Phase front curvature

Fig. 5.126. Gaussian beam with beam waist wo and phase-front curvature R(z) [5.24]...

For the fundamental or TEMooq mode, the Hermite polynomials are unity. X is the free-space radiation wavelength, d is the mirror separation, and q + 1 the number of half-wavelengths between the mirrors. The beam waist, Wg, is the distance from the center of the cavity to the 1/e points of the field strength. The beam waist is a maximum when d = R, the confocal arrangement, and falls to zero when d = 0 or d = 2R. Using the radius of curvature of one of our mirrors, 84 cm, and a mirror separation of 70 cm, the beam waist at 10 GHz is 12.6 cm. The k/>2/2R factor accounts for curvature of the wave front arising because of the curved mirrors. The phase front is... 

The phase fronts of the fundamental modes inside a confocal resonator close to the resonator axis can be described as spherical surfaces with a z-dependent radius of curvature. For za) = R/2 R = R. This means that at the... 

Dunkle (Ref 7) stated that accdg to the NDZ theory the first part of the deton wave, sometimes called the von Neumann spike, is an almost ideal shock wave in which very little chem reaction takes place. While the pressure at the spike is ca twice the C-J pressure, the temperature is ca half the C-J temp. This initial pressure and temp rise occurs entirely within ca 10 5cm thickness of the deton front. The 2nd phase of the deton wave is a gradual decrease in the pressure and an increase in the temp concurrent with the completion of the chem reactions. The length of the reaction zone can be detd experimentally from the minimum diam of a rod of explosive which propagates a steady-state detonation also from the changes in the deton velocity when this rod is surrounded by an inert casing material of varying thickness or from the decrease in the deton velocity when the deton wave is made to go around a bend of known radius of curvature... 

The near-field correction is calculated conceptually by repositioning the detector positions from a linear arrangement to a spherical arrangement that matches the curvature of the incoming wavefront. The theoretical phase delay from a point source at normal incidence is subtracted from the measured phase. As shown in those Figures, the curvature of the phase (indicating a curved wave front) is removed by the near-field correction, yielding a linear dependence of phase on detector position. The slopes of the near-field-corrected phase versus detector position plots indicate the direction to the source. 

FIGURE llA (Left) Phase diagram of the transportable region given by the velocity and the inclination angle of the wave front (o) transported, (x) not transported. (Right) Model of the rolling cylindrical gel on the peristaltic gel snrface (Rc, radins of curvature W, load of the PAAm gel b, contact half-width. 

It can be shown [5.1,5.24] 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 Rf of each mirror at the position zo 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 wave front - in the approximation outlined above. 

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