### 2.2 From incident wave to outgoing light: light transformation in the GW interferometers

To proceed with the analysis of quantum noise in GW interferometers we first need to familiarize ourselves with how a light field is transformed by an interferometer and how the ability of its mirrors to move modifies the outgoing field. In the following paragraphs, we endeavor to give a step-by-step introduction to the mathematical description of light in the interferometer and the interaction with its movable mirrors.

#### 2.2.1 Light propagation

We first consider how the light wave is described and how its characteristics transform, when it propagates from one point of free space to another. Yet the real light beams in the large scale interferometers have a rather complicated inhomogeneous transverse spatial structure, the approximation of a plane monochromatic wave should suffice for our purposes, since it comprises all the necessary physics and leads to right results. Inquisitive readers could find abundant material on the field structure of light in real optical resonators in particular, in the introductory book [171] and in the Living Review by Vinet [154].

So, consider a plane monochromatic linearly polarized light wave propagating in vacuo in the positive direction of the -axis. This field can be fully characterized by the strength of its electric component that should be a sinusoidal function of its argument and can be written in three equivalent ways:

where and are called amplitude and phase, and take names of cosine and sine quadrature amplitudes, and complex number is known as the complex amplitude of the electromagnetic wave. Here, we see that our wave needs two real or one complex parameter to be fully characterized in the given location at a given time . The ‘amplitude-phase’ description is traditional for oscillations but is not very convenient since all the transformations are nonlinear in phase. Therefore, in optics, either quadrature amplitudes or complex amplitude description is applied to the analysis of wave propagation. All three descriptions are related by means of straightforward transformations:
The aforesaid means that for complete understanding of how the light field transforms in the optical device, knowing the rules of transformation for only two characteristic real numbers – real and imaginary parts of the complex amplitude suffice. Note also that the electric field of a plane wave is, in essence, a function of a single argument (for a forward propagating wave) and thus can be, without loss of generality, substituted by a time dependence of electric field in some fixed point, say with , thus yielding . We will keep to this convention throughout our review.

Now let us elaborate the way to establish a link between the wave electric field strength values taken in two spatially separated points, and . Obviously, if nothing obscures light propagation between these two points, the value of the electric field in the second point at time is just the same as the one in the first point, but at earlier time, i.e., at : . This allows us to introduce a transformation that propagates EM-wave from one spatial point to another. For complex amplitude , the transformation is very simple:

Basically, this transformation is just a counterclockwise rotation of a wave complex amplitude vector on a complex plane by an angle . This fact becomes even more evident if we look at the transformation for a 2-dimensional vector of quadrature amplitudes , that are:
where
stands for a standard counterclockwise rotation (pivoting) matrix on a 2D plane. In the special case when the propagation distance is much smaller than the light wavelength , the above two expressions can be expanded into Taylor’s series in up to the first order:
and
where stands for an identity matrix and is an infinitesimal increment matrix that generate the difference between the field quadrature amplitudes vector after and before the propagation, respectively.

It is worthwhile to note that the quadrature amplitudes representation is used more frequently in literature devoted to quantum noise calculation in GW interferometers than the complex amplitudes formalism and there is a historical reason for this. Notwithstanding the fact that these two descriptions are absolutely equivalent, the quadrature amplitudes representation was chosen by Caves and Schumaker as a basis for their two-photon formalism for the description of quantum fluctuations of light [39, 40] that became from then on the workhorse of quantum noise calculation. More details about this extremely useful technique are given in Sections 3.1 and 3.2 of this review. Unless otherwise specified, we predominantly keep ourselves to this formalism and give all results in terms of it.

