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Since ampli cation appears to be independent of the pump-probe beam angle in the above formula, 4WM interaction could permit to amplify complex incident and conjugate wavefront carrying spatial informations However, it will be shown that with real materials there often exist an optimum range of pump-probe beam angles due to the physical mechanisms involved for the generation of the photoinduced grating index modulation This angular bandpass will limit the number of transverse modes that can be ampli ed or conjugated, and it determines the space bandwidth product of this new type of image ampli er As an example, stimulated Brillouin scattering and photorefractive phenomena at zero applied eld are more ef cient for high spatial frequency gratings arising from interference of contrapropagating beams On the other hand, other effects will require large grating periods due a low beam angle between probe and pump.

An important consequence of the 4WM interaction is its capability of realizing a laser cavity including a phase conjugate mirror having a continuous gain as demonstrated rst by Feinberg and Hellwarth [19] with BaTiO3 crystal and shown in Fig 15 A classical mirror of amplitude re ectivity r is introduced on the path of the incident probe; and due to the phase conjugate properties of the nonlinear mirror whose re ectivity can be higher than unity, there can be a stable oscillation buildup in the cavity The beam oscillating in the cavity starts from the coherent noise due to the pump beams E1 and E2 Its frequency is equal to the pump frequencies v0 in the case of a degenerate interaction and is independent of the distance between the mirror and the nonlinear medium.

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, Steffenburg, S, & Kyllerman, M (2004) Motor impairments, neurological signs, and developmental level in individuals with Angelman syndrome Developmental Medicine and Child Neurology, 46, 239 243 Beebe, D W, Ris, M D, Armstrong, F D, Fontanesi, J, Mulhern, R K, Holmes, E, et al (2005) Cognitive and adaptive outcome in low-grade pediatric cerebellar astrocytomas: Evidence of diminished cognitive and adaptive functioning in national collaborative studies Journal of Clinical Oncology, 23, 5198 5204 Beilock, S L, Rydell, R J, & McConnell, A R (2007) Stereotype threat and working memory: Mechanisms, alleviation, and spillover Journal of Experimental Psychology: General, 136, 256 276 Beitchman, J H, Brownlie, E B, Inglis, A, Wild, J, Ferguson, B, Schachter, D, et al (1996) Seven-year follow-up of speech/language impaired and control children: Psychiatric outcome.

can be considered for analyzing the behavior of a system Among these different properties or views, there are again parallelism and non-determinism As for temporal logics, two families of relations of behavioral equivalences exist: the equivalences of traces family, which considers the execution of a system to be the set of its sequences of executions (linear temporal logics) and the bisimulations family, which considers the execution of a system to be a tree of executions (branching time temporal logics) Again, these two families can also be de ned either by using the states (state-based temporal logics) or the events (event-based temporal logics) that constitute the execution To present these approaches and families, the rst section informally introduces various possible solutions that can be used to compare the behaviors of two systems The second section presents the concepts of bisimulation and simulation, together with the associated decision procedures.

This experiment is generally realized with interfering pump and probe beams having the same state of polarization If this condition is not satis ed due to the optical components that introduce both spatial aberrations and depolarizations, a special experimental arrangement is required to process the two polarization states for obtaining vectorial phase conjugation 123 Self-pumped phase conjugation.

The third section addresses the notion of weak equivalences to compare systems described at various levels of abstraction Finally, the last section shows the links that exist between the behavioral approach and the logical approach: in particular it introduces the HML [HEN 85] logic, which gives a modal characterization of the bisimulation relation It will also present Browne s results [BRO 88], which give a behavioral characterization of temporal logic CT L The last part of this chapter will analyze the decidability: a) of evaluating temporal logic formulas on Petri nets; and b) of checking bisimulation of a marked net using a labeled transitions system More precisely, it will establish that, for a propositional temporal logic, the evaluation is undecidable both for the CT L and the LT L fragments This result can be extended to event-based arborescent logic.

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