Large-scale gravitational-wave detectors currently under construction such as the LIGO detectors use multiple-mirror resonant optical systems containing several surfaces at which the relative phase of interfering light beams must be controlled. We describe a tabletop experiment that demonstrates a scheme for extracting signals in such an interferometer corresponding to deviations from perfect interference

Raab, F. J.

Regehr, M. W.

Whitcomb, S. E.

Caltech Authors - Main

July 1, 1995 / Vol. 20, No. 13 / OPTICS LETTERS 1507Demonstration of a power-recycled Michelson interferometer
with Fabry–Perot arms by frontal modulation
M. W. Regehr,* F. J. Raab, and S. E. Whitcomb
LIGO Project, California Institute of Technology, Pasadena, California 91125
Received March 22, 1995
Large-scale gravitational-wave detectors currently under construction such as the LIGO detectors use multiple-
mirror resonant optical systems containing several surfaces at which the relative phase of interfering light
beams must be controlled. We describe a tabletop experiment that demonstrates a scheme for extracting
signals in such an interferometer corresponding to deviations from perfect interference.Gravitational radiation from astrophysical sources pro-
duces a strain in space transverse to its direction of
propagation. Because this strain has opposite sign
along orthogonal axes, variants of the Michelson inter-
ferometer are well suited for detection of these waves.
Such interferometers with kilometer-scale arm lengths
are now under construction by the LIGO1 and VIRGO2
collaborations.
Each LIGO interferometer will be a Michelson
interferometer with partially transmitting mirrors
interposed in the arms between the beam splitter and
the end mirrors, forming long Fabry–Perot cavities,
and with a recycling mirror between the laser and the
beam splitter3,4 (see Fig. 1). The Fabry–Perot cavi-
ties amplify the change in phase of the light returning
to the beam splitter for a given displacement of the end
mirrors. Optimum sensitivity is obtained when light
returning from the cavities interferes constructively
at the beam splitter in the direction toward the laser.
The recycling mirror improves the signal-to-shot-noise
ratio by ref lecting this beam back into the interfer-
ometer, thus increasing the power incident upon the
beam splitter. Antisymmetric motions of the cavity
end mirrors (such as those produced by a gravitational
wave) alter the interference and cause light to exit the
interferometer through the beam splitter.
High-sensitivity operation of such an interferometer
requires that light beams interfering at each of the
four partially transmitting surfaces have the correct
relative phase. Errors in the relative phases must be
sensed and corrected by a control system that adjusts
the mirror positions and the laser wavelength. A
simple technique that uses phase-modulated light has
been developed to solve this problem for a single cavity5
and serves as a basis for the work described here.
Previously constructed recycled interferometers6,7 have
relied on an external modulation technique8 to generate
one or more of the necessary signals. This technique
requires the introduction of one additional optical
surface whose position needs to be controlled.
In this Letter we describe the demonstration of a
signal extraction scheme that does not increase the
number of degrees of freedom requiring control. This
scheme is a variant of an idea proposed by Schnupp
(Ref. 9); Drever10 has proposed a similar scheme.
As shown in Fig. 2, the laser light is phase modu-
lated at frequency fmod (typically a few tens of mega-0146-9592/95/131507-03$6.00/0hertz) with a Pockels cell (PM). The Fabry–Perot
cavities in the arms of the interferometer are located at
distances lI and lP ­ lI 1 d from the recycling mirror.
The nominal arm length is L ­ sLI 1 LP dy2 (note that
LI . LP ). Light is extracted in three places and de-
tected with photodiodes (PD’s). The signal from each
photodiode is demodulated with a reference signal that
is either in phase (I) or 90– out of phase (Q) with the ap-
plied phase modulation. The nominal cavity lengths
were chosen such that, when the carrier resonates in
both the arms and the recycling cavity, the phase-
modulation sidebands resonate in the recycling cavi-
