An exact solution to Maxwell’s equations including an evanescent field for a gapped three layered optical system

Abstract

We have investigated the electromagnetic characteristics of an optical configuration consisting of two dielectric regions of the same n>1 with a thin flat gap of air/vacuum of width d between them. Based on an analytical and exact solution to Maxwell’s equations including an evanescent field in the vacuum gap, we have obtained the transmission property of a four terminal optical circuit that consists of two input lights aimed toward the gap from both sides of the dielectric regions and two output lights from the gap, where all of the four terminals have the same angle to a plane that is vertical to the boundary plane between the dielectric and the vacuum. The results show several unique possible applications of the optical circuit elements such as a variable ratio power splitter, phase to amplitude field modulator, and optical mode selector.

Optics of Conducting Materials: An Electromagnetic Potential Perspective

“…To complete the theoretical setting, one phenomenologically relates (linearly or nonlinearly) the excitation fields to the field strengths by postulating appropriate frequency and wave vector dependent (linear or nonlinear) response tensors, ,,− which are to be determined a priori, either by experimental means or, following the Lorentz program, by an independent microscopic theory (preferably now a quantum mechanical one) of polarizable charges and currents within the matter. This conceptually eminent paradigm has remained the foundation of most theoretical studies (in the classical domain) on optical properties of complex materials for over a century now, as evident from the contemporary discourse by various authors on this subject. ,− ,,− Within this paradigm, a number of important theoretical studies on optics of complex materials have been reported, all of them enriching our understanding of a variety of experimentally accessible phenomena of classical material optics. A significant contribution along this direction is due to Levine and Schwinger, who provided an integral equation based approach (employing a dyadic Green’s function) to the problem.…”

“…This approach has been further extended and used in optical research . Most recent studies on this subject have been focused on direct numerical integration of Maxwell’s equations (in conjunction with simplified models for the constitutive relations that define the response tensors for the materials of interest) to determine the field strengths E⃗ ( r⃗ , t ), B⃗ ( r⃗ , t ) and other observables of interest, though there have also been a number of analytical studies on simple model systems (within the scalar wave regimes), intended to capture the essential physics behind the phenomena of interest. ,− While the direct determination of E⃗ ( r⃗ , t ) and B⃗ ( r⃗ , t ) from the Maxwell’s equations is pedagogically appropriate, there are basic limitations in such mathematical formulations insofar as practical applications are concerned. This is primarily due to the fact that the vector fields E⃗ ( r⃗ , t ) and B⃗ ( r⃗ , t ), because they have to satisfy the Maxwell’s equations, also carry redundant information (leading to an unnecessarily larger set of independent field variables to be determined), which eventually manifests as the well-known electromagnetic gauge degree of freedom …”

“…A representative example of the analytical complexity involved here can be seen in the classic works of Mie − and Debye , on the electromagnetic scattering by a homogeneous spherical grain of metal, where the formal solution in terms of an infinite series, though mathematically elegant, remains slowly convergent as the grain size increases relative to the wavelength of light. ,− In the context of scattering of electromagnetic waves with complex materials, there are three fundamentally distinct length scales involving the size of the material object and the wavelength of lightthe most complex being the situation when the dimension a of the material object is of the order of the incident electromagnetic wavelength λ, as the recent experiments ,,,− have also recognized. Theoretical understanding of the optics of complex materials in this subwavelength regime ( a ∼ λ) usually requires a complete solution of the vector field equations of Maxwell and hence remains a formidable intellectual challenge, notwithstanding the numerical and analytical advances of recent years. ,− ,− Given the outstanding importance of this subject, particularly in light of contemporary experimental interests, it is immensely urgent to develop new classes of mathematical solutions which are valid for arbitrary curvilinear geometrical configurations of the material systems. While a purely numerical solution (such as the FDTD and its variants), in a particular instance, is always useful, it is also important, for a general physical insight, that the mathematical formalism, besides being formally exact, must easily lead to a controlled sequence of various well-defined approximation regimes where a purely analytical treatment also remains tractable and yields physically meaningful results.…”