A. I. Mahan

Far-Field Diffraction Patterns of Single and Multiple Apertures Bounded by Arcs and Radii of Concentric Circles*†

Some equations are derived for computing the form of the far-field diffraction pattern of an aperture bounded by arcs and radii of concentric circles whose dimensions are large compared with the wavelength of the incident radiant flux and over which the amplitude and phase are constant. These equations are derived in such a way that it is possible not only to compute the form of the far-field diffraction pattern of individual apertures bounded by arcs and radii of concentric circles, but also for combinations of such apertures over which the amplitudes and phases are constant but differ from aperture to aperture. With the aid of an IBM 7090 computer, these equations have been used to compute the forms of the far-field diffraction patterns of a 60° sector aperture, a semicircular aperture, and an aperture formed by arcs and radii from concentric circles, and three-dimensional models have been constructed to describe the radiant flux distribution. A start has also been made on the more complicated problem of multiple apertures by computing the form of the far-field diffraction pattern of two adjacent semicircular apertures, over which the amplitudes and phases are constant in the two apertures, but the amplitudes in the two apertures have been assumed to be equal and the phases different. Some corresponding photographed far-field diffraction patterns are also included for comparison purposes and to illustrate the wide variety possible.

Far-Field Diffraction and Boresight Error Properties of a Two-Dimensional Wedge*†

An optical diffraction theory has been developed which can be used to predict the forms of the far-field diffraction patterns and boresight error effects of a two-dimensional wedge made from two semi-infinite plane parallel plates. The solution is strictly an optical one, since the wavelength has been assumed to be small compared with the dimensions of the rectangular diffracting aperture, and also because the rectangular diffracting aperture is assumed to lie in the “near-field” of the wedge. In tracing the directly transmitted and all the multiply internally reflected wave fronts through the wedge to the plane of the diffracting aperture, amplitude losses due to reflection and absorption, phase changes due to increases in optical path, and changes in width of the various wave fronts are all evaluated in the plane of the diffracting aperture. When once the amplitude and phase distributions are known in the plane of the diffracting aperture, the Kirchhoff integrals are readily written and integrated. The UNIVAC computer can then be used to evaluate the forms of the far-field diffraction patterns characteristic of the wedge and diffracting aperture. These far-field diffraction patterns are in general non-symmetric and have forms which are functions of the plane of polarization of the incident electric vector and nine other variables. For arbitrary values of the variables, the forms of these far-field diffraction patterns cannot be understood in terms of conventional plane parallel plate theory, but in certain limiting cases plane parallel plate theory is useful. Boresight error curves have been evaluated as functions of many of the variables, and the magnitudes and signs of these errors are in agreement with the previously computed diffraction patterns. “Qualitative” checks on the theory have also been made by comparing the calculated forms of the far-field diffraction patterns for a particular glass wedge with those which were photographed experimentally. All of the calculated changes in form of these far-field diffraction patterns have been observed experimentally, and there is good reason to believe that a quantitative check might also be possible under more favorable experimental conditions.

Some Macroscopic Properties of Dielectric, Absorbing, and Active Cylinders*†

Solutions to the wave equation and Maxwell’s equations in cylindrical coordinates were found that satisfy the boundary conditions at the interface between an infinitely long absorbing or active cylinder and an arbitrary outside medium. These solutions are exponentially decreasing or increasing TE, TM, or hybrid waves traveling along the direction of the axis. The components of the complex Poynting vector make it possible to determine the time average of the excess of the inflow or outflow of radiant flux along each of the coordinate directions and over the surfaces of an arbitrary cylindrical volume element. These equations show for the first time, in the simpler case of TE and TM waves, that cylinders of large radii possess unique modes, which simultaneously exhibit stimulated emission and absorption along different coordinate directions. In each case, the pumped-in radiant flux either reappears as stimulated emission along other coordinate directions or disappears by absorption introduced by the conductivity and complex dielectric constant. The theory is of a linear, resonant type, and is therefore limited to weak pumping fields for which fluorescence is absent.

