Spatial coherence of the generalized diffraction-filtered resonator copper vapor laser

The results of a study on the spatial coherence of a generalized diffraction-filtered resonator (GDFR) copper vapor laser (CVL) for various magnifications are presented. The coherence width and output power are compared with that of unstable resonators (UR's) of equivalent magnifications. It is established, by use of reversal shear interferometry, that the GDFR CVL beam has better spatial coherence and average power characteristics than the UR CVL beam for equivalent resonator magnifications.     

Measuring characteristic length scales of eigenfunctions of Sturm–Liouville equations in one and two dimensions

The spatial resolution of eigenfunctions of Sturm–Liouville equations in one-dimension is frequently measured by examining the minimum distance between their roots. For example, it is well known that the roots of polynomials on finite domains cluster like O(1/N ²) near the boundaries. This technique works well in one dimension, and in higher dimensions that are tensor products of one-dimensional eigenfunctions. However, for non-tensor-product eigenfunctions, finding good interpolation points is much more complicated than finding the roots of eigenfunctions. In fact, in some cases, even quasi-optimal interpolation points are unknown. In this work an alternative measure, ℓ, is proposed for estimating the characteristic length scale of eigenfunctions of Sturm–Liouville equations that does not rely on knowledge of the roots. It is first shown that ℓ is a reasonable measure for evaluating the eigenfunctions since in one dimension it recovers known results. Then results are presented in higher dimensions. It is shown that for tensor products of one-dimensional eigenfunctions in the square the results reduce trivially to the one-dimensional result. For the non-tensor product Proriol polynomials, there are quasi-optimal interpolation points (Fekete points). Comparing the minimum distance between Fekete points to ℓ shows that ℓ is a reasonably good measure of the characteristic length scale in two dimensions as well. The measure is finally applied to the non-tensor product generalized eigenfunctions in the triangle proposed by Taylor MA, Wingate BA [(2006) J Engng Math, accepted] where optimal interpolation points are unknown. While some of the eigenfunctions have larger characteristic length scales than the Proriol polynomials, others show little improvement.     

Modelling non-Gaussian random fields using transformations and Hermite polynomials

In the context of modelling residual roughness on nominally flat moderately polished metal surfaces, a method is proposed for solving problems related to sample function properties and/or special points such as maxima, minima, saddle points for random fields having non-Gaussian height distributions by recasting them in terms of the corresponding problems for the much more tractable Gaussian random fields by means of transformations. Special reference is made to the expansion of the transformations in series of Hermite polynomials. While the use of Hermite polynomials in connection with transformations of random fields and the useful results they yield with regard to covariance functions are well known, this paper derives the most general explicit formula for the expectation of any product of several Hermite polynomials in correlated Gaussian arguments thereby allowing their application to the higher moments of the transformed random field, in particular, to the third moment, which may be used to measure skewness.     

Optical axis perturbation in folded planar ring resonators

A mathematical model of a four-sided folded planar ring resonator is established. The model can be modified into a triangular ring resonator, a square ring resonator, and a four-sided folded ring resonator, all of which are widely used for ring laser gyroscopes by changing certain design parameters such as incident angle Ai and side ratio H. By use of the extended matrix formulation, the optical axis perturbation, including optical axis decentration and optical axis tilt, in those planar ring resonators is analyzed in detail resulting in some novel findings. It has been determined that the longer the mirror radius, the larger the mode volume, the higher the sensitivity of optical axis decentration and the lower the sensitivity of optical axis tilt. The same mirror misalignment value, mostly the misalignment induced by optical axis decentration in the x and y components, has the conventional ratio of 1:[cos(Ai)](2) for the symmetrical points of the resonator. Details of the effect of Ai and H on the optical axis tilt have also been determined. The difference in optical axis tilt between different kinds of ring resonator is disclosed. The sensitivity of optical axis tilt was found to undergo singular rapid change along with the right edge of the second stable area. This singular behavior is useful for those resonators that have a small incident angle, such as Ai=15 degrees , because those resonators have a second stable region. These interesting findings are important for cavity design, cavity improvement, and alignment of planar ring resonators.     