ty but are nearly antiresonant in the arms. This is
achieved when L . sk 1 1/2dcys2fmodd and slI 1 lP d .
sn 1 1/2dcys2fmodd (where k and n are integers).
We calculated the response of the interferometer con-
figuration of Fig. 2 to small changes in the positions of
the mirrors, using the approximation that the position
changes are slow compared to the equilibration time for
the light in the cavities. According to this quasi-static
model, the dependence of the outputs on the distances
between the mirrors and the laser wavelength is, near
resonance (see Fig. 2),
V1 ~ sdLI 1 dLP d 1 «1sdlI 1 dlP d ,
V2 ~ sdLI 1 dLP d 1 «2sdlI 1 dlP d ,
V3 ~ sdLI 2 dLP d 1 «3sdlI 2 dlP d ,
V4 ~ sdlI 2 dlP d 1 «4sdLI 2 dLP d .
Seismic noise drives lI , lP , LI , and LP so that they
have comparable deviations from perfect resonance.
The ek are small coefficients with absolute values typi-
cally between 0.1 and 0.001, depending on the mirror
Fig. 1. Simplified diagram of a LIGO interferometer.
Light beams are depicted with line thicknesses that indi-
cate relative optical power levels. 1995 Optical Society of America
1508 OPTICS LETTERS / Vol. 20, No. 13 / July 1, 1995Fig. 2. Signal extraction scheme for a power-recycled
interferometer that uses frontal modulation.
parameters. A computer program based on the quasi-
static model solves for the complex amplitude of each of
the three frequency components of the light (the carrier
and the two nearest rf sidebands produced by the
phase modulation) everywhere in the interferometer,
finds the corresponding demodulator outputs, and
differentiates these with respect to mirror positions.
The control system must be designed to accommo-
date the coupling of the different lengths in the output
signals. Ideally, linear combinations of these four
signals could be formed to adjust the four con-
trollable degrees of freedom independently. One
potential diff iculty is immediately apparent from ex-
pressions (1). Of the common-mode and differential
combinations of lI , lP , LI , and LP , the dependence on
the recycling cavity length slI 1 lP dy2 is not dominant
in any output, and the performance of the servo
depends on the accuracy of the coefficients of this
linear combination. An alternative technique, used in
this experiment, is to operate the servo loop feeding
back to the laser wavelength with much higher gain
than the others.11 Using V1 to generate feedback to
the laser then drives dLI 1 dLP . 2Lsdlyld suffi-
ciently close to zero that V2 is nearly proportional to
s«2 2 «1d sdlI 1 dlP dy2. In most cases of interest,
the absolute value of s«2 2 «1d is of the same order of
magnitude as «1 and «2 themselves, so this factor does
not significantly degrade the sensitivity to sdlI 1 dlP d.
A tabletop interferometer was assembled and its
response was compared to the model’s predictions.
Light from an argon-ion laser (Coherent Innova 100)
was phase modulated, spatially filtered, and power
stabilized before entering the interferometer. A
12.33-MHz phase modulation was used. We satisfied
the resulting requirement for cavity lengths of approxi-
mately 6 m sk ­ n ­ 0d on our 3.7-m table by folding
the recycling cavity and both arm cavities in half. An
asymmetry d of 30 cm was used. The Fabry–Perot
cavity input mirrors had ref lectivities of 91%. Round-
trip losses in the arm cavities were measured to be
between 0.2% and 0.25% and were caused mainly by
dust in the air and on the mirrors. The recycling
mirror ref lectivity was 82%, and the ratio of optical
power in the recycling cavity to the incident laser
power was 4, corresponding to round-trip losses in
the recycling cavity (including power def lected by the
pickoff and loss in the Fabry–Perot cavities) of 16%.The length control system is shown schematically in
Fig. 3. Interferometer mirror positions were adjusted
with piezoelectric transducers; the laser frequency
was adjusted with piezoelectric transducers on the
laser mirrors and an extracavity Pockels cell for fast
correction of the phase of the light. The demodulator
phases and amplifier gains were adjusted empirically
by optimization of the stability of the locked state.
When servo loop 4 was disabled and the perpendic-
ular Michelson arm length lP was varied slowly back
and forth, the remaining three servo loops occasion-
ally acquired lock until continued motion disrupted the
