Equivalent Fresnel Number Research Articles

Misalignment Sensitivity of Unstable Resonators with Spherical Mirrors

Abstract The effects of misaligning the spherical mirrors of unstable resonators on the losses, and the near and far field have been investigated both theoretically and experimentally. The theoretical description is based on the Kirchhoff integral equation of the equivalent symmetric resonator with shifted integration limits, which was solved numerically for G-parameters smaller than 2·5 and equivalent Fresnel numbers in the range up to three. Experiments were performed with a pulsed Nd-YAG system. Intensity distributions in the near and far field were recorded with an image-processing system and the losses were measured absolutely by using a previously developed method for loss measurement. Furthermore, a comparison of the misalignment sensitivity of stable and unstable resonators indicates that the latter are promising candidates for resonators, even in solid-state lasers.

Unstable resonator semiconductor lasers. Part 1: Theory

We review the theory necessary to describe the class of semiconductor lasers with nonplanar mirrors whose resonator geometry is of the unstable resonator configuration. We present formulas from geometrical unstable resonator theory and calculate quantities relevant to semiconductor lasers, including quantum efficiency and the width limitation due to total internal reflection. We describe the mirror coupled mode formalism for analysing the modes of unstable res onator semiconductor lasers and present near-field patterns and calculations of loss as a function of length, width, and lateral index step. We show that the periodicity of losses as a function of length is determined by a quantity analogous to the equivalent Fresnel number of macroscopic unstable resonators, which is determined by the lateral index step within the laser.

Unstable resonators with negative equivalent Fresnel numbers

The equivalent Fresnel numbers of two unstable resonators of current interest for application in free-electron lasers are negative. For a resonator of this class, the equivalent collimated round-trip propagation of Siegman’s canonical formulation is a time-reversed propagation, and the modes and eigenvalues are the complex conjugates of those corresponding to a resonator having a positive equivalent Fresnel number with the same magnitude.

Unstable optical resonators with linear magnification

Bare cavity unstable axisymmetric resonators with magnifications that are linear functions of the radial coordinate are investigated. Some properties similar to those of ordinary resonators are found. The geometrical theory predicts modes and eigenvalues in reasonable agreement with physical optics results. Interleaving of radial mode eigenvalues with increasing equivalent Fresnel number is observed. One form of scaling in size is found. The number of parameters involved and the lack of equivalence relations limit the scope and generality of the investigation.

Distributions of the electromagnetic field and of the gain in a plane-parallel resonator of a CO2gasdynamic laser at high values of the Fresnel number

An investigation is reported of changes in the characteristic lengths of the processes of exchange and their relaxation in an inverted CO2–N2–H2O medium on the structure of modes in a plane-parallel resonator of a CO2 gasdynamic laser. A four-temperature kinetic model is used in the calculations. The problem of the ultimate geometric-optics distribution of the field is considered. It is shown that for sufficiently large values of the equivalent Fresnel number the modes with an intensity distribution close to that predicted by geometric optics have a minimum in the gain distribution in the downstream direction.

Effects of unstable resonator misalignment in the cusping domain

Computational investigations were made of the power loss (output coupling) of misaligned strip unstable resonators as a function of the equivalent Fresnel number for several values of the misalignment parameter in a domain where cusping occurs. The behavior of power loss was found to agree with predictions based on consideration of a separate value of equivalent Fresnel number associated with each side of the resonator axis. Effects of even small misalignments can be quite pronounced.

Wave-approximation calculation of an unstable resonator with field rotation

An analysis is made of a method for numerical calculation of the parameters of an unstable resonator with field rotation. The wave approximation is used and the three-dimensional diffraction problem is reduced to two dimensions. This makes it possible to analyze the spatial structure of the field for large equivalent Fresnel numbers allowing for the influence of some forms of distortion in the resonator. By way of example, a calculation is carried out of the required precision of the relative alignment of mirrors in corner-cube retro reflectors used in unstable resonators with field rotation.

Resonator mode analysis using linear prolate functions

The modes of an unstable aligned empty strip resonator are found using the kernel expansion technique with the linear prolate functions as a basis set. Calculations of the modes are used to examine the accuracy of the asymptotic methods at low equivalent Fresnel numbers and to demonstrate the orthogonality of the cavity modes.