Dynamic behaviour of sandwich structure containing spring-mass resonators

The behaviour of wave propagation in a sandwich beam with internal resonators is studied analytically and experimentally. It is found that, with the occurrence of a bandgap, waves generated by actuations in certain frequency range are forbidden to propagate into the sandwich beam without attenuation. By tailoring the local resonance frequency of the resonator, the range and the location of the bandgap can be selected. Also, from the experimental and FEA results, it is seen that waves can be significantly attenuated in a sandwich beam with a finite number of internal resonators if the forcing frequency is near the local resonance frequency. It is concluded that the resonators can be used as a mechanism for suppressing the flexural motion in a sandwich beam.     

Image description with generalized pseudo-Zernike moments

A new set, to our knowledge, of orthogonal moment functions for describing images is proposed. It is based on the generalized pseudo-Zernike polynomials that are orthogonal on the unit circle. The generalized pseudo-Zernike polynomials are scaled to ensure numerical stability, and some properties are discussed. The performance of the proposed moments is analyzed in terms of image reconstruction capability and invariant character recognition accuracy. Experimental results demonstrate the superiority of generalized pseudo-Zernike moments compared with pseudo-Zernike and Chebyshev-Fourier moments in both noise-free and noisy conditions.     

Simplified transformation circle theory in analyzing a laser resonator

Transformation circle theory is simpler than other methods for analyzing laser resonators. In our analysis only sigma circles and simple mathematical knowledge are used to analyze the stability and calculate the parameters of the laser resonator, which further simplifies the transformation theory. The results agree well with the well-known matrix theory. Two- and three-mirror (including a thermal lens) laser resonators are used as examples to present the stability formula and the Gaussian beam dimensions at the mirrors. Furthermore, we apply the commonly used example in which the laser medium is close to the cavity mirror.     

Unstable resonator modes: a beam expansion approach

A novel method of computing the modes of unstable resonators is presented. By writing the modes as a series expansion in a set of orthogonal functions, the problem is reduced to finding how the expansion coefficients change in a circuit of the resonator. The eigenvalues of the transfer matrix are shown to be the mode eigenvalues, while the eigenvectors are the expansion coefficients defining the mode in question. The technique is demonstrated in one transverse dimension using Hermite–Gaussian functions as the basis. The approach offers significant advantages over other methods.     

Triangular laser resonators with astigmatic compensation

The magnitudes and locations of the beam waists in both the sagittal and the tangential planes have been found by means of the ABCD matrix method for a triangular resonator. Equilateral and isosceles resonators are discussed, and curves are given from which resonators with astigmatism-free beams can be designed. A frequency-doubled triangular Nd ring laser has been constructed after this design, and it is demonstrated that this laser emits a single longitudinal mode with a circular TEM(00) Gaussian beam.     

An X-ray double crystal topographic assessment of defects in quartz resonators

This paper describes an X-ray double crystal topographic study of defects in eighteen quartz resonators designed to operate at 1.4 MHz. The types of defects found in quartz are described, together with their reported effects on resonator performance. The mode of operation of the bulk resonator and the technique of X-ray double crystal reflection topography are outlined. Topographs reveal the electrode structures and surface features of the resonators together with the presence of growth defects such as dislocation cells, sub-boundaries, growth striations and growth sector boundaries. Spurious flexure modes in two resonators are also shown. It is demonstrated that a correlation exists between the presence of growth striations (and probably sub-boundaries) and a higher equivalent series resistance of the resonators. It is shown that such defects change the contributions to the losses, possibly by changing the nature of the vibration pattern of the resonator.     

Single-pulse time-resolved comparative study on the performance of a master-oscillator,power-amplifier copper-vapor-laser system with generalized diffraction-filtered and unstable resonators as master oscillators

We present results of comparative time-resolved coherence studies on a single pulse from amaster oscillator power amplifier (MOPA) copper vapor laser with generalized diffraction-filtered and unstable resonators as master oscillators. It is shown that, unlike the conventionally used unstable-resonator MOPA reported in literature, the coherence of a generalized diffraction-filtered resonator MOPA pulse is fairly independent of the delay between the oscillator and the amplifier. It also remains constant throughout the pulse, with the result that the flux is constant over a large range of the delay.     

Theory of 1- N-way phase-locked resonators

A 1-N-way resonator based on beam splitting and beam combining effects in rectangular cross-sectional multimode waveguides was recently proposed. Such a resonator structure offers a valuable way in which N low-power laser elements may be combined in a coherent fashion. We examine the case of passive 1-N-way resonators. We develop a theory of these 1-N-way structures to show that there is only one possible mode of these resonators. The theory is used to give a scaling law for the design tolerances of the beam splitting and beam combining region of the resonator.     