resonance. This served as the basis for an automatic
lock-acquisition circuit. When the circulating power
in either arm cavity fell below a preset threshold, this
circuit disabled the feedback to loop 4 and swept lP
with a triangle wave. Once the power in both arms
was again above threshold, the triangle wave sweep
was removed and the loop 4 servo was reengaged. An
example of this process is shown in Fig. 4.
The closed-loop response of the interferometer was
measured for loops 2, 3, and 4 of Fig. 3 and com-
pared with that of the quasi-static model. We took
each measurement by injecting a swept-frequency sine
wave into a summing node in one of the high-voltage
amplif iers driving a piezoelectric actuator. A dynamic
Fig. 3. Control loop configuration used for this experi-
ment. An optical isolator (OI) is used to extract the beam
ref lected from the recycling mirror. The feedback signal
shown schematically as driving the beam splitter actually
drove a folding mirror on the in-line arm.
Fig. 4. Time record of acquisition transient. The traces
showing stored power levels have been offset for clarity.
July 1, 1995 / Vol. 20, No. 13 / OPTICS LETTERS 1509Table 1. Sampled Values (in decibels)
of Experimental and
Calculated In-Resonance Responses a
Loop Measured
Loop Driven 2 3 4
2 19.0 22.1 26.9 24.6 23.6 23.4
3 9.86 16.4 38.3 37.7 15.4 17.8
4 27.3 31.8 24.6 25.8 36.3 36.9
aThe experimental values are in boldface type; the calcu-
lated values are in regular type. In all cases the driving
signal was a 2-kHz sine wave.
Fig. 5. Possible LIGO control configuration.
signal analyzer (Hewlett-Packard 3562A) was used to
measure the transfer function from the output of the
amplifier to the (amplified) output of the mixer. We
calculated the expected response by incorporating the
quasi-static optical model into a closed-loop control
model, using standard multivariable servo modeling
software.12 In addition to mirror ref lectivities and po-
sitions, the interferometer response depends on a num-
ber of factors such as the optical power, the efficiency
of the photodiodes, and the response of the piezoelec-
tric transducers. For each of the loops we measured
the product of these factors by misaligning mirrors
within the interferometer so that only a single Fabry–
Perot cavity was resonating. The response of this cavi-
ty together with the factors to be calibrated was then
measured and divided by the known response of a
Fabry–Perot cavity. The experimental and calculated
responses of the interferometer at a test frequency of
2 kHz are tabulated in Table 1. The agreement is rea-
sonable given the estimated experimental uncertainty
of 63 dB.
The control loop configuration (Fig. 3) that was used
in the prototype interferometer was adequate for proof-
of-principle tests and model verif ication. In fact, a
more balanced control configuration, shown schemati-
cally in Fig. 5, is better suited for use in a gravitational-
wave detector. The quadrature phase output from the
isolator is used instead of the quadrature signal from
the pickoff, since the signal at the isolator is larger
relative to the shot noise. The gravitational wave sig-
nal is taken directly from control loop 3. Additionalmodulation frequencies may also be used to obtain fur-
ther decoupling of the individual feedback loops.13
In summary, we have demonstrated a means of
extracting signals from a power-recycled Fabry-Perot
interferometer by introducing phase modulation of the
laser light and an asymmetry in the positions of the
arm cavities. The response of a tabletop prototype
incorporating this scheme agrees well with the predic-
tions of a simple model.
We thank our colleagues on the LIGO project for
their encouragement and helpful discussions during
this work. This work is supported by the National
Science Foundation under Cooperative Agreement
PHY-9210038.
*Present address, Ortel Corporation, 2015 West
Chestnut Street, Alhambra, California 91803-1542.
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Demonstration of a power-recycled Michelson interferometer with Fabry-Perot arms by frontal modulation

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Demonstration of a power-recycled Michelson interferometer with Fabry-Perot arms by frontal modulation

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