Advances in Imaging and Electron Physics, Vol. 103 by Peter W. Hawkes (Ed.)

By Peter W. Hawkes (Ed.)

Advances in Imaging and Electron Physics merges long-running serials--Advances in Electronics and Electron Physics and Advances in Optical & Electron Microscopy . The sequence good points prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photograph technology and electronic photograph processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in these kind of domain names.

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3. The Real Field The real PB field is given by taking the real part of (80) [see (71)]. To clarify the structure of the real field we shall express it in terms of the real waveform f y (t) defined via (see (23)) Clearly, f,,( t ) decays as the parameter y increases: On the beam axis y = 0 and fy = f is strongest, but as the distance from the axis increases, y increases and the waveform f,,weakens. Substituting (87) in (81) and taking the real part, using also A(z) = AR i A I , the real PB field has the form + where y ( x , z ) is given in (83).

Noting from (45) that the far-field pattern is related to the spectral pattern via 161 + sine, we may express the beamwidth of the radiation pattern in terms of the spectral width of (60) + sin@ = D (64) Under the well-collimated conditions (54)-(53, we have a narrow spectral spread, that is, @zD<<1. (65) D. Special Case: Well-CollimatedCondition Under the well-collimated conditions (54)-(55), we can evaluate the timedependent plane-wave integral (31) with (56) in closed form. Noting from (65) that D << 1 here, we may approximate here [ = 2: 1 The resulting integral obtains the form of the inverse transform (31) for z = 0 applied d w SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS + to Go of (56) with a!

E. , t ) = 2 8 coseiib((, t)l+nl, (45) where iio is calculated from uo(x0, t ) via (33) and 8 is the observation angle relative to the z-axis. Thus, the time-dependent radiation pattern is determined by the time-dependent plane wave iio((, t) that propagates in the spectral direction a = 3. In view of (29), Eqs. (44)-(45) can readily be identified as a Fourier transform into the time domain of the time-harmonic radiation pattern representation in (18)-( 19). Equations (44)-(45) may be derived alternatively from either the time-dependent Green's function integral (25) or the time-dependent plane-wave integral (32), thereby explaining the properties of these integrals for points in the far zone.

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