We present a new optical control scheme for a laser interferometric gravitational wave detector that has a high degree of tolerance to interferometer spatial distortions and noise on the input light. The scheme involves resonating the rf sidebands in an interferometer arm cavity

Camp, J. B.

Kells, W.

Mason, J.

McClelland, D. E.

Whitcomb, S. E.

English

Caltech Authors - Main

1014 OPTICS LETTERS / Vol. 24, No. 15 / August 1, 1999Arm cavity resonant sideband control for laser interferometric
gravitational wave detectors
D. E. McClelland
Department of Physics, The Faculties, The Australian National University, Science Road, Canberra 0200, Australia
J. B. Camp, J. Mason, W. Kells, and S. E. Whitcomb
LIGO Project, California Institute of Technology, MS 51-33, 1200 East California Boulevard, Pasadena, California 91125Received May 3, 1999
We present a new optical control scheme for a laser interferometric gravitational wave detector that has a high
degree of tolerance to interferometer spatial distortions and noise on the input light. The scheme involves
resonating the rf sidebands in an interferometer arm cavity.  1999 Optical Society of America
OCIS codes: 120.3180, 120.2230.The scientif ic goal of direct measurement of gravi-
tational radiation will be pursued in the next decade
with the use of laser interferometry. Long-baseline
interferometers are under construction at a number of
locations1,2 and are expected to become operational in
the next few years. In such first-generation instru-
ments, light from a frequency and amplitude stabilized
laser is injected into an interferometer configured as
a recycled Michelson interferometer with Fabry–Perot
arm cavities; rf sideband phase modulation techniques
are used to provide a phase reference for measuring
small differential length changes and to extract length
control signals.3 The sidebands and the carrier un-
dergo different filterings in the interferometer as they
propagate to the gravitational wave (gw) signal port,
where length changes of the arm cavities that are due
to a gravitational wave may be observed.
The interferometer sensitivity is directly related to
the spatial and temporal stability of the carrier and
sideband frequencies. In the control scheme employed
by the Laser Interferometer Gravitational Wave Obser-
vatory1 (LIGO) and VIRGO,2 which we will call the
singly resonant sideband scheme (SRS), the rf side-
bands propagate only in the nearly degenerate power
recycling cavity (PRC), whereas the carrier is resonant
in both the PRC and the arm cavities. Therefore, as
explained below, the carrier receives spatial and tem-
poral filtering that the sidebands do not receive. As a
result, most gw signal noise sources related to the laser
light are dominated by sideband noise, instability, or
both. These sources include thermal lensing from ab-
sorption in the recycling cavity,4 rf intensity noise,5 and
in-band frequency and intensity noise.6
We consider here an alternative optical control
scheme that would reduce the coupling of all the above
noise terms to the gw signal output. The new ap-
proach involves resonating the rf sidebands in an arm
cavity. This achieves spatial and temporal filtering
of the sidebands, thus providing a more robust, stable
configuration.
For the analysis presented here we use the nominal
LIGO parameters: recycling mirror transmission and
radius of curvature, 3% and 10 km, respectively; in-0146-9592/99/151014-03$15.00/0put test mass transmission, 3%; optic loss, 50 parts
in 106. For the SRS system a Michelson asymmetry
d of 0.4 m is chosen to permit maximum transmis-
sion of the 25-MHz sideband frequency to the gw sig-
nal port. As shown in Fig. 1, four length degrees of
freedom (dof ’s) need to be controlled about their nom-
inal values: the arm cavity and recycling cavity com-
mon mode lengths, L1  Lx 1 Ly and l1  lx 1 ly , and
the arm cavity and recycling cavity differential mode
lengths, L2  Lx 2 Ly and l2  lx 2 ly  d, respec-
tively. Setting l1  m 1 12cfsb, where m is an in-
teger and fsb is the sideband frequency, ensures that
when the carrier resonates in the arms and in the re-
cycling cavity the sidebands will resonate only in the
recycling cavity. Control signals are obtained when
cavity length deviations from resonance cause differ-
ential phase shifts of the carrier relative to the rf
sidebands that interfere at the photodetectors as a
beating at the rf. Demodulation of the detector output
into in-phase and quadrature-phase components rela-
tive to the modulation phase then gives cavity length
control signals.
Extensive analysis has been carried out on the stan-
dard SRS scheme. Apparent gw signal noise caused
Fig. 1. Schematic layout of the LIGO interferome-
ter: ITM’s, input test masses; ETM’s, end test masses;
BS, beam splitter; RM, recycling mirror. The lengths lx,
ly , Lx, and Ly are controlled. Light is shown exiting the
interferometer at the gw signal port. 1999 Optical Society of America
August 1, 1999 / Vol. 24, No. 15 / OPTICS LETTERS 1015by coupling of laser and oscillator noise to arm cav-
ity rms length f luctuations 10213 m and imper-
fect matching of the interferometer arm ref lectivities
0.5% set requirements on the laser and oscilla-
tor stability. Column 2 of Table 1 summarizes the
requirements for LIGO to achieve a sensitivity of
10220 mHz1/2 at 100 Hz.6 Fluctuations at the gw sig-
