This paper is a review of the theory-of laser beams and resonators. It is meant to be tutorial in nature and useful in scope. No attempt is made to be exhaustive in the treatment. Rather, emphasis is placed on formulations and derivations which lead to basic understanding and on results which bear practical significance.

Laser Beams and Resonators

We study the transmission properties of a guide consisting of a dielectric rod with rectangular cross section, surrounded by several dielectrics of smaller refractive indices. This guide is suitable for integrated optical circuitry because of its size, single-mode operation, mechanical stability, simplicity, and precise construction. After making some simplifying assumptions, we solve Maxwell's equations in closed form and find, that, because of total internal reflection, the guide supports two types of hybrid modes which are essentially of the TEM kind polarized at right angles. Their attenuations are comparable to that of a plane wave traveling in the material of which the rod is made. If the refractive indexes are chosen properly, the guide can support only the fundamental modes of each family with any aspect ratio of the guide cross section. By adding thin lossy layers, the guide presents higher loss to one of those modes. As an alternative, the guide can be made to support only one of the modes if part of the surrounding dielectrics is made a low impedance medium. Finally, we determine the coupling between parallel guiding rods of slightly different sizes and dielectrics; at wavelengths around one micron, 3-dB directional couplers, a few hundred microns long, can be achieved with separations of the guides about the same as their widths (a few microns).

Dielectric rectangular waveguide and directional coupler for integrated optics

The field configurations and propagation constants of the normal modes are determined for a hollow circular waveguide made of dielectric material or metal for application as an optical waveguide. The increase of attenuation due to curvature of the axis is also determined. The attenuation of each mode is found to be proportional to the square of the free-space wavelength λ and inversely proportional to the cube of the cylinder radius a. For a hollow dielectric waveguide made of glass with v = 1.50, λ = 1μ, and a = 1 mm, an attenuation of 1.85 db/km is predicted for the minimum-loss mode, EH 11 . This loss is doubled for a radius of curvature of the guide axis R ≈, 10 km. Hence, dielectric materials do not seem suitable for use in hollow circular waveguides for long distance optical transmission because of the high loss introduced by even mild curvature of the guide axis. Nevertheless, dielectric materials are shown to be very attractive as guiding media for gaseous amplifiers and oscillators, not only because of the low attenuation but also because the gain per unit length of a dielectric tube containing He-Ne “masing” mixture at the right pressure can be considerably enhanced by reducing the tube diameter. In this application, a small guide radius is desirable, thereby making the curvature of the guide axis not critical. For λ = 0.6328μ and optimum radius a = 0.058 mm, a maximum theoretical gain of 7.6 db/m is predicted. It is shown that the hollow metallic circular waveguide is far less sensitive to curvature of the guide axis. This is due to the comparatively large complex dielectric constant exhibited by metals at optical frequencies. For a wavelength λ = 1μ and a radius a = 0.25 mm, the attenuation for the minimum loss TE 01 mode in an aluminum waveguide is only 1.8 db/km. This loss is doubled for a radius of curvature as short as R ≈ 48 meters. For λ = 3μ and a = 0.6 mm, the attenuation of the TE 01 mode is also 1.8 db/km. The radius of curvature which doubles this loss is approximately 75 meters. The

Hollow metallic and dielectric waveguides for long distance optical transmission and lasers

We describe the design, construction, and operation of a new type of microwave spectrograph which allows the measurement of the resonant transitions of transient or otherwise short‐lived species. The spectrograph is composed of three parts: a Fabry–Perot cavity, a pulsed supersonic nozzle as a source for the sample, and the pulsed microwave Fourier transform method. Following a detailed discussion of the three above components in the spectrograph, the operation of the entire system is described and several examples are given.

Fabry–Perot cavity pulsed Fourier transform microwave spectrometer with a pulsed nozzle particle source

This paper is a review of the theory of laser beams and resonators. It is meant to be tutorial in nature and useful in scope. No attempt is made to be exhaustive in the treatment. Rather, emphasis is placed on formulations and derivations which lead to basic understanding and on results which bear practical significance.

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Laser beams and resonators

A theoretical investigation has been undertaken to study diffraction of electromagnetic waves in Fabry-Perot interferometers when they are used as resonators in optical masers. An electronic digital computer was programmed to compute the electromagnetic field across the mirrors of the interferometer where an initially launched wave is reflected back and forth between the mirrors. It was found that after many reflections a state is reached in which the relative field distribution does not vary from transit to transit and the amplitude of the field decays at an exponential rate. This steady-state field distribution is regarded as a normal mode of the interferometer. Many such normal modes are possible depending upon the initial wave distribution. The lowest-order mode, which has the lowest diffraction loss, has a high intensity at the middle of the mirror and rather low intensities at the edges. Therefore, the diffraction loss is much lower than would be predicted for a uniform plane wave. Curves for field distribution and diffraction loss are given for different mirror geometries and different modes. Since each mode has a characteristic loss and phase shift per transit, a uniform plane wave which can be resolved into many modes cannot, properly speaking, be resonated in an interferometer. In the usual optical interferometers, the resolution is too poor to resolve the individual mode resonances and the uniform plane wave distribution may be maintained approximately. However, in an oscillating maser, the lowest-order mode should dominate if the mirror spacing is correct for resonance. A confocal spherical system has also been investigated and the losses are shown to be orders of magnitude less than for plane mirrors.

A. G. Fox

Papers

Resonant modes in a maser interferometer