Open Resonator

an oscillatory system formed by a set of mirrors in which weakly damped electromagnetic oscillations in the optical and superhigh-frequency (SHF) ranges can be excited and maintained, with radiation into free space. An open resonator is used as the oscillatory system, or resonator, of an optical quantum generator (laser), and also in some instruments in the millimeter and submillimeter ranges, such as the orotron.

Cavity resonators, which are widely used in the SHF range and have dimensions of the order of λ, are difficult to use for wavelengths λ < 0.1 cm because of their small size and the large energy losses at the walls. However, the use of cavity resonators with dimensions greatly exceeding λ is also impossible, since a large number of vibrational modes that are close together in frequency is excited in such a resonator. As a result, the resonance lines overlap and the resonance properties virtually disappear. However, when some of the walls of such a cavity resonator are removed, nearly all the resonator’s modes of vibration become strongly damped, and only a small portion of them (if the remaining walls are of appropriate shape) is weakly damped. As a result, the spectrum of the vibrational modes of an open resonator formed in this manner is strongly “thinned out.”

Figure 1. Distribution of the fluxes onto the surface of a rectangular mirror for the oscillations E_21q^x and E_21q^y

The first open resonators, in the form of two plane parallel mirrors, were proposed in 1958 by A. M. Prokhorov and later by the American scientists R. H. Dicke, A. L. Schawlow, and C. H. Townes. If a plane wave is assumed to propagate between two plane mirrors separated by a distance L, then a standing wave forms in the space between the mirrors as a result of reflection from them. The resonance condition has the form L = qλ/2, where q is an integer, called the longitudinal index of the oscillation. The mode frequencies of the open resonator form an arithmetic progression with the spacing c/2L (an equidistant spectrum). In reality, the edges of the mirrors distort (perturb) the field of the plane wave, leading to the appearance of oscillations with different transverse indexes m and n, which give the number of oscillations of the field in the transverse directions and the current density distribution on the surface of the mirrors (Figure 1). The larger the indexes m and n, the larger the number of oscillations and the greater their damping caused by radiation into space—that is, in essence, by diffraction at the mirror edges. The spectrum of the natural frequencies of a planar open resonator has the form shown in Figure 2. Since the attenuation factor increases with increasing transverse indexes m and n more quickly than does the frequency interval between adjacent oscillations, the resonance curves corresponding to large m and η overlap, and the corresponding oscillations do not appear. The attenuation factor caused by the radiation depends both on the indexes m and η and the number N of Fresnel zones visible on a mirror of diameter R from the center of another mirror located at a distance L, that is N = R²/2Lλ. When N ∼1, one or two oscillations accompanying the fundamental oscillation remain.

Figure 2. Frequency spectrum of an open resonator

Open resonators with plane mirrors are sensitive to deformations and misalignments of the mirrors, which limits their use. Open resonators with spherical mirrors, are free of this shortcoming, since the rays, upon repeated reflection from the concave mirrors, do not pass outside of an enveloping surface—the caustic. Caustic surfaces are formed only in a certain range of values of L and of the radii of curvature R₁ and R₂ of the mirrors (Figure 3). Since the wave field is attenuated rapidly outside the caustic surface as it moves away from it, radiation from a spherical open resonator with a caustic surface is much smaller than the radiation from a plane open resonator. In this case thinning out of the spectrum takes place because the dimensions of the caustic surface bounding the field increase with m and n. For oscillations with large m and n, the caustic is located near the edge of the mirrors or does not form at all, and the oscillations radiate strongly. Such spherical open resonators are said to be stable, since they are insensitive to small mirror misalignments and displacements. Stable open resonators are used in gas lasers.

Figure 3. (a) Formation of the caustic surface in an open resonator with spherical mirrors, (b) graphic representation of the conditions for existence of caustic surfaces for different ratios of the mirror radii R₁ and R₂ and the distance L between mirrors: the white areas correspond to the presence of caustic surfaces, and the cross-hatched areas to strong radiation damping. The points corresponding to resonators with flat mirrors (F) and concentric mirrors (CC) lie on the boundary between the white and crosshatched areas; (C) con-focal mirror, (C’) one flat and one concave mirror (half of a confocal resonator).

Unstable open resonators in which an outer caustic surface cannot form are sometimes used in solid-state lasers. The outer caustic surface in such resonators cannot form because a ray passing near the resonator axis at a small angle to it moves continuously away from the axis on successive reflections. On the boundary between stable and unstable open resonators (Figure 3) are the confocal open resonators, in which the foci of the two mirrors (separated by distances R₁/2 and R₂/3 from the corresponding mirror) coincide; these include the telescopic open resonator, which consists of a small convex mirror and a large concave mirror. Unstable open resonators have larger radiation losses than do stable open resonators, but the losses are much greater for oscillations of higher types than for the fundamental oscillation. This makes it possible to achieve single-mode operation of the laser and the associated high directivity of the radiation.

There are various supplementary methods of thinning out the spectrum that entail changes in the profile of the mirror edges, the use of lenses, and so on. Thinning-out of the spectrum of an open resonator with respect to the longitudinal indexes q is achieved by using coupled open resonators or special optical filters. Open ring resonators, dielectric open resonators, and open resonators with intermediate mirrors (Figure 4) are used in addition to open resonators with two mirrors.

Figure 4. Complex types of resonators

Although the term “open resonator” came into use comparatively recently, open resonators essentially have long been known in physics and engineering. All musical instruments and a number of acoustic and electronic devices, such as the acoustic (Helmholtz) resonator, the tuning fork, and antenna vibrators, are open resonators. However, the radiation of these devices does not significantly affect the spectrum of their natural frequencies, whereas the radiation of mirror-type open resonators is the main cause of the thinning-out of the spectrum.