Analytic solution for the static fields in presence of the dielectric wedge is available. Analytic solution for dynamic fields scattered by the dielectric wedge is, however, unknown to date because it is non-separable due to two different wave numbers inside and outside of the wedge, respectively.
By expanding the fields in a power series of ρ, where ρ is the distance between the field point and the edge of the wedge, Meixner showed that the edge condition gives the leading terms in the limit of ρ goes to zero approaching static field behavior. This series is shown to diverge, however, when the wedge angle is a rational multiple of π. The series may be modified to be free of diverging coefficients but numerical calculations employing hypersingular integral equations could not show that the dynamic fields near the edge of dielectric wedge approaches the singular behavior of the static fields at the edge.
Analytic solutions for the diaphanous (isorefractive) wedge shows that the dynamic solution agrees with the static solution at the edge. Preliminary analytic approaches for the isorefractive wedge shows that there may be two limiting values of fields at the edge, one static and the other one dynamic. It is still an open question for the scattered fields near the edge of the dielectric wedge.
Two extinction integrals in the mathematically complementary region of the free space and the dielectric wedge, respectively, gives the correction fields to the physical optics approximation in its far fields. This half analytic and half numerical method is used for the calculation of the fields scattered by the dielectric wedge of arbitrary wedge angles and permittivities in the case of plane wave incidence. It is shown that the dynamic fields approach to the corresponding static solutions both for their angular as well as distance dependence if the distance between the field point and the edge tip is smaller than $10^{-2}$ times the free space wavelength either for the singular or the non-singular edge fields. Calculated results also show that the dynamic fields go through a smooth transition to the far scattered fields for the distance larger than $10^{-1}$ times the wavelength, where the ratio of the transverse electric and magnetic field becomes almost a constant.
For the better converging series in the far field, a modified Neumann series satisfying the static limit edge condition and the radiation condition and having two different wave numbers of air and dielectric for its arguments is used for the surface currents correcting the edge diffraction of the physical optics solution. This representation gives accurate solution over the whole region including the grazing incidence of the plane waves upon the dielectric wedge of large permittivities, where the accuracies of the solution may be checked from the numerical smallness of the solutions that should be zero in the complementary regions.
