Interferometer Techniques for GravitationalWave Detection
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Abstract
Several kmscale gravitationalwave detectors have been constructed world wide. 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
Beam Splitter Gaussian Beam Light Field Beam Parameter Michelson Interferometer1 Introduction
1.1 The scope and style of the review
The historical development of laser interferometers for application as gravitationalwave detectors [47] 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, 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 [23, 5], Advanced Virgo [4] and LCGT [36] 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. The present review, in its first version, does not consider quantum effects but reserves them for future updates.
The 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 either timedependent or frequencydomain behaviour of interferometers and their component systems is Finesse [22, 19]. This was developed by one of us (AF), 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
As set out in very many works, gravitationalwave detectors strive to pick out signals carried by passing gravitational waves from a background of selfgenerated noise. The principles of operation are set out at various points in the review, but in essence, the goal has been to prepare many photons, stored for as long as practical in the ‘arms’ of a laser interferometer (traditionally the two arms are at right angles), so that tiny phase shifts induced by the gravitational waves form as large as possible a signal, when the light leaving the appropriate ‘port’ of the interferometer is detected and the resulting signal analysed.
The evolution of gravitationalwave detectors can be seen by following their development from prototypes and early observing systems towards the Advanced detectors, which are currently in the final stages of planning or early stages of construction. Starting from the simplest Michelson interferometer [18], then by the application of techniques to increase the number of photons stored in the arms: delay lines [31], FabryPérot arm cavities [16, 17] and power recycling [15]. The final step in the development of classical interferometry was the inclusion of signal recycling [41, 30], 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 also lownoise actuation, is described in many works. As the interferometer optics become more complicated, the resonance conditions, i.e., the allowed combinations of intercomponent path lengths required to allow the photon number in the interferometer arms to reach maximum, become more narrowly defined. It is likewise necessary to maintain angular alignment of all components, such 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, and the requirement can be in the low nanoradian range for kmscale detectors [44, 21]. 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 can consist of up to around seven highly sensitive components and so there can be of order 20 degrees of freedom to be measured and controlled [3, 57].
Although the light fields are linear, 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. However, the fields do vary linearly or at least 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, 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.
1.3 Overview of the physics of the primary interferometer components
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 first version of the 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.
1.4 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.
1.5 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 FabryPérot 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 reffected or transmitted laser beam. However, in some cases, one of these surfaces has been treated with an antireffection (AR) coating to minimise the effect on the transmitted beam.
The terms mirror and beam splitter are sometimes used to describe a (theoretical) optical surface in a model. We define real amplitude coefficients for reflection and transmission r and t, with 0 ≤ r, t ≤ 1, so that the field amplitudes can be written as The π/2 phase shift upon transmission (here given by the factor i) refers to a phase convention explained in Section 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: 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. Section 2.6.
2.2 The twomirror resonator
The linear optical resonator, also called a cavity is formed by two partiallytransparent mirrors, arranged in parallel as shown in Figure 5. This simple setup makes a very good example with which to illustrate how a mathematical model of an interferometer can be derived, using the equations introduced in Section 2.1.
2.3 Coupling matrices
2.3.1 Coupling matrices for numerical computations
2.3.2 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 × 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
2.4.1 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 optical system, 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.
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 [29]: one is the macroscopic ‘length’ L, defined as the multiple of a constant wavelength λ_{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, λ_{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 λ in any equation refers to the actual wavelength at the respective location as λ = λ_{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 ≈ 2.8 · 10^{14} Hz, radio frequency (RF) sidebands used for interferometer control with frequencies (offset to the laser frequency) of f ≈ 1 • 10^{e} to 150 • 10^{e} Hz, and the signal sidebands at frequencies of 1 to 10,000 Hz^{3}. 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 [39].
3.6 Phase modulation through a moving mirror
Several optical components can modulate transmitted or reflected light fields. In this section we discuss in detail the example of phase modulation by a moving mirror. Mirror motion does not change the transmitted light; however, the phase of the reflected light will be changed as shown in Equation (11).
3.7 Coupling matrices for beams with multiple frequency components
3.8 Finesse examples
3.8.1 Modulation index
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 ω_{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 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 ω_{0} ± ω_{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
5 Basic Interferometers
The large interferometric gravitationalwave detectors currently in operation are based on two fundamental interferometer topologies: the FabryPérot and the Michelson interferometer. The main instrument is very similar to the original interferometer concept used in the famous experiment by Michelson and Morley, published in 1887 [42]. The main difference is that modern instruments use laser light to illuminate the interferometer to achieve much higher accuracy. Already the first prototype by Forward and Weiss has thus achieved a sensitivity a million times better than Michelson’s original instrument [18]. In addition, in current gravitationalwave detectors, the Michelson interferometer has been enhanced by resonant cavities, which in turn have been derived from the original idea for a spectroscopy standard published by Fabry and Pérot in 1899 [16]. The following section will describe the fundamental properties of the FabryPérot 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 FabryPérot interferometer

