This paper will review a very general and useful formalism that has recently been developed for defining the transverse beam quality as well as other propagation parameters of arbitrary real laser beams. We also describe a convenient new instrument for measuring the propagation parameters of high-power laser beams, and summarize some results of beam quality measurements on various types of laser devices.

A few examples are presented to show that the possibilities of developing new promising resonators based on standard optical elements with plane or spherical surfaces are far from being exhausted. Methods of constructing field-rotation linear resonators using a modified Sagnac interferometer are discussed; the properties and application potential of the simplest resonators with spatial filtration of radiation are considered; and a method of constructing multi-pass unstable resonators with a radically reduced sensitivity to misalignment is presented.

Since its first demonstration in 1976 the free-electron laser has evolved rapidly, producing coherent radiation over a spectral range extending from the millimeter to the ultraviolet. Recent establishment of user facilities is enabling its use as an investigative tool in a number of scientific disciplines. Its ability to generate tuneable, high power radiation over a wide range of wavelengths interests medical, industrial and military users. However, exploiting the many features of this intrinsically high power laser poses new challenges in optical resonator design. We review significant advances in optical resonators for free-electron lasers including the influence of this peculiar gain medium on resonator designs.

The Continuous Electron Beam Accelerator Facility (CEBAF) is a prime example of the progress that SRF technology has made. CEBAF was designed to operate at 1500 MHz due to the needs of nuclear physics users and the relative maturity of the Cornell cavity design on which it is based. CEBAF will be operational in 1994 and is designed for 4 GeV of energy at 200 microamps of CW average current. The specified emittance (< 1 nm at 1 GeV) and energy spread (< 10^-4) are extremely tight although the design peak current is too low to give significant gain in an FEL. The CEBAF cavities have a specified gradient of 5 MeV/m but delivered gradients have been significantly higher, recently exceeding an average of 8.5 MeV/m for approximately 100 production cavities. The CEBAF injector has been operated at energies up to 85 MeV and has met all designs specifications in over 3000 hours of around-the-clock operation. Two FELs utilizing a high charge injector have been designed. The CW nature of the beam results in high average powers (order 1 kW) from the IR FEL in the region from 4 to 20 microns and from the UV FEL in the 0.15 to 0.25 micron region. Tunable radiation at this power level places extreme demands on the optical systems. The approaches developed to resolve the optics issues will be examined and progress in implementation and testing is presented.

The Rocketdyne free-electron laser (FEL) being presently developed for operation in the visible to one-micron regime is described, with particular attention given to some of the principal optics and atmospheric propagation issues. The paper describes the system assembly and discusses the performance requirements for power beaming, the resonator design, and the basic ideas and calculations involved in the beam propagating through the atmosphere and tilt corrections. This FEL will be capable of an average output of greater than 1 kW in the near infrared. The laser system has an ability of scaling to power levels required for beaming power to space platforms.

The dependence of the output laser beam characteristics on the magnification factor of supergaussian unstable resonators applied to a XeCl laser has been investigated. It has been found that high magnification factors and supergaussian mirrors with a high peak reflectivity are required to get a good mode control of the laser oscillating radiation.

Two simple annular resonators, each consisting of two total reflecting annular mirrors with radially concave curvatures, are described. The beam is guided to a coupling aperture in azimuthal direction by the azimuthal mirror shape. A resonator with tilted toric mirrors works like an azimuthal unstable resonator with two beams starting from a core oscillator of this resonator towards the coupling aperture. The function of this resonator is akin to conventional positive branch unstable strip resonators wrapped to an annular geometry. A resonator with helix-shaped mirrors emits one well defined beam. A numerical model of the resonators using a 3D-diffraction code and including effects of misalignment and saturable gain is presented. Numerical beam propagation techniques based on different approximations are compared. Due to the high azimuthal Fresnel number geometrical optics becomes applicable and turns out to be useful for a rough estimation of resonator design parameters. Experiments have been carried out using a diffusion cooled transverse RF-excited coaxial CO2-laser-gas discharge. Experimental data are compared with theoretical results concerning near-field and far-field patterns as well as misalignment sensitivity. Theoretical and experimental data correlate well. Both resonators yield more than 10 percent efficiency.