#### 2.2.2 Modulation of light

Above, we have seen that a GW signal displays itself in the modulation of the phase of light, passing through the interferometer. Therefore, it is illuminating to see how the modulation of the light phase and/or amplitude manifests itself in a transformation of the field complex amplitude and quadrature amplitudes. Throughout this section we assume our carrier field is a monochromatic light wave with frequency , amplitude and initial phase :

##### Amplitude modulation.
The modulation of light amplitude is straightforward to analyze. Let us do it for pedagogical sake: imagine one managed to modulate the carrier field amplitude slow enough compared to the carrier oscillation period, i.e., , then:

where and are some constants called modulation depth and relative phase, respectively. The complex amplitude of the modulated wave equals to
and the carrier quadrature amplitudes are, apparently, transformed as follows:
The fact that the amplitude modulation shows up only in the quadrature that is in phase with the carrier field sets forth why this quadrature is usually named amplitude quadrature in the literature. In our review, we shall also keep to this terminology and refer to cosine quadrature as amplitude one.

Illuminating also is the calculation of the modulated light spectrum, that in our simple case of single frequency modulation is straightforward:

Apparently, the spectrum is discrete and comprises three components, i.e., the harmonic at carrier frequency with amplitude and two satellites at frequencies , also referred to as modulation sidebands, with (complex) amplitudes . The graphical interpretation of the above considerations is given in the left panel of Figure 4. Here, carrier fields as well as sidebands are represented by rotating vectors on a complex plane. The carrier field vector has length and rotates clockwise with the rate , while sideband components participate in two rotations at a time. The sum of these three vectors yields a complex vector, whose length oscillates with time, and its projection on the real axis represents the amplitude-modulated light field.

The above can be generalized to an arbitrary periodic modulation function , with . Then the spectrum of the modulated light consists again of a carrier harmonic at and an infinite discrete set of sideband harmonics at frequencies ():

Further generalization to an arbitrary (real) non-periodic modulation function is apparent:

From the above expression, one readily sees the general structure of the modulated light spectrum, i.e., the central carrier peaks at frequencies and the modulation sidebands around it, whose shape retraces the modulation function spectrum shifted by the carrier frequency .

##### Phase modulation.
The general feature of the modulated signal that we pursued to demonstrate by this simple example is the creation of the modulation sidebands in the spectrum of the modulated light. Let us now see how it goes with a phase modulation that is more related to the topic of the current review. The simplest single-frequency phase modulation is given by the expression:

where , and the phase deviation is assumed to be much smaller than 1. Using Eqs. (14), one can write the complex amplitude of the phase-modulated light as:
and quadrature amplitudes as:
Note that in the weak modulation limit (), the above equations can be approximated as:
This testifies that for a weak modulation only the sine quadrature, which is out-of-phase with respect to the carrier field, contains the modulation signal. That is why this sine quadrature is usually referred to as phase quadrature. It is also what we will call this quadrature throughout the rest of this review.

In order to get the spectrum of the phase-modulated light it is necessary to refer to the theory of Bessel functions that provides us with the following useful relation (known as the Jacobi–Anger expansion):

where stands for the -th Bessel function of the first kind. This looks a bit intimidating, yet for these expressions simplify dramatically, since near zero Bessel functions can be approximated as:

Thus, for sufficiently small , we can limit ourselves only to the terms of order and , which yields:

and we again face the situation in which modulation creates a pair of sidebands around the carrier frequency. The difference from the amplitude modulation case is in the way these sidebands behave on the complex plane. The corresponding phasor diagram for phase modulated light is drawn in Figure 4. In the case of PM, sideband fields have constant phase shift with respect to the carrier field (note factor in front of the corresponding terms in Eq. (22)); therefore its sum is always orthogonal to the carrier field vector, and the resulting modulated oscillation vector has approximately the same length as the carrier field vector but outruns or lags behind the latter periodically with the modulation frequency . The resulting oscillation of the PM light electric field strength is drawn to the right of the PM phasor diagram and is the projection of the PM oscillation vector on the real axis of the complex plane.

