High Loss Region Research Articles

Power loss and flux density distributions in the T-joint of a three phase transformer core

A three phase transformer test facility has been used for measuring the localized power loss and flux distribution within the core. The flux and loss distributions in the T-joint of a core with simple double overlap joints have been investigated. The flux density distribution was measured by an array of search coils wound on individual laminations, and the localized power loss was found by measuring the initial rate of temperature rise using miniature thermocouples. For measurements made in the range of flux densities from 1.1 to 1.6 T, the inner edges of the yoke and limb laminations were the highest loss areas, up to 30% higher than the mean core loss. These regions of high power loss were found to be due to a high third harmonic flux density component being present. The fundamental component of flux density tended to follow the rolling direction of both yoke and limb laminations; however, some deviation from the rolling direction was present in the case of other harmonics.

The eigenmodes of a spatially inhomogeneous laser

The laser eigenmodes are determined from an eigenvalue equation, which includes spatial inhomogeneities of the inversion, of the density of atoms, and of the local loss. The solutions are found iteratively, starting from the empty-cavity modes. Axial modes are found to be superposed by their off-axial neighbours, which oscillate with the same frequency due to a locking of their phase differences. The eigenfunctions are focused into regions of high inversion and small loss thus lowering their threshold.

Modes of Stable and Unstable Optical Resonators

A method is developed for the approximate determination of the normal modes of stable and unstable optical resonators and the associated resonant frequencies and power losses. The method is based on replacing the finite integration limits in the integral equation for the normal modes by infinite limits and, subsequently, finding a differential equation whose solutions coincide with or approximate the solutions of this integral equation. When the end reflectors of the resonator are conical surfaces, a differential equation is found which corresponds exactly to the integral equation with infinite limits. Moreover, the equivalent differential equation is found to be of the same form as the wave equation for a monochromatic transverse electric wave propagating in an inhomogeneous medium of infinite extent with the inhomogeneity being transverse to the direction of propagation, showing the correspondence between the modes of a homogeneously filled conical resonator and the eigenmodes of an infinite inhomogeneous mdium. For the stable, low loss (convergent) region the solutions of the differential equation are readily found. For the unstable, high loss (divergent) region the solutions are found by using the principle of analytic continuation. The specific example of parabolic end reflectors is treated in more detail, and solutions for the eigenvalues and eigenfunctions are given for the case of infinite strip and circular geometries.

Modes and Instabilities of Modes of the Flat-Roof Open Resonator

The problem of modes and losses of a resonator terminated by flat-roof reflectors is approached with the Fox and Li method of iteration. Field distributions, phaseshifts, and losses for some of the low-order modes are derived as functions of the roof angle and mirror aperture. A good agreement is found with the approximate theory previously developed by Toraldo di Francia. High loss regions are also observed. The behavior of different modes in such regions is investigated in detail. Design guidelines are given too. Experimental texts carried out on an X-band model confirm the theoretical predictions.

Propagation of Gaussian beams in anisotropic media

The form of the Gaussian beam in an anisotropic medium has been found by solution of Maxwell's equations. For mathematical convenience a cylindrical beam (two-dimensional) is considered, with the field constant in one cross-sectional dimension and varying as the Gaussian function in the other. For uniaxial crystals, and for biaxial crystals where the propagation is in one of the principal planes, two solutions are obtained, corresponding to the ordinary and extraordinary waves. The Gaussian beam for the ordinary wave is identical to that for an isotropic medium. The extraordinary Gaussian beam is modified by double refraction and by an increase in the beam spread of 11/33. The properties of the extraordinary Gaussian beam are derived and applied to the problem of an optical resonator filled with an anisotropic medium. The low- and high-loss regions are found for the resonator together with its resonant frequencies. Two kinds of anisotropic cavity are distinguished: those where the optic axis of the medium is at 0? or 90? to the optical axis of the cavity and those where the optic axis is at 45? to the cavity axis. The three-dimensional Gaussian beam is treated by assuming that it can be represented by two orthogonal two-dimensional beams. The effect of anisotropy on the extraordinary beam gives rise to nonspherical wavefronts and elliptical cross sections. Some applications of the theory to laser problems are outlined.

Wall Stall in Compressor Cascades

The “wall stall” in compressor cascades has been observed by many research workers. There is a region of high loss in the corner bounded by the end wall and the suction surface of a blade. It is the separation of the flow in this region which is largely responsible for the increase in the axial velocity across the midspan section of the blades when there is no end-wall suction. It is important to differentiate between this wall stall and the two-dimensional stall of the blading. The work on wall stall described in the paper was carried out at the University of Cambridge and the University of Liverpool in the period 1955-1962.