There is much work devoted to the design of new resonator configurations which could optimize efficiency of energy extraction and beam quality of high-power lasers with large- volume gain media. In general, operation with stable resonators at high beam quality limits filling of the gain volume and leads to a reduced efficiency. Unstable resonators exhibit large modal volume and excellent transverse mode discrimination. These are of primary concern in high-power laser systems. Output beams from unstable resonators with spheric mirrors have an annular intensity profile. Beam quality as well as the outcoupling increase as magnification increases. If the gain is not sufficiently high, the increase in the outcoupling reduces laser efficiency. From this it turns out, that in general the maximum beam quality and the maximum efficiency cannot be reached at the same time. This limits the use of these unstable resonators in lasers with low or medium gain. The problem can be overcome using the toric resonators. The beam quality of toric resonators is higher than in the case of unstable resonators with spheric mirrors and is not influenced by the magnification. Therefore, in the case of toric resonators the outcoupling can be matched to the gain in order to optimize the efficiency of energy extraction at high beam quality. Following theoretical analyses and design considerations, different configurations of toric unstable resonators were established for a fast axial flow CO₂ laser with two rf-discharge tubes. The efficiency, the beam quality and the alignment sensitivity of toric resonators with different configurations were studied.

A novel computer model of an injection-seeded pulsed CO₂ laser is used to predict the intrapulse frequency chirp as a function of laser resonator and gain medium parameters. A new mechanism for causing intrapulse frequency sweeping is predicted. It is found that nonuniform and time-varying gain saturation changes the transverse distribution of the intracavity electric field during the laser pulse, which leads to a change in the oscillator frequency. This mechanism is predicted to be much more pronounced in low-magnification graded-reflectivity unstable resonators as compared to low-Fresnel-number stable resonators. It is also shown that to obtain quantitative estimates of chirp in large aperture unstable resonators lasers, it is necessary to couple the intracavity electric field with realistic laser kinetics and index-changing mechanisms.

In this paper we are interested in the effects of statistical phase and gain distortions. We will present experimental results, which show the strong dependencies of the beam quality of a 13 kW CO₂ laser with an unstable resonator on the gas flowing state and on the rf-excitation power. These dependencies are mainly attributed by the phase distortion (stationary and statistical) of the active medium, resulting from the gas discharge, from the turbulence of the high mass rate flow and from the interaction between the turbulence and the discharge. In order to estimate the influences of the statistical phase and gain distortions on beam quality and output power, a theoretical model has been developed, where the phase and gain distortions are simulated using two statistical phase sheets and two gain sheets, respectively, located near the both resonator mirrors. Using the model the dependencies of beam quality, the Strehl ratio and output power on the dimension and the strength of the statistical phase and gain distortions were investigated. The numerical results will be discussed.

Rigrod theory was used to model outcoupled power from a low-gain laser with good accuracy. For a low-gain overtone CW HF chemical laser, Rigrod theory shows that a higher medium saturation yields a higher overall overtone efficiency, but does not necessarily yield a higher measureable power (power in the bucket). For low absorption/scattering loss overtone mirrors and a 5 percent penalty in outcoupled power, the intracavity flux and hence the mirror loading may be reduced by more than a factor of two when the gain length is long enough to well saturate the medium. For the UIUC overtone laser which has an extensive data base with well characterized mirrors for which the Rigrod parameters g(0) and I(sat) were firmly established, the accuracy to which the reflectivities of high reflectivity overtone mirrors can be deduced using measured mirror transmissivities, measured outcoupled power and Rigrod theory is approximately +/- 0.07 percent. This method of accurately deducing mirror reflectivities may be applicable to other low-gain laser systems which use high reflectivity mirrors at different wavelengths. The use of a non-homogeneous gain saturation law indicated that a gain saturation law parameter of m = 1.2 models UIUC SSL fundamental data more accurately. A completely inhomogeneous saturation law (m = 2) models UIUC SSL overtone data more accurately than a completely homogeneous gain saturation law.

An oscillator-amplifier configuration is described in which the amplifier's spontaneous emission (ASE) feeds partially collimated light from the amplifier into the oscillator that is driving it. The resonator is confocal, and its axis is decentered so that the output is shifted laterally. The partial collimation of the input light causes a buildup of ASE in the resonator so that a small amount of power entering it can have a significant effect on the beam quality of the output.

A three-wave mixing crystal is an anisotropic material. To model beam propagation through such a medium, one must have a propagator which includes the anisotropic effects, such as walkoff, in addition to the usual diffraction effect. This requirement leads to the specific exponential forms appearing in the set of coupled differential equations we use to model the nonlinear coupling. In this paper, three coupled beams in two transverse dimensions (i.e., with arbitrary initial cross sections) are propagated through an anisotropic three-wave mixing crystal with their bulk absorption and surface losses included. We present the theory of anisotropic propagation and the field coupling of three wave-mixing, including phase matching/mismatching. Results from the full numerical simulations are compared with experimental data. Our numerical approach avoids a number of simplifications common in analytical treatments, thereby providing more realistic modeling of experimental situations.

