When a plane wave interacts with a diffraction grating, it gives rise to diffracted waves. Diffraction is the bending or spreading of waves as they encounter an obstacle. This phenomenon, widely known in the field of optics, has been a cornerstone of various applications, from spectral analysis to optical imaging. However, at a low grazing angle of incidence, a peculiar behavior emerges – the high-order diffracted waves vanish, and only specular reflection occurs, resulting in a dark “shadow.” This curious phenomenon poses a challenge in defining diffraction efficiency using conventional diffraction amplitudes. Traditionally, diffraction efficiency is calculated based on the diffraction amplitude, but at low grazing angles, this approach becomes inadequate. It is in this context that the shadow theory, initially devised by Nakayama, introduced a novel description of electromagnetic fields using scattering factors. This innovative framework enables the definition of diffraction efficiencies not only at low grazing angles but also for all complex angles of incidence. Nakayama’s theory initially focused on perfectly conducting gratings but laid the groundwork for further exploration.
Indeed, the interaction of light with periodic structures has been a subject of profound fascination and extensive research. In a new study conducted by Professor Hideaki Wakabayashi, Professor Jiro Yamakita from Okayama Prefectural University, and Professor Masamitsu Asai from Kindai University, published in the Journal of the Optical Society of America A, has made significant strides in shedding light on a particular facet of this intricate field. Their work explored the scattering problem of composite dielectric gratings embedded with conducting strips, offering novel insights into the behavior of electromagnetic waves in such structures.
To thoroughly investigate the behavior of electromagnetic waves in dielectric gratings, the researchers developed the Matrix Eigenvalues Method (MEM). This new method extracts first-order matrix differential equations directly from Maxwell’s equations and obtains numerical solutions through matrix eigenvalue calculations. Building upon this foundation, they applied the shadow theory to MEM, extending the novel electromagnetic field description from the incident region to all regions, including the grating layers.
The authors successfully reported the resolution of cases where eigenvalues degenerate to zero at low grazing angles and in the middle regions. While these phenomena manifested only in a few points in practical numerical calculations, the utility of scattering factors remained unproven until now. Moreover, the researchers put forward two compelling applications of scattering factors that could potentially transform the way we understand electromagnetic wave interactions: firstly, response to Surface Current Density: Scattering factors were discovered to represent the response to unit surface electric or magnetic current density, directly connected to spectral-domain Green’s functions. Using these spectral-domain Green’s functions derived from scattering factors, the researchers formulated secondary fields generated by induced currents on periodic conducting strips. However, this work was limited to conventional diffraction amplitudes, not the scattering factors. Secondly, numerical Accuracy Criterion: The reciprocity theorem of scattering factors emerged as a rigorous criterion for assessing numerical accuracy, surpassing the convergence of solutions. While previously considered valid only in lossless regions, this study demonstrated its applicability even in cases involving lossy media, challenging the established notion of energy conservation.
In their study, the authors focused on the scattering problem of composite dielectric gratings embedded with conducting strips. These conducting strips are so thin compared to the wavelength that their thickness is effectively neglected. The study approaches the total scattering fields as a superposition of primary and secondary fields, culminating in a method to derive scattering factors as fundamental quantities for grating analysis. Moreover, the researchers applied the shadow theory to describe the primary fields in MEM, resulting in scattering factors for the primary fields. Furthermore, the authors expressed secondary fields using spectral-domain Green’s functions derived from scattering factors and surface electric currents induced on the conducting strips . They also employed the Galerkin procedure to determine the surface electric currents on the conducting strips from the resistive boundary conditions, yielding scattering factors for the secondary fields.
The authors introduced scattering factors for the total fields as final solutions, facilitating the definition of diffraction efficiencies based on scattering factors and normalized Joule losses for various types of incidences, including propagating and evanescent waves. The researchers provided numerical results for an asymmetric multiple resistive plane grating consisting of conducting strips. The authors’ findings offered several noteworthy observations: firstly, low Grazing Limit: At a low grazing limit of incidence, Joule losses in each plane grating approached zero, mirroring the behavior of secondary fields of all orders. Secondly, was particularly prominent in the case of evanescent waves.
The study demonstrated symmetries in diffraction efficiencies and scattering factors. Notably, in the presence of a resistive plane grating with inherent loss, the scattering factors upheld reciprocity for any transverse propagating constant, including evanescent waves.
In conclusion, the research conducted by Professors Wakabