The free-electron laser simulation code FRED, developed at LLNL primarily to model single-pass free-electron laser (FEL) amplifiers driven by induction linear accelerators, is described. The main emphasis is on the modelling of optical propagation in the laser, and on the differences between the requirements for modelling rf-linac-driven vs. induction-linac-driven FEL's. Examples of optical guiding and mode cleanup are presented for a 50 pm FEL.

The essential features of generalized numerical models developed at Rocketdyne to compute rigorously the 3 D transverse mode structure of free-electron laser oscillators are presented. The oscillator may consist of a conventional standing or a traveling wave cavity, or may be more complex, and contain intracavity grazing optical elements. Numerical resolution needed to (1) compute the gain due to a spatially distributed electron beam and (2) propagate the electromagnetic field when coupled to this gain are discussed with illustrative examples.

A two-dimensional model of power-saturable gain has been developed to represent the interactions in an RF linac free electron laser oscillator. The model is included in a wave optics resonator code, and provides convenient analyses of various free electron laser resonator configurations.

A model of HF-DF chemical lasers which includes a wave optics treatment of the resonator, a flowing gain medium, all important gas phase kinetics processes, and rotational non-equilibrium is discussed. The effects of rotational cross-relaxation rate on the predicted spectra for both stable and unstable resonators are shown, as well as the interaction of this effect with the placement of the resonator optical axis within the mode. The effect of partial aperture lasing and means for its suppression using cross-relaxation rate and/or optical axis placement are discussed. Some comparison with experiment is presented.

Bare cavity eigenmode analysis is used in this paper to evaluate the diffractive interaction of the forward and reverse modes of an unstable ring laser. The particular case analyzed here is an HF chemical laser, which has a multiline inhomogeneously broadened gain medium. The simplified model presented here provides extremely good agreement with experimental data and gives increased understanding of reverse mode suppression in multiline chemical lasers with unstable ring resonant cavities.

A large signal model of a InGaAsP semiconductor laser is presented. The model allows the simulation of the multimode laser behavior and includes statistical photon fluctuations and self-pulsation options.

Two configurations for coupling two Fabry-Perot resonators are discussed, one with no loss in the coupling path and one with loss. The lossy configuration is found to give discrimination between supermodes and, for certain cases, equal supermode frequency. With inhomogeneously broadened gain, the lossy configuration is found to suppress one supermode, while the lossless configuration does not. A locking range is predicted for the lossy configuration.

A general formalism describing the supermodes of an array of N identical, circulantly coupled resonators is presented. The symmetry of the problem results in a reduction of the N coupled integral equations to N decoupled integral equations. Each independent integral equation defines a set of single-resonator modes derived for a hypothetical resonator whose geometry resembles a member of the real array with the exception that all coupling beams are replaced by feedback beams, each with a prescribed constant phase. A given array supermode consists of a single equivalent resonator mode appearing repetitively in each resonator with a prescribed relative phase between individual resonators. The specific array design chosen for example is that of N adjoint coupled confocal unstable resonators. The impact of coupling on the computer modeling of this system is discussed and computer results for the cases of two- and four-laser coupling are presented.

A theory is developed for an index-guided semiconductor laser array. This theory is based on the expansion of the laser field in terms of the passive array eigenmodes, and on the use of semiclassical laser theory to treat the interaction of these modes among themselves and with the active medium. It is valid for all values of coupling between lasers and for all array sizes. To illustrate the application of this theory, we use it to investigate the lockbands of a two-laser array.

We present a self-consistent numerical model of light propagation in diode laser arrays. This model has been used to study the free-running modes of a five-element gain-guided array operated just below threshold. The resulting calculations show that, in contrast to index-guided arrays, the modeshapes in general bear little or no resemblance to the "supermodes" obtained previously from linear perturbation theories. In addition, we find more than 5 modes for an 5-element device, so that the intensity maxima in the near-field are not necessarily centered over the stripes. These observations are explainable in terms of a simple plane wave model of the array cavity.

A semiconductor diode laser array model is applied to the case of the optimization of a twin channel laser (TCL) design. Results substantially in agreement with experiment are obtained as to optimum general geometry, active layer thickness, cladding layer thickness to either side of the coupled channels (wing region), and threshold current. Differential efficiency predictions are too high relative to experiment, probably due to leakage currents in the experimental device. A current-induced mode instability at high current is predicted.

A homogeneously broadened system of two coupled lasers is studied using the longitudinal double-cavity supermodes as a basis set for expanding the field and polarization of the system. This approach is valid for strong as well as weak coupling of the lasers. A system of two self-consistent field equations for the field amplitude En and phase On of the nth supermode are obtained. Assuming only one supermode in the system, these equations are solved in steady-state to obtain the steady-state field amplitude and operating frequency of that supermode. The system is defined as phase-locked if neighboring supermodes, arising from small deviations for steady-state, do not grow with time. Tuning curves and locking ranges calculated with these methods were found to be in good agreement with those obtained from an experiment with a system of two coupled CO₂ lasers.

A two-dimensional model using wave optics has been developed for real-time simulation of the four wave mixing process in nonlinear optical media. Because of the additional dimension to represent the intensity and phase variation across the fields, the code is capable of demonstrating the effects such as beam misalignment, self-focusing, and phase conjugation fidelity. The reflectivity obtained in the short-pulse regime is in good agreement with analytical theory.

Predictions of a wave optics model of stimulated Brillouin scattering (SBS) have been validated by comparison with experimental measurements of reflectivity and beam quality. The code also models beam combination by SBS.

Efficient (> 50%) large aperture (> 10 cm), high energy (> 1 kJ) harmonic conversion has been demonstrated with short pulse (< 5 ns) lasers, notably atomic iodine (1315 nm) and Nd:glass (1064 and 1053 nm). Efficient harmonic conversion utilizes phase matching techniques within selected birefringent crystal media. Considerable interest has arisen to apply these techniques to CW-pumped lasers to enhance their wavelength versatility. When phase matching techniques are applied directly to either long pulse or CW lasers, serious problems arise owing to thermal detuning and crystal damage. Promising approaches using thin harmonic conversion crystals for efficient conversion of quasi-CW lasers are described.

Stokes seed production in a Raman generator has been modeled for a pump of long duration and with many longitudinal modes. The model uses three-dimensional wave optics and includes pump depletion. The modeling is similar to that of Lewenstein in its use of stochastic c-number equations. The model simulates the spontaneous Stokes source with effective Stokes fields supplied in a prescribed manner. The analysis to support this simulation and examples of computer results will be presented.

The efficiency of a first Stokes Raman amplifier may be seriously reduced by the generation of a second Stokes beam. The dominating process for this generation is often a Raman four-wave mixing interaction which couples the pump and second Stokes beams. A one-dimensional model for this effect has a closed-form solution, simply described in terms of an equivalent input, under conditions where high first Stokes conversion efficiency is possible. Excellent agreement with numerical integration results has been obtained.

A Hilbert Space formalism is shown to greatly simplify the problem of Raman conversion of high power multimode pump laser(s) by an injected Stokes 'seed'. The case of (a) a single pump and (b) two pumps with bisecting Stokes can be solved using this formalism if diffraction effects are neglected. If they are included, the formalism greatly facilitates code development by reducing the number of variables for the problem.

Two sources of phase aberrations during amplification by stimulated Raman scattering have been studied. Both are caused by intensity nonuniformity in the incident pump laser beam. First, nonuniform population transfer of molecules undergoing the Raman transition results in a cumulative refractivity nonuniformity associated with the difference between the polarizabilities of the two molecular states. As shown experimentally by Butylkin et al. and Baklushina et al., the fractional polarizability difference for the Q(l) vibrational transition of H₂ is ≈0.15-0.20. As a result, this effect becomes large at a fluence nonuniformity level >33/cm². The dependence on fluence, rather than intensity, arises from the cumulative population transfer that takes place when the relaxation of the upper level during the laser pulse only weakly affects the population difference. Second, collisional relaxation of the excited population during the pulse causes heating and a nonuniform density distribution; this effect becomes more important as the pulselength is increased or the spatial scale size of the intensity gradients is decreased. The equations coupling the pump and Stokes fields to the refractivity and density changes are solved in the ray-optics limit of collimated beams and the effect on beam quality is determined. It is shown that in the limit of weak refractivity effects and negligible heating the beam-quality loss varies as the square of the rms fluence modulation.

We present the results of calculations of the beam quality obtainable in a collimated, single-beam Raman beam cleanup system, in which the pump laser contains two lines whose frequency separation is enough to cause significant dispersive "slip" between them in the Raman amplifier. A specific example is the 351-353 nm line pair of an XeF laser. We start from equations coupling the various modes of the lines and their corresponding Stokes modes. Refractivity effects caused by nonuniform Raman--induced population transfer are specifically accounted for. The impact on beam quality is assessed as a function of the pressure in a H₂ Raman amplifier, and it is shown that the minimum effect occurs at a pressure of 2.4 Atm, for which the motionally narrowed Raman linewidth takes its minimum value. The two-line coupling causes the population transfer due to the stronger of the two lines to adversely affect the beam quality of the weaker line. For an rms percent fluence modulation of 15 percent, the two-line coupling lowers the Strehl ratio (a measure of beam quality) by ~15 percent for a 1-μs XeF laser pulse length.

