PHYSICAL REVIEW A VOLUME 43, NUMBER 11 1 JUNE 1991 Fabry-Perot resonators with oscillating mirrors S. Solimeno, F. Barone, C. de Lisio, L. Di Fiore, L. Milano, and G. Russo Dipartimento Scienze Fisiche, Uniuersita degli Studi di Napoli, Mostra d Oltremare, Padiglione l9, I-80125 Napoli, Italy (Received 28 January 1991) In this paper we derive analytic expressions of the field on the mirror surfaces of a pendular and misaligned Fabry-Perot resonator by taking into account longitudinal and transverse optical modes. In particular, we obtain analytic expressions of the spectral components of the power reAected by these devices for oscillating mirrors and modulated laser beams. In addition, we discuss the equa- tions of motion of the mirror holders and analyze the onset of instabilities induced by the radiation pressure, by accounting for transverse optical modes and torsional oscillations of the multiple pen- dula to which the mirrors are suspended. I. INTRODUCTION Fabry-Perot resonators are playing a major role in long baseline interferorneters used for detecting gravitational waves (GW). Following the idea first illustrated in a pa- per by Gertsenshtein and Pustovoit' in 1963 and ana- lyzed by Weiss in 1972, Drever built in 1980 a GW detector by placing two Fabry-Perot (FP) cavities in the arms of a Michelson interferometer. Presently, two pro- totype Fabry-Perot interferometers are working at the University of Glasgow, Scotland and at California Insti- tute of Technology, Pasadena, California. In order to reduce the seismic noise, the mirrors of projected GW an- tennas will be fixed to suitably suspended masses. Con- sequently, typical features of these detectors will be lerigths of about 3 km. and mirrors attached to mechani- cal pendula. 4 The mirrors of these long FP resonators must be kept aligned to high precision with the laser beam in order to achieve a projected accuracy of 10 ' in the measure of the relative variation of the distance of the test masses. Proper alignment of the input laser beam means that it couples completely to the fundamental spatial mode of the cavity and not at all to the higher-order ones. Trans- verse displacement and mismatch of the beam with respect to the cavity axis and waist size give rise to in- phase coupling to the first- and second higher-order modes of the cavity. On the other hand, angular misalignment and waist translations couple these modes in quadrature. All these effects reduce the detection sen- sitivity of GW signals. The solution to the alignment problem of FP cavities is provided by servo systems using alignment and matching error signals obtained by modulating the laser beam and/or the mirror positions at suitable frequencies and monitoring with coherent detection techniques the inten- sity of the rejected and/or transmitted beams. A systematic analysis of the excitation of the cavity modes can be carried out by representing the field inside the cavity and that reAected from it as a combination of Gauss-Hermite modes. The beams incident on and reQected from or transmitted through the FP cavity are represented by vectors (E), whose components give the amplitudes of the above modes. In this framework, the FP cavity is represented by the matrices Xi and Xz, which transform the incident vector (e) into the vectors (E', ', K~2+') relative to the beams incident on mirrors M, and M2 respectively. The components of Xi and X2 can be calculated by means of scalar diffraction theory, which takes into account the deviation of the mirror surfaces from the ideal profiles, their finite sizes, and misalign- ments. The radiation pressure in a Fabry-Perot cavity pro- vides a spring action, observed experimentally by Dorsel et al. and discussed by Meystre et al. , which either acts against or reinforces any perturbation. A motion of the mirrors produces not only a phase change on the light emerging from it, but also an intensity change inside the cavity. The resultant change in radiation pressure will act back on the mirrors. Braginsky and Manukin first pointed out that the radiation pressure in a cavity that is not perfectly resonant will provide a spring action that acts against any perturbation, while the changing part of it tends to destabilize the system. Instability will result if this dominates the damping effect of the mirror suspen- sion. The analysis of the dynamical consequences of radiation-pressure changes has been carried out by Tourrenc, Aguirregabiria, Deruelle, Bel, and Boulanger' ' by considering a simple model of cavity with plane mirrors. They have shown that there exists a threshold power of the laser beam above which the cavity becomes unstable. In particular, Bel et aI. ' ' have asso- ciated a nonlinear difFerential equation to the retarded system, whose solution approximates asymptotically the exact one. More recently, Meers and MacDonald' have analyzed this problem by taking into account the stabiliz- ing efFects of the electronic system controlling the mirror position. All these authors have treated the cavity modes as sim- ple plane waves. Now, the question arises, to what extent do their analyses remain valid for cavities with spherical mirrors? The present paper addresses this problem by considering both longitudinal and tilting displacements of the mirror holders. For the sake of simplicity we consid- 43 6227 1991 The American Physical Society