Three gain saturation laws are frequently used to describe the light energy propagation in laser amplifiers: two assigned to Rigrod, referring to homogeneous and inhomogeneous laser line. The third assigned to Frantz-Nodvick law. We show that under lossless medium conditions, and for both small signal and saturation regimes the laws can be presented uniformly as: Y_out equals Y_in(DOT)exp(g_o(DOT)L) where Y is a function of the normalized intensity I, which is the ratio of light intensity to saturation light intensity. The saturation intensity and the dependence of Y on I are different for each of the laws, but for all cases Y approaches I in the small-signal regime. We provide approximate analytical expressions, for the inverse function I(Y) for the two Rigrod laws, facilitating a one-step integration of the gain saturation laws.

We review the study of hole-coupled resonators for broadly tunable free electron laser (FEL) applications. The mode profiles inside and outside the cavity, the diffraction losses at the mirror edges and intracavity apertures, the amount of useful power coupled through the holes, and the FEL gain are calculated for several dominant azimuthal and radial modes. The FEL interaction is taken into account by constructing a propagator similar to the Fresnel integral for free space propagation. It is found that non-confocal resonators can provide efficient hole coupling over a broad wavelength range, as long as the mode beating caused by a degeneracy in the round trip loss can be avoided. The degeneracy between the azimuthally symmetric class of modes is removed by FEL interaction, and the azimuthally asymmetric modes can be suppressed by means of intracavity apertures. Therefore, in a non-confocal configuration, a hole-coupled resonator can be designed that is tunable over a broad range of wavelength by employing an adjustable intracavity aperture. On the other hand, confocal resonators are not suitable for hole coupling; although mode beating does not occur in a confocal resonator, the hole coupling is difficult because the modes tend to avoid the hole. We provide a simple physical understanding of the difference in the performance of the confocal and non-confocal resonators. We also calculate and analyze the mode content of an empty resonator under continuous external mode injection. Such calculation is useful in interpreting experiments testing the hole coupling performance using CW lasers.

We report on ongoing experiments and simulations which model the performance of hole- coupled resonators. We have previously studied a hole-coupled resonator which was well inside the stable region (stability parameter g equals 0.8). In the far field, good agreement between experiment and simulation was obtained for both the intra-cavity and outcoupled mode-profile. The present study involves a resonator with a stability parameter of -0.987, identical to the stability parameter of the proposed Infrared Free Electron Laser (IRFEL) at Lawrence Berkeley Laboratory. The experiments were carried out with a frequency stabilized CW-HeNe laser beam at a wavelength of 632.8 nm. Both intra-cavity and outcoupled mode profiles and power levels were measured. The simulations were done using the code HOLD, which is based on the Fresnel approximation for the Huygens kernel. Within the experimental uncertainties, magnified due to the 1/(1+g) dependence of the mode characteristics on errors in measured resonator parameters, we have obtained fair agreement between experiment and simulation.

A design for a hole outcoupling experiment for the Mark III FEL at Duke University is presented. The theoretical results of a matrix formulation, comprised of a gain matrix describing the FEL gain interaction and a loss matrix describing the mode mixing and losses at the cavity end mirrors, are applied to determine useful hole sizes and to predict the optical outcoupling percentage, the optical intracavity mode structure, and the diffractive losses.

The structure and some probable limitations of lenses grown into semiconductor lasers or amplifiers are discussed. As an example, an afocal telescope that converts a collimated beam with a width of 20 microns to a collimated beam with a width of 220 microns requires a length of 1.25 mm. Its design uses a transverse structure that is based on previous work but is less conservative as far as reflection and the conversion of light to radiating waves at the lens surfaces.

We describe the motivation, design principles, and operating characteristics of broad-area semiconductor lasers with lateral mode control provided by a variety of hybrid resonator and amplifier configurations.

Brightness is power per emitting area per solid angle. Obviously, to achieve high brightness, one needs to have a source with power and the smallest combination of effective emitting area and solid angle. Consider a square with side "d" emitting power "p". For a plane wave in the aperture diffraction theory tells us that the light emitted will diverge with an angle on the order of Aid. This yields; B = p/((d x d) x (A/d) x (A/d)) = P1 (A x A) . Consider the same aperture , but this time the wavefront is curved with a radius "r". The solid angle is now d x d/ (r x r) which can be much larger than A x t/(d x d) , but the effective area is no longer d x d. Diffraction theory says the effective size of the source to fill the aperture is r x Aid. This again yields B = p/(A x A). This leads to the criteria for a useful high brightness source; that it be spatially coherent across the emitting aperture and be correctable to an effectively planar wavefront at some reference plane.

