Spatial Field Distributions in Open Resonators

The mode configurations of open resonators can be obtained by an iterative 

The attenuation factor C, which does not depend on x and y, represents the diffraction losses and can be expressed by 

Inserting (5.20) into (5.19) gives the following integral equation for the stationary field configurations 

Because the arrangement of successive apertures is equivalent to the plane 

The general integral equation (5.22) cannot be solved exactly and one 

In Sect. 2.1 we have seen that any stationary field configuration in a closed cavity (called a mode) can be composed of plane waves. Because of diffraction, plane waves cannot give stationary fields in open resonators, since the diffraction losses depend on the coordinates (x, y) and increase from the z-axis of the resonator towards its edges. This imphes that the distribution A x,y), which is independent of X and y for a plane wave, will be altered with each round-trip for a wave traveling back and forth between the mirrors of an open resonator until it approaches 

The mode configurations of open resonators can be obtained by an iterative procedure using the Kirchhoff-Fresnel diffraction theory [307]. Concerning the diffraction losses, the resonator with two plane square mirrors can be replaced by the equivalent arrangement of apertures with size (2aand a distance d between successive apertures (Fig. 5.7). When an incident plane wave is traveling into the z-direction, its amplitude distribution is successively altered by diffraction, from a constant amplitude to the final stationary distribution An x, y). The spatial distribution An(x, y) in the plane of the nth aperture is determined by the distribution An-i (x, y) across the previous aperture. 

Because the arrangement of successive apertures is equivalent to the planemirror resonator, the solutions of this integral equation also represent the stationary modes of the open resonator. The diffraction-dependent phase shifts 0 for the modes are determined by the condition of resonance. They are chosen in such a way that the diffracted wave reproduces itself after each round trip through the resonator. 

The general integral equation (5.28) cannot be solved analytically and one has to look for approximate methods. For two identical plane mirrors of quadratic shape (2a), (5.28) can numerically be solved by splitting it into two one-dimensional equations, one for each coordinate x and y, if the Fresnel number N = a /(dA) is small compared with (d/a), which means if a (d A)i/. Such numerical iterations for the infinite strip resonator have been performed by Fox and Li [5.18]. They showed that stationary field configurations do exist, and computed the field distributions of these modes, their phase shifts and their diffraction losses. 

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An open volume, high isolation RF system suitable for pulsed NMR and EPR spectrometers with reduced dead time has been described. It comprises a set of three RF surface coils disposed with mutually parallel RF fields and a double-channel receiver (RX). Theoretical and experimental results obtained with a prototype operating at about 100 MHz have been reported. Each surface RF coil (diameter 5.5 cm) was tuned to 100.00 0.01 MHz when isolated. Because of the mutual coupling and the geometry of the RF coils, only two resonances at 97.94 MHz and 101.85 MHz were observed. These were associated with two different RF field spatial distributions. In continuous mode operation the isolation between the TX coil and one of the RX coils (singlechannel) was about —10 dB. By setting the double-channel RF assembly in subtraction mode the isolation values were optimised to about —75 dB. The described system was selected as a model for potential applications in solid-state NMR and in free radical EPR spectroscopy and imaging. 

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