A Scalar Resonator Theory for Optical Frequencies

A scalar resonator theory for optical or near-optical frequencies is developed. For an arbitrary resonator system a general integral equation is derived which represents a solution of Helmholtz's equation between the reflectors and satisfies Maxwell's equations along the surface of the reflectors. To meet the consequences of the boundary conditions an eigenmode is defined as an energy distribution which when launched from an arbitrary mathematical surface between the reflectors reproduces itself within an amplitude factor after a number of bounces given by the periodicity of the system under consideration. The eigenvalues of the integral equation determine the diffraction losses and the resonance spectrum. The mode patterns between the reflectors can be deduced from the corresponding eigenfunction. The general theory is discussed in connection with resonator systems with spherical reflectors. This reveals a physical picture of the low-loss and high-loss regions as the reflector spacing is varied. With the example of the Fabry-Perot interferometer it is demonstrated that the present resonator theory agrees with geometrical optics as the Fresnel number becomes very large. The scalar formulation to be presented accounts for the phenomenon of diffraction and is consistent with the microwave and the antenna theories. The general theory when applied to a periodic sequence of thin lenses suggests that it may equally well be the basis of a theory for beam waveguides.

An Extension to the High Loss Region of the Solution of the Confocal Fabry-Perot Resonator Integral Equation (Correspondence)

An Extension to the High Loss Region of the Solution of the Confocal Fabry-Perot Resonator Integral Equation (Correspondence)

Unstable optical resonators for laser applications

A simple geometrical analysis has been developed which describes the lowest-order transverse mode of any large-Fresnel-number optical resonator located in the unstable or high-loss region of the resonator mode chart. Such resonators include, for example, resonators in which one or both of the mirrors are divergent spherical surfaces. The lowest mode in such a resonator is assumed to consist of two oppositely traveling divergent spherical waves which uniformly illuminate the end mirrors. The centers of curvature of these spherical waves do not, in general, coincide with the mirror centers of curvature, but are found by requiring that each center be the image of the other upon reflection from the appropriate mirror. The resonator losses are found from purely geometrical considerations, and are given by simple analytical expressions. These losses turn out to be independent of the mirror sizes, so that hyperbolic universal equiloss contours can be drawn on the resonator mode chart. The losses agree well with more exact results obtained by Fox and Li for a few specific cases. Experimental results in good agreement with the analysis have been obtained using a ruby laser rod having a divergent spherical surface ground directly onto one end of the laser rod. Unstable resonators, particularly the Cassegrainian unstable configuration used in the experiments, appear potentially useful for diffraction output coupling applications, and possibly also for transverse mode control, in ruby and other high-gain lasers.

Modes in a maser interferometer with curved and tilted mirrors

Fabry-Perot interferometers have played an important role in the conception and realization of optical masers. The authors have previously made a study of the idealized interferometer. In this paper they present some results of a continued study of the effects of certain simple forms of aberration. The first is represented by tilted plane mirrors and the second by curved mirrors. Tilting the mirror causes the mode patterns to become asymmetric and the diffraction loss to become greater. It also tends to equalize the losses of the two lowest-order modes thus causing a beating phenomenon which has been observed experimentally. In the case of interferometers with mirrors of arbitrary radii of curvature, there exist regions of low loss and high loss as the mirror spacing is varied. The loss function can be represented by a contour map of a three-dimensional model which has certain symmetry properties. This model is useful in choosing proper mirror spacing for low loss operation of optical masers.

Generalized Confocal Resonator Theory

The theory of the confocal resonator is extended to include the effect of unequal aperture size and unequal radii of curvature of the two reflectors. The latter is equivalent to a periodic sequence of lenses with unequal focal lengths. This treatment is in Cartesian coordinates as previously used. In an appendix the modes and resonant formulas are written in cylindrical coordinates. The effect of unequal aperture size of the two reflectors is shown to produce mode patterns of unequal size on the two reflectors of a confocal resonator. The previous computations for diffraction losses are found to be applicable. Generalization of the theory to the case of reflectors of unequal curvature shows the existence of low-loss regions and high-loss regions as the reflector spacing is varied. One of the high diffraction loss regions occurs when the reflector spacing is between the two unequal radii of curvature. Such a region is interpretable in terms of instabilities in a periodic sequence of lenses of unequal focal length. An estimate of diffraction losses is obtained for the low-loss regions. The presence of a high diffraction loss region or unstable region should be of importance in the design of resonators or of a periodic sequence of lenses.

Recovery Ratio—A Measure of the Loss Recovery Potential of Compressor Stages

Abstract Certain types of axial-flow-compressor stage designs have the ability to add extra energy to high-loss regions of the inlet flow and thereby tend to smooth out inlet-flow distortion rapidly. A recovery ratio, which is believed to give a quantitative measure of this ability for radial (axisymmetric) distortions, is defined and expressed in terms of familiar stage parameters. It is found that the flow coefficient (meridional velocity/blade speed) should have a value near 0.5 or less, and that the stage loading should be light.