Various applications of coupled-cavity configuration in controlling diode laser properties are reviewed. Coupling to a second cavity gives an additional degree of freedom that can be used to control the diode laser behavior. Two broad categories of coupled-cavity diode lasers can be defined: external-cavity and monolithic, depending whether the second cavity is discrete or monolithically integrated with the laser. Both schemes can be applied in a variety of applications, including longitudinal-mode control, transverse-mode control, wavelength tuning, static and dynamic linewidth control, mode locking, and optical bistability.

High power density can be obtained from vertical-cavity surface emitting semiconductor lasers (VCSELs), but the emitting area must be kept small if spatial coherence is desired. We present the result of a theoretical and calculational study of the possibility of placing the gain regions of VCSELs in external cavities to provide mode control over a larger area. A Fabry-Perot or stable resonator with very low loss is required. Several configurations are discussed and evaluated. A ring cavity is the most practical due to its relative lack of sensitivity to misalignment of the mirrors.

A stability criterion and growth rates are derived for filaments in a semiconductor optical amplifier with a cylindrically expanding beam. The instantaneous growth rates are the same as they would be for a collimated geometry, but the development of the filaments is slowed because the spatial frequency is changing as the beam expands.

Basic features of laser resonators with field rotation are discussed. Main types of schemes of resonators providing high performance and low sensitivity to misalignment are represented. Results of investigation, development and application of resonators with field rotation to various laser systems are reported.

We prsnt thi tIwnry of cnmpTLet1y passive additive pulse mode locking of soild-stat 1asrs with Iong-1iv1n: upper level. Antlitica1 solution for the laser pulse shape and its relttion to thf: main parameters of iaser resonator hav been obtained.

Different stable-unstable resonator designs with one-side coupling for tunable free electron lasers have been analyzed. The eigenmode analysis of bare cavities shows that using a self-imaging resonator configuration enhances the mode discrimination against higher-order modes and results in better output intensity and phase profiles. It is also suggested to improve the near-field and far-field beam quality by using a combining and relayed optics for the output beam.

A tapered wiggler is used in a FEL oscillator to improve the saturation efficiency. During signal buildup the tapered wiggler does not provide optimum phase synchronism between the electron beam and the electromagnetic wave, resulting in an appreciable loss in small-signal gain. This problem can be ameliorated by using a multicomponent wiggler, which is a combination of a uniform wiggler and a tapered section. During buildup, gain is primarily contributed by the linear element, and at high power levels the gain and efficiency are enhanced by the taper. Ideally, one would like to have the optical waist location near the linear section at small-signal levels and at near the tapered section at high power levels. Placing the FEL in a symmetrical confocal resonator approaches this desired effect automatically since it has the unique characteristic that a stable mode exists for all locations of the waist of a Gaussian beam along the axis of the interferometer.

A canonical formulation is applied to analyze the roof-top resonators. When the optical resonator loses some symmetries, some higher order transverse modes will be eliminated. Then good transverse mode selectivities occur.

Laser resonators of a new type are suggested in this paper. The main distinction separeting them from the common ones is their property or abiiity for self reconstrustion points for some intracavity plane. it means that the image of every point of this plane is placed in the same point after a full pass of radiation through the resonator. Theoretical investigation of these cavities with self reconstructed points is fulfilled with the help of matrix optics method12. The ability of both dispersive and nondispersive cavities construction is shown. The conditions are found for self reconstruction of both plate and long intracavity objects.

Resonators capable of extracting highly coherent energy from DF and HF chemical laser annular gain media have been under investigation for weapon application since 1974. This survey article traces the background of interest in these devices, describes the various concepts that have been experimentally and analytically investigated, and discusses the issues associated with their operation. From the discussion of issues, preferred concepts are selected. Applicability of these concepts to high-power operation is addressed through discussions of past and ongoing high-power demonstration programs and the issues facing their application to weapon sized devices capable of strategic and tactical missions such as ballistic missile defense (BMD), theater missile defense (TMD), and anti satellite (ASAT).

Eigenmodes characterized by distinct polarization states for the resonator configuration known as the High-Extraction Decentered Annular Ring Resonator (HEXDARR) have been found. The effects of internal annular optical elements (waxicons, reflaxicons, and cones) as well as coatings and mode rotation are included. For a good design, the low-order mode polarization states are nearly circular, with the polarization ellipse orientation and phase angle slightly spatially dependent across the near field of the resonator output. The two lowest-loss eigenmodes are separated by only a small loss which is dependent on the coating properties, one being principally right-hand circularly polarized and the other principally left-hand circularly polarized.

