Huygens source

Figure 14a. Provided the wavefront is not interrupted, it moves out uniformly in a radial direction in accordance with the accepted notion of the rectilinear propagation of light. Suppose we now have a series of plane wavefronts incident on a slit with each successive wavefront being constructed from the envelope of secondary wavelets from the preceeding wave-front as shown in Figure 14b. If the width of the slit is small relative to the wavelength of light, then the portion of the wavefront at the slit may be considered as a single Huygens source radiating in all directions to produce...

The wave function v /ex (r) of electrons at the exit face of the object can be considered as a planar source of spherical waves according to the Huygens principle. The amplitude of diffracted wave in the direction given by the reciprocal vector g is given by the Fourier transformation of the object function, i.e. 

The distribution of light intensity in Figure 13 can be computed by application of Huygens principle which allows us to calculate the shape of a propagating wavefront provided the wavefront at an earlier instant is known. According to this principle, every point of a wavefront may be considered as a source of secondary waves (often called a wavelet) which spread out in all directions, i.e., all points on a wavefront are point sources for the production of spherical secondary wavelets. The new wave front 2 is then found by constructing a surface tangent to all the secondary wavelets as shown in... 

The above discussion implicitly obeys Huygens 97 principle, that each point on a spherical wavefront can be regarded as the source of a secondary wavelet (another spherical wave), as well as Fermat s98 principle of least time. 

We will test the consistency of our solution by evaluating the diffraction field of a Gaussian beam from a reference plane defined by 2 = 0. We will use the Huygens-Fresnel construction (Born and Wolf, 1980, pp. 370-386), where we treat each point on the wavefront in the reference plane as the source point for a secondary wavefront of the form exp(tk r)/r and sum over all source points. If the diffracted field has the same functional form as the incident field, then we will have demonstrated that our solution is useful even in the presence of diffraction. 

As one can see from relation (13.112), the wavefield at the point r D may be viewed at the moment of time t as the sum of elementary fields of point and dipole sources distributed over the surface S with densities dP r,t)/dn and P r,t) respectively. The interference of these fields beyond the domain D results in complete suppression of the total wavefield. Thus, the Kirchhoff integral formula can be treated as the mathematical formulation of the classical physical Huygens-Fresnel principle. 

Source Linda J. Spilker, ed. Passage to a Ringed World The Cassini-Huygens Mission to Saturn and Titan. NASA Special Publication SP-533. Washington, D.C. National Aeronautics and Space Administration, October 1997, Chapter 3. 

This means that, just as mentioned above about Huygen s principle, every point where the wave function is non-zero at time t = 0 serves as a source for the wave function at a later time. Eq.(3.2) thus describes the propagation of the wave in free space between two different points and times, thereby giving the Green s function its second name the propagator. But what happens if we introduce a perturbing potential We assume that the potential is a point-scatterer (that is, it has no extension) and that it is constant with respect to time, and denote it by 1, which is a measure of its strength. If we now also assumes that we only have one scatterer, located at iq, the time development of the wave becomes ... 

Consider a plane wave whose direction of propagation is perpendicular to a wall containing two small holes (Fig, 3,4). According to Huygens principle, these holes subsequently become sources of spherical waves which then interfere. At a distance which is large with respect to A and to d, the spherical wave in the direction of observation s can be treated as a plane wave (Fraunhofer s approximation). The path difference A between the two waves and 2 the direction s is the projection of d onto s ... 

A strange pattern appears on the screen a number of high concentrations of traces are separated by regions of low concentration. This resembles the interference of waves e.g., a stone thrown into water causes interference behind two slits an alternation of high and low amplitudes of water level. Well, but what does an electron have in common with a wave on the water surface The interference on water was possible because there were two sources of waves (the Huygens principle) — that is, two slits. 

Huygens construction (Huygens principle) Every point on a wavefront may itself be regarded as a source of secondary waves. Thus, If the position of a wavefront at any instant is known, a simple constmction enables its position to be drawn at any subsequent time. The constmction was first used by Christiaan Huygens. 

Huygens-Fresnel principle Every point on a primary wavefront serves as the source of secondary wavelets such that the wavefront at some later time is the envelope of these wavelets. 

According to the above equations every point on a wavefront serves as the source of spherical wavelets. The field amplitude at any point is the superposition of the complex amplitudes of all these wavelets. The representation of a field as a superposition of many elementary wavelets is known as the Huygens principle, since it was formulated in his Traits de la Lumi6re in 1690. Eresnel completed the description of this principle with the addition of the concept of interference. 

How is the amplitude of the sound field distributed from a planar source The Huygens representation of a wave source (point-like source radiating spherical waves, i.e., wavelets) is helpful to resolve such a question (Fig. 6). 

FIGURE 13.19 Geometry of an elementary aperture a Huygen s source. 

D. E. Merewether, R. Fisher, and F.W. Smith, On implementing a numeric Huygen s source in a finite differenee program to illustrate scattering bodies, IEEE Trans. Nucl. Sci. NS-27 (1980), 1829-1833. 

---