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.