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IEEE Transactions on Control Systems Technology

Frequency Locking of an Optical Cavity Using a Time-Varying Kalman Filtering Approach

S. Z. Sayed Hassen, Ian R. Petersen

Published Jun 21, 2013• 9 citations
Publisher

Abstract

The cavity frequency locking problem, which arises in the field of quantum optics, involves matching the resonant frequency of an optical cavity to that of an incoming laser. Although this problem is investigated using linear control techniques for small perturbations, in this brief, we address the nonlinear control problem of frequency locking an optical cavity. Models of the measurement nonlinearities inherent in optical cavities are derived using a singular perturbation approach and instantaneous bounds are placed on the observation errors. A convex set within which the error between the laser frequency and the cavity resonant frequency lies is then determined at each time step by solving two high-order polynomial equations. Based on a reduced-order model of the cavity, a time-varying Kalman filter is designed and simulation results are presented to validate our controller on a full-model of the cavity in the nonlinear region of operation.

DOI
10.1109/tcst.2013.2266693

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In this paper, we address the problem of frequency locking an optical cavity which arises in the field of quantum optics. The cavity frequency locking problem involves matching the resonant frequency of an optical cavity to that of an incoming laser beam. We model the measurement nonlinearity inherent in the optical cavity and place instantaneous bounds on the observation errors. A convex set within which the error between the laser frequency and the cavity resonant frequency lies is determined and we synthesize a time-varying integral LQG controller which uses information from this set. Simulation results are shown to validate our design in the nonlinear region of operation.
This paper describes a new optical cavity controller test bed system for implementing modern quantum control techniques, with an emphasis on the control of optical cavities. One such quantum control task is the frequency locking of an optical cavity. Locking an optical cavity refers to the process of matching the input laser frequency to the cavity's resonant frequency. Any deviation in the two frequencies, characterized in terms of the detuning, is undesirable. The test bed comprises an input laser, a three-mirror ring cavity, the associated optics, and a dSPACE digital signal processing system. The detuning in the system is measured in the form of an error signal, which is fed to a controller. The controller provides a suitable control input to a piezoelectric actuator mounted on one of the mirrors, altering the resonant frequency of the cavity to achieve zero detuning. The dynamics of the cavity and the piezoelectric actuator are modeled using system identification methods, an integral linear quadratic Gaussian controller is designed and implemented in dSPACE, and experimental results are presented.
This paper considers the application of Linear Quadratic Gaussian (LQG) optimal control theory to a problem of cavity locking in quantum optics. Using an equivalence between quantum LQG control and classical LQG control, this problem is reduced to a standard LQG control problem. The cavity locking problem involves controlling the error between the laser frequency and the cavity frequency. A model for the cavity system is set up and an LQG controller is synthesized. The resulting control system implemented on an experimental cavity.
Systematic approaches to the design of quantum optical systems will become more and more important as their complexity grows. One such systematic technique is the Linear Quadratic Gaussian (LQG) methodology. Here we describe the design and implementation of a LQG feedback controller in a quantum optical experiment—frequency‐locking an optical cavity to a laser. The successful implementation of the LQG design procedure to this particular problem lays the groundwork for the application of other modern control techniques to quantum optical systems. Our results are promising for future, more complex stabilisation problems in quantum optics, as the described approach is inherently multi‐variable and naturally incorporates multiple sensors and actuators as well as nested loops.
The coupling of an electric field to an atomic beam prepared in a phase-locked coherent superposition of energy eigenstates is examined for a ring-laser cavity. The active medium is assumed to be a collection of noninteracting homogeneously broadened two-level atoms whose transition is electric-dipole allowed. The atom-field interaction is treated semiclassically and the atomic coherence injected into the cavity is assumed to be phase matched to the unidirectional plane-wave cavity mode. The polarization is derived for the case in which the electric-field-mode envelope has at most linear phase variations in time and is applied to the problem of frequency locking in a cavity. The presence of the atomic coherence causes the active medium to possess an inherent polarization, i.e., a field-independent contribution. This enables an optical cavity to support laser oscillation over the entire range of relative excitations, in particular, even for strongly absorbing media. A simple linear stability analysis is performed on the electric-field envelope by describing the medium with a slowly varying polarization valid when the time scales of the atomic response are much smaller than those of the variations in the envelope. The stability of frequency-locked oscillation depends on the detuning of the cavity from line center and the strength of the injected coherence relative to that of the injected population difference.