We analyze the possible operation modes of a degenerate optical parametric generator or oscillator that includes a saturable loss element in its cavity. In the case of the optical generator, we identify a domain of parameters for which the threshold for frequency downconversion is characterized by the emergence of a self-pulsing sub-harmonic mode. This mode can occur before the usual threshold for a cw subharmonic field; it can also lead to a domain of bistability between the generator and the oscillator regimes. We study the oscillator mode of operation only in the good-cavity limit, reducing the description of the system to only two real field equations. Six different bifurcation diagrams, involving monostable steady states, bistable steady states, and periodic solutions, are identified, and their domains of existence in parameter space are completely defined by inequalities involving the fixed parameters of the system. In the limit of a good frequency downconverter, passive Q switching occurs. We study the pulsed solution analytically and obtain a fairly simple equation that gives the pulse peak intensity and repetition rate. Good agreement is obtained with numerical solutions for a moderate input field.
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