Intracavity adaptive optic compensation of phase aberrations III: Passive and active cavity study for a large N_eq resonator

The results of a numerical study of both the passive and the active mode-structure properties and far-field behavior of a linear, positive-branch, confocal unstable resonator with a large equivalent Fresnel number in the ideal unaberrated, intracavity phase-aberrated, and intracavity phase-corrected states are presented. A simple, saturable, gain-medium model of a homogeneously broadened CO2–electron-beam discharge laser system is employed in the active cavity study. The active cavity results presented here show that, at least for the level of gain saturation considered, the passive cavity theory of phase-aberration sensitivity remains applicable in this more-physical situation. Furthermore, the ideal correction to phase-tilt and curvature aberrations, derived from passive cavity considerations, remains exact in the presence of saturable gain. Finally, the influence of a multidithered, zonal deformable mirror on the cavity-mode structure and far-field behavior is analyzed in order to account for recent experimental tests of the intracavity adaptive optics concept.

Influence of a nonlinear active medium on the structure of natural oscillation modes of a Fabry–Perot interferometer

The results are given of a parametric calculation of the mode structure of a circular Fabry–Perot resonator of a solid-state laser. It is shown that an increase in the equivalent Fresnel number of the resonator (when its value is sufficiently large) alters the mode field distribution in some cases so that it resembles the modes of increasing order occupying the main part of the resonator, whereas in other cases the mode field is concentrated in a decreasing region near the resonator axis. The existence of several different solutions for the same mode is demonstrated. In the case of resonators with an equivalent Fresnel number much greater than unity the distributions of the intensity are close to geometric in the main volume of the resonator and estimates are given of the angular divergence of the output radiation.

Diffractive effect of a scraper mirror in an unstable resonator

In an unstable optical resonator, a scraper mirror located in front of and displaced from the feedback mirror clips the feedback beam twice on each round trip through the resonator. The increased diffractive scattering produced by this double application of the scraper aperture enhances the peaking of the intracavity intensity and raises the feedback ratio. This increased diffractive scattering is proportional to the square root of the scraper displacement and to the equivalent Fresnel number of the resonator.

Unstable resonator with aspherical mirrors

The geometric analysis of unstable optical resonators is extended to include mirrors with aspherical surfaces. This leads to a functional equation for the wave front of the geometric mode in terms of the mirror surface figures. In an example case, a solution is obtained for the aspherical feedback mirror required to produce a desired aspherical-output wave front, and the equivalent Fresnel number is calculated as a function of mirror aperture. Various possible uses of aspherical mirrors in unstable resonators are suggested.

Selective properties of a linear resonator with an asymmetric dispersive component

Theoretical and experimental investigations were made of the selective properties of a linear resonator with an asymmetrically located dispersive component in the case when the dimensions of the end reflectors are limited by stops. When the equivalent Fresnel number is less than unity, the resonator band depends strongly on the dimensions of the stops, position of the dispersive component in the resonator, telescopic coefficient, and mirror curvature. A good agreement between the theory and experiment shows that the proposed calculation method can be used in the optimization of the parameters of small-aperture dispersive resonators frequently employed in dye lasers.

Intracavity adaptive optic compensation of phase aberrations II: Passive cavity study for a small N_eq resonator

The results of a numerical study of the passive mode structure properties and far-field behavior of a linear positive-branch, confocal unstable resonator cavity with a small equivalent Fresnel number in the ideal unaberrated, intracavity phase-aberrated and intracavity phase-corrected states are presented. The analysis of Part I of this paper [ J. Opt. Soc. Am.71, 862– 872 ( 1981)] is employed throughout to illuminate the numerical results obtained here. These numerical results illustrate both the limitation of the approximate geometrical theory to small applied aberration strengths and the powerful corrective capabilities afforded by an ideally configured intracavity adaptive optic element.

Attainment of the theoretical minimum input power for injection locking of an unstable-resonator KrF laser

We report attaining a ratio of output to injected power as high as 10(8) in a KrF system, the limit being reached when injected power equals the noise per mode in the resonator. We find that, because of coherence effects, adjustment of the cavity length for correct value of the equivalent Fresnel number can change the threshold power for frequency locking by at least 2 orders of magnitude. The laser's locked bandwidth is <0.1 A, and the line can be tuned over a range exceeding 10 A. The beam divergence is near (i.e., 1.5x) the diffraction limit.

