General Spherical Resonators

It can be shown [5.1,23] that in nonfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can also be described by the Gaussian profile (5.32). The confocal resonator with d = R can be replaced by other mirror configurations without changing the field configurations if the radius Rj of each mirror at the position Zq equals the radius R of the wavefront in (5.37) at this position. This means that any two surfaces of constant phase can be replaced by reflectors, which have the same radius of curvature as the wavefront - in the approximation outlined above. 

These resonators are equivalent, with respect to the field distribution, to the confocal resonator with the mirror radii R and mirror separation d = R. 

The beam radius Wg of the TEMqo mode at the mirror surfaces (called the spot size) is according to (5.31 and 38) with Zq = d/2 for a symmetric resonator with the mirror separation d and the mirror radii R 

The second factor is a function of d and becomes maximum for d = R. This reveals that of all symmetrical resonators with a given mirror separation d the confocal resonator with d = R has the smallest spot sizes at the mirrors. 

The diffraction losses of a resonator depend on its Fresnel number N = a /dX (Sect 5.2.1) and also on the field distribution A(x, y,z = d/2) at the mirror. The fundamental mode, where the field energy is concentrated near the resonator axis, has the lowest diffraction losses, while the higher transverse modes, where the field amplitude has larger values toward the mirror edges, exhibit large diffraction losses. Using (5.31) with z = djl the Fresnel number j dX) can be expressed as 

The beam radii w (z) on the spot size w z) can be obtained from (5.31) and 

The mode waist Wq(z = 0) is minimum for g = 0, i.e., d = R. The confocal resonator has the smallest beam waist. Also, the spot sizes wj = wj are minimum for g = 0. We therefore obtain the following result  

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Resonators with two curved mirrors (a general spherical resonator, see Figure 3.5b) include a range of different configurations, depending on the radii of curvature of the mirrors and their relative... 

The analysis has been extended by Boyd, Gordon, and Kogelnik to resonators with confocally-spaced spherical mirrors [5.20,5.21] and later by others to general laser resonators [5.22-5,30]. For the symmetric confocal case (the two foci of the two mirrors with equal radii R — — R coincide, i.e., the mirror... 

Since this dipolar interaction remains in isotropic media as similarly found for the contact shift (see sect. 2.1), it is often termed as the pseudo-contact shift. For the general case of a complex possessing an anisotropic magnetic susceptibility tensor, Kemple et al. (1988) show that the pseudo-contact shift spherical coordinates of the resonating nucleus in an arbitrary axes system with the lanthanide metal ion i (III) located at the origin (fig. 1)... 

Like its energy, the angular momentum of the valence electron also becomes spherically averaged. It rotates in spherical mode and its total angular momentum appears as quantum torque. Like a spinning electron this orbital quantum torque sets up a half frequency wave field that resonates non-locally with the environment. The most general solution to the wave equation of a spherically confined particle therefore is the Fourier transform of the spherical Bessel function... 

Dynamic Theories, Dynamic theories take into account the scattering of acoustic waves from individual inclusions and generally include contributions from at least the monopole, dipole, and quadrupole resonance terms. The simpler theories model only spherical inclusions in a dilute solution and thus do not consider multiple scattering. To obtain useful algebraic expressions from the theories, the low concentration and the low frequency limit is usually taken. In this limit, the various theories may be readily compared. 

Any deviation from an ideal spherical shape will lead to a non-vanishing value of NM H) and, in general, to a distribution of and resonance... 

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