Standard Green's Function Derivation of the Thermal Blooming Compensation Instability Equations

摘要

An eigenvalue equation for the temporal growth rate of the thermal blooming compensation instability has been derived recently. We present a rigorous and concise derivation of all the Green's function matrix elements for the case of arbitrary axial variation of the propagation medium's flow velocity and of the thermal blooming strength. We start with the perturbation growth equation of the high-power beam in an arbitrary Galilean reference frame, include axially varying parameters at the beginning, and employ the standard scalar Green's function method for solving the high-power and beacon propagation equations. As an added benefit, we obtain the precise definition of the elements of a high-power Green's function matrix; they are the scalar Green's function and various partial derivatives of it. While the high-power beam Green's function cannot be expressed in closed form (except in special cases, such as, constant velocity and blooming strength) the general Green's function solution serves as a useful rigorous basis for further approximations such as the WKB approximation. Finally, we obtain the losed loop compensation equation from the Green's functions of the three sub-systems: (1) the high-power beam, (2) the low-power beacon (vacuum propagation), and (3) the compensation system. (ERA citation 13:022292)