Ruby as a Macroscopic Fluorescing and Laser Material

We consider ruby as a linear, uniaxial, maxwellian medium, which, in its absorbing state, has axially dependent material constants. It is made active by pumping with electric fields, in the region of shorter-wavelength absorption bands, which, by means of an energy transfer, induce source-type electric fields having constant amplitudes in the R1 and R2 regions, and make possible absorption, spontaneous emission, and stimulated emission. Experiments indicate that the absorption, fluorescence, and laser action are all maximum for electric fields perpendicular to the optic axis, and minimum when the electric field is parallel to the optic axis; the relative irradiances and shapes of the R1 and R2 bands in all cases depend upon the orientation of the plane of polarization of the emerging radiant flux relative to the optic axis. Attempts are made to compute the spectral distributions in the R1 and R2 bands for the ordinary and extraordinary waves, neglecting all boundary-value problems. Agreement was found to be possible only in the region below the threshold of laser action.

Far-Field Diffraction Properties of a Plane-Parallel Plate When Placed Partially in Front of a Rectangular Diffracting Aperture*†

It has been possible, using Kirchhoff-type integrals, to develop some optical equations for evaluating the form of the Fraunhofer diffraction patterns characteristic of a plane-parallel plate and rectangular diffracting aperture, when the plane-parallel plate has arbitrary positions and orientations in front of the rectangular diffracting aperture. The solution is an optical one, because the rectangular diffracting aperture is assumed to be large compared to the wavelength, diffraction effects in the plane of the rectangular diffracting aperture due to the edges of the plane-parallel plate are negligible, and multiple reflections between the edges of the rectangular diffracting aperture and plane-parallel plate have been neglected. Within these initial assumptions it is rigorous, for the directly transmitted and all the higher-order internally reflected wavefronts which make contributions to the amplitude and phase in the plane of the diffracting aperture have been considered. A UNIVAC 1103A-type computer has been used to calculate the forms of many of these diffraction patterns for particular choices of the nine possible variables. Attempts were also made with a 21-ft Jarrell-Ash Spectrograph to observe the corresponding experimental forms of these diffraction patterns. When an interferometer plate was moved across, and in front of, the rectangular diffracting aperture, the changes in form of the diffraction pattern were slow and could be followed in detail by corresponding theoretically calculated diffraction patterns. When, however, the interferometer plate was rotated in front of the rectangular diffracting aperture about an axis passing through its center, the changes became very rapid and required a very large number of calculated diffraction patterns to follow all the detailed changes. The theory in general seems to be quite useful in predicting the possible forms of the corresponding experimentally observed diffraction patterns.

Reflection and Transmission Properties of Cylindrically Guided Electromagnetic Waves

Cylindrically guided hybrid, TE and TM electromagnetic waves, which satisfy the wave equation, Maxwell’s equations, and certain boundary conditions at the curved surface of a homogeneous, isotropic cylinder, are permitted to be incident normally on plane interfaces between the cylinder and arbitrary outside media. The laws of reflection and transmission are then derived as boundary-value problems, using electromagnetic theory. These laws, for all three types of waves, differ from each other and from Fresnel’s equations, and depend upon the material constants of both media, the frequency, the radius of the cylinder, the form of the incident electromagnetic wave, and on whether the cylinder is experiencing stimulated emission or absorption in the direction of the axis. At interfaces outside the cylinder, these laws recover some of their Fresnel-like properties and depend only upon the material constants and frequency, but there are still differences between the three types of waves and Fresnel’s equations. There are also corresponding differences in the radiant-flux distributions for all three types of waves in each of the media.

