A method is developed to determine the propagation characteristics of a high power laser beam using only Plexiglas<sup>TM</sup> burn patterns. Under the assumption of an embedded Gaussian in the laser beam and using ABCD ray analysis, the fundamental laser beam characteristics, independent of the optical array, are determined. The method can be used to approximate the beam irradiance profile, peak irradiance, and beam characteristic size along the beam path, the cavity induced curvature applied to the beam, and the laser beam Rayleigh range. Finally, the beam far-field spreading angle and beam quality relative to an ideal beam are calculated. The method developed defines gross characteristics of aberrated beams generated by both stable and unstable laser designs.
The location of focal planes and waists can be calculated using the equivalent Gaussian beam method. Predictions can be made for both Gaussian and top-hat (flat phase, flat irradiance profile) beams. A Fresnel number relationship is given where the waist of top-hat beams can be either within or outside the far field region. Experimental implications of this focal plane shift for top-hat and gaussian beams are discussed. The most important implication is that calculated reference power-in-the-bucket curves, used to calculate beam quality, may be off as much as 5% for top-hat beams, when not taking into account focal shifts for some optical systems.
Laboratory-scale negative-branch unstable ring resonators were designed to simulate large Fresnel number, large-mode cross section positive-branch high-power ring lasers. These laboratory-scale lasers can be built having small-mode cross sections, long lengths, and large Fresnel numbers by incorporating imaging systems within the resonators. Large variations of Fresnel numbers can also be obtained by simple positioning of the output aperture. To verify that the diffractive modes of the high-power lasers are accu rately si mulated, ba re cavity mode discri mi nation properties of these ring resonator designs were calculated by various 3-D diffractive models. Modeling was completed for resonators with equivalent Fresnel numbers up to 1 1 and with equivalent Fresnel numbers of 10 and above using a fast Fourier transform (FFT) and a virtualsource code, respectively. Comparison of the FFT and the virtual source code results, at moderate Fresnel numbers, indicated good agreement between the different modeling techniques. Good mode discrimination was observed at half-integral equivalent Fresnel numbers, whereas poor mode discrimination was observed at integral equivalent Fresnel numbers. These results show that less computational intensive codes, such as the virtual source code, can quickly and cheaply model a resonator having a wide range of Fresnel numbers. These results also show that the equivalent Fresnel number of a near-imaging negative-branch ring resonator is a reasonably valid parameter for determining resonator diffractive phase and irradiance profiles even for resonators having large Fresnel numbers. The modeling results show that the modes of positive- and negative-branch resonators, having the same Fresnel number, behave the same to within a complex conjugate.
A one-to-one correspondence between the parameter spaces of propagated top-hat (plane wave, flat top) irradiance profiles and Gaussian beams has been proposed. The diffractive 'size' of the propagated top hat is estimated by calculating that of an equivalent Gaussian beam with the same Fresnel number. Given the propagation Fresnel number, one also knows the detailed intensity and phase profiles of the propagated top hat within the Gaussian envelope; a library of plots of beam profiles is provided for both rectangular and circular top hats propagated over a large spectrum of Fresnel numbers. For Fresnel numbers less than approximately 10, the Gaussian envelope is shown to enclose roughly 90 percent of the top hat's total power. The formalism thus allows one to perform the simple matrix manipulations of Gaussian beam propagation to determine the propagated top-hat beam envelope and then use the look-up tables of beam profiles to determine the detailed intensity and phase of the plane as it propagates through a paraxial optical train. The equivalent Gaussian method allows one to include lowest order diffraction effects when designing an optical system instead of relying solely on geometrical optics. For beams that are not significantly different from top hats, one can approximate their propagated profiles by using this method with a modified wavelength, lengthened to account for non-ideal beam spreading. Examples include propagation through a focus and a one-to-one imaging system, both encountered in ring resonator designs, and design and implementation of a multiwavelength imaging laser diagnostic optical train. A final example is extension to non-orthogonal optical systems using the analysis of J. A. Arnaud to propagate the Gaussian beam.
The ground to space laser encounters the entire atmosphere and the laser power transmittable with good beam quality is ultimately limited by this propagation. For CW lasers, the ultimate limit is thermal blooming and this problem is addressed. The concept of a critical laser power is used which is the optimum laser power which leads to the highest far-field intensity. A perfect beacon is assumed and a perfect wave front sensor is also assumed to determine the maximum intensity achievable with adaptive optics phase conjugation of the thermal blooming. An analytic model is derived and its predicted results are compared with a wave optics computer code results. The agreement is quite good and shows the parametric dependence of the critical power on laser beam size, wavelength and atmospheric absorption and path length.
The unfolded single-aperture near-imaging negative branch confocal unstable ring-resonator design is considered. Diffractive properties and mode characteristics of the moderate Fresnel-number baseline-resonator design with Neq = - 10.28 are calculated using three-dimensional FFT, FHT, and virtual source codes. The FFT results are in agreement with those from positive branch unstable resonators having the same Fresnel number. Similar modeling for resonator designs where Neq is varied from - 6.6 to - 12.0 is analyzed, and good mode discrimination is found between the dominant 1 = 0 modes and higher order modes at half integral Fresnel numbers. Both FFT and FHT diffractive codes exhibit increasing loss of accuracy when applied to resonator designs as equivalent Fresnel numbers increase above Neq = 12 - 15. The virtual-code results agree well with the FFT and FHT modeling results for Neq = - 10 to - 25.
Coupling of unstable resonators in circulant arrays is considered, in which the array is invariant with respect to the interchange of any pair of resonators. A proof-of-concept experiment performed with six series coupled CO(2) lasers with confocal unstable resonators is reported. Adjoint coupling was used with two symmetrically placed coupling apertures in each resonator output. The results of measurements of the locking range, supermode content, and beam quality for standing wave and ring resonator arrays are presented and interpreted using simplified models. It is pointed out that adjoint coupled unstable resonators provide a modular scaling configuration with modest phase-locking length control requirements.