# Laser Frequency Noise in Next Generation Gravitational-Wave Detectors

Craig Cahillane  1,* Georgia L. Mansell  1,2 and Daniel Sigg  1  1LIGO Hanford
Observatory, Richland, WA 99352, USA
2LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA<EMAIL_ADDRESS>
###### Abstract

Future ground-based gravitational-wave detectors are slated to detect black
hole and neutron star collisions from the entire stellar history of the
universe. To achieve the designed detector sensitivities, frequency noise from
the laser source must be reduced below the level achieved in current Advanced
LIGO detectors. This paper reviews the laser frequency noise suppression
scheme in Advanced LIGO, and quantifies the noise coupling to the
gravitational-wave readout. The laser frequency noise incident on the current
Advanced LIGO detectors is $8\times 10^{-5}~{}\mathrm{Hz/\sqrt{Hz}}$ at
$1~{}\mathrm{kHz}$. Future detectors will require even lower incident
frequency noise levels to ensure this technical noise source does not limit
sensitivity. The frequency noise requirement for a gravitational wave detector
with arm lengths of 40 km is estimated to be $7\times
10^{-7}~{}\mathrm{Hz/\sqrt{Hz}}$. To reach this goal a new frequency noise
suppression scheme is proposed, utilizing two input mode cleaner cavities, and
the limits of this scheme are explored. Using this scheme the frequency noise
requirement is met, even in pessimistic noise coupling scenarios.

††journal: oe

##  1 Introduction

Gravitational-wave (GW) astronomy is now a well established field, with
regular detections of compact binary coalescences from the ground-based
Advanced LIGO and Virgo detectors[1, 2, 3, 4, 5]. Current interferometers are
converging on their design sensitivities with incremental upgrades, while the
design of a new generation of gravitational-wave detectors is currently under
consideration. Proposed future ground-based detectors include Cosmic
Explorer[6, 7], a USA-led effort with a Michelson interferometer topology and
40 km long arms, and the Einstein Telescope[8], a European-led effort with
triangular interferometer topology and 10 km long arms. These next generation
detectors, in conjunction with the LISA mission[9], seek to answer questions
about the universe by observing GW sources out to cosmological distances with
extreme precision[10]. To achieve their proposed sensitivities, headway must
be made in understanding and reducing technical noises which currently limit,
or lie just below, the noise floor of current detectors.

Laser frequency noise is one technical noise fundamental to laser
interferometer GW detectors. The current Advanced LIGO input optic design is
motivated largely by frequency noise suppression, including a 16 m suspended
input mode cleaner[11, 12]. At GEO600, two 3.9 m input mode cleaners are
employed in series, primarily for additional spatial stability[13].

In this paper we present the current understanding of laser frequency noise in
Advanced LIGO, which lies within a factor of two of the O3 Advanced LIGO noise
floor above a few kHz[14]. We propose a new laser frequency noise suppression
scheme which would reduce this otherwise limiting noise below the proposed
Cosmic Explorer sensitivity. The proposed scheme utilizes two suspended
cavities to mitigate frequency noise incident on the interferometer.

##  2 Frequency noise in Advanced LIGO

In this section we explore how frequency noise affects GW sensitivity in
Advanced LIGO. Here we overview the frequency noise control loops employed,
the coupling of frequency noise to differential arm motion (DARM), and the
measured Advanced LIGO frequency noise budget.

This information will inform our frequency noise requirements for third-
generation interferometers (Section 3), and motivate our proposed input optics
design which satisfy those requirements (Section 4).

###  2.1 Frequency noise suppression

Figure 1:  Layout of the laser frequency noise suppression system in Advanced
LIGO. Laser beams are shown in red. Electronics for the reference cavity are
shown black, electronics for the input mode cleaner are dark green, and
electronics for the common-arm length of the interferometer are mint green.

Frequency noise must be suppressed to lock the laser to the interferometer,
and to avoid frequency noise limiting the GW sensitivity. The laser frequency
noise suppression scheme in Advanced LIGO consists of three main optical
elements: the reference cavity, the input mode cleaner, and the common arm
length. Each optical element is locked via the Pound-Drever-Hall (PDH)
reflection locking technique [15]. The suppression scheme is shown in Fig. 1.

####  2.1.1 Reference cavity

A non-planar ring oscillator (NPRO) laser is amplified to 80 W before passing
through a pre-mode cleaner (PMC). In transmission of one of the PMC’s highly
reflective mirrors, a low-power beam double passes an acousto-optic modulator
(AOM), acquiring a frequency offset relative to the main interferometer light.
The frequency-shifted beam then resonates in a 20 cm fixed-spacer reference
cavity. The reflected light contains the reference cavity length signal, which
is fed back to the NPRO frequency actuators: the crystal temperature, a piezo-
electric transducer (PZT), and a phase correcting Pockels cell (EOM). The NPRO
laser frequency is locked to the reference cavity length with a 500 kHz
bandwidth servo, achieving a frequency noise level of about $0.01~{}{\rm
Hz}/\sqrt{\rm Hz}$.

####  2.1.2 Input mode cleaner

Most of the power transmitted through the PMC passes through an electro-optic
modulator (EOM), where phase-modulation RF sidebands are added to the beam.
One set of RF sidebands is used to lock the input mode cleaner (IMC), while
another set is transmitted through the IMC and used to lock the common mode of
the main interferometer.