Let us now generalize the obtained results to an arbitrary modulation function :

In the most general case of arbitrary modulation index , the corresponding formulas are very cumbersome and do not give much insight. Therefore, we again consider a simplified situation of sufficiently small . Then one can approximate the phase-modulated oscillation as follows:
When is a periodic function: , and in weak modulation limit , the spectrum of the PM light is apparent from the following expression:
while for the real non-periodic modulation function the spectrum can be obtained from the following relation:
And again we get the same general structure of the spectrum with carrier peaks at and shifted modulation spectra as sidebands around the carrier peaks. The difference with the amplitude modulation is an additional phase shifts added to the sidebands.

#### 2.2.3 Laser noise

Thus far we have assumed the carrier field to be perfectly monochromatic having a single spectral component at carrier frequency fully characterized by a pair of classical quadrature amplitudes represented by a 2-vector . In reality, this picture is no good at all; indeed, a real laser emits not a monochromatic light but rather some spectral line of finite width with its central frequency and intensity fluctuating. These fluctuations are usually divided into two categories: (i) quantum noise that is associated with the spontaneous emission of photons in the gain medium, and (ii) technical noise arising, e.g., from excess noise of the pump source, from vibrations of the laser resonator, or from temperature fluctuations and so on. It is beyond the goals of this review to discuss the details of the laser noise origin and methods of its suppression, since there is an abundance of literature on the subject that a curious reader might find interesting, e.g., the following works [119, 120, 121, 167, 68, 76].

For our purposes, the very existence of the laser noise is important as it makes us to reconsider the way we represent the carrier field. Apparently, the proper account for laser noise prescribes us to add a random time-dependent modulation of the amplitude (for intensity fluctuations) and phase (for phase and frequency fluctuations) of the carrier field (13):

where we placed hats above the noise terms on purpose, to emphasize that quantum noise is a part of laser noise and its quantum nature has to be taken into account, and that the major part of this review will be devoted to the consequences these hats lead to. However, for now, let us consider hats as some nice decoration.

Apparently, the corrections to the amplitude and phase of the carrier light due to the laser noise are small enough to enable us to use the weak modulation approximation as prescribed above. In this case one can introduce a more handy amplitude and phase quadrature description for the laser noise contribution in the following manner:

where are related to and in the same manner as prescribed by Eqs. (14). It is convenient to represent a noisy laser field in the Fourier domain:
Worth noting is the fact that is a spectral representation of a real quantity and thus satisfies an evident equality (by we denote the Hermitian conjugate that for classical functions corresponds to taking the complex conjugate of this function). What happens if we want to know the light field of our laser with noise at some distance from our initial reference point ? For the carrier field component at , nothing changes and the corresponding transform is given by Eq. (16), yet for the noise component
there is a slight modification. Since the field continuity relation holds for the noise field to the same extent as for the carrier field:
the following modification applies:
Therefore, for sideband field components the propagation rule shall be modified by adding a frequency-dependent phase factor that describes an extra phase shift acquired by a sideband field relative to the carrier field because of the frequency difference .

#### 2.2.4 Light reflection from optical elements

So, we are one step closer to understanding how to calculate the quantum noise of the light coming out of the GW interferometer. It is necessary to understand what happens with light when it is reflected from such optical elements as mirrors and beamsplitters. Let us first consider these elements of the interferometer fixed at their positions. The impact of mirror motion will be considered in the next Section 2.2.5. One can also refer to Section 2 of the Living Review by Freise and Strain [59] for a more detailed treatment of this topic.

Mirrors of the modern interferometers are rather complicated optical systems usually consisting of a dielectric slab with both surfaces covered with multilayer dielectric coatings. These coatings are thoroughly constructed in such a way as to make one surface of the mirror highly reflective, while the other one is anti-reflective. We will not touch on the aspects of coating technology in this review and would like to refer the interested reader to an abundant literature on this topic, e.g., to the following book [71] and reviews and articles [154, 72, 100, 73, 122, 49, 97, 117, 57]. For our purposes, assuming the reflective surface of the mirror is flat and lossless should suffice. Thus, we represent a mirror by a reflective plane with (generally speaking, complex) coefficients of reflection and and transmission and as drawn in Figure 5. Let us now see how the ingoing and outgoing light beams couple on the mirrors in the interferometer.

