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.. ::::YS::CA .& ..V::.W PARTICLES AND FIELDS

THIRD SERIES, VOLUME 38, NUMBER 8 15 OCTOBER 1988

Recycling in laser-interferometric gravitational-wave detectors

Brian l. Meers Department ofPhysics and Astronomy, Uni Uersity of Glasgow, Glasgow 612 8QQ, Scotland

(Received 31 March 1988)

Laser interferometers may detect gravitational waves by sensing the strain in space produced by their passage. The resultant change in intensity of an interference fringe must be observable against a background noise due to the statistical fluctuations in the number of detected photons. Optimiza- tion of the detector sensitivity thus involves devising an optical system which both maximizes the signal and minimizes the noise. This is attempted in the various arrangements known collectively as light recycling. Here, the performance of these systems is quantitatively assessed. Standard or broadband recycling functions essentially by making efficient use of the available light, but it is shown that it may also be made to further enhance the sensitivity within a narrow bandwidth, becoming tuned recycling. This works, as do all the narrow-band variants, by arranging for both the laser light and a gravitational-wave-induced sideband to be resonant in the optical system. The original narrow-band system, resonant recycling, can also be made broadband; the various sensitivity-bandwidth combinations, together with the tuning properties of such a system, are dis- cussed. Furthermore, a new optical arrangement, dual recycling, is proposed. Its optical layout is an extension of standard recycling and its strength lies in its flexibility. It is shown that, relatively simply, it may be made into either a broadband or a narrow-band system, in each case with the same performance as the best of the other schemes. It may be tuned more efficiently and easily over a wide range of frequencies. Uniquely, optimum performance may be obtained with dual recycling without the requirement that the storage time of the optical elements in each arm of the interferom- eter be comparable with the period of the gravitational wave. This may allow the operation of delay line interferometers down to much lower gravitational-wave frequency and will provide great opera- tional flexibility. Finally, it is shown that dual recycling, together with resonant recycling, is rela- tively insensitive to imperfections in the geometrical quality of the optical system. When imple- mented on interferometers with lengths greater than about a kilometer, recycling should allow the attainment of the sensitivity required in order to observe gravitational waves and open up a new window to the Universe.

INTRODUCTION Photodetector

Gravitational-wave detectors based upon the use of laser interferometry to monitor the separation of widely spaced free masses are under development around the world. The kilometer scale interferometers currently be- ing proposed should open up a new field of astronomy. '1,2 An integral feature of these detectors will be the use of variants of the optical technique known as recycling, an idea first proposed by Drever. Here we quantitatively analyze these variants in a unified way, showing how they improve the sensitivity level of the detector set by photon-counting Auctuations, and propose a new variant, which may have considerable practical and operational advantages. The basic arrangement of a laser interferometric

gravitational-wave detector is shown in Fig. 1 (without mirror Mo ). Principles of operation are reviewed in Refs. 1 and 3.

Laser Seam „BS

L

cavities OF'

Delay Lines

aFIG. 1. A laser-interferometric gravitational-wave detector is essentially a Michelson interferometer. A gravitational wave produces opposite phase shifts on the light in the two arms of the interferometer; when interference occurs at the beam- splitter, the result is an intensity change on the light traveling to the photodetector.

38 2317 1988 The American Physical Society

2318 BRIAN J.MEERS 38

4lr 51 d /l cl) . g s i~g&g~2 (2)

where A, and u are the wavelength and angular frequency of the light, respectively. So the signal is a maximum if the storage time v; is equal to half the gravitational-wave

A gravitational wave (amplitude /i) of optimum polar- ization induces opposite length changes 5/ in each arm of the Michelson interferometer of magnitude

5l/1 =—,'h cosco t, where 1 is the length of the interferometer arm and cog is the angular gravity-wave frequency. The resultant phase change 5P on a single beam of light spending a time r, in the interferometer is

period —the expected result since the gravitational wave reverses its sign with period —,' vg. In practice, multipass optical delay lines or cavities are

used to match the storage time to the gravitational-wave period. ' The choice of Fabry-Perot cavities makes the analysis more complex, for there are contributions to the signal from light with storage times of a11 multiples of 2/lc. However, standard techniques for the analysis of Fabry-Perot interferometers may be used as long as the signal, while arbitrarily fast, is assumed to produce a small phase change on the light and any large offset from resonance is assumed to be slow compared with the cavi- ty storage time. Thus, the field Ez reflected off a cavity with incident field Ep, input mirror of (amplitude) transmission T„reflectance R„ far mirror reflectance R2, and free spectral range vp=c/2/ is