The beam then propagates to the input mode cleaner, which provides the second
stage of frequency noise suppression. The laser frequency is locked to the
input mode cleaner length at 100 kHz bandwidth, with “fast” actuation applied
to the AOM on the reference cavity path. The performance of the IMC is shown
as the blue trace in Fig. 2, where a frequency noise level of about $1\times
10^{-4}~{}{\rm Hz}/\sqrt{\rm Hz}$ is achieved.

####  2.1.3 Common-arm cavity length

The common-arm (CARM) coupled cavity is formed by the arm cavities and the
power-recycling mirror in the main interferometer. The CARM length is the
average length of the two arms $L=(L_{x}+L_{y})/2$. CARM is detected via the
reflection from the interferometer. The CARM length signal forms the final
stage of laser frequency actuation with a servo bandwidth of 15 kHz.

There are two feedback paths in the CARM loop, the “slow” IMC length path and
the “fast” laser frequency path. Below 100 Hz the CARM feedback goes to an
input mode cleaner optic suspension. This forms the “slow” CARM feedback loop,
which indirectly actuates the laser frequency via the input mode cleaner
servo. This “slow” feedback allows the laser light for the interferometer to
pass through the IMC as well.

At frequencies above 100 Hz, the CARM feedback is summed with the input mode
cleaner error signal. This forms the “fast” CARM feedback loop, which actuates
the laser frequency directly. The function of the “fast” loop requires that
the RMS frequency correction be less than the IMC bandwidth of 8.8 kHz, in the
case of Advanced LIGO.

A cavity is a frequency sensor below the cavity pole, but a phase sensor
above. The CARM double cavity pole $\omega_{\rm cc}$ of the Advanced LIGO
interferometer is 0.6 Hz, which makes the reflection detector a phase sensor
over the entire frequency band of interest. This low double cavity pole
degrades the interferometer’s shot-noise limited sensitivity to frequency
noise as the audio frequency $f$ increases. The performance of the CARM shot
noise is shown as the red trace in Fig. 2, where a phase noise level of about
$3.5\times 10^{-9}~{}{\rm rad}/\sqrt{\rm Hz}$ is achieved.

###  2.2 Frequency noise coupling to DARM

An ideal Michelson interferometer with equal arm lengths will reject any
frequency noise on the input light away from anti-symmetric port where the GW
signal light exits the interferometer. This property of Michelson
interferometer is known as “common-mode rejection”. Advanced LIGO employs a
dual recycled Michelson interferometer with arm cavities. An imbalance in the
arm cavity storage times or an imbalance in their reflectivity will weaken the
common-mode rejection[16, 17, 18, 19, 20].

The couplings due to interferometer imbalances are propagated by the
fundamental optical mode in the interferometer and experience the double
cavity pole as well as the extraction through the signal recycling cavity. If
the coupling is expressed in displacement sensitivity, the latter term
cancels, since a differential arm length change has to go through the same
signal recycling cavity to reach the photodetector at the antisymmetric port.
In Advanced LIGO, at frequencies below 20 Hz additional terms arise from
radiation pressure noise, but they tend to be easy to suppress.

The above coupling terms can be written as:

$\dfrac{\Delta L_{-}}{\Delta\nu}(f)=\dfrac{\lambda}{2\omega_{\rm
c}}\dfrac{1+r_{\rm a}}{r_{\rm a}^{\prime}}\dfrac{1}{1+s_{\rm cc}}\left[-\delta
r_{\rm a}-\dfrac{\delta\omega_{\rm c}}{\omega_{\rm c}}(1+r_{\rm
a})+\dfrac{l_{\rm sch}r_{\rm a}\omega_{\rm c}}{c}\left(1-\dfrac{s_{\rm
c}}{r_{\rm a}}\right)(1+s_{\rm c})\right]+k_{\rm HOM}$ (1)

where $L_{-}=L_{x}-L_{y}$ is the differential arm length, $\Delta\nu$ is the
amplitude of the laser frequency noise, $\lambda$ is the laser wavelength,
$r_{\rm a}\approx 1$ is the average arm cavity reflectivity, $r_{\rm
a}^{\prime}$ is the derivative of the arm reflectivity with respect to the
round trip phase, $\omega_{\rm c}$ is the average angular frequency of the arm
cavity pole, $\delta r_{\rm a}$ is the difference in reflectivity between the
two arm cavities, $\delta\omega_{\rm c}$ is the difference in the arm cavity
pole frequency, $l_{\rm sch}$ is the Schnupp asymmetry[21], and $k_{\rm HOM}$
describes the coupling through higher order optical modes. The frequency
dependence is contained in the cavity pole terms, $s_{\rm c}$ and $s_{\rm
cc}$, which are defined as

$\displaystyle s_{\rm c}$ $\displaystyle=i\dfrac{\omega}{\omega_{\rm c}},$
$\displaystyle\quad s_{\rm cc}$ $\displaystyle=i\dfrac{\omega}{\omega_{\rm
cc}}$ (2) $\displaystyle\omega_{\rm c}$ $\displaystyle=f_{\rm
FSR}\log\left(\dfrac{1}{r_{i}r_{e}}\right),$ $\displaystyle\quad\omega_{\rm
cc}$ $\displaystyle=f_{\rm
FSR}\log\left(\dfrac{1+r_{p}r_{i}}{r_{i}r_{e}+r_{p}r_{e}(r_{i}^{2}+t_{i}^{2})}\right)$
(3)

where $\omega$ is the audio angular frequency, $\omega_{\rm cc}$ is the double
cavity pole, $r_{p}$ is the power recycling mirror amplitude reflectivity,
$r_{i}$ is the input test mass amplitude reflectivity, and $r_{e}$ is the end
test mass amplitude reflectivity[17, 18, 19, 20]. Parameters for LIGO Hanford
during the third observation run are listed in Table 1 [22, 14].

