Expressions are summarized for a sample of propagation effects when the refractive index fluctuations deviate from Kolmogorov statistics but are described by a spectrum with a general power law. These effects include a representative amplitude effect, the log amplitude variance, and a phase effect, the coherence length. The expressions provide the numerical coefficients, wavelength dependence and path weighting in terms of the power law exponent. The numerical coefficient relating the structure function and the spectrum is also given in terms of the arbitrary power law exponent. In all these cases, the range of power laws where the relations are valid is explicitly stated. The methodology is also extended to the case of finite outer scale. The techniques can be readily extended to other propagation effects and presents a formalism for evaluating the importance to propagation of deviations from Kolmogorov statistics.

Basing laser radar systems aboard aircraft provides capability for accurate tracking of ground, airborne, and ballistic missile targets from long range. The atmospheric effects of long propagation paths impact the performance of laser radar systems in two important ways -- molecular and aerosol extinction and optical turbulence. This paper provides quantitative assessment of the impact of these effects for laser radar wavelengths beyond 1.4 microns. The treatment of round-trip turbulence on direct detection systems is believed to be the first estimate of this effect. A more complete system analysis is underway and will be described in a forthcoming paper.

A wave-optics simulation has been developed which can model and optimize the performance of three-dimensional laser guidestar systems. Important parameters of such a system are discussed. The simulation is described and used to illustrate the difference between zero-, two- and three-dimensional beacons. A number of runs are made to show the effect of varying various simulation and system parameters.

A comparison of observed and modeled isoplanatic angles using synoptic meteorological data is performed over an annual cycle during 1993 - 1994. Time series of isoplanatic angles were observed during 58 night-time episodes at the Phillips Laboratory (PL) Starfire Optical Range. Isoplanatic angles were also computed for corresponding times by integration of modeled Cn² vertical profiles derived from the Dewan and Hufnagel models. Both models used adjacent Albuquerque rawinsonde data for model input. Results show significantly smaller model error associated with the Dewan isoplanatic angles than associated with the Hufnagel isoplanatic angles. The Dewan model is also sensitive to tropopause height which is used as part of its scale length determination. Synoptic weather comparisons are accomplished for the winter period of December 1993 - February 1994. The comparisons tend to support the correlation of low isoplanatic angle with disturbed and higher isoplanatic angle with undisturbed synoptic weather patterns, respectively.

Other fluid dynamic phenomena, in addition to isotropic turbulence, can produce index of refraction fluctuations within the atmosphere. The stratified conditions in the upper atmosphere lead to phenomena such as thin shear layers of the order of a few hundred meters thick and few tens of kilometers in horizontal extent, as well as gravity wave action. Because of the large scale associated with these phenomena, their effects are often dismissed in optical propagation calculations since the major optical effect is assumed to be tilt. Fluid dynamic modeling of the coherent structure within a shear layer has been carried out for a simulated altitude of 12 km. The outer scale associated with this coherent structure is shown to have a dominant effect on the index of refraction structure function even when Kolmogorov turbulence is present. The phase distortion resulting through propagation within the fluid density field is determined. Results of these calculations are presented as well as the effect of these phase distortions on the energy distribution in the focal plane.

Predictions of atmospheric transmittance in desert aerosol environments using MODTRAN code diverge significantly from measured data. Good prediction of the desert particulate size distribution is required in order to predict atmospheric scattering and absorption parameters. It is also essential to the prediction of the aerosol atmospheric modulation transfer function which is often the dominant component of the overall atmospheric MTF. Recently, an effort to predict statistics but not size distribution according to simple weather parameters has been made for coarse desert aerosols. A quantitative analysis of the desert particulate size distribution models was also performed. In this research the size distribution parameters measured by optical counters are related to weather parameters. Known statistical and analytical models such as MODTRAN relate the size distribution parameters only to relative humidity for continental atmospheres. Although humidity has a significant role in the prediction of aerosol size statistics, other weather parameters are seen here to strongly influence also the size distribution parameters. Comparisons such as the above can be used to predict under which conditions the MODTRAN aerosol models have good or poor accuracy. It is also hoped that they will lead to improvements in MODTRAN, improving the accuracy of the humidity dependence as well as by incorporating other meteorological parameters into the MODTRAN prediction models.