Injection-locking optimization in unstable resonators

The problem of transverse-mode matching between the lowest-loss mode of a confocal unstable resonator and a Gaussian beam, which is injected into the resonator through a coupling hole in its back mirror, is analyzed. It is found that in resonators having equivalent Fresnel number >0.5, there is a value of the coupling hole diameter for which a mode transition occurs. Below this value the coupling coefficient between the mode and the injected signal increases with the hole diameter. The optimal coupling hole diameter for injection locking lies somewhat below the transition value, where the low-order mode is still dominant and the coupling coefficient is high (~0.25-0.30). The power loss through the coupling hole is negligible (~2%) compared to the output coupling, which is ~50-70%.

Eigenvalues for unstable resonators with slightly misaligned strip mirrors

Very small misalignments in unstable strip resonators may cause detachment of the low-loss eigenmode at a lower equivalent Fresnel number N(eq) and introduce different periodicities into the eigenvalue curves as a function of N(eq). Using the resonance equation developed previously from waveguide mode theory, this behavior is explained in physical terms by coupling between mode fields with even and odd symmetry, which are uncoupled in the perfectly aligned configuration. A simplified explicit equation derived for the eigenvalues of the detached mode is found to predict correctly the periodicities of the eigenvalue oscillations. The waveguide method therefore continues to provide an explicit and reliable procedure for rapidly locating the maxima and minima of the detached-mode eigenvalues over broad ranges of N(eq).

Annular (HSURIA) resonators: some experimental studies including polarization effects

A repetitively pulsed CO(2) laser facility was developed for testing annular resonators. The large-aperture device exhibits generally uniform gain over an annular region of 18-cm o.d. and 10-cm i.d. The half-symmetric unstable resonator with internal axicon (HSURIA) was tested at equivalent Fresnel numbers up to 4.5. This resonator design incorporates a W-axicon mirror beam compactor that transforms a cylindricalmode region into an annular-mode region. Two HSURIA configurations were evaluated: (a) with a conical end mirror and (b) with a flat end mirror in the annular leg. With the conical end mirror, the aligned resonator produced a predominantly higher-order azimuthal mode with an on-axis null in the far field. The output was strongly linearly polarized with the electric-field vector tangential to the optic axis in both the near and far fields. The higher-order tangentially polarized mode appears to be the result of a geometric polarization scrambling effect caused by the conical end mirror. The boundary conitions for the conical or W-axicon mirrors imply that the radial electric field has a 180 degrees phase shift on reflection, whereas the tangential component is unchanged. Thus, a tangentially polarized mode is self-reproducing, but a linearly polarized mode is not. To eliminate the polarization scrambling effect in the HSURIA, the conical end mirror was replaced with a flat end mirror. The HSURIA with a flat end mirror produced a central spot in the far field that indicated an l = 0 mode with no spatial variations in polarization. Beam quality was measured in terms of the ratio n(2) of the theoretical (geometric-mode) power transmitted through an aperture of the central lobe diameter to the observed power; n(2) values as low as 1.2 were obtained. The variation of beam quality with tilt of the flat end mirror indicated a factor of 2 degradation in n(2) for a 20-microrad tilt, which is in good agreement with theory.

Calculation of equivalent Fresnel numbers for unstable resonators with scraper mirrors

Calculation of equivalent Fresnel numbers for unstable resonators with scraper mirrors

A canonical formulation for analyzing multielement unstable resonators

A simple but general formulation for the unified analysis of many different varieties of unstable optical resonators is obtained by expressing the round-trip Huygens' integral for the resonator in terms of the round-trip paraxial ray matrix. Complicated multielement resonators, ring resonators, and unstable resonators with variable-reflectivity output mirrors are thereby all reduced to a single equivalent collimated-beam diffraction problem. Essential assumptions in the analysis are that all elements in the resonator obey paraxial ray theory and that there is only one significant output coupler or limiting aperture per round trip inside the resonator. The only parameters needed to describe an arbitrary unstable resonator in this formulation are the round-trip magnification M and either the equivalent Fresnel number N eq or else an equivalent collimated Fresnel number N_{c} = [2M^{2}/(M^{2} - 1)] N_{eq} , plus for variable-reflectivity mirrors a specification of the mirror reflectivity profile.