It is possible to construct summations of Laguerre-Gaussian modes which have the appearance of a zero order fundamental Gaussian but which, in fact, have no zero order content. These examples have circulated informally as a warning against trusting a single beam profile measurement as to the indication of the modal content of a given beam. These 'non-Gaussian' Gaussian beams also turn out to be extremely revealing of the fundamental assumptions upon which all modal decompositions and modal-based beam quality measures are based upon. Due to the contrived nature of these beams, they are also subject to some very subtle but important theoretical errors. This paper will rigorously examine a 'non-Gaussian', Gaussian beam in terms of its amplitude and phase characteristics, propagation behavior, M<sup>2</sup> and what it reveals about modal decompositions and modal beam quality measures in general.
The ISO 11146:1999 standard has been published for 6 years and set forth the proper way to measure the M<sup>2</sup> parameter. In spite of the strong experimental guidance given by this standard and the many commercial devices based upon ISO 11146, it is still the custom to quote M<sup>2</sup> measurements without any reference to significant figures or error estimation. To the author's knowledge, no commercial M<sup>2</sup> measurement device includes error estimation. There exists, perhaps, a false belief that M<sup>2</sup> numbers are high precision and of insignificant error. This paradigm causes program managers and purchasers to over-specify a beam quality parameter and researchers not to question the accuracy and precision of their M<sup>2</sup> measurements. This paper will examine the experimental sources of error in an M<sup>2</sup> measurement including discretization error, CCD noise, discrete filter sets, noise equivalent aperture estimation, laser fluctuation and curve fitting error. These sources of error will be explained in their experimental context and convenient formula given to properly estimate error in a given M2 measurement. This work is the result of the author's inability to find error estimation and disclosure of methods in commercial beam quality measurement devices and building an ISO 11146 compliant, computer- automated M<sup>2</sup> measurement device and the resulting lessons learned and concepts developed.
We describe two methods for the spectral measurement of nonlinear absorption and refraction in reverse-saturable absorber materials. In the first, we use a picosecond optical parametric oscillator to perform Z-scan at many different wavelengths to measure excited state refraction and absorption cross sections throughout the visible. The second methods uses a chirped-pulse amplification scheme to produce 100 fs pulses at 840 nm. Focusing these into sapphire generates a white light continuum that is used as a probe in an excite-probe experiment. The excitation beam is derived from the second harmonic of the remaining 840 nm light. By measurement of the transmission spectrum of the probe as a function of excite- probe delay time, we can determine the spectral dependence of the excited-state absorption cross section. Moreover, by use of Kramers-Kronig relations, the excited state refraction can also be extracted from this data. We describe our measurements using both methods in a Zn:tetrabenzporphyrine derivative (TBP). The fact that both methods give excellent agreement not only verifies the utility of continuum measurements, but also reveals some interesting properties of the excited states of TBP.
This course covers definitions and applications of common measures of beam quality, including Brightness, Power-in-the-bucket, M<sup>2</sup>, 'times diffraction limited', Strehl ratio, beam parameter product etc. Special emphasis will be given to choosing an appropriate beam quality metric, tracing the metric to the application of the laser system, and to various conceptual pitfalls which arise in this field. This course is especially applicable to novel lasers that may not have Gaussian modes, especially high energy lasers or unstable resonators. Material presented will come from general scientific literature as well as original work done by Dr. Sean Ross and Dr. William Latham, both from the Air Force Research Laboratory Directed Energy Directorate.