nal port arise because of beating of the rf sidebands
with audio band noise of the carrier and beating of
carrier light with audio band noise on the rf compo-
nents.6 As the carrier light is resonant in both the
recycling cavity and the arm cavities, it propagates
in a coupled cavity with a linewidth given by fcc 
fc1 1 rrr02, where fc is the arm cavity linewidth, rr
and r0 are the recycling mirror and arm cavity ampli-
tude ref lectivities for the carrier, respectively, and fcc
is the coupled-cavity linewidth. This formula gives a
half-linewidth of 1 Hz. As a result, audio band noise
on the carrier light above 1 Hz is attenuated. The rf
sidebands, however, are not resonant in the arm cavi-
ties; they propagate only in the recycling cavity, which
has a linewidth of 75 kHz. Hence in the SRS scheme
it is the audio band noise on the rf sidebands beating
with the carrier light that dominates the noise budget.
In the initial phase of the LIGO, 6 W of laser power
at 1.06-mm wavelength will be incident upon the power-
recycling mirror. Measurements of the optics put the
absorption in the input test mass and beam splitter
substrates at the order of 5 parts in 106 per centime-
ter and coating absorption at 0.5 part in 106. Absorp-
tion in the input test mass induces a transverse and
longitudinal spatial variation in the refractive index
of the fused silica, which then acts as a lens on the
propagating laser beam. This phenomenon, referred
to as thermal lensing,4 alters the wave-front curvature
of the beam so that it no longer matches the curva-
ture of the recycling cavity. As the recycling cavity is
almost degenerate g1g2  0.997 and highly suscep-
tible to wave-front errors, the result is degraded power
buildup and transmission of sidebands. This effect
has been simulated with a computer model of the inter-
ferometer that uses a fast Fourier transform– (FFT–)
based optical propagation code,7 including the surface
figure of all optical components and optical path-length
difference maps of substrates with finite absorption.
As a measure of the effect of thermal lensing, we
consider the sideband power buildup in the recycling
cavity, given by the recycling gain Gsb, which we op-
timize by varying the Michelson asymmetry. For the
nominal parameters of the LIGO, Gsb is 30 in the ab-
sence of thermal lensing. As shown in Fig. 2, the in-
clusion of thermal lensing in the input test mass yields
a sideband gain of less than 2 for the nominal recy-
cling mirror radius of curvature (10 km). This result
presents severe difficulties for both interferometer con-
trol and signal extraction and cannot be compensated
for by a simple increase in the modulation depth. Fig-
ure 2 also shows that it is possible to recover most
of the circulating power if the curvature of the recy-
cling mirror can be altered to compensate for the dis-
torted field. This approach, however, is optimal for
a restricted range of laser power and recycling mirror
curvature. We note that a curvature variation of 5%from this value causes a further drop in Gsb of 20%.
We also find that a change of input power of just 61 W
results in a 40% change in Gsb, accompanied by a 10%
decrease in the interferometer strain sensitivity, after
the modulation depth is optimized.
It is important to appreciate that the carrier field
resonant buildup is not subject to these effects.7 The
carrier resonance in the nondegenerate arm cavities
g1g2  0.3 effectively stabilizes its TEM00 mode
such that the net double resonance is highly stable.
Extensive FFT modeling shows this clearly: For the
parameters of Fig. 2, the carrier recycling gain varies
by less than 2% in the presence of input test mass
thermal lensing. A modal picture of the mechanism
for this is evident: The overcoupled arm cavities select
the desired resonant mode TEM00 by introducing a p
phase shift on the ref lected wave. All other non-arm-
resonant modes are then antiresonant in the double-
cavity system. In contrast, at the sideband frequency
no special mode is selected by resonance in the arms, so
all modes resonate in the recycling cavity, leading to a
corresponding loss of power from the TEM00 mode.
Extending the arm cavity mode selectivity to the
sideband light renders the rf sidebands similarly im-
mune to spatial distortions. To do this we allow the
rf sideband pair to resonate in one of the arm cavities
(allowing the rf sidebands to resonate in both arm cavi-
ties8 will not provide enough signals for length con-
trol) and set l1  m 1 12c2fsb. We refer to this
Table 1. Noise Requirements at 100 Hz for
10–20 mHz1/2 gw Sensitivity
Noise SRS Value SOAS Value
Laser
frequency 2 3 1027 HzHz1/2 6 3 1026 HzHz1/2
Laser
amplitude 1027Hz1/2 3 3 1026Hz1/2
Laser rf
amplitude 8 3 10210Hz1/2 2 3 1028Hz1/2
Oscillator
amplitude 1027Hz1/2 3 3 1026Hz1/2
Oscillator
phase 2 3 1024 radHz1/2 1026 radHz1/2
Fig. 2. Sideband gain versus recycling mirror radius of
curvature in the presence of end test mass thermal lensing.
One can recover the reduced gain at the nominal recycling
mirror’s curvature (10 km) by altering the recycling mir-
ror’s curvature to compensate for the distorted field.
1016 OPTICS LETTERS / Vol. 24, No. 15 / August 1, 1999scheme as a single one-arm sideband (SOAS) scheme.
Using the FFT code, we find that, with no thermal
lensing, the SOAS sideband gain in the PRC is 22
with a Michelson asymmetry chosen to maximize the
