**Published:** Jan 1988

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**Source: **STP1028-EB

Thermally induced optical distortions severely degrade the quality of a laser beam thus reducing the irradiance on target. The objective of this paper is to formulate procedures for evaluating the impact of spherical aberrations generated by optically transparent cylindrical components subjected to CW laser radiation; specifically, the paper concerns edge- and face-cooled optical elements and assesses how beam shape and cooling strength affect the performance. The author first reviews the physics of thermal lensing and discusses how optical distortion coefficients relate the spatial temperature distribution to the wavefront error functions δϕ^{+} and δϕ^{-}. For semiconductors with positive thermo-optic coefficients, the analysis can be carried out on a scalar diffraction basis, which implies that the degradation in focal irradiance associated with primary spherical aberrations originates from a single quartic term, δϕ^{4}ρ^{4}. On assuming that it is a good approximation to describe the beam-induced temperature rise by means of a fourth-order even polynomial, it is shown that the amplitude-weighted variance of δϕ^{4}ρ^{4} provides a suitable measure of the degradation, which leads to the concept of a spherical aberration factor S = δT^{4}√var[ρ^{4}].The heat-flow equations for both edge- and face-cooled cylindrical components can be formulated non-dimensionally and solved exactly thus demonstrating that (a) temperature variations causing optical distortion scale with βP/K, *i.e.*, the total power deposited per unit length and the inverse thermal conductivity of the medium; (b) with the possible exception of high-truncation-parameter Gaussians, fourth-order polynomials yield reasonably accurate temperature profiles; and (c) apodization can be very useful in mitigating the deleterious effects of thermal lensing. For edge-cooled optical elements, the heat-transfer coefficient has no influence on the temperature profile, and the aperture diameter does not affect the performance. An appropriate materials figure of merit is K/(χβ^{V}),which includes the thermal conductivity K, the distortion coefficient χ, and the absorption coefficient β^{V}. The treatment of face cooling requires a more elaborate analysis and involves the Nusselt number Nu = hL^{C}/K, the sympol L^{C} referring to the characteristic length (D/2)^{2}/L. For weak cooling (Nu.⩽ 1), the spherical aberration factors are similar to those of edge cooling, but Nusselt numbers Nu ≳ 10 result in a significant reduction of the distortion, which has important implications for selecting proper operating conditions.

**Keywords:**

apodization parameter, cylindrical component, edge cooled, face cooled, laser beam, Nusselt number, optical distortion, spherical aberration, thermal lensing, truncation parameter

**Author Information:**

Klein, CA *Raytheon Company, Lexington, Massachusetts*

**Paper ID:** STP18542S

**Committee/Subcommittee:** E13.09

**DOI:** 10.1520/STP18542S