We measured the energy density profile and wavefront of the laser beam through the near resonator sodium heat pipe using a linear sensor camera and a Shack-Hartmann type wavefront sensor. Development of the wavefront was obtained as a function of laser wavelength, and self-focusing and -defocusing were observed for a small laser beam diameter. The energy density profile of the laser beam shows good agreement with the result of our simulation code based on the coupled Maxwell-Density Matrix equations including transverse and time- dependent variations. For the laser beam with large diameter, change of wavefront from the convex to the concave in the shorter wavelength region was observed.

A system of flexible beam expanders is presented which allows a complete control of the focal parameters of a high power laser beam such as the diameter of the focus and the focal position towards the focusing element, independent of the distance between laser and work station. The system is based on two optical systems each of them containing one adaptive copper mirror. The first system, the beam expander which is located close to the laser, allows the control of the beam diameter on the focusing optic. In the focusing optic itself the second adaptive mirror controls the focal position. The independent interaction of these adaptive elements allows the variation of the focus radius while the focal position can be changed nearly independently around the geometrical focus position. In order to evaluate the accuracy of the calculated beam propagation data the influence of apertures and of misaligned optical elements on the one hand and the effects of thermally induced deformations of the used optical elements on the other hand have to be considered with respect to the actually measured beam propagation properties.

We have successfully deployed several fast steering mirror systems in the Uranium Atomic Vapor Laser Isotope Separation (U-AVLIS) facility at Lawrence Livermore National Laboratory. These systems employ 25 mm to 150 mm optics and piezoelectric actuators to achieve microradian pointing accuracy with disturbance rejection bandwidths to a few hundred hertz.

A Fresnel error-reduction algorithm is compared with a stationary phase method as the method for designing the mirror surface profile of a laser beam forming system. We reshaped a Gaussian laser beam into a more uniform-intensity circular beam using a fabricated aspherical mirror.

To obtain on-line information about the different processes in laser machining like cutting, welding and material removal, optical sensors are integrated in the working head of a high power CO₂ laser machining system. In order to detect the dynamic light or plasma intensity fluctuation during machining, these sensors provide the real-time signal of the vapor or plasma flame intensity in the wavelength spectrum of 200 nm to 1100 nm. Moreover, the real-time intensity of the laser power is measured in real-time with a pyroelectrical sensor. The aim is to analyze each process. With the knowledge of the specific characteristics of the process, a closed-loop control is set up. Distinguishing between continuous and pulsed processes, different control algorithms have been developed and tested. Thus, a control system based on micro-controller hardware and its theoretical background for failure detection and closed loop control is explained.

The Kurtosis is introduced as a new tool parameter to characterize the laser beam with respect to the requirements of materials processing with lasers. Especially laser welding shows that the conventional laser parameters beam radius and beam quality are not sufficient to describe the processing result appropriately. The possibility to adjust the Kurtosis as a tool parameter independent of beam radius and beam quality is demonstrated and the design of appropriate optical systems for adjustment is discussed. Experimental investigations and diagnostic measurements for CO₂-laser beams as well as the results of welding experiments performed with various differently adjusted Kurtosis curves along the beam divergence confirms the discussion. Especially the influence on the welding seam geometry and the welding depth are discussed in detail.

The paper presents applications of newly developed room-temperature mercury based photo- detectors (single-element and array type) in the measurement of infrared CO₂ laser radiation during material processing. A specially designed measuring device to determine the intensity profile of high-power CO₂ lasers in two perpendicular cross-sections with a time- resolution up to 10 kHz for a whole line-scan is presented. The data-acquisition and analyzing equipment based on a digital-signal-processor (DSP) enables us to calculate parameters which characterize the temporal stability of the laser mode on-line. Experimental results illustrate the possibilities of the device. Moreover, the time-resolved measurement of laser beam parameters during material processing serves as a tool to investigate the influence of the laser beam on the dynamic process behavior. Optical signals obtained from the laser beam welding process are correlated with laser beam power variations to determine the grade of interaction.

The minimum spotsize, beam quality or M², and strehl ratio of a focused laser beam provide different measures of the performance of the laser/optic system. Focusing lenses typically used to provide irradiances sufficient to cause melting and/or vaporization of metals or ceramics typically exhibit considerable spherical aberration, and thus limit the minimum spotsize attainable for a given lens at a specific laser power. The purpose of this work it to quantify the increase in the minimum spotsize and decrease in strehl ratio of focused materials processing cw Nd:YAG laser caused by (1) laser cavity heating and (2) spherical aberration introduced by the focusing lens. Minimum spotsize was determined by making several measurements of spotsize along the propagation direction using a scanning aperture system, and fitting the data to the laser propagation equation. These measurements were performed for 6 plano-convex lenses of different focal lengths, using laser power ranging from 500 to 1500 watts. A nonlinear variation of spotsize with laser power and with focal length was observed for the lenses and power levels tested.

