We have shown that such a scalar wave is purely electric and has no magnetic component.

41) that the scalar wave ([LAMBDA]) distribution induces a charge density, [[rho].

are exact NtD maps corresponding to the scalar problem, for each scalar wave equation with velocities [v.

The function f ([LAMBDA]) is the NtD map corresponding to the optimal PML for the scalar wave equation.

The resulting optical field is denoted by a

scalar wave function,

This description above of in-and out-waves is almost identical to the quantum waves of the electron that can be obtained rigorously using a scalar wave equation in Section H.

Wolff, (6,7) Mead, (8) and Haselhurst (13) explored the Scalar Wave Equation and found that its solutions form a quantum-wave structure, possessing all the electron's experimental properties, eliminating the paradoxes of quantum mechanics and cosmology.

The scalar wave functions U discussed in this paper all denote the complex disturbance at a point of observation P(x, y, z) in the diffraction pattern of a perfectly conducting, infinitesimally thin, semi-infinite screen that occupies the half plane x > 0, z = 0 of a cartesian coordinate system, as depicted in Fig.

Although the squared magnitudes of scalar wave functions are commonly identified with the irradiance of the field, the data plotted in Figs.

Quantum matter waves exist in space and are solutions of a

scalar wave equation.

In these equations, U is a scalar wave function, S is a closed surface containing a plane aperture A located in the xy-plane of a cartesian coordinate system as indicated in Fig.

The Kirchhoff and Rayleigh-Sommerfeld integral equations (1) and (2) are alternative forms of the theorem of Helmholtz (5), which expresses Huygens' principle in terms of a scalar wave function U and its normal derivatives without assuming specific attributes of this function, except that it is continuous and twice differentiable with continuous derivatives and obeys the homogeneous wave equation,