T,R2 e' R,R—2 i/lC0 . /tCoeE„/Ep =R,— 1— sincogt (1—R,R2) 1+F'sin 5/2

i(5—co /vo) r

5—COg /vp1+F'sin 2

/i coe

4cog

i(5+co /Uo)

5+cog /vp1+F'sin (3)

5$= a, h COg7 s cog [1+(cp r, }']' '

where the storage time is

2F/ 2/(R, Ri)'is n.c c (1—R iR2) (5)

where e is the speed of light and a„ the factor by which the field emerging from the cavity is larger than the in- cident field, is

where F'=4F /&=4RiR2/(1 —RiR2); F is known as the finesse, 5 is the phase offset (from resonance) for a sin- gle traverse of the interferometer. The different terms can be viewed as embodying the way in which the un- modulated laser light and the sidebands produced by the phase modulation of the gravitational wave resonate in the cavity. The static terms describe the effective reflectivity of the cavity, while the fluctuating terms give the signal. While the general form of (3) is useful (and shall be encountered again), simple interferometers operate with the cavities on resonance, at 5=0. The phase change of the light is then determined by the size of the fluctuating, imaginary part of the field compared with the static, real part (i.e., relative sideband ampli- tude). This may be determined by straightforward alge- braic manipulation of (3). If the length of the interferom- eter is small compared with a gravitational wavelength (so that co /vp «1), the phase shift of the emerging light 1S

T)R2 1—R )R2 1+A /T2

where A is the loss coeScient of each mirror ( T +R +A = 1 }. Note that the field inside the cavity is enhanced by cc, /T, . From now on it shall be assumed that both R i and R2 are close to 1. So for low loss cavities, as envisaged, the phase change

(4) on the light is the same for cavity with co r, & 1 as for a delay line r, =—,'vg. They will thus have equal potential sensitivities: if the smallest detectable intensity change (produced by the interference of two beams with phase fluctuation 5$) is set by photon-counting statistics, then the smallest detectable gravitational-wave amplitude is

' 1/2

/i (7) MA, b vg Vg eIpc sin(co r, l2)

fib, [l+(co r, ) ]b,vg 4m.eIpcr,

DL

(8)

where hvg is the measurement bandwidth, e, Io is the effective laser power, and A is the reduced Planck's con- stant.

BROADBAND OR STANDARD RECYCLING

It is important to realize that only a small fraction of the incident optical power is absorbed in the simple detectors considered so far. A delay line with mirrors of amplitude reflectance R

( A « 1} and N reflections has an overall intensity reflectance of

Michael Mauel

Michael Mauel

38 RECYCLING IN LASER-INTERFEROMETRIC. . .

RDL ——1—R —1——~ A2 2% C 2I while that of a cavity is [cf. (3) + algebra]

(9)

4FA /m. 1+F'sin 8/2

(10)

Thus, with /=1 km, A =10, and ~, =—,'~ =0.5 ms, the losses only total about 1.5%. With the interferometer working on a dark fringe the remaining light travels back toward the laser and, in a simple interferometer, is wast- ed. The simplest version of recycling (sometimes called "broadband" or "standard" recycling) consists of placing another mirror Mo in the beam (see Fig. 1) with the correct position to coherently send light back to the inter- ferometer. The recycling mirror may also be regarded as an impedance matching device which ensures efficient transfer of power. The resonant enhancement of the laser intensity in this recycling cavity reduces the significance of photon-counting errors: the power in- creases by the effective number of times the light is recy- cled, the shot-noise-limited sensitivity is enhanced by the square root of this factor. The increase in power inside the recycling system is limited by the losses in the system, principally the absorption and scattering of the cavity mirrors but with possible contributions from waveform distortions (limiting the efficiency with which the light from the two arms interferes to travel back toward Mo). Specifically, the maximum power gain P is

P= 1 I—R,~

Kvg

4voA

' —1/2

' 1/2 vg

1 kHz

' 1/2 I

1 km (12)