Page 283 - Introduction to Information Optics

268 5. Transformation with Optics

Let us consider, among the whole set of the electromagnetic wavelets, which
are translated in 4D space-time, two wavelets that are translated by (f'i,c,)
and (t' 2, ! 2)- respectively, as two events. Their squared interval in 4D space time
is

2
2
2
dl,2 = c [(t - t\) - (t - t' 2)-] ~~ |(r - I,) - (r - 1 2)1 ,
=
2
2
which is Lorentz invariant. When c (t' 2 — t\} = |£ 2 — £il 2 anc * ^1,2 ^ one
wavelet propagating at the speed c results in the other wavelet. The two
wavelets are the same event. This lightlike separation is not relevant. When
2
d .2 > 0, one can Lorentz-transform the two events into a rest reference frame
where <^ = | 2> so that the two wavelets are at the same location, but are
observed at different instances t\ and t' 2. This is the timelike separation. When
d-1,2 < 0, one can Lorentz-transform the two events into a rest reference frame
where t\ — t' 2. One observes the two wavelets at the same instant. This is the
spacelike separation. Since the Euclidean region E supports full spatial trans-
lations, the wavelets are spacelike separated. They are all observed at the same
time. There should be no time translation of the wavelets, t' = 0, in the wavelet
transform of Eq. (5.17).

5.5.3. ELECTROMAGNETIC WAVELET TRANSFORM AND
HUYGENS DIFFRACTION
We now apply the wavelet transform to a monochromatic optical field.

f(r,t) = E(r)e J<00t

with the positive temporal frequency co 0 > 0. The unconstrained Fourier
transform of the monochromatic field is
f
3
f(r, ct) 0) = 2(o)/c) E(r) exp( -jp • x)d r,

The wavelet transform of the monochromatic field is then, according to Eqs.
(5.16) and (5.13),

s&
W f(l, s) = E(l}e jiaot 'e((.a Q$)e~ \ (5.1.8)
where t' = 0, if we consider the wavelet family is spacelikely separated, as
discussed in the preceding. We take the temporal Fourier transform in both