Page 279 - Introduction to Information Optics

264 5. Transformation with Optics

5.5. PHYSICAL WAVELET TRANSFORM

The wavelet transform is a multiresolution local signal analysis and was
developed mainly in the 1980s. Historically, the term wavelet was first intro-
duced by Huygens in 1678 to describe the secondary waves emitted from the
wavefront in his interpretation of diffraction. It is of interest to seek a possible
fundamental link [15] between Huygens wavelets and the wavelet transform
developed some 300 years later. At first glance, the Huygens wavelets and the
mathematical wavelets are different. The latter are designed to be local in both
the space and frequency domains and to permit perfect reconstruction of the
original signal. The wavelets must satisfy the admissibility condition and have
regularity.
However, as we shall show in the following, the Huygens wavelets are, in
fact, electromagnetic wavelets, proposed recently by Kaiser [16], and the
Huygens -Fresnel diffraction is, in fact, a wavelet transform with the elec-
tromagnetic wavelets [17]. The electromagnetic wavelets are solutions of the
Maxwell equations and satisfy the conditions of the wavelets. The optical
diffraction described by the fundamental Huygens -Fresnel principle is a
wavelet transform with the electromagnetic wavelet.
In the following subsection we introduce the electromagnetic wavelet and
show that the Huygens-Fresnel diffraction is the wavelet transform with the
electromagnetic wavelet. (Understanding this subsection requires a basic
knowledge of electrodynamics [18].)
5.5.1. ELECTROMAGNETIC WAVELET

An electromagnetic field is described in the space-time of 4D coordinates
4
r = (r, r 0) e R , where r is the 3D space vector and r () = ct, with c the speed of
light and t the time. In the Minkowsky space-time the Lorentz transform
invariant inner product is defined as

4
The corresponding 4D frequencies are the wave-number vector p = (cp, p 0)eR ,
where p the 3D spatial frequency. For free-space propagation in a uniform,
isotropic, and homogeneous dielectric medium the Maxwell equations are
reduced to the wave equations, and it is easy to show that the solutions of the
wave equation are constrained on the light cone C in the 4D frequency space:
2 2 2
P — Po — c \p\ = 0, so that p Q = ±co with co/c = \p\, where co is the temporal
frequency.
The electromagnetic wavelet is defined in the frequency domain as [16]

2 sm/c ip
H^(p) =2(co/c) e(sa)/c)e~ e~ ^ (5.13)