Far-Field Diffraction and Polarization Properties of a Three-Dimensional Hollow, Homogeneous, Isotropic Cone*†

A brief description is given of a theory for computing the form of the far-field diffraction and polarization properties of a hollow, homogeneous, isotropic cone, when it is placed between arbitrarily oriented polarizer and analyzer. For this theory, the incident plane wavefront and circular aperture are normal to the axis of the cone. The solution is an “optical one” for which the circular aperture is large compared with the wavelength, the diffraction properties of the cone in the plane of circular aperture are negligible, and multiple reflections between the cone, circular aperture, and focusing system are neglected. All the multiple reflections which arise inside the cone walls and make contributions to the amplitude and phase in the circular aperture are, however, included. Within these limits, it is possible to compute the radiant flux and form of the far-field diffraction pattern for a cone of arbitrary angle, wall thickness, and optical properties, when it is placed between arbitrarily oriented polarizer and analyzer. In all cases, the symmetry properties of these far-field diffraction patterns can be predicted from the forms of the equations. Computations on the detailed forms of these far-field diffraction patterns have been carried out with an IBM 7090 computer, and three-dimensional models constructed and photographed to show their symmetry properties. It has not been possible experimentally to check these computations because of the problems arising in the making and testing of a cone of the required interferometer quality.

Absorption, Spontaneous Emission, Stimulated Emission, and Maxwell’s Equations*†

In this paper, four different ways are described in which homogeneous, isotropic, nonferromagnetic, Maxwellian media at rest may exhibit absorption and stimulated emission. These four ways may be identified by a positive or negative conductivity, a sink or source-type electric field, a complex dielectric constant, and a complex permeability. The forms of the electromagnetic waves, the refractive index, and the positive or negative extinction coefficient are determined for each process for both absorbing and active media. Spontaneous emission is associated with source-type electric and magnetic fields which generate conduction or displacement currents in the medium with improper phases to contribute to the radiant flux in the incident electromagnetic wave. Stimulated emission is characterized by exponentially increasing incident electric and magnetic fields and an excess of the time average of the outflow of radiant flux over the surfaces of an arbitrary volume. Stimulated emission appears when the time average of the rate at which the source-type electric and magnetic fields do work in carrying certain conduction or displacement currents in directions opposite to the incident electric and magnetic fields is greater than the time average of the rate at which the incident electric and magnetic fields dissipate energy by doing work in carrying other conduction or displacement currents in directions parallel to the incident electric and magnetic fields.

Oscillator Strengths for the Liquid Phase

The following paper gives a discussion of the methods involved in evaluating the oscillator strengths from both the dispersion and absorption equations for liquids. The resulting equations are then applied to a set of infra-red bands on which the absorption and dispersion were measured.

Absorbing and radiating cylinders as boundary-value problems*

The exact Maxwell forms of the hybrid E and H fields and radiant-flux distributions in and outside of an absorbing cylinder of fixed length, when the cylinder is irradiated from below by E and H fields characteristic of particular modes, have been determined as a boundary-value problem in electromagnetic theory. The continuity of the tangential components of these E and H fields and the normal components of the radiant flux across the curved surface lead to the well-known Hondros equation, whose roots determine the propagation constants of the cylindrically guided waves. The simpler TE radiant fluxes above and inside the cylinder exhibit non-Fresnel-type periodic dependences on the length of the cylinder and the position of the observation plane inside the cylinder, which are functions of the material constants of both media, the frequency, the radius of the cylinder, and the TE mode being used. Approximate solutions, with cylinders of large radii, indicate the TE radiant fluxes at the curved surface of the cylinder and at the interface between the illuminated and unilluminated regions above the cylinder have continous normal components, discontinuous axial components, and reversed angular components. The reversals of the angular components are due to reversals of the directions of the radial electric fields associated with the Sommerfeld and Hondros nebenwellen, due to the charges and currents on these interfaces. Approximate as well as rigorous solutions are helpful in considering these problems; the primary difficulty in obtaining exact solutions is to find roots of the Hondros equation.