Comparison of Talbot and 1-to- N-way phase-locked array resonators

Phased-array resonators provide an important basis for achieving high output powers from arrays of low-power elements. We have recently proposed a novel form of 1-to-N-way phased-array resonator based on the beam splitting and regeneration characteristics of rectangular sectioned multimode waveguides. We compare its performance with that of the widely used, yet problematic, Talbot resonator. Our design is found to have significant advantages over the Talbot resonator in terms of improved modal stability, unique photon-mixing characteristics, and near- and far-field outputs of quasi-Gaussian form.     

Hybrid stable-unstable resonators for diffusion-cooled CO(2) slab lasers

We present the numerical and experimental study that we carried out to compare the performances of two hybrid stable-unstable resonators for diffusion-cooled CO(2) slab lasers. The two resonators are designed to fit a 320 mm × 60 mm ×2 mm rf-discharge channel and are both guided in the narrow transverse direction. They differ in the other transverse direction, consisting of a positive- or a negative-branch unstable resonator scheme. The two solutions have been characterized in terms of modal structure, power extraction, stability, and quality of the extracted beam.     

计及压电效应时矩形振子的三维振动研究

在计及压电效应的情况下，本文利用解析方法对有限尺寸压电陶瓷矩形振子的三维振动进行了研究，推出了振子耦合振动的频率方程，并对振子的振动模式进行了分析。理论研究表明，利用本文中的解析法研究振子的耦合振动，计算简单、物理意义明显。与传统的一维理论分析方法及数值方法相比，由于本研究考虑了振子的压电效应以及不同振动模式不同的相互耦合，因此，振子的理论计算频率与测量值更加符合。     

Analysis of the rutile-ring method of frequency-temperature compensating a high-Q whispering gallery sapphire resonator

The rutile-ring method of dielectrically frequency-temperature compensating a high-Q whispering gallery (WG) sapphire resonator is presented. Two and three-dimensional finite element (FE) analysis has been implemented to design and analyze the performance of such resonators, with excellent agreement between theory and experiment. A high-Q factor of 30 million at 13 GHz, and compensation temperature of 56 K was obtained. It is shown the frequency-temperature compensation can occur either because the rutile adds a small perturbation to the sapphire resonator or because of a mode interaction with a resonant mode in the rutile. The characteristics of both of these methods are described, and it is shown that for high frequency stability, it is best to compensate perturbatively     

Nonplanar ring resonator modes: generalized Gaussian beams

Several findings on the modes of a nonplanar ring resonator have been carried out by utilizing the matrix method of generalized Gaussian beams. The widely used nonplanar ring resonator--a four-equal-sided nonplanar ring cavity with two curvature mirrors and two planar mirrors--has been chosen as an example. The overall stability maps of this typical nonplanar ring cavity with L/R ranging from 0 to infinity are calculated and described, while the total cavity length is L and the radius of the curvature mirrors is R. The stability map has also been found to have asymmetry with total image rotation rho ranging from 0 degrees to 360 degrees and 02. The Gaussian modes of the nonplanar ring cavity have been found to have different characteristics with different design parameters such as rho and L/R. The azimuth angles of the major and minor axes of the spot size have been found to be variable with different design parameters such as rho and L/R and variable under different location of the nonplanar cavity. These interesting findings are totally different from the behavior of conventional planar stable resonators and are important to the cavity designs of nonplanar ring resonators.     

Scattering analysis of two-port SAW resonators

Linear equations derived from the scattering matrix approach to the two-port resonator were solved, and analytical expressions for the normalized SAW amplitudes were obtained. Asynchronous and synchronous resonators were analyzed numerically. It was shown that the output of the two-port resonator is a sum of two signals. In the case of the asynchronous resonator, these signals are in phase at a resonance frequency; for the synchronous resonator, they are in phase quadrature, which causes the higher insertion loss of the synchronous resonator     

Comparison of resonator-originated and external beam shaping

The spatial shaping of laser beams is a subject of research in modern optics. Recently the introduction of diffractive elements in laser resonators has offered an alternative to external beam-shaping optics by mode shaping within the resonator. We describe the specification of the laser resonator mirrors to obtain by means of internal mode shaping a desired beam outside the resonator. Modal discrimination of the modified resonator and the mirror alignment sensitivity is discussed. Basic features of resonator-originated and external beam shaping are compared.