Investigation of Structure of Transformation Phase Space and Chaotic Instabilities in Tunable Cr⁴⁺:Y₃Al₅O₁₂ Laser

Abstract

Carried out preliminary investigations of the Cr⁴⁺:YAG material have shown very good prospects for using it in the master oscillator to obtain Kerr mode-locked. For this reason we have investigated theoretically and by computer simulation the formation of ultra-short pulses in our laser system using the idea of the description of ultra-short coherent optical pulses as temporal Gaussian beams analogous to complex Ermit-Gaussian beams⁴ with displaced center which are well known in the spatial beam optics. The impulse passes trough optical elements of the cavity in our model were presented by the Gaussian function. Each element of the cavity were described by a formulas of transformation of five-dimensional space: A _o, t_o, α_o, R, and ω with parameters of elements:

$$\left\{ {{A_{o(n + 1)}},{\tau _{o(n + 1)}},{\alpha _{o(n + 1)}},{R_{(n + 1)}},{\omega _{(n + 1)}}} \right\} = {f_i}\left\{ {{A_{o(n)}},{\tau _{o(n)}},{\alpha _{o(n)}},{R_{(n)}},{\omega _{(n)}}|\left[ {parameters} \right]} \right\}$$

(1)

where f_i — is the function relates the output and input of the five-dimensional space for i — element of cavity. The computer simulation have shown the presence of asymptotically stable stationary point in behavior of temporal Gaussian beam similar spatial mode structure in the resonators, when the temporal mode does not change passing through all dispersion element in laser. Our calculations show that the sign of dispersion is very important for formation of phase portrait in our laser system