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Unformatted text preview: JOURNAL OF MODERN OPTICS, 1987, VOL .34, NO .2, 227-255 Quantum theory of high-resolution length measurement with a Fabry-Perot interferometer M . LEY and R. LOUDON Physics Department, Essex University, Colchester C04 3SQ, England (Received 17 November 1986) Abstract. 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...
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This note was uploaded on 12/30/2011 for the course PHYSICS 7353 taught by Professor Dowling during the Fall '07 term at LSU.
- Fall '07