Numerical solution of the exact cavity equations of motion for an unstable optical resonator

Abstract

We solve numerically, we believe for the first time, the exact cavity equations of motion for a realistic unstable resonator with a simple gain saturation model. The cavity equations of motion, first formulated by Siegman ["Exact Cavity Equations for Lasers with Large Output Coupling," Appl. Phys. Lett. 36, 412-414 (1980)], and which we term the dynamic coupled modes (DCM) method of solution, solve for the full 3-D time dependent electric field inside the optical cavity by expanding the field in terms of the actual diffractive transverse eigenmodes of the bare (gain free) cavity with time varying coefficients. The spatially varying gain serves to couple the bare cavity transverse modes and to scatter power from mode to mode. We show that the DCM method numerically converges with respect to the number of eigenmodes in the basis set. The intracavity intensity in the numerical example shown reaches a steady state, and this steady state distribution is compared with that computed from the traditional Fox and Li approach using a fast Fourier transform propagation algorithm. The output wavefronts from both methods are quite similar, and the computed output powers agree to within 10%. The usefulness and advantages of using this method for predicting the output of a laser, especially pulsed lasers used for coherent detection, are discussed.

<title>Effects of laser resonator parameters and gain saturation on the frequency chirp of pulsed CO<formula><inf><roman>2</roman></inf></formula> lasers</title>

“…The details of this transformation will be reported elsewhere.7 The resulting equations of motion for eh individual transverse-and longitudinal-mode expansion coefficient are similar to those derived in Ref. [5] with an additional driving term due to the injected signal. These equations represent a set of coupled first-order differential equations in time.…”

“…The two equations in Eq. [5] were solved by fmite differencing and using the Lax scheme for numerical stability. Since the time scale for changes in the gas dynamics are much longer than the kinetic and cavity field time scale, the change in the thermodynamic quantities were not updated at every kinetic time step, but rather at larger time steps.…”

Modeling of transverse-mode competition in unstable resonators with large discharge current using the exact cavity equations of motion with dynamic gain

“…A general derivation of the DCM method for application to arbitrary output mirror configuration can be found in previously reported work. 3 The transverse eigenmodes may be decomposed into azimuthal components u lp (r, z) = u lp (r, z) exp (ilθ), where l and p denote the azimuthal and radial mode index respectively. Assuming that the output coupler reflectivity and phase are exclusively radial dependent, each component is found to satisfy…”

“…3 The dynamic coupled modes (DCM) method expands the loaded cavity field into the bare cavity eigenmodes previously described by Siegman 4 and obtains the mode competition in a CO 2 laser with an unstable resonator. Although a great deal of information of the laser steady state can be obtained with this approach, the temporal evolution of the system was of secondary interest and a simple model was used for the gain temporal dependence.…”

“…Our work extends and consolidates previous studies on temporal behavior of optical resonators. 3,4,[9][10][11][12][13][14] …”