In an effort to understand multimode laser beam propagation characteristics for better development of laser material processing applications, beam diagnostic experiments were performed on a 1800 watt cw Nd:YAG laser. Beam diameter data were acquired at approximately 12 positions along the beam optical axis about the minimum waist created by a long focal length single element lens at several power levels. These data were then used to evaluate the laser output beam characteristics using two differing techniques. For the ISO technique, two data points from the beam diameter data were used in determining the output laser beam characteristics. These points were the beam minimum waist diameter and the diameter at a point along the beam optical axis where the beam diameter had increased to approximately 0.7 times that of the beam minimum waist diameter. The second analysis technique involved fitting the entire data set to theoretical equations used to describe the multimode laser beam propagation and points from the fitted curve fit were then used to determine the output beam characteristics from the laser. For all power levels evaluated, calculated results predicting the laser beam minimum waist location were in agreement with measured values and more consistent using the curve-fit technique than the two-point evaluation technique.

The practical aspects of measuring and defining the beam parameter products of laser beams from high-average-power solid state lasers with unstable resonators are presented. In particular, measurements at kW-class Nd:YAG slab laser are discussed. Emphasis is on experimental techniques, measurement errors, and the influence of thermo-optical aberrations. Numerical Kirchhoff-integral calculations of the unstable resonator modes of a slab laser with thermo-optical aberrations are presented and discussed.

In this paper we propose a calibration procedure that allows beam characterization based on power content (or any other reasonable) beam widths. Once the characterization is done, the beam width behind ABCD-systems can be predicted with sufficient accuracy. This general characterization procedure is based on two fundamental ideas. Firstly, it is assumed that the free-space propagation of any beam width is hyperbolical with deviation near the waist. The second starting point is the fact that the structure of the intensity distribution that a given beam can take on behind an ABCD-system depends only on the quantity B/A (where A and B are elements of the geometrical ray matrix). We present a simple set-up, consisting of two parts, that allows the measurement of both the three parameters and the calibration function.

Methods of characterizing laser beams for specific laser applications and lasers as components of these systems are discussed including traditional beam divergence methods, the proposed ISO standard, and more modern methods using detector arrays and sophisticated data reduction methods.

Methods for determining the position and diameter of a laser beam are discussed. Of particular interest is a method that makes use of optical filters having transmittance that varies with one or more spatial coordinates. With the appropriate distribution of transmittance, a simple measurement of the fractional power transmitted by a particular filter can, after proper scaling, directly give beam position and beam diameter.

We have developed a wavefront analyzer for measuring the complex field distribution of a laser beam using dynamic shearing interferometry. The instrument includes two orthogonal lateral shearing interferometers, CCD cameras to capture the fringe patterns from the interferometers, and a small computer for data capture, analysis and display. The complex field distribution of the laser is found using standard fringe-scanning-interferometer analysis techniques, and can be displayed as a wavefront profile or as an intensity profile, or it can be transformed mathematically to the far field for display as the point spread function of the laser and optical system. In addition to the graphical display, the instrument provides numerical data that characterizes the laser beam, including the intensity distribution, the wavefront shape, and the far-field divergence. We have used the instrument to evaluate the performance and cavity alignment of continuous wave (cw) and pulsed lasers, and as a diagnostic tool to optimally align external optics to a diode laser.

By comparing the measured width of an optical test patten to the known width, the absolute error of a clip-level profiler is determined to be (-0.1 +/- 0.3)%. An expanded fundamental mode beam illuminates a pair of opposed knife edges (a wide slit) to generate the test pattern by Fresnel diffraction. Analysis of the diffraction pattern gives 18.2% as the appropriate clip level to read the geometrical shadow width between edges (with additional small adjustments for illumination non-uniformity and the finite size of the scanning aperture). The separation between the edges is determined by mechanical translation edge to edge through a focused beam. 3

The standard of beam space quality evaluation should reflect the space coherence and intensity magnitude (power or energy) of this laser beam. In the paper are discussed purity of modes, coherence concept of M² factor and three methods of determining beam spot size are compared.

