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Nonlinear Dynamics in Optical Systems

2 Dimensional Transverse Patterns in Optical Bistability

W. J. Firth, Graham S. McDonald, Andrew J. Scroggie, Luigi A. Lugiato, R. Leféver

Published Jan 1, 1992• 0 citations
Publisher

Abstract

Pioneered by Moloney and collaborators [1], transverse effects of nonlinear passive systems in a ring cavity have been the object of extensive study. Analytical investigations with diffractive coupling in one transverse dimension have demonstrated [1] that static and dynamic transverse structures occur in this system. Extension of the mean-field dispersive model developed from optical bistability to include diffraction [2] independently demonstrated qualitatively similar features, again in one transverse dimension. Investigations in two transverse dimensions have been largely confined to numerical studies by the Moloney group [1], showing for gaussian beam illumination ring structures which develop through azimuthal instability to a pattern of spots which typically execute a slow chaotic motion with a progressive loss of symmetries.

DOI
10.1364/nldos.1992.wa6

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Pioneered by Moloney and collaborators [1], transverse effects of nonlinear passive systems in a ring cavity have been the object of extensive study. Analytical investigations with diffractive coupling in one transverse dimension have demonstrated [1] that static and dynamic transverse structures occur in this system. Extension of the mean-field dispersive model developed from optical bistability to include diffraction [2] independently demonstrated qualitatively similar features, again in one transverse dimension. Investigations in two transverse dimensions have been largely confined to numerical studies by the Moloney group [1], showing for gaussian beam illumination ring structures which develop through azimuthal instability to a pattern of spots which typically execute a slow chaotic motion with a progressive loss of symmetries.
Transverse effects are known to play a central role in nonlinear-optical systems.Since experiments are generally performed with lasers with finite transverse dimensions, the transverse nature of the field must be considered.Experimental evidence, for example, indicates that mechanisms in the form of diffusive and diffractive coupling within the medium, free-space diffraction, or some combination of these, can create transverse structures which are important in optical bistability and self-focusing/defocusing.In particular, transverse coupling is attributed to such phenomena as radial dependence of the switch up times of bistable loops, spatial hysteresis, and the formation of solitary waves in passive systems.The dynamical effects of free-space diffraction in a unidirectional ring cavity containing a thin resonant absorber driven by an external coherent signal is considered.Minimal focusing effects exist in this system by incorporating a thin medium whose width is small compared to the cavity length and Raleigh range.There is no restriction on the medium response time relative to the round trip time; i.e. the time-dependent polarization is not adiabatically eliminated from the equations of motion.Free-space diffraction effects are isolated and diffraction and diffusive coupling within the medium is negligible.In this model where free-space diffraction is the dominant mechanism for changing the field profile in the system, shifting, deformation, and radial variation of the bistable loop and spatial hysteresis is attributed to an inherent dispersion induced by the free-space diffraction.
A ring-cavity system, filled with an isotropic nonlinear medium and driven by a linearly polarized coherent input field, has been the subject of intense study over the last decade. This system can show a number of non-equilibrium phase transitions, including optical bistability (OB), and transverse pattern formation. The instability responsible for pattern formation then simply serves to drive the system from the unstable lower branch to the stable upper branch. This conclusion, however, is based on the assumption that the internal field in the nonlinear ring-cavity preserves its polarization state as that of the input field. Here we extend the mean field model to include polarization effects.>
A nonlinear oscillator model is used as a prototypical medium for intrinsic optical bistability.1 The phenomenon of intrinsic optical bistability is so named because the medium does not require the optical feedback of a cavity. In this model we retain dispersive effects and introduce a diffractive coupling of the electromagnetic field amplitude. The threshold of the optical bistable device is insensitive to the Fresnel number, which determines the coupling between field amplitudes at different transverse positions.2 This is useful for obtaining reproducible operating characteristics of an array of devices. The medium can be introduced inside a cavity to understand the role of dispersive and diffractive effects in a more fundamental way. The threshold intensity is again insensitive to the Fresnel number. Furthermore, we observe instabilities in the cavity configuration, but no chaotic solutions were found. The periodic instabilities are sensitive functions of the Fresnel number.
Two-dimensional (2D) ``chaotic'' localized structures are observed in the transverse profile of light with a delay model of a ring cavity with strongly dispersive two-level atoms. Near nascent optical bistability, the 2D homogeneous solution undergoes a subcritical transition to a spatiotemporal chaos. The inhomogeneous pattern coexisting with the homogeneous solution is filled with large-intensity peaks, oscillating with time. In the subcritical domain, localized structures are observed, with isolated peaks more stable and more coherent than the ones in the filled structure. The order is easily recovered by a filtering procedure that reduces the wave-vector bandwidth.
Of the Lorenz-type instabilities expected in optical systems, the single-mode instability in optical bistability has been the subject of detailed theoretical and experimental work by our groups. A treatment employing the plane-wave and mean-field limits of the Maxwell-Bloch equations predicts a rich variety of higher-order bifurcations including period doublings and intermittency.1 An extension of this work to include the transverse dependence of the intracavity field reveals no significant qualitative differences between the plane-wave theory and the Gaussian beam theory as regards the stability boundaries obtained from a linearized analysis.2 The instability persists in the presence of a Gaussian transverse profile and the boundaries are not radically modified compared with those obtained from the plane-wave theory. However, numerical integration of the Maxwell-Bloch equations in the single transverse mode approximation does not show the wealth of higher-order dynamic states present in the plane-wave case. An experimental investigation of optical bistability for two-level atoms in a single transverse, single longitudinal mode cavity exhibits limit cycle oscillation with stability boundaries that agree well with the boundaries obtained from the work in Ref. 2. However, for the range of values C < 300, |Δ| < 5, |θ| < 50, X ~ 102, the experimental results do not show the diversity of higher dynamic states expected from the plane-wave theory, in agreement with our numerical results. In addition the measurements support the existence of a single transverse mode oscillation. The sum of our theoretical and experimental work provides a quantitative statement for the role of the transverse effects in optical instabilities.