Table 1:  Interferometer parameters for Advanced LIGO and Cosmic Explorer. Most values are design parameters with the exception of the rows marked by (*), where the Advanced LIGO parameters were measured with the detector at the LIGO Hanford Observatory, and where the Cosmic Explorer parameters are estimates.  Parameter | Symbol | Values | Units
---|---|---|---
LIGO | CE
Arm length | $L$ | 4 | 40 | km
Arm cavity pole | $\omega_{\rm c}/2\pi$ | 43.9 | 4.2 | Hz
Arm cavity pole difference∗ | $\delta\omega_{\rm c}/2\pi$ | 2.4 | 0.04-0.2 | Hz
Double cavity pole | $\omega_{\rm cc}/2\pi$ | 0.6 | 0.06 | Hz
Arm reflectivity | $r_{\rm a}$ | 0.9989 | 0.9936 | –
Arm reflectivity difference∗ | $\delta r_{\rm a}$ | 31 | <5000 | ppm
Arm reflectivity derivative wrt phase | $r_{\rm a}^{\prime}$ | 272 | 284 | –
Schnupp asymmetry | $l_{\rm sch}$ | 8 | 1.4 | cm
HOM coupling∗ | $k_{\rm HOM}$ | 0.2–1 | 0.2–1 | fm/Hz
Differential readout pole | $\omega_{\rm rse}/2\pi$ | 410 | 825 | Hz

The coupling through the difference in the arm cavity pole frequency tends to
dominate over the difference in reflectivity or the Schnupp asymmetry.
Experimentally at LIGO Hanford in O3, we found that the difference in the arm
cavity poles leads to a good approximation of the laser frequency noise
coupling below 1 kHz.

At higher frequencies a flat term dominates the coupling which is due to
higher order optical modes. Higher order modes on the input light are not
resonant in the CARM cavity and do not experience any frequency dependence due
to the double cavity pole. Higher order modes can be transformed back to the
fundamental mode by a mode mismatch between the two arm cavities. A
differential mode mismatch will inject this light directly into the signal
recycling cavity, where it will experience the same extraction efficiency to
the anti-symmetric port as light from a differential Michelson arm length
change. This light also has the same frequency dependence as a differential
arm length change, so this effect will show as an additional flat term, when
the coupling is expressed in displacement sensitivity. The correct coupling
factor is difficult to predict, since it depends on the exact optical mode
composition of the input light as well as the modal difference between the two
arm cavities. Experimentally at LIGO Hanford in O3, we found this term to be
between $0.2-1\times 10^{-15}$ m/Hz, with the exact number depending on the
thermal state of the interferometer which we can partially control[23].

###  2.3 Frequency noise budget

Figure 2:  The LIGO Hanford observation run three frequency noise budget. The
measured frequency noise incident on the interferometer, as measured by the
common mode control signal, is the blue curve. The measured frequency error
signal suppressed by the common mode servo is the green curve. Red shows the
shot noise estimate for the interferometer reflection sensor, while orange
shows the shot noise estimate for the input mode cleaner. Input mode cleaner
shot noise is witnessed by the interferometer as true frequency noise, and is
suppressed to the level seen in purple. Interferometer reflection shot noise
is the ultimate limit for the laser frequency control loop. The dashed brown
curve depicts the amount of frequency noise that would produce a signal in the
gravitational wave readout channel that is equivalent to the Advanced LIGO
design sensitivity. The true frequency noise needs to be significantly below
this curve to prevent any degradation to the sensitivity. The dashed pink
curve represents an estimate of the frequency noise requirement in Cosmic
Explorer from Eq. 4. It includes a factor of ten safety margin.

Fig. 2 shows the frequency noise measured in the Advanced LIGO interferometer
[22]. The frequency noise incident on the interferometer is suppressed by the
input mode cleaner servo to a level around $8\times 10^{-5}~{}{\rm
Hz}/{\sqrt{\rm Hz}}$, limited by the shot noise of the input mode cleaner
photodetector. The common mode servo suppresses the frequency noise further to
a phase noise level of about $5\times 10^{-9}~{}{\rm rad}/{\sqrt{\rm Hz}}$,
limited by shot noise on the interferometer reflection photodetector.

Fig. 2 shows a curve representing the frequency noise that corresponds to the
Advanced LIGO design sensitivity. It is based on the ratio of the measured
frequency to DARM transfer function in $\mathrm{m/Hz}$ and the estimated
Advanced LIGO DARM amplitude spectral density in $\mathrm{m/\sqrt{Hz}}$. In
reality, we would like the laser frequency noise to be a good margin below the
Advanced LIGO sensitivity, so that it becomes negligible in the gravitational
wave readout channel. In Advanced LIGO, the design requirement for technical
noises, such as laser frequency noise, is to be ten times below the detector
sensitivity.

##  3 Requirements and Challenges in Next Generation Detectors

One of the main features of next generation gravitational wave observatories
is their size. Einstein Telescope[24] is a triangular design with 10 km long
arm cavities. Cosmic Explorer[10] is an L-shaped detector with 40 km long arm
cavities. The longer arm cavities will lead to better sensitivity to
gravitational waves.