##### Mirrors:
From the general point of view, the mirror is a linear system with 2 input and 2 output ports. The way how it transforms input signals into output ones is featured by a matrix that is known as the transfer matrix of the mirror :

Since we assume no absorption in the mirror, reflection and transmission coefficients should satisfy Stokes’ relations [139, 15] (see also Section 12.12 of [99]):
that is simply a consequence of the conservation of energy. This conservation of energy yields that the optical transfer matrix must be unitary: . Stokes’ relations leave some freedom in defining complex reflectivity and transmissivity coefficients. Two of the most popular variants are given by the following matrices:
where we rewrote transfer matrices in terms of real power reflectivity and transmissivity coefficients and that will find extensive use throughout the rest of this review. The transformation rule, or putting it another way, coupling relations for the quadrature amplitudes can easily be obtained from Eq. (27). Now, we have two input and two output fields. Therefore, one has to deal with 4-dimensional vectors comprising of quadrature amplitudes of both input and output fields, and the transformation matrix become -dimensional, which can be expressed in terms of the outer product of a matrix by a identity matrix :
The same rules apply to the sidebands of each carrier field:

In future, for the sake of brevity, we reduce the notation for matrices like to simply .

##### Beam splitters:
Another optical element ubiquitous in the interferometers is a beamsplitter (see Figure 6). In fact, it is the very same mirror considered above, but the angle of input light beams incidence is different from 0 (if measured from the normal to the mirror surface). The corresponding scheme is given in Figure 6. In most cases, symmetric 50%/50% beamsplitter are used, which imply and the coupling matrix then reads:

##### Losses in optical elements:
Above, we have made one assumption that is a bit idealistic. Namely, we assumed our mirrors and beamsplitters to be lossless, but it could never come true in real experiments; therefore, we need some way to describe losses within the framework of our formalism. Optical loss is a term that comprises a very wide spectrum of physical processes, including scattering on defects of the coating, absorption of light photons in the mirror bulk and coating that yields heating and so on. A full description of loss processes is rather complicated. However, the most important features that influence the light fields, coming off the lossy optical element, can be summarized in the following two simple statements:

1. Optical loss of an optical element can be characterized by a single number (possibly, frequency dependent) (usually, ) that is called the absorption coefficient. is the fraction of light power being lost in the optical element:
2. Due to the fundamental law of nature summarized by the Fluctuation Dissipation Theorem (FDT) [37, 95], optical loss is always accompanied by additional noise injected into the system. It means that additional noise field uncorrelated with the original light is mixed into the outgoing light field in the proportion of governed by the absorption coefficient.

These two rules conjure up a picture of an effective system comprising of a lossless mirror and two imaginary non-symmetric beamsplitters with reflectivity and transmissivity that models optical loss for both input fields, as drawn in Figure 7.

Using the above model, it is possible to show that for a lossy mirror the transformation of carrier fields given by Eq. (30) should be modified by simply multiplying the output fields vector by a factor :

where we used the fact that for low loss optics in use in GW interferometers, the absorption coefficient might be as small as . Therefore, the impact of optical loss on classical carrier amplitudes is negligible. Where the noise sidebands are concerned, the transformation rule given by Eq. (31) changes a bit more:
Here, we again used the smallness of and also the fact that matrix is unitary, i.e., we redefined the noise that enters outgoing fields due to loss as , which keeps the new noise sources and uncorrelated: .

#### 2.2.5 Light modulation by mirror motion

For full characterization of the light transformation in the GW interferometers, one significant aspect remains untouched, i.e., the field transformation upon reflection off the movable mirror. Above (see Section 2.1.1), we have seen that motion of the mirror yields phase modulation of the reflected wave. Let us now consider this process in more detail.

