Quantum theory of high-resolution length measurement with a Fabry-Perot interferometer

The quantum limits on measurements of small changes in the length of a Fabry-Perot cavity are calculated . The cavity is modelled by a pair of dissimilar mirrors oriented perpendicular to a one-dimensional axis of infinite extent . The continuous spectrum of spatial modes of the system is derived, and the electromagnetic field is quantized in terms of a continuous set of mode creation and destruction operators . Coherent state and squeezed vacuum-state excitations of the field are characterized by energy flow, or intensity, variables . The determination of small changes in the cavity length by observations of fringe intensity is considered for schemes in which the cavity is simultaneously excited by coherent and squeezed vacuum-state inputs . The contributions to the limiting resolution from photocount and radiation-pressure length uncertainties are evaluated . These properties of the Fabry-Perot cavity are compared with the corresponding results for the Michelson interferometer . 1 . Introduction Interest in the limiting resolutions of interferometers for measurements of small changes in length has been greatly stimulated in recent years by the development of optical methods for the detection of gravitational waves [1-3] . Most of the detailed theoretical work on the limiting length resolution has been concerned with the Michelson interferometer [4-7], but practical systems that use the Fabry-Perot interferometer are also being developed [8-10] . The main content of the present paper is a study of the quantum theory of the Fabry-Perot interferometer and its application to the measurement of length. The interferometer is here treated in isolation, and we do not consider its incorporation into a gravitational-wave detecting system . The Fabry-Perot cavity is modelled by a pair of plane high-reflectivity mirrors oriented perpendicular to a one-dimensional axis . No boundaries are placed on the axis, and the spatial modes of the cavity system accordingly have a continuous distribution of wave-vectors . The mirror reflectivities are in general allowed to be different, and the mode structure derived here generalizes earlier work [11, 12] in which one of the mirrors was taken to be perfectly reflecting . The electromagnetic field is quantized by the association of creation and destruction operators with these spatial modes. For a spatial axis of infinite extent, it is natural to work with the energy flow, or intensity, of the field rather than the photon-number variables often used in quantum optics theory . The flow variables also correspond more closely to what is measured in experimental determinations of fringe intensity, and we express the results from a simple model of photodetection in terms of these variables . 228 M. Ley and R. Loudon It is assumed throughout that the cavity is excited through one of its mirrors by a beam of coherent light with a narrow spread of frequencies . It has been pointed out by Caves [6] that the length resolution of a Michelson interferometer can in principle be improved by the injection of squeezed vacuum-state light through the normally unused input channel. We accordingly consider the effects of simultaneous excitation of the Fabry-Perot cavity through its other mirror by a beam of squeezed vacuum-state light obtained from a degenerate parametric amplifier . Small changes in the cavity length produce small changes in the Fabry-Perot fringe intensities . We treat length measurement schemes in which photodetectors are placed on both sides of the cavity with intensity data taken from one, or the other, or from the difference of the two detector readings . The inaccuracy of the length determination is produced by two factors . The first of these is the uncertainty or fluctuation in the photocount rate that occurs for the coherent input light . Its magnitude can in principle be reduced without limit by increasing the intensity of the coherent input and by increasing the degree of squeezing of the auxiliary input light . However, both these increases have the counter-effect of increasing the second contribution to the length measurement inaccuracy, which is caused by fluctuations in the cavity length associated with fluctuations in the radiation pressure . An appropriate balance between the two contributions produces a minimum length uncertainty equal to a standard quantum limit that has the same value for a range of length-measuring schemes . The main results of the paper are summarized in its final section, where a comparison is made of the length-measuring capabilities of the Fabry-Perot and Michelson interferometers . 2. Cavity model and field modes The optical system is treated as purely one-dimensional with plane-wave propagation parallel to the z-axis. The Fabry-Perot interferometer consists of two partially reflecting mirrors whose planes are at right angles to the z-axis . The details of the optical propagation within the mirrors are not important for the present study . These details can be suppressed by representing each mirror as a dielectric slab of thickness e and real dielectric constant x, taken in the limit where (-+O and K--* 00 in such a way that μ=xe (1) remains finite . The appropriate limits of standard results for a dielectric slab then give the complex amplitude reflection and transmission coefficients in the forms r=ikp/(2-iky) and t=2/(2-ikit), (2) where k is the optical wave-vector . These coefficients satisfy the usual amplitude and phase requirements for a symmetrical mirror, Ir12+It12=1 (3) and rt*+r*t=0 or argr-argt=2n . (4) They also have the additional properties t-r=1, t+r=exp(2iargt) (5) and the optical phase changes on reflection and transmission are approximately argr : nj tj and argt x 217r-1tI . (8) The Fabry-Perot cavity, represented in figure 1, has different mirrors of characteristic constants μl and μ 2 placed respectively at coordinates -2L and 22L . The cavity is conveniently specified by the position-dependent relative permittivity : K(z)=1 +μ18(z+2L)+μ25(z-2L) . (9) The mirror reflection and transmission coefficients, denoted r 1 , t 1 and r2, t2 , are defined by equations similar to (2) in terms of the mirror constants μl and μ 2 , respectively . For a fixed linear polarization, Maxwell's equations have solutions in which the electric field has a time-dependence exp (ickt) and a spatial variation described by a mode function Uk(z) that satisfies the wave equation (d 2Uk(z)/dz 2 ) + k2K(z) Uk (z) = 0 . (10) There are two solutions for each wave-vector magnitude k . We choose them so that one mode, with function Uk(z), is purely outgoing on the right of the cavity at positive z, while the other mode, with function Uk(z), is purely outgoing on the left of the cavity at negative z . These modes correspond respectively to illumination of the cavity from the left and from the right . The spatial dependences of the two kinds of mode are taken to be exp (ikz) + Rk exp (ikz) -oo<z<-4L Uk(z)= Ik exp (ikz) + J, exp(-ikz) -ZL<z<1L (11) Tk exp (ikz) Uk i I I I I4 I Rk 1L 2 <z 00 Figure 1 . Geometry of the Fabry-Perot cavity showing the two kinds of mode and notation for the mode coefficients . the Ik W i Tk Jk I I I -fL I I Ik I I J I k ' R k' I Nz Quantum theory of the Fabry-Perot interferometer 229