In the case of Cosmic Explorer, the double cavity pole $\omega_{\rm cc}$ will
be an order of magnitude lower than Advanced LIGO. Looking at the frequency to
DARM coupling function in Eq. 1, the longer arms and reduced double cavity
pole will cancel out at GW detection band frequencies. Hence, the laser
frequency noise couplings due to the imbalances of the two arm cavities will
stay the same—assuming all other relevant parameters are unchanged. The
coupling through higher order optical modes is insensitive to the double
cavity pole and one might expect the coupling to stay the same as well.

While one might expect improvements in the optical surface quality for future
generation detectors, the test masses have to be significantly larger to
accommodate the larger optical beam size. For a design like Cosmic Explorer
the optical beam size in an arm cavity is expected to be about twice the size
of Advanced LIGO[25, 7]. Furthermore, higher circulating power and larger beam
sizes will make thermal lensing on optics in CE similar to Advanced LIGO.
Hence we do not expect significant improvements in the optical beam quality
for future detectors which employ high laser power and fused silica test
masses. We can estimate a laser frequency noise requirement for Cosmic
Explorer assuming the same coupling of $0.2-1\times 10^{-15}$ m/Hz as Advanced
LIGO.

Cosmic Explorer has a projected displacement sensitivity that is about a
factor of two lower than Advanced LIGO at high frequencies. Therefore, the
laser frequency noise requirement will be a factor of two lower than in
Advanced LIGO. An additional safety factor of ten is incorporated into the CE
requirement to ensure that frequency noise does not limit displacement
sensitivity.

If we assume the highest measured value $k_{\rm HOM}=1\times 10^{-15}~{}\rm
m/Hz$ for the frequency noise coupling through higher order modes, we estimate
the laser frequency noise requirement of Cosmic Explorer:

$F_{\rm in}(f)\leq 7\times 10^{-7}\frac{\rm Hz}{\sqrt{\rm
Hz}}\times\left|1+\frac{if}{825{\rm~{}Hz}}\right|.$ (4)

The requirement in Eq. 4 is valid for $f>10$ Hz as long as cavity pole
imbalance between the two arm cavities is not larger than 1%. Fig. 2 shows the
frequency noise requirement for Cosmic Explorer based on the parameters in
Table 1.

In Advanced LIGO, a high bandwidth servo loop is actively suppressing
frequency noise by taking the Pound-Drever-Hall error signal in reflection of
the interferometer and feeding it back to the laser[11]. There are two
problems with this suppression scheme for third-generation detectors. First,
the sensing noise for the CE reflection photodetector will be quite high for
$f>100$ Hz, which limits the amount of suppression one can achieve. Second,
the unity gain frequency of this servo is limited to about half the free
spectral range of the arm cavity[26]. In Advanced LIGO, this is about 15 kHz,
while for Cosmic Explorer the servo bandwidth would be limited to about 1.5
kHz.

Figure 3:  Proposed laser frequency noise suppression scheme for 3rd
generation gravitational wave detectors, utilizing two suspended input mode
cleaners. The pre-stabilized laser consists of a NPRO amplified to 500 W of
optical power. This beam acquires phase modulation sidebands via two EOMs. The
first input mode cleaner (IMC1) is critically coupled, the reflected beam
passes through an additional mode cleaner cavity (IMC refl) to filter out
higher order modes. The laser frequency is locked to IMC1 using a high
bandwidth servo, the actuator is the PZT inside the laser head. The second
mode cleaner (IMC2) longer than IMC1, so the two cavities have different free
spectral ranges. The length of IMC2 is locked to the laser frequency, with the
feedback applied to one of the optic suspensions.

This requires a significant change in how to suppress frequency noise from the
laser. Since we can no longer rely on an active servo to clean up the
frequency noise in the kHz region, we need to improve the filtering that is
provided by the input mode cleaner. However, the frequency noise of a laser is
very large and cannot be suppressed by a realistic optical filter alone. We
will still require active suppression with a high bandwidth servo loop. Fixed
spacer cavities are good references at very low frequencies, but are limited
in length and suffer from mechanical modes at higher acoustic frequencies that
limit their usefulness.

For future generation detectors we propose two suspended input mode cleaners
in series. The first input mode cleaner is then used to stabilize the laser
frequency with a high bandwidth servo loop, while the second input mode
cleaner is locked to the incident light with a low bandwidth servo—providing
the necessary filtering at higher frequencies. A proposed layout is shown in
Fig. 3.

We also need to achieve lower sensing noise in the photodetector in reflection
of the interferometer. Inspecting Fig. 2 we see that the sensing noise of the
Advanced LIGO photodetector mostly meets the Cosmic Explorer requirement below
100 Hz, but falls short at 1 kHz—significantly so with the high coupling for
higher order modes. To meet the requirement below 100 Hz the arm cavity pole
frequencies $\omega_{\rm c}$ of Cosmic Explorer must be matched at the 1%
level. This is five times better than the worst case we have seen in Advanced
LIGO, but in agreement with the best case.

In Advanced LIGO, the power recycling mirror is not exactly matched to the
losses of the arm cavities. This leads to a static carrier field in reflection
that does not contribute to the signal but is responsible for most of the shot
noise. Cosmic Explorer might need a set of power recycling mirrors with
different reflectivities, so one can choose the proper one to match the
losses.

Another way to improve the signal strength in reflection is to implement a
non-resonant RF sideband for the Pound-Drever-Hall reflection locking signal.
A non-resonant RF sideband will have a reflectivity close to unity and will
not suffer from the modulation index strength limitations of resonant
sidebands that leak to the antisymmetric port. Higher order modes contribute
to shot noise, but they can be stripped away with an additional mode cleaner
cavity in the reflection path.

