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:
- 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:
- 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.