Spectral correlation and phase locking properties of Raman conversion in inhomogeneous media are analyzed with a multi-mode model.

The effects of linear dispersion and four-wave mixing mechanisms on conversion efficiency and beam quality in Raman conversion of two-line XeF pump sources in H₂ are evaluated. Gain enhancement and gain cancellation occur as the lines propagate in and out of phase. An additional effect, that of phase compensation of the Stokes fields to the k-vector mismatch, has been analyzed and shown to be a function of gain and degree of dispersion. Beam cleanup in parallel and crossed pump beam configurations is dependent on gain saturation, interaction distance relative to the Fresnel number and scale size of the pump aberration, and angle of injection.

The role of numerical methods to simulate the several physical processes (e.g., diffraction, self-focusing, gain saturation) that are involved in coherent beam propagation through large laser systems is discussed. A comprehensive simulation code for modeling the pertinent physical phenomena observed in laser operations (growth of small-scale modulation, spatial filter, imaging, gain saturation and beam-induced damage) is described in some detail. Comparisons between code results and solid state laser output performance data are presented. Design and performance estimation of the large Nova laser system at LLNL are given. Finally, a global design rule for large, solid state laser systems is discussed.

Historically, wave optics computer codes have been paraxial in nature. Folded systems could be modeled by "unfolding" the optical system. Calculation of optical aberrations is, in general, left for the analyst to do with off-line codes. While such paraxial codes were adequate for the simpler systems being studied 10 years ago, current problems such as phased arrays, ring resonators, coupled resonators, and grazing incidence optics require a major advance in analytical capability. This paper describes extension of the physical optics codes GLAD and GLAD V to include a global coordinate system and exact ray aberration calculations. The global coordinate system allows components to be positioned and rotated arbitrarily. Exact aberrations are calculated for components in aligned or misaligned configurations by using ray tracing to compute optical path differences and diffraction propagation. Optical path lengths between components and beam rotations in complex mirror systems are calculated accurately so that coherent interactions in phased arrays and coupled devices may be treated correctly.

In this paper, we describe a wave optics model we have developed for predicting the performance of high-energy laser systems, with particular emphasis on its application to the cylindrical, source-flow HF chemical laser. The structure of the code is based on the 'lumped equivalent optical train' concept, in which any continuous spatial effect on the optical field, such as mirrors, apertures, and gain medium, is approximated numerically by a finite number of transfer functions on the field. Free space propagation between elements is achieved by using Fast Fourier Transforms. The gain is modeled as a series of gain sheets, where the spatial dependence of the gain is calculated from a detailed aerokinetic treatment of the interaction of the intensity field with the flow in the laser cavity. Resonator calculations will be described for two different gain models. In the rotational nonequilibrium (RNE) model, the effects of disequilibrium in the rotational distribution of the individual vibrational levels are accounted for explicitly by solving an evolution equation for each vibro-rotational state of the lasing molecule. The second gain model, referred to as the single line (SL) model, is based on two assumptions, namely that the rotational levels in each vibrational level are in thermal equilibrium, and that the gain on the lasing lines is identical.

In 1983, Rocketdyne Division of Rockwell International constructed the OASIS code in support of the Air Force to meet increasing needs for diffractive optical system analysis. The OASIS code (Optical Analysis Software for Interactive Systems) was developed to meet these needs in the context of new supercomputer environments. From its inception, the code was anticipated to be a highly interactive analysis tool featuring the basic required physics, plus generality and flexibility. A valuable teaching tool, the code is command line driven and offers complete documentation.

The recent availability of high performance, low cost microcomputers with extensive memory and disk storage capacity has made possible the development of sophisticated analysis tools that were previously limited to batch processing on mainframe computers. In this paper, we describe a quick response, interactive wave optics model we have developed for analyzing a wide variety of optoelectronic systems including laser resonators, amplifiers, beam trains, and adaptive optics. The model, as developed for the IBM PC AT, permits interactive or batch system definition, and provides both screen and printed graphical output of the intensity, phase, and power distribution throughout the system. The model provides considerable flexibility to the user in defining the system to be analyzed including initial beam setup, wavelength, system Fresnel number, and component order. The structure of the code is based on the 'lumped equivalent optical train' concept, in which any continuous spatial effect on the optical field, such as mirrors, apertures, and gain medium, is approximated numerically by a finite number of transfer functions on the field. Free space propagation between elements is achieved by using Fast Fourier Transforms. The gain is modeled as a series of gain sheets, where the spatial dependence of the gain is calculated from either a detailed aerokinetic calculation or from a simple threshold gain versus saturated intensity relationship. The model presented here has proven to very useful and accurate for parametric resonator power extraction analysis and design and for determining the effects of various aberrations on system performance. Results from this model are presented and compared with more sophisticated mainframe models along with run time comparisons.