##  4 Input Optics Design

An input mode cleaner[27] is an optical resonator with equal reflectivity of
its input and output coupler to maximize the power throughput. For our
purpose, the cavity can be parameterized by its length, $L$, and the
reflectivity of its input and output couplers, $R$. Alternatively, we can
characterize the cavity by its free-spectral-range frequency $f_{\rm{FSR}}$,
cavity pole frequency $f_{\rm{pole}}$, and finesse $\mathcal{F}$:

$f_{\rm{FSR}}=\frac{c}{2L}\quad{\rm and}\quad
f_{\rm{pole}}=\frac{f_{\rm{FSR}}}{2\mathcal{F}},\quad{\rm
with}\quad\mathcal{F}=\pi\frac{\sqrt{R}}{1-R}.$ (5)

The cavity pole frequency $f_{\rm{pole}}$ is related to the storage time
$\tau$ through $f_{\rm{pole}}=1/2\pi\tau$ and determines the frequency
response. Input laser noise transmitted through the cavity is filtered above
the cavity pole. Increasing the finesse or increasing the cavity length will
both decrease the cavity pole frequency. However, mirror losses in the cavity
[28] will ultimately limit the finesse, since they degrade the power
throughput. Fig. 4 in [29] shows the best achievable mirror losses as function
of spot size.

If we approximate the cavity round-trip losses, $\delta_{\rm RT}$, with a
power law:

$\delta_{\rm
RT}(L)\approx\delta_{0}\left(\dfrac{L}{1~{}\mathrm{m}}\right)^{\frac{1}{3}}\quad{\rm
and}\quad\delta_{0}\approx 10~{}\mathrm{ppm},$ (6)

the total loss in transmission of an input mode cleaner becomes

$\delta_{\rm tot}(L)\approx\frac{\mathcal{F}}{\pi}\delta_{\rm RT}(L).$ (7)

Figure 4:  Plots of the cavity finesse, pole, maximum power density, and
frequency noise at 1 kHz, as a function of cavity length. The blue, orange,
and green traces correspond to total transmission losses of 1, 3 and 10%,
respectively, assuming realistic mirror losses. The bottom graph shows the
radiation pressure at a fixed frequency of 1 kHz expressed in units of
frequency noise assuming 500 W of input power and mirrors that weigh 2.92 kg.
It also shows a brown trace representing the frequency noise induced by
thermal noise estimated at a frequency of 1 kHz. The shaded region is excluded
by the frequency noise requirement of future detectors. The bottom right graph
shows the power density on each mirror assuming a cavity with the minimum spot
size for the given length.

Fig. 4 shows the cavity pole frequency and the finesse of a cavity, assuming
realistic mirror losses, as function of cavity length. Since the losses
increase with spot size, a longer cavity with its larger beam size ends up
with a lower finesse than a shorter one, if we keep the power throughput the
same. For example, if we tolerate 1% transmission loss, a cavity of a 100 m
length will have a finesse of approximately 700 with a corresponding cavity
pole near 1 kHz.

Advanced LIGO is designed for arm cavity powers of 800 kW, which requires
about 120 W of light at the input. Future detectors want to quadruple the arm
powers. We assume that they will need around 500 W of input light through the
input mode cleaners. Frequency noise introduced at the input mode cleaner,
including radiation pressure noise, thermal noise, controls noise, must be
below the frequency noise requirement.

###  4.1 Radiation pressure noise

Fluctuations in photon momentum cause random forces on the test masses. This
radiation pressure noise can be written as:

$F_{\rm rad}(f)=4\frac{\sqrt{2P_{\rm cav}\frac{hc}{\lambda}}}{ML\lambda
f^{2}},$ (8)

where $P_{\rm cav}$ is the intra cavity power, $\lambda$ the wavelength of the
light, and $M$ the mass of a mirror. We obtain 2.92 kg for a fused silica
mirror that is 150 mm in diameter and 75 mm thick. Circulating powers will be
quite high with $P_{\rm cav}=\frac{\mathcal{F}}{\pi}P_{\rm in}$. If we limit
the finesse to 3000, we get numbers in the $50-500$ kW range. Radiation
pressure noise needs to be below the frequency noise requirement above 1 kHz,
since there will be no high-bandwidth CARM servo gain to suppress it. The
bottom left graph of Fig. 4 shows the radiation pressure induced frequency
noise at 1 kHz as function of cavity length assuming an input power of 500 W.
It is clearly below the requirement for any reasonable cavity length and we
conclude radiation pressure is not a relevant noise source.

The critical power for angular instabilities[30] is around 2 MW with mirrors
of the above size and mass. So, this configuration is stable. We can also
calculate the maximum power density on each mirror assuming we choose an
optical beam that minimizes the beam size on the mirrors. This is shown in the
bottom right graphs of Fig. 4. The Advanced LIGO input mode cleaner is
designed to run at a power density of 350 kW/cm2 or less. We have observed
laser induced damage on the main test masses in LIGO at much lower power
densities, around 5 kW/cm2. For future detectors, we want to keep power
densities below a MW/cm2 and preferably not higher than in Advanced LIGO.

