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Journal of Modern Optics

Mode Transformation Properties and Quantum Limits for a Fabry-Perot Interferometer

Alfredo Luis, Luis Lorenzo Sánchez-Soto

Published May 1, 1991• 9 citations
Publisher

Abstract

A full quantum theory of the Fabry-Perot interferometer is presented. The quantum limits on the determination of small changes in the cavity length are considered, studying the case in which the interferometer is simultaneously excited by coherent and squeezed-vacuum inputs. The contributions to the limiting resolution from photon noise and radiation pressure uncertainty are evaluated in a unified calculation by means of a unitary transformation which relates the input modes to the output modes. These results are compared with the corresponding results for the Michelson interferometer.

DOI
10.1080/09500349114550951

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A full quantum theory of the Fabry-Perot interferometer is presented. The quantum limits on the determination of small changes in the cavity length are considered, studying the case in which the interferometer is simultaneously excited by coherent and squeezed-vacuum inputs. The contributions to the limiting resolution from photon noise and radiation pressure uncertainty are evaluated in a unified calculation by means of a unitary transformation which relates the input modes to the output modes. These results are compared with the corresponding results for the Michelson 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.
We find an upper bound for the sensitivity of Fabry-P\'erot interferometric length measurements using inefficient photodetectors. This bound applies to all possible quantum-mechanical states of the light entering the interferometer. It applies to standard photodetection, differenced photodetection, and homodyne detection. In the limit of large photon number we identify the optimal detection scheme and input states. They are homodyne detection with a coherent state in one input port, and a nonvacuum quadrature squeezed state in the other. No other quantum states of the input light give better sensitivity, provided that the interferometer output is measured by either photodetection or homodyning, and that the photodetectors are inefficient.
In the present study we extend and generalize previous results for coherent-state and squeezed-state Michelson interferometer quantum mechanical uncertainties (or fluctuations), which are commonly referred to as `quantum noise'. The calculation of photon counting (PC) fluctuations in the squeezed-state interferometer is extended to fourth-order correlation functions used as the measured signal. We also generalize a `unified model' for treating both PC and radiation pressure fluctuations in the coherent-state interferometer, by using mathematical methods which apply to Kerr-type interactions. The results are more general than those reported previously in two ways. First, we obtain exact expressions, which lead to previous results under certain approximations. Second, we deal with cases in which the responses of the two mirrors to radiation pressure are not equal.