}] = 0\), which is a surprising result seeing as the functions can be any arbitrary values. In practice it is found that commutation errors are only present if the functions are not described using enough higherordermodes. If significant amount of information is lost in modes that are not considered, commutation errors are likely to occur.
11.10 Clipping by finite apertures
Another spatial effect present in real interferometers is the finite size of the optics. Often in simulations with Gaussian modes or plane waves there is some intrinsic assumption that the optics are infinite. In reality the size of the optics is carefully chosen, optimising between large optics to contain the power of the incident beams and smaller optics to reduce the impact of thermal noise.
Generic analytic coupling coefficients describing clipping at a circular aperture is available in Vinet and the Virgo Collaboration (2001).
11.11 Cavity modes of many shapes
This example is illustrative of a more general effect: the fact that the resonant modes of a distorted cavity will differ from a perfect Gaussian mode. We also note that the finite size of the cavity mirrors makes the situation more complex, for example it affects the orthogonality of the cavity eigenmodes (Siegman 1979).
Footnotes
 1.
This equation only considers shotnoise, which is one aspect of quantum noise, see Sect. 6 for a more detailed description of quantum noise.
 2.
In many implementations of numerical matrix solvers the input vector is also called the righthand side vector.
 3.
Note that in other publications the tuning or equivalent microscopic displacements are sometimes defined via an optical pathlength difference. In that case, a tuning of \(2\pi \) is used to refer to the change of the optical path length of one wavelength, which, for example, if the reflection at a mirror is described, corresponds to a change of the mirror’s position of \(\lambda _0/2\).
 4.
The signal sidebands are sometimes also called audio sidebands because of their frequency range.
 5.
The term effective refers to that amount of incident light, which is converted into photoelectrons that are then usefully extracted from the junction (i.e., do not recombine within the device). This fraction is usually referred to as quantum efficiency \(\eta \) of the photodiode.
 6.
Please note that in the presence of losses the coupling is defined with respect to the transmission and losses. In particular, the impedancematched case is defined as \(T_1=T_2\times \mathrm {Loss}\), so that the input power transmission exactly matches the light power lost in one roundtrip.
 7.
The term ‘main signals’ refers to the optical signal providing the readout of the interferometric measurement, for example, of a position or length change. In addition, other output signals exist: for example, the light power reflected back into the West port can be recorded for monitoring the interferometer status.
 8.
Derivations of the accumulated phase can be found in many works, a simple example is presented in Bond (2014).
 9.
It is proportional to a factor \(\chi \), the photodiodes quantum efficiency, which states how many Amps per Watt of incident power is output by the photodiode. We will assume here the efficiency is perfect, \(\chi =1\), for simplicity.
 10.
The tuned case is slightly special, because the integrated area is half compared to the others, because the plot shows only the positive half of the total linewidth seen by signal sidebands.
 11.
The typical light wavelength is \({\sim } 10^{6}\,\)m while all groundbased interferometers built or planned have target displacement noise spectral densities below \({\sim } 10^{19}\,{\mathrm m}/\sqrt{\mathrm Hz}\) in a frequency band of order 100 Hz wide.
 12.
To minimise thermal noise in suspensions, lowloss materials and techniques are employed to avoid dissipation. The resonant modes of these suspensions are seen in the frequency domain as narrow spectral features, or lines. The violin modes are transverse oscillations of the stretched suspension fibres that support the mirrors, which vibrate much like a violin string. The frequencies of these modes typically lie between 300 and 800 Hz, and they are often conspicuous in the spectra of signals from gravitationalwave detectors.
 13.
Also known as the farfield angle or the divergence of the beam.
 14.
Please note that this formula from Siegman (1986) is very compact. Since the parameter q is a complex number, the expression contains at least two complex square roots. The complex square root requires a different algebra than the standard square root for real numbers. Especially the third and fourth factors can not be simplified in any obvious way: \(\left( \frac{q_0}{q(z)}\right) ^{1/2} \left( \frac{q_0q^*(z)}{q_0^*q(z)}\right) ^{n/2} \ne \left( \frac{q_0^{n+1}{q^*}^n(z)}{q^{n+1}(z){q_0^*}^n}\right) ^{1/2}\)!
 15.
Siegman (1986) states that the indices must obey the following relations: \(0\le l\le p\). However, that is not the case.
 16.
Here we refer to the input power as the power injected into the central Michelson interferometer, after the input mode cleaner and other input optics.
 17.
 18.
34.5 km is the focal length when modelled as an individual lens in a vacuum, the approach in this document. Sometimes quoted is 50 km corresponding to the lens when modelled inside the fused silica substrate of the ITM.
 19.
Many more such documents exist for Advanced LIGO and other gravitational wave detectors. This selection is based on our familiarity with the described work.
 20.
A list of scientific papers and reports citing Finesse is provided in Freise (2015).
Notes
Acknowledgements
We would like to thank our colleagues in the GEO 600 project and in the LIGO Scientific Collaboration for many useful discussions over the years. We would like to thank GariLynn Billingsley for providing us with data and advice regarding Advanced LIGO mirror maps. We thank Miguel Dovale Álvarez, Anna Green, Daniel Töyrä and Haixing Miao for their time to read many of the new sections in this article and for the helpful discussions. AF, CB and DB acknowledge support from the University of Birmingham. KS acknowledges support from the University of Glasgow and the Albert Einstein Institute, Hannover. Some of the illustrations have been prepared using the component library by Alexander Franzen. This document has been assigned the LIGO Laboratory document number LIGOP1500233.
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