###  4.2 Thermal noise

Another noise source is thermal noise, which is dominated by coating Brownian
noise [31, 32]. We can write:

$F_{\rm
coating}(f)=\frac{\sqrt{2}c}{L\lambda}\sqrt{\frac{4k_{B}T}{\pi^{2}f}\frac{(1+\sigma_{s})(1-2\sigma_{s})}{E_{s}}\frac{d}{w^{2}}\phi_{c}},$
(9)

where $\sigma_{s}=0.17$ is the Poisson ratio in silica, $E_{s}=71$ GPa the
Young modulus in silica, $w$ is the laser beam spot size, $d$ is the thickness
of the coating, and $\phi_{c}$ is the loss angle. If we assume a loss angle of
$\phi_{c}\approx 4\times 10^{-4}$ and a coating thickness of $d\approx
4.5~{}\mu$m, we can compute the frequency noise induced by the coating
Brownian noise at 1 kHz. We can see from the bottom left plot of Fig. 4 that
for the above coating parameters this requires a mode cleaner that is at least
50 m long to meet the frequency noise requirements.

###  4.3 Sensing noise

We typically use the Pound-Drever-Hall technique[15] to lock a cavity in
reflection. The Pound-Drever-Hall technique imposes phase modulated RF
sidebands onto the input light and detects the reflected light with a
photodiode. Since the cavity acts as an FM-to-AM converter when off resonance,
demodulating the photodetector signal at the RF frequency yields an error
signal which is proportional to the laser frequency deviation, $F(f)$. For
$f\ll f_{\rm FSR}$, the signal of a Pound-Drever-Hall locking signal can be
written as

$P_{\rm PDH}(f)=2\,\Gamma\,P_{\rm
in}\mathcal{F}\,\frac{F(f)}{f_{\rm{FSR}}}\,\frac{1}{1+if/f_{\rm{pole}}},$ (10)

where $P_{\rm in}$ is the input power, and $\Gamma$ is the modulation depth of
the RF sidebands.

An ideal input mode cleaner is critically-coupled, with equal input and output
coupler reflectivities $R$. 100% of incident carrier power is transmitted
through the input mode cleaner, with no carrier light in reflection.

In reality, cavity losses will lead to a small amount of reflected carrier
light. In Advanced LIGO, the reflected carrier light amounts to about
$\epsilon=1-2\%$ of the input light. The shot noise limited sensitivity is
then given by:

$S_{\rm PDH}(f)=\sqrt{\frac{2hc}{\lambda P_{\rm
in}}}\sqrt{\frac{3}{2}+\epsilon\frac{2}{\Gamma^{2}}}\,\left|f_{\rm{pole}}+if\right|.$
(11)

At input powers of tens or hundreds of watts, the amount of light incident on
the photodetector becomes a problem. If we adjust the modulation depth so that
the RF sideband power is roughly equal to the carrier power in reflection, we
get $\epsilon\frac{2}{\Gamma^{2}}\approx 1$. If we then further assume that
future generation input mode cleaners will be no worse than the current
generation, i.e., $\epsilon\lesssim 2\%$, the power on the photodetector
reaches $\lesssim\\!20~{}{\rm mW}$ with 500 mW of input power. Most of this
excess carrier power in reflection is due to higher order optical modes on the
input light that are rejected by the cavity.

The actual total power on the photodetector must remain below around $50~{}\rm
mW$; we have assumed $25~{}\rm mW$ for the sensing noise projections in Fig.
5. If we want significantly higher input power $P_{\rm in}$ without
overloading the photodetector, we will need an output mode cleaner in the
reflected beam path to strip away the higher order modes. Without it, any
increase of the input power requires an optical attenuator in the reflected
beam path, which destroys any increase in shot-noise limited sensitivity we
may have enjoyed. Based on these assumptions we write the following estimates:

$\displaystyle S_{\rm PDH}(f)$
$\displaystyle\approx\left\\{\begin{array}[]{ll}1\times 10^{-6}\dfrac{\rm
Hz}{\sqrt{\rm Hz}}\times\sqrt{\dfrac{1\,\rm W}{P_{\rm in}^{\,\rm
eff.}}}\times\dfrac{f_{\rm{pole}}}{1\,{\rm kHz}}&{\rm for}\quad
f<f_{\rm{pole}}\\\ 1\times 10^{-9}\dfrac{\rm rad}{\sqrt{\rm
Hz}}\times\sqrt{\dfrac{1\,\rm W}{P_{\rm in}^{\,\rm eff.}}}&{\rm for}\quad
f>f_{\rm{pole}}\end{array}\right.$ (14)

where $P_{\rm in}^{\,\rm eff.}$ is the effective input power of the $\rm
TEM_{00}$ carrier light incident on the mode cleaner which makes it through
the reflection output mode cleaner. We point out that the Cosmic Explorer
requirement from Eq. 4 will require $P_{\rm in}^{\,\rm eff.}\approx 10$ W.
Fig. 3 shows an example layout with a bowtie output mode cleaner, called IMC
refl, in reflection of the first IMC.

##  5 Controls and Performance Projections

The frequency noise in transmission of a mode cleaner cavity can be written as
function of the incoming laser frequency noise, $F_{\rm in}(f)$, the mirror
displacement noise, $F_{\rm disp}(f)=F_{\rm rad}(f)+F_{\rm coat}(f)$, the
sensing noise of the Pound-Drever-Hall reflection locking signal, $F_{\rm
sens}(f)$, and the feedback compensation network, $G(f)$.