when T_{1} < T_{2} the cavity is called undercoupled

when T_{1} = T_{2} the cavity is called impedance matched

when T_{1} > T_{2} the cavity is called overcoupled
5.2 Michelson interferometer
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 Section 6.4.
5.3 Finesse examples
5.3.1 Michelson power
5.3.2 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.
6 Interferometric Length Sensing and Control
In this section we introduce interferometers as length sensing devices. In particular, we explain how the FabryPérot 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 and transfer functions, which are used to describe most essential features of length sensing and control.
6.1 Error signals and transfer functions
In general, we will call an error signal any measured signal suitable for stabilising a certain experimental parameter p with a servo loop. The aim is to maintain the variable p at a 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. By definition a transfer function describes only the linear coupling of signals inside a system. This means a transfer function is independent of the actual signal size. For small signals or small deviations, most systems can be linearised and correctly described by transfer functions.
Experimentally, network analysers are commonly used to measure a transfer function: one connects a periodic signal (the source) to an actuator of the plant (which is to be analysed) and to an input of the analyser. A signal from a sensor that monitors a certain parameter of the plant is connected to the second analyser input. By mixing the source with the sensor signal the analyser can determine the amplitude and phase of the input signal with respect to the source (amplitude equals one and the phase equals zero when both signals are identical).
Mathematically, transfer functions can be modeled similarly: applying a sinusoidal signal sin(ω_{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 ±ω_{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 ω_{s}, which can be extracted, for example, by means of demodulation (see Section 4.2).
6.2 FabryPérot 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.
6.3 The PoundDreverHall length sensing scheme
6.4 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 FabryPérot cavities within the Michelson. This construction of new topologies by combining Michelson and FabryPérot interferometers will be described in detail in a future version of this review.
6.5 The Schnupp modulation scheme
Similar to the FabryPérot 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 ΔL/λ = (2N + 1) • 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 [49].
One approach to make use of the advantages of the dark fringe operating point is to use an operating point very close to the dark fringe at which the optical gain is not yet zero. In such a scenario a careful tradeoff calculation can be done by computing the signaltonoise with noises that must be suppressed, such as the laser amplitude noise. This type of operation is usually referred to as DC control or offset control and is very similar to the similarlynamed mechanism used with FabryPérot cavities.
6.6 Finesse examples
6.6.1 Cavity power and slope
Figure 33 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.
6.6.2 Michelson with Schnupp modulation
Figure 39 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 Figure 36, 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.
7 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 [35]. Siegman has written a very interesting historic review of the development of Gaussian optics [52, 51] and we use whenever possible the same notation as used in his textbook ‘Lasers’ [50].
7.1 The paraxial wave equation
7.2 Transverse electromagnetic modes
7.3 Properties of Gaussian beams
Such a beam profile (for a beam with a given wavelength λ) can be completely determined by two parameters: the size of the minimum spot size ω_{0} (called beam waist) and the position z_{0} of the beam waist along the zaxis.
7.4 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 [46]: the tangential plane, defined as the x–z plane and the sagittal plane as given by y and z.
Remember that these HermiteGauss 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 ω_{0,x} = ω_{0,y} and z_{0,x} = z_{0,y} but all the results can also be applied directly to a pair of parameters.
7.5 Higherorder HermiteGauss modes
7.6 The Gaussian beam parameter
7.7 Properties of higherorder HermiteGauss modes

