Switching times in optically bistable devices P. Kinsler, MSc Thesis, University of Auckland, (1989). Email: Dr.Paul.Kinsler@physics.org (a) Abstract (b) Table of contents (c) Conclusions Abstract ======== The mean switching times of two different bistable optical devices are calculated. The first case is optical bistability in a purely dispersive, slow medium and the next is the quantum degenerate parametric oscillator above threshold. In the dispersive case, the switching due to weakly coloured phase diffusion and amplitude noise in the pump is considered. The semiclassical equations describing the system give rise to Langevin equations on the addition of suitable noise terms. A Fokker-Planck equation is subsequently developed and then Kramers method is used to obtain an expression for the mean switching time. It is shown that as the intensity of the pump laser moves further from a critical intensity, the mean switching time increases exponentially. It is also found that an increase in noise correlation time causes a rise in the mean switching time. The theoretical expression is then shown to agree with numerical calculations and computer simulations. These results are also compared to previously published work, and an apparent disagreement is explained. The starting point of the degenerate parametric oscillator calculations are previously derived Ito stochastic differential equations. These purely quantum mechanical equations are for the subharmonic mode of the oscillator with the pump mode adiabatically eliminated. The equations are transformed to give constant noise terms, and a Fokker-Planck equation is then developed. A multidimensional form of Kramers method is then applied to obtain a formula for the mean noise induced switching time for the oscillator wen above threshold in the small noise limit. This limit is equivalent to large threshold photon number. The result shows an exponential dependence on both the noise strength, and on how far the device is above threshold. Table of Contents ================= Chapter 0 - Front matter Acknowledgements i Abstract ii List of Figures and Graphs iii Chapter I - Introduction I.1 Outline of thesis 1 I.2 Optical Bistability 3 I.3 Effects of Noise 6 I.4 Fokker-Planck Equations 7 Chapter II - Dispersive Optical Bistability II.1 The model 11 II.2 Laser noise 13 II.3 Rotating Frame 16 II.4 Adiabatic Elimination of the Intracavity Field 17 II.5 Expansion to First Order in Noise 18 II.6 Steady State Solutions 19 II.7 Stability of the Steady States 22 Chapter III - The Fokker-Planck Equation III.1 Drift and Diffusion Coeeficients 25 III.2 van Kampens method 29 III.3 Expansion to Second Order in Fluctuations 32 III.4 Diffusion Coefficient 35 III.5 Drift Coefficient 43 Chapter IV - Switching Times IV.1 Switching time formulae 45 IV.2 Intermediate Calculations 47 IV.3 Switching Time Results 56 IV.4 Numerical Simulations 60 IV.5 Further Comments 66 Chapter V - The Degenerate Parametric Oscillator V.1 The Degenerate Parametric Oscillator 69 V.2 Constant Diffusion 75 V.3 Fokker-Planck Equation for the DPO 79 V.4 DPO Switching Time 82 V.5 Comments 86 Chapter VI - Conclusion 89 Appendices A van Kampens Method: a summary of the derivation 91 B Delta Correlated Noise Terms in Numerical Simulations 97 C Device and Pump Laser Parameters 98 References 101 Conclusion ========== Two main results have been derived in this thesis. Firstly, in the case of dispersive opbtical bistability in a slowly respondong medium, and expression for the noise induced switching time was found for small noise levels when the device was close to its critical point. The result showed that the mean switching time was exponentially dependent on the difference between the pump laser intensity and the critical intensity of the device. Noise with a short correlation time was also allowed for, and an increse in the switching time with increasing noise correlation time was noted. The formula was then compared in chapter four with numerically calculated switching times and computer simulations, and was shown to be in agreement with both. In addition, the apparent disagreement between my results and previously published work [4] was discussed, and the differences explained. The second result was a calculation of the switching time in an above threshold degenerate parametric oscillator due to the effect of quantum noise. This utilised a bounded manifold of the phase space of the oscillator discovered by Wolinsky and Carmichael [19] which reduced the number of degrees of freedon in the problem, thus making it easier to treat. The variable describing the oscillator were then transformed to give a Fokker-Planck equation with constant diffusion, and a multidimensional form of Kramers method was applied. A result valid in the limit od small quantum noise was obtained. This limit is equivalent to that of large threshold photon number in the oscillator. The switching time wa then shown to be exponentially dependent on both noise strength and the pump intensity. In addition, the stability of the manifold was investigated and it was found to have unstable regions [A]. However, these regions were unlikely to significantly affect the behaviour of the oscillator in the small noise limit. [4] P. Filipowicz, J.C. Garrison, P. Meystre, E.M. Wright, Phys. Rev. A35 (1983) [19] M. Wolinsky, H.J. Carmichael, Phys. Rev. Lett. 60, 1836 (1988). [A] CORRECTION: Subsequently the manifold was in fact shown to be be stable everywhere.