Methods are considered to improve performance of phased array systems by analyzing the effects of higher order aberrations on piston estimation and control. The analysis confirms that the use of wavefront sensor measurements to augment the piston sensor readings allow accurate estimation of the true piston difference in an edge sampling array. In addition, average phase at the sampler can be estimated very accurately even in the presence of WFS errors. Finally, using a three element phased array in an OTF non-redundant geometry, one can, after image reconstruction, significantly increase resolution compared to a single telescope of comparable entrance pupil area.

The use of grazing incidence optics in resonators alleviates the problem of damage to the optical elements and permits higher powers in cavities of reasonable dimensions for a free electron laser (FEL). The design and manufacture of a grazing incidence beam expander for the Los Alamos FEL mock up has been completed. In this paper, we describe the analysis of a bare cavity, grazing incidence optical beam expander for an FEL system. Since the existing geometrical and physical optics codes were inadequate for such an analysis, the GLAD code was modified to include global coordinates, exact conic representation, raytracing, and exact aberration features to determine the alignment sensitivities of laser resonators. A resonator cavity has been manufactured and experimentally setup in the Optical Evaluation Laboratory at Los Alamos. Calculated performance is compared with the laboratory measurements obtained so far.

A summary is presented of possible phased array and shared aperture concepts for free electron lasers. The advantages and disadvantages of the various configurations are discussed.

The degradation of image quality in a turbid medium is analyzed within the framework of the small-angle approximation, the diffusion approximation, and a rigorous two-dimensional radiative transfer equation. These three approaches allow us to emphasize different aspects of the imaging problem when multiple scattering effects are important. For a medium with a forward-peaked phase function, the separation of multiple scattering into a series of scatterings of various order provides a fruitful technique. The use of the diffusion approximation and transport theory extends the determination of the modulation transfer function to a turbid medium with an arbitrary degree of anisotropy.

An integral equation for the coherent field in a collection of large point scatterers is formulated. It differs from preceding development in that to good approximation all higher-order correlations between particles are included. The equation is developed from the self-consistent equations for multiple scattering from point scatterers in the "Twersky approximation," which ignores scatterings from particle 1 to particle 2 and then back to particle 1 everywhere. The four lowest-order terms in binary-correlation expansion correspond with those obuqned from Tsolakis, Besieris, and Kohler [Radio Sci. 20, 1037-1052; 1985]. Analysis of the region of validity reveals that the equation is a extension of previous work that is significant for small particles and higher densities.

On the basis of a phase-space approach utilizing the Wigner distribution function, a systematic derivation is undertaken of the equation governing scattering-dominated radiative transfer for a model of uncorrelated, isotropic, point scatterers.

Over a snow-covered field, I have measured time series of the turbulent fluctuations in temperature and humidity with a fine-wire platinum resistance thermometer and a Lyman-alpha hygrometer. From these time series I computed temperature and humidity spectra and temperature-humidity cospectra. When properly summed, these spectra and cospectra yielded refractive index spectra for visible and millimetre wavelengths, the first such spectra ever collected over snow.

A new method is described for the numerical simulation of atmospheric turbulence effects on optical propagation. The method is based on a convolution approach which offers improved mathematical accuracy and convenience for numerical implementation, as compared with the more commonly used Fourier synthesis technique. Random phase screens generated with the new method are shown to have the correct r^5/3 dependence for the phase structure function for all separations r within the specified domain of validity. Typical results are also presented which show the use of the numerical turbulence simulation model in wave optics code calculations of laser beam propagation through the atmosphere.

The description of the mean field in randomly inhomogeneous media can be made using the "effective dielectric permittivity" tensor. This well known approach implies the interpretation of the mean electric field as a regular field in some "effective" medium with spatial dispersion, where the electric induction and field vectors are related by formula

In the present paper a connection between the approximate Huygens-Fresnel formulae and the exact path-integral solutions is studied. We consider the approach which makes it possible to separate from the path integral the contribution that corresponds to the phase approximation of the extended Huygens-Fresnel principle. This approach is used to derive the corrections to the phase approximation.

The intensity statistics of a laser light passing through a double phase screen are measured and compared with those predicted by the I-K distribution. The phase screens are produced in the laboratory by turbulent mixing of the convective hot air flow above a gas flame and the surrounding cocler air. Measurements are made over actual distances up to 60 feet behind the first phase screen with various separation distances between the two screens. The normalized intensity moments were found to be similar to those occuring in extended turbulence conditions, but here the normalized second moment never exceeded 3. In all cases we found the I-K distribution with parameter α = 3/2 to be an excellent theoretical model.