There are two ways to achieve laser resonance in an optical cavity. First, we
can lock the laser to the cavity by actuating on the laser frequency. Second,
we can lock the cavity to the laser by actuating on the cavity length. The
loop gain and bandwidth achievable is vastly different between the two types
of locks, as are the input, displacement, and sensing noise couplings through
the mode cleaner. We refer to Appendix A for the full details.

Advanced LIGO uses an NPRO laser as its master oscillator. The NPRO free-
running frequency noise can be estimated by the following heuristic equation:

$F_{\rm NPRO}(f)=100~{}\frac{\rm Hz}{\sqrt{\rm Hz}}\times\frac{100~{}{\rm
Hz}}{f}.$ (15)

This is a good approximation in the frequency range from a few Hz up to a MHz.
In the MHz region excess noise can be observed, but this tends to be
irrelevant, since it is high above the observation band.

The Advanced LIGO laser is first locked to a fixed spacer reference cavity
with a bandwidth of 500 kHz. This setup provides a frequency noise of $F_{\rm
laser}(f)\approx 0.01~{}{\rm Hz}/\sqrt{\rm Hz}$ in the band from 100 Hz to 10
kHz. At lower frequencies, we observe excess noise due to acoustic
excitations, whereas at higher frequencies, mechanical modes of the fixed
spacer reference cavity can show up prominently. There is no real need for
reference cavity and one could lock the master oscillator directly to the
first input mode cleaner. However, this will require a mode cleaner servo with
very high gain and also high bandwidth.

For our noise projections, we define a simplified feedback compensation
network, $G(f,f_{\rm UGF},n)$, that is parameterized by the unity gain
frequency, $f_{\rm UGF}$, and the number of low frequency boost gain stages,
$n$. We write

$G(f,f_{\rm UGF},n)=\left(\frac{f_{\rm
UGF}}{if}\right)^{n+1}\left(\frac{1+\frac{5if}{f_{\rm
UGF}}}{\left|1+5i\right|}\right)^{n}\frac{\left|1+\frac{i}{5}\right|}{1+\frac{if}{5f_{\rm
UGF}}}\left(1+i\frac{f}{f_{\rm pole}}\right)$ (16)

where we added a pole at five times the unity gain frequency for a faster roll
off. We also added a zero at the cavity pole frequency to compensate for the
low pass filtering provided by the optical resonator. For long cavities with a
free spectral range not far from the unity gain frequency, we will also need
notches at the free spectral range frequencies to keep the servo stable. The
requirement of a bandwidth in the 100 kHz range makes input mode cleaners
longer than 300 m impracticable—at least for the first step of laser noise
suppression.

Figure 5:  Incident frequency noise projections for the proposed input optic
configuration with one (top) and two (bottom) input mode cleaners. The cavity
length of for the first, or only, IMC is 100 m, whereas the second IMC has a
cavity length of 330 m. The finesse is 700 for all IMCs. Solid curves are
noises associated with the first, or only, IMC, where the dashed curves are
noises from the second IMC. The purple band represents the frequency noise
requirement at the input of the Cosmic Explorer interferometer, as defined in
Eq. 4. Below a few hundred hertz the output frequency noise of the mode
cleaner(s) does not meet the requirement. A common mode servo with a low
bandwidth of around 200 Hz is required to achieve the frequency noise
requirement.

Fig. 5 shows the projected frequency noise at the input of the interferometer
after passing through one and two input mode cleaners, respectively. The
length of the first input mode is 100 m and has a free spectral range of 1.5
MHz. The second input mode cleaner is 330 m and has a free spectral range of
0.5 MHz.

The laser is locked to the first or sole input mode cleaner with a unity gain
frequency of 100 kHz and three boost gain stages. The second input mode
cleaner is locked to the laser and has a unity gain frequency of 30 Hz with
two boost gain stages. Below 1 kHz the displacement noise due to thermal noise
and radiation pressure from the first or sole input mode cleaner dominates.

Adding a second input mode cleaner greatly reduces noise at very high
frequencies. The frequency noise projection indicates that for the detection
of gravitational waves the second input mode cleaner is not strictly
necessary. In both instances, a relatively slow feedback path from the main
interferometer with a unity gain frequency around 200 Hz will be sufficient to
keep the interferometer locked.

The main interferometer is an inferior frequency sensor above 1 kHz compared
to the first input mode cleaner. Since the double cavity pole of the
interferometer is around 0.05 Hz, the interferometer common mode is
effectively a phase detector. Furthermore, the reflected carrier power is
determined by the losses in the arm cavities, which are difficult to predict
exactly, as well as optical higher order modes. However, the sensing noise of
the interferometer keeps improving towards lower frequencies, whereas the
sensing noise of the input mode cleaner stays constant. If we assume that the
input mode cleaner photodetector can be operated at about 20 times less light
power attenuation, the two sensing noise curves cross over around 200 Hz.

A second input mode cleaner has many other advantages. It will provide enough
filtering at RF frequencies to make the pre-mode cleaner unnecessary. In
Advanced LIGO the pre-mode cleaner is used to cleanup intensity and frequency
noise near the RF sidebands that are used by the Pound-Drever-Hall reflection
locking technique. A second input mode cleaner also provides additional jitter
suppression of the input beam which will most likely be required in future
detectors. Lastly, it simplifies the intensity noise stabilization, since the
two input mode cleaners provide double filtering above their cavity pole
frequencies, which is around 1 kHz for the case above.