We report on the use of a self referencing cyclic radial shear interferometer to obtain the single shot measurement of the phase front of a 150 fs titanium sapphire laser. Examples are also presented on the use of the device in conjunction with a probe laser to obtain nS resolution of the optical density of a z-pinch plasma, a high pressure pulsed gas jet, and sub- picosecond resolution of the evolution of plasma in 35 TW laser/solid interaction experiments where its ease of use makes it superior to other interferometric techniques.

The Wigner Distribution Function (WDF) has been well known in quantum mechanics since the thirties. Applied to paraxial optics it can be considered as a phase-space representation of quasi-monochromatic, partially coherent beams. For a two-dimensional beam (one transverse dimension) it depends on one spatial and one angular coordinate. Due to some properties of the WDF, it may be considered as an intensity distribution of geometrical rays, depending on position and direction, although this analogy is limited. Once the WDF of a laser beam in one transverse plane is known, the intensity distribution behind any optical system can be derived from it. The measurement of the WDF cannot be connected with a single intensity measurement as can be shown easily. Several suggestions for measurement procedures involving different kinds of apertures have been made. But all of them suffer from the influences of those apertures. Here we present a method which uses only a single focusing lens and several intensity measurements. It is based on a known mathematical procedure called the inverse radon transformation. An experimental result of applying this method to a laser beam emerged by an unstable resonator is presented, too.

An advanced M2 simulator for both multiplexed 16 point M2 measurements and single point beam waist M2 measurements has been developed and compared to actual data acquired with a multiplexed M2 optic. The simulator allows ideal beam characteristics to be modified and the predicted data set displayed in beam image form, as a plot of diameter versus propagation distance, and in tabular form of M2, beam waist, divergence, Rayleigh range and astigmatism. The real data captured can then be updated on a shot to shot (or cw) basis to allow changes in beam propagation characteristics to be monitored in real time. Both multiplexed 16 point and single point M2 data show better than 5% absolute agreement and can be used for optimization of laser power/energy versus beam quality. The simulator and beam analysis software operate under windows.

Experimental results are presented of the angular position stability measurement of the radiation beams emitted by lamp or diode-pumped cw Nd:YAG lasers with frequency doubling. Passive stabilization method based on the use of fiber optical wave guides is proposed and experimentally evaluated.

The use of space-angle intensity moments in description and characterization of a paraxial laser beam with small-scale inhomogeneities of its transverse shape is presented. The connection between the moments and customary beam characteristics is discussed. The equations of moments evolution in lens-like optical media with small-scale scattering inhomogeneity are derived and analyzed, on the basis of which some recommendations for fast and suitable medium parameters determination are made.

A nitrogen cooled high precision CCD-camera is used to measure the propagation characteristics of laser beams, where in particular the influence of camera noise on the measured data is investigated. As a consistency test, all measurement procedures, which are proposed by the ISO committee, are simulated with camera based data and the results are compared with measurements, obtained by using the real ISO-measurement setups. Furthermore, the influence of finite apertures on laser beam propagation is investigated systematically. The experimental investigations are complemented by numerical studies, which show the consistency of independently measured data, and which give some additional results on the influence of apertures on the beam parameters.

The propagation parameters of CO₂ laser beams have been investigated using second moment and knife-edge width measurement techniques. The characteristics of two laser types have been measured: a low power stabilized single frequency instrument and a prototype 3 kW laser. The propagation parameters have been estimated by using two commercial beam analyzing instruments: a rotating drum knife-edge device and a 2D array scanner. The propagation of errors through the analysis procedures has been investigated. The experiments were designed to assess the viability of the draft ISO standard for the measurement of beam width and propagation characteristics of real laser beams. Uncertainties in the estimated beam parameters, resulting from the propagation of errors, are taken into account when assessing the relative merits of the width measurement procedures. It was found that, for the high quality laser beam investigated, there were some small but systematic variations in estimating the width of waists of subject laser beams. In the case of the lower quality, high power laser beam, both the knife-edge and second moment techniques produced estimates of the input beam waist properties that were within the experimental uncertainty limits but again some inconsistency was displayed. It is suggested that the source of the inconsistency is diffraction by hard-edge apertures in the beam path. Nevertheless, the ISO standard procedures are judged to be suitable for the measurement of important beam parameters with an accuracy that is sufficient for the majority of industrial applications.

