Page 284 - Introduction to Information Optics

5.5. Physical Wavelet Transform

sides of the inverse wavelet transform shown in Eq. (5,17), and let the
time-frequency oj = o» 0

f(t,t)e~ Jiaot dt = d£ds dtW f(£,s)hfj(r,t)e- J<oat . (5.19)
£

On the right-hand side of Eq. (5.19) only the wavelet h^ s(r,t) is a function
of time t. The temporal Fourier transform of the electromagnetic wavelet
described in Eq. (5.15) can be computed by the contour integral or by the
Laplace transform that yields

2
t
Ji- (v t\0~3™ dt — : #f\YnVn />~' sa> ^S OfU
'*r sI'•> ' /*•" ***- ~~~ ~A 7 5*~. j " ^~i V{ijUJj{.U t , \ */«^,v/1
,/! jr ^ /-i •»* i \» /"
Combining Eqs. (5.18), (5.19), and (5.20) we obtain
1

The integration with respect to s may be computed as

ds6(sa) Q)e~ 2s(ao = -—

because co () > 0. Hence, we have finally

-^ e
|r - cl

where the wavelength x 0 = 2nc/a) 0. According to Eq. (5.21) the complex
amplitude E(r) of a monochromatic optical field is reconstructed from a
superposition of the monochromatic spherical wavelets, the centers of which
are at the points r — 1 and the amplitudes of which E(£) vary as a function of
3D space translation |. The coherent addition of all those spherical wavelets
forms the monochromatic electromagnetic field E(r). Equation (5.21) is indeed
the expression of the Huygens principle. Only the directional factor in the
Huygens-Fresnel formula is absent in Eq. (5.21). Another difference from the
Huygens- Fresnel formula is that Eq. (5.21) describes the convergent and
divergent spherical wavelets, which are incoming to and outgoing from the
localization point <*. The incoming wavelets were not considered in the
Huygens-Fresnel formula.