A nonlinear hyperbolic problem arising from a question of nonlinear optics,Part II

Summary A modified analytical and numerical method is presented for a preliminary study of the problem of Graffi-Cesari in Nonlinear Optics [1, 12]. This problem concerns electromagnetic laser wave propagation in a nonlinear (quartz) crystal slab embedded between two linear media, and is a suitably simplified version of the experimental situation of Franken, Ward and co-workers (1961). A new version of an existence and uniqueness theorem is proved in a functional class which is chosen following Cesari's works on quasilinear hyperbolic systems in the Schauder canonic form, under a set of restrictions upon the admissible slab widtha of the crystal. The present method consists essentially in centering the relevant functional ball at the linear solution, and yields numerical values fora of the order of 2 mm, with a good agreement with values of quartz crystal widths used in experiments.

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Sommario Si presenta una trattazione analitica e numerica modificata, relativa al problema di Graffi-Cesari in Ottica Non Lineare, studiato in [1] e [12]. Si dimostra un teorema di esistenza, unicità e dipendenza continua (dall' onda laser incidente) per il campo elettromagnetico entro la lamina di cristallo, seguendo, come in [1], lo schema teorico elaborato da L. Cesari in [8–12]. Il presente metodo fornisce valori numerici per lo spessore ammissibile della lamina di cristallo (di quarzo) dell' ordine di 2 mm, in buon accordo con valori dello spessore delle lamine (di quarzo) usate negli esperimenti.

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Research supported by the GNFM of the CNR.