Bistability, self-pulsing and chaos in optical parametric oscillators

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

“…However, the presence of homoclinic orbits in the phase space of the model (1) is also one likely mechanism responsible for the occurrence of a strange attractor seen numerically at O(1) values of ε. In particular, a period-doubling route to such an attractor was computed in [4], and it is not inconceivable that aŠilnikov mechanism could be responsible for this route. (In this respect, see the exposition and references to the orginal works in [18].)…”

“…These include bistability, stable periodic pulsations, and chaos [1][2][3][4], which were observed both numerically and also experimentally. Thus, it seems appropriate to investigate mathematical properties of the model (1) that could shed light on these types of behavior, and the chaotic dynamics in particular, which we do in this letter.…”

“…where A 1 and A 2 are the slowly-modulated amplitudes of the fundamental color and its second harmonic, κ 1 and εκ 2 are the frequency mismatches, εαA 1 and εαA 2 are the losses of light in the ring cavity, and εγ 2 is the external pumping of the second-harmonic mode [1][2][3][4]. In order to derive this model, one must assume that the crystal in the cavity does not have a cubic symmetry.…”

“…Moreover, the crystal must be short, so that all the spatial effects in it, such as the modulational instability, can safely be neglected [5]. One must also assume that no modes other than a fundamental frequency of the light and its second harmonic can be present in the cavity [1,4]. Finally, one must assume that the cavity is of extremely high quality, so that both the pumping and the losses can be taken as relatively small and considered as perturbation terms.…”

Resonant cascaded down-conversion

“…(2.5) with ζ i = 0, i ∈ [0,4], which are integrated numerically using a fourth-order Runge-Kutta routine similar to the method given in [23]. The first and second nonlinearities were taken to be equal, i.e., χ 1 = χ 2 .…”