sideband throughput. With the introduction of ther-
mal lensing, the gain falls less than 3%. Changing
the radius of curvature from 10 to 13 km has 1% ef-
fect: the SOAS gain and throughput are essentially
the same. We note that with an impedance-matched
carrier the sideband throughput obtained is 70% be-
cause of the additional losses encountered in the arm
cavity.
Clearly, using the SOAS scheme overcomes the side-
band thermal lensing problem; with the rf sidebands
resonating in an arm cavity, the recycling cavity will
no longer appear as a degenerate f lat mirror cavity.
In addition, the system is more stable against recycling
cavity geometric errors such as curvature mismatch.
FFT runs show that error tolerance on the radii of cur-
vature of the PRC mirrors can be relaxed to 50%.
The above results were calculated for an input power
of 6 W. Further research is needed to determine how
much optical power can be supported with this ar-
rangement, but we expect that a factor-of-10 increase
should be sustainable. The use of SOAS’s would there-
fore offer significantly greater robustness to an in-
crease in input power.
Furthermore, the rf sidebands in the SOAS scheme
propagate in a coupled cavity with a half-linewidth of
4 Hz (larger than the SRS carrier linewidth because
of the loss that is due to the Michelson asymmetry out-
coupling of the sidebands.) As a result, audio noise on
both the carrier and the sideband light is strongly fil-
tered above this frequency. Column 2 of Table 1 sum-
marizes the stability requirements on the laser and
the rf oscillator for the SOAS scheme for the LIGO to
achieve a sensitivity of 10220 mHz1/2. We estimated
these results by applying the technique of Ref. 6 to
the SOAS configuration. Filtering of the rf sidebands
leads to a relaxation of the requirements on the au-
dio band frequency and amplitude stability of the laser
and the amplitude stability of the RF source by a fac-
tor of 30 in comparison with the SRS method. We find
a similar factor-of-30 relaxation in the requirement of
laser rf intensity noise. However, the required phase
stability of the rf oscillator has increased by a factor of
200. This requirement arises from phase noise gener-
ated at the demodulation. In the SRS case, the phase
noise of the demodulation is precisely followed by the
phase variation of the modulation sideband frequency;
however, in the SOAS scheme the sideband noise is fil-
tered, leaving the demodulation noise to produce gw
signal noise. This requirement, which is at the perfor-
mance level of the best currently available commercial
oscillators, couples to a sideband frequency offset from
an arm cavity resonance of 10 Hz.
An important aspect of the SOAS scheme is that
signals are now no longer pure quadrature- or in-
phase signals. This is so because of the imbalance
of the arms of the interferometer for a set of rf side-
bands, which mixes the portion of the signal that
is due to phase shifts to the rf sidebands into bothin-phase and quadrature demodulations. The conse-
quence of this mixing, which depends on interferometer
parameters such as cavity coupling and optical gain,
are twofold: First, the control matrix derived from
the use of one set of rf sidebands is no longer block
diagonal (as in SRS), as common mode and differen-
tial signals are now mixed. Second, the demodula-
tion phase that maximizes or minimizes a given of
is no longer either 0 or 90 deg, and these respective
phases are different for every dof. Although it is pos-
sible to extract control signals by use of the SOAS
scheme, the effects of mixing of common and differen-
tial mode signals are not well understood. Block di-
agonalization may be recovered by introduction of a
second set of sidebands that are resonant in the op-
posite arm, which provides differential mode signals
with the same magnitude but opposite sign, and com-
mon mode signals with the same sign. We do this
by setting fsb1 1 fsb2  2m 1 1c2l2, fsb1 1 fsb2 
n 1 1cl1, and fsb1 2 fsb2  p 2 12cl1. Taking
sums and differences of the demodulated outputs then
gives back separation of the common and differential
dof ’s. However, the constraint on arm cavity length
matching to ensure signals of the same magnitude is of
the order of several micrometers, which will require ac-
tive control. Finally, the ability to tailor the demodu-
lation phase for a particular dof allows cross coupling
to other dof ’s to be minimized.
In summary, a new optical control scheme for inter-
ferometric gw detectors has been proposed, which in-
volves resonating the rf sidebands in an arm cavity.
This procedure achieves a high degree of tolerance to
interferometer spatial distortions and noise on the in-
put light.
This work was supported by the National Science
Foundation under Cooperative Agreement PHY-
9210038.
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Arm cavity resonant sideband control for laser interferometric gravitational wave detectors

Analysis of light noise sources in a recycled Michelson interferometer with Fabry–Perot arms,’’ submitted to

Modelling the performance of interferometric gravitational wave detectors with realistically imperfect optics,’’ Ph.D. dissertation

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Arm cavity resonant sideband control for laser interferometric gravitational wave detectors

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