Table 2:  Proposed Cosmic Explorer input optics controls design parameters. The following parameters are used to estimate the CE frequency noise budgets in Fig. 5. The reference cavity feedback is optional for the final frequency noise suppression scheme, as the bandwidth of the IMC1 loop can be increased to 500 kHz to achieve the required frequency noise suppression. The common-arm feedback goes directly to the laser, but must adjust both IMC lengths such that they accept the laser frequency required for the interferometer, as in Advanced LIGO.  Cavity | Length | Bandwidth | Feedback
---|---|---|---
Reference cavity | 20 cm | 500 kHz | Laser (optional)
Input mode cleaner 1 | 100 m | 100 kHz | Laser
Input mode cleaner 2 | 330 m | 30 Hz | IMC2 suspension
Common-arm cavity | 40 km | 200 Hz | Laser and both IMC suspensions

##  6 Conclusions

We have shown that future gravitational wave detectors such as Cosmic Explorer
require a different approach to laser frequency stabilization compared to
current detectors. The much lower free spectral range of the arm cavities will
make it impossible to use a fast laser servo loop from the common mode of the
main interferometer to meet the frequency noise requirements. Instead, future
detectors will have to implement one or two input mode cleaners with a length
in the range of 50 m to 300 m to suppress and filter the incoming laser
frequency noise. A single input mode cleaner of 100 m length can meet the
frequency noise requirement of future detectors. However, a second input mode
cleaner of 330 m length has many advantages in providing additional filtering
of the input light, in particular for beam jitter and intensity noise.

## Acknowledgments

LIGO was constructed by the California Institute of Technology and
Massachusetts Institute of Technology with funding from the National Science
Foundation, and operates under Cooperative Agreement No. PHY-0757058. Advanced
LIGO was built under Grant No. PHY-0823459. The authors also gratefully
acknowledge the support of the LIGO Scientific Collaboration Fellows program.

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##  Appendix A Appendix

Figure 6:  The left graph shows the servo diagram for locking a laser to a
cavity, whereas the right graph shows the corresponding servo diagram for
locking a cavity to the incoming laser light. $C_{\rm LP}(f)$ describes the
low pass filtering provided by the cavity, whereas $C_{\rm HP}(f)$ is the
corresponding high pass. $G(f)$ is the feedback compensation network designed
for locking.

Here we overview the two ways to lock a laser to a cavity via Pound-Drever-
Hall feedback. In Fig. 6, the block diagram on the left shows the control loop
with feedback going to the laser. The diagram on the right shows the feedback
going to a cavity mirror suspension. With $F_{\rm in}(f)$ the incoming laser
frequency noise, $F_{\rm disp}(f)$ the mirror displacement noise and $F_{\rm
sens}(f)$ the sensing noise, the frequency noise in transmission of the mode
cleaner can then be written as:

$F_{\rm trans}(f)=a_{\rm in}(f)F_{\rm in}(f)+a_{\rm sens}(f)F_{\rm
sens}(f)+a_{\rm disp}(f)F_{\rm disp}(f),$ (17)

For locking the laser to the cavity, we get

$\begin{split}a_{\rm in}(f)&=\frac{C_{\rm LP}(f)}{1+C_{\rm
LP}(f)G(f)},\quad\quad\quad a_{\rm sens}(f)=\frac{C_{\rm LP}(f)G(f)}{1+C_{\rm
LP}(f)G(f)},\\\ a_{\rm disp}(f)&=\frac{C_{\rm LP}(f)G(f)}{1+C_{\rm
LP}(f)G(f)}C_{\rm LP}(f)+C_{\rm HP}(f),\end{split}$ (18)

and for locking the cavity to the laser, we get

$\begin{split}\overline{a}_{\rm in}(f)&=\frac{C_{\rm LP}(f)(1+G(f))}{1+C_{\rm
LP}(f)G(f)},\quad\,\,\quad\overline{a}_{\rm sens}(f)=\frac{C_{\rm
HP}(f)G(f)}{1+C_{\rm LP}(f)G(f)},\\\ \overline{a}_{\rm disp}(f)&=\frac{C_{\rm
HP}(f)}{1+C_{\rm LP}(f)G(f)},\end{split}$ (19)

where

$\begin{split}C_{\rm LP}(f)&=\frac{1-R}{1-Re^{-2\pi
i\frac{f}{f_{\rm{FSR}}}}}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny$f\\!\\!\ll\\!f_{\rm
FSR}$ }}}}{{\enspace\longrightarrow\enspace}}}\frac{1}{1+if/f_{\rm pole}}\\\
C_{\rm HP}(f)&=1-C_{\rm
LP}(f)\quad\enspace\,\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny$f\\!\\!\ll\\!f_{\rm
FSR}$ }}}}{{\enspace\longrightarrow\enspace}}}\frac{if/f_{\rm
pole}}{1+if/f_{\rm pole}},\quad{\rm with}\\\
R&=\left(\frac{\sqrt{4\mathcal{F}^{2}+\pi^{2}}-\pi}{2\mathcal{F}}\right)^{2}\approx
1-\frac{\pi}{\mathcal{F}}.\end{split}$ (20)

The Pound-Drever-Hall reflection locking signal will experience some complex
phase dynamics at the free spectral range frequencies[33]. We can rewrite
Equation 10 as

$P_{\rm PDH}(f)=2\,\Gamma\,P_{\rm
in}\mathcal{F}\,\frac{F(f)}{f_{\rm{FSR}}}\,C_{\rm LP}(f).$ (21)

This becomes important when the unity gain frequency of the locking servo
approaches the free spectral range. The feedback compensation network will
require a steep cut-off filter or notches to keep the servo stable.