A method is presented by which the feedback and effective magnification of a high extraction efficiency decentered annular ring resonator (HEXDARR) can be changed, with the introduction of a flux-redistribution irradiance mapping optical pair, into the compact leg of the resonator. The mirror pair shapes were derived, and, using ray-trace methods, it is verified that the desired redistribution is effected. It is also shown that the specification of the mirror surfaces given by the equations derived meet the requirements for reducing the effective magnification and introducing no phase errors.

We consider resonator designs with a range of effective magnifications (1.2 to 2.95), and show how these resonators perform. Specifically, we analyze power extraction performance, internal power gains and losses, and resonator wavefront beam quality for this range of effective magnifications. As perturbations, we consider the effect of struts and compact leg misalignments. We find that, at the lower levels of magnification, the resonators produce more power but are more sensitive to perturbations.

The paper discusses the design features of four low-power laser (LPL) concepts (the external LPL concept, the annular LPL, the compact beam LPL, and the rear cone LPL) for use in the alignment of a high-power laser (HPL) for space applications and describes a specific high-power ring resonator, the High Extraction Decentered Annular Ring Resonator. Particular attention is given to the comparison of the four LPL concepts. It is concluded that the external LPL is not desirable due to the added alignment steps required to boresight the LPL to the HPL optical axis and that the annular LPL has an unacceptably large threshold gain for gain generator areas less than 3000 sq cm. While the compact beam LPL concept was found to be the most flow efficient, the rear cone LPL was found to maximize the gain area and thus the output power for the 60-cm maximum gain length determined by the spacecraft envelope requirements.

The on axis intensity of a focused beam is a critical measurement of laser wavefront quality. This paper describes a simple method and device for measuring the Strehl ratio. The device requires a collimated beam of known input size and wavelength. The experimental Strehl ratio is determined by the power within the central 1/5 of the Airy disc diameter. Error sources are discussed, as well as experimental measurements.

This paper provides the results of a parametric study conducted to assess the relative performance advantages derived from the use of coolants other than water to improve the peformance of the cooled optic or crystal. Cryogenic coolant and liquid gallium were investigated for their effectiveness in temperature and distortion control of an optical surface with an underlying pin post heat exchanger. The heat exchanger configuration remained the same but included changes in entrance and exit areas to accommodate the different coolants studied. The synergism of cooled beam line optics and high energy laser cooled element sees multiple passes by the beam before the beam is outcoupled, and, while smaller amounts of energy are absorbed, the resulting distortion is significant because of the accumulation of wave front distortion from the multiple passes. In the beam line application, large amounts of energy are absorbed but the number of passes is limited to one, and the resulting distortion occurs from the one pass. Thus, the heat exchanger design analysis tools developed for the laser apply to the beam line case without significant modification for the application. The experience base includes anchored analysis techniques that are low-cost and contribute a streamlined design approach toward achieving the desired cooled optical element performance.

An investigation was made of a loop stimulated Briliouin catLring (SBS) laser whose output radiation may contain a mode propagating in the opposite direction to the exciting radiation and modes rotated relative to this direction. It was demonstrated theoretically rind experimentally that, or resonator lengths corresponding to a phase shirt q) ' IC, the lasing threshold o the central mode wa higher than the thresholds o the first rotated modes.

At Los Alamos National Laboratory, we are preparing to image submicrometer-size features using the Free Electron Laser (FEL) operating at 248 nm. This article describes the optical transfer systems that were designed to relay the ultraviolet (UV) optical output of the FEL, resulting in expected imaged feature sizes in the range 0.3 - 0.5 micrometers . Nearly all optical subsystems are reflective, and once the coatings were optimized any optical wavelength could be used. All refractive optics were UV-grade fused silica. The optical design, engineering, and manufacture of the various component systems are described along with some experimental results.

Two versions of the 'Phase-Step Mirror' (PSM), a novel optical component that prevents the formation of sidebands in a Free-Electron Laser (FEL) were tested on the Los Alamos National Laboratory (LANL) APEX FEL. Sideband suppression and frequency control with high extraction efficiency and single line, transform limited operation were demonstrated. The results of our LANL experiments and computer simulations showed that for very high gain applications, the first-order sideband is completely suppressed, but the laser gain is so strong that on about pass 300 the sideband at the second-order or next free spectral range of the PSM appears. This second-order sideband may be suppressed by designing a PSM with grooves having two alternating depths, one chosen to suppress the first-order sideband, and the other, the second-order sideband.