The experimental testing of baseline clipping algorithms was carried out on a purposely constructed test bench. Three different lasers were used for the tests including HeNe and collimated laserdiode. The beam profile intensity distribution was measured using a CCD camera at various distances from a reference lens. Results were analyzed on an 486 PC running custom developed software written in Turbo Pascal. This allows very fast evaluation of the algorithms to be performed at rates of several times per second depending upon computational load. Tables of beam width data were created and then analyzed using Mathematica to see if the data confirmed ABCD propagation laws. Values for the beam waist location, size, and propagation constant were calculated.

We calculated irradiance distributions resulting from clipping a Gaussian beam with hard- edged slits transmitting 99 or 95 percent of the incident power. We determined the widths -- second moment (with 8- and 12-bit resolution), scanning slit, and knife-edge -- of these profiles at several distances from the source, both with and without a focusing lens. When using a lens, we had the option of determining the characterization parameters from two beam width measurements and one axial distance, or from a least-squares fit of several beam width measurements and axial distances. Characterization parameters determined by these two options can differ significantly, and under the conditions of this study, beam divergence is more accurately determined from a single beam width measurement than from a least-squares fit of several measurements. Finite resolution of irradiance, inherent in any measurement device, truncates integrals necessary for calculation of the second moment of irradiance distributions.

The procedures suggested in the standardization draft ISO/CD 11146 for measurement for beam widths and beam radii are tested critically in application to several laser systems common in industrial use. It is shown from measured intensity profiles that the above mentioned procedures require data filtering and introduction of a cutoff. Examples of filtering and cutoff are given. The influence of filtering and cutoff on the results is shown. The use of the presented additional data processing is demonstrated in several measured examples. A slight modification of the definition of beam parameters is invented and a fitting routine for their calculation is specified. Application to beams of typical stable and unstable resonator systems as well as coaxial resonator systems is presented. The results of the paper recommend a supplement of the standardization draft.

This paper deals with computer-generated diffractive elements which transform diverging Gaussian beams of CO₂ lasers into user-defined intensity distributions in certain focal domains. For a special case of both scientific and technical interest -- a uniformly filled-in rectangle (`flattop') as user-defined intensity distribution -- the full cycle of calculation, micro- technological preparation and testing of the element is described. For this element the focus depth and the stability of the beam-shaping with respect to changes of size and divergence of incident Gaussian beams have been studied by computer modeling, as well as diffractive efficiency and accuracy of shaping. Calculated flattop intensity distributions have been compared with corresponding experimental results, measured by an IR camera. On the occasion we could demonstrate a good conformity between experimental and theoretical results of intensity distributions I(x,y,z) respectively, in desired focal plane as well as for planes before and behind focal plane. Likewise, there is such a conformity for the relative amount of power focused into the desired rectangle. For a further beam-forming element transforming a Gaussian TE₀₀-beam into a ring-shaped intensity distribution the calculated phase function and -- exemplary -- one of the binary masks are presented. First results about focus depth derived by computer modeling are displayed.

Thermal and stress lensing effects have been measured in a Cr,Er:YSGG rod by observing a transmitted 1064 nm Nd:YAG beam diverging from an operating Cr,Er:YSGG laser. The results compare favorably with theory and estimated thermal-optic properties of YSGG, which is intermediate between YAG and GSGG. Numerical simulations of the laser agree substantially with the threshold and power observed and show a heat generation rate which is consistent with our observations of lensing. Thermal fracture of the rod has been observed on several occasions, leading to an estimate of the thermal fracture figure of merit which is also intermediate between YAG and GSGG. Back focal distances of less than 20 cm occur in the vicinity of half the thermal rupture limit.

Measurement principles of laser beams parameters such as intensity, power, the beam waist and its profile utilizing orientational phenomena in liquid crystals induced by electromagnetic fields are presented.

The problem of compensation for atmospheric distortions of a wavefront has been studied for a sufficiently long time. The first papers on this subject were published in the mid 1960s. At that time, however, the engineering base gave no way for designing the efficient devices for compensating for atmospheric distortions. In recent years much progress has been made in developing wavefront sensors and correctors and then fitting the optical facilities operating under atmospheric distortions with these devices. In this connection there has been an increased interest in theoretical works concerning the optical design and configuration of the wavefront corrections.