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

The phase distribution of HermiteGauss modes shows the curvature (or radius of curvature) of the beam. The curvature depends on z but is equal for all higherorder modes.
7.8 Gouy phase
7.9 LaguerreGauss modes
7.10 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 Section 7.11). 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).
7.11 ABCD matrices
7.11.1 Transmission through a mirror:
7.11.2 Reflection at a mirror:
The matrix for reflection is given by The reflection at the back surface can be described by the same type of matrix by setting C = 2n_{2}/R_{C}.
7.11.3 Transmission through a beam splitter:
7.11.4 Reflection at a beam splitter:
7.11.5 Transmission through a thin lens:
7.11.6 Transmission through a free space:
8 Interferometer Matrix with HermiteGauss Modes

the propagation through free space has to include the Gouy phase shift, and

upon reflection or transmission at a mirror or beam splitter the different HermiteGauss modes may be coupled (see below).
The Gouy phase shift can be included in the simulation in several ways. For example, for reasons of flexibility the Gouy phase has been included in Finesse as a phase shift of the component space.
8.1 Coupling of HermiteGauss modes

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

if the beam size or shape at the second cavity does not match the beam shape and size of the (resonant) fundamental eigenmode (q_{1}(z_{cav}) ≠ q_{2}(z_{cav})), the beam is then not modematched to the second cavity, i.e., there is a mode mismatch.
The above misconfigurations can be used in the context of simple beam segments. We consider the case in which the beam parameter for the input light is specified. Ideally, the ABCD matrices then allow one to trace a beam through the optical system by computing the proper beam parameter for each beam segment. In this case, the basis system of HermiteGauss modes is transformed in the same way as the beam, so that the modes are not coupled.
For example, an input beam described by the beam parameter q_{1} is passed through several optical components, and at each component the beam parameter is transformed according to the respective ABCD matrix. Thus, the electric field in each beam segment is described by HermiteGauss modes based on different beam parameters, but the relative power between the HermiteGauss modes with different mode numbers remains constant, i.e., a beam in a u_{00} mode is described as a pure u_{00} mode throughout the entire system.
8.2 Coupling coefficients for HermiteGauss modes
HermiteGauss modes are coupled whenever a beam is not matched to a cavity or to a beam segment or if the beam and the segment are misaligned. This coupling is sometimes referred to as ‘scattering into higherorder modes’ because in most cases the laser beam is a considered as a pure TEM_{00} mode and any mode coupling would transfer power from the fundamental into higherorder modes. However, in general, every mode with nonzero power will transfer energy into other modes whenever mismatch or misalignment occur, and this effect also includes the transfer from higher orders into a low order.
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 HermiteGauss 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 HermiteGauss modes and can be expressed by coupling coefficients that yield the change in the light amplitude and phase with respect to mode number.
8.3 Finesse examples
8.3.1 Beam parameter
8.3.2 Mode cleaner
This example uses the ‘tem’ command to create a laser beam which is a sum of equal parts in u_{00} and u_{10} modes. This beam is passed through a triangular cavity, which acts as a mode cleaner. Being resonant for the u_{00}, the cavity transmits this mode and reflects the u_{10} mode as can be seen in the resulting plots.
8.3.3 LG33 mode
Footnotes
 1.
^{1} In many implementations of numerical matrix solvers the input vector is also called the righthand side vector.
 2.
^{2} 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 2π 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 λ_{0}/2.
 3.
^{3} The signal sidebands are sometimes also called audio sidebands because of their frequency range.
 4.
^{4} 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 η of the photodiode.
 5.
^{5} 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} · Loss, so that the input power transmission exactly matches the light power lost in one roundtrip.
 6.
^{6} Also known as the farfield angle or the divergence of the beam.
 7.
^{7} Please note that this formula from [50] 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: \({\left({{{{q_0}} \over {q(z)}}} \right)^{1/2}}{\left({{{{q_0}{q^{\ast}}(z)} \over {q_0^{\ast}q(z)}}} \right)^{n/2}} \neq {\left({{{q_0^{n + 1}{q^{\ast n}}(z)} \over {{q^{n + 1}}(z)q_0^{\ast n}}}} \right)^{1/2}}\)!
 8.
^{8} [50] states that the indices must obey the following relations: 0 ≤ l ≤ p. However, that is not the case.
Notes
Acknowledgements
We would like to thank our colleagues in the GEO 600 project for many useful discussions over the years. AF acknowledges 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.
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