Consider the mirror described by the matrix , introduced above. Let us set the convention that the relations of input and output fields is written for the initial position of the movable mirror reflective surface, namely for the position where its displacement is as drawn in Figure 8. We assume the sway of the mirror motion to be much smaller than the optical wavelength: . The effect of the mirror displacement on the outgoing field can be straightforwardly obtained from the propagation formalism. Indeed, considering the light field at a fixed spatial point, the reflected light field at any instance of time is just the result of propagation of the incident light by twice the mirror displacement taken at time of reflection and multiplied by reflectivity :

Remember now our assumption that ; according to Eq. (19) the mirror motion modifies the quadrature amplitudes in a way that allows one to separate this effect from the reflection. It means that the result of the light reflection from the moving mirror can be represented as a sum of two independently calculable effects, i.e., the reflection off the fixed mirror, as described above in Section 2.2.4, and the response to the mirror displacement (see Section 2.2.1), i.e., the signal presentable as a sideband vector . The latter is convenient to describe in terms of the response vector that is defined as:

Note that we did not include sideband fields in the definition of the response vector. In principle, sideband fields also feel the motion induced phase shift; however, as far as it depends on the product of one very small value of by a small sideband amplitude , the resulting contribution to the final response will be dwarfed by that of the classical fields. Moreover, the mirror motion induced contribution (35) is itself a quantity of the same order of magnitude as the noise sidebands, and therefore we can claim that the classical amplitudes of the carrier fields are not affected by the mirror motion and that the relations (30) hold for a moving mirror too. However, the relations for sideband amplitudes must be modified. In the case of a lossless mirror, relations (31) turn:

where is the Fourier transform of the mirror displacement

It is important to understand that signal sidebands characterized by a vector , on the one hand, and the noise sidebands , on the other hand, have the same order of magnitude in the real GW interferometers, and the main role of the advanced quantum measurement techniques we are talking about here is to either increase the former, or decrease the latter as much as possible in order to make the ratio of them, known as the signal-to-noise ratio (SNR), as high as possible in as wide as possible a frequency range.

#### 2.2.6 Simple example: the reflection of light from a perfect moving mirror

All the formulas we have derived here, though being very simple in essence, look cumbersome and not very transparent in general. In most situations, these expressions can be simplified significantly in real schemes. Let us consider a simple example for demonstration purposes, i.e., consider the reflection of a single light beam from a perfectly reflecting () moving mirror as drawn in Figure 9. The initial phase of the incident wave does not matter and can be taken as zero. Then and . Putting these values into Eq. (30) and accounting for , quite reasonably results in the amplitude of the carrier wave not changing upon reflection off the mirror, while the phase changes by :

Since we do not have control over the laser noise, the input light has laser fluctuations in both quadratures that are transformed in full accordance with Eq. (31)):
Again, nothing surprising. Let us see what happens with a mechanical motion induced component of the reflected wave: according to Eq. (36), the reflected light will contain a motion-induced signal in the s-quadrature:
This fact, i.e., that the mirror displacement that just causes phase modulation of the reflected field, enters only the s-quadrature, once again justifies why this quadrature is usually referred to as phase quadrature (cf. Section 2.2.2).

It is instructive to see the spectrum of the outgoing light in the above considered situation. It is, expectedly, the spectrum of a phase modulated monochromatic wave that has a central peak at the carrier wave frequency and the two sideband peaks on either sides of the central one, whose shape follows the spectrum of the modulation signal, in our case, the spectrum of the mechanical displacement of the mirror . The left part of Figure 9 illustrates the aforesaid. As for laser noise, it enters the outgoing light in an additive manner and the typical (though simplified) amplitude spectrum of a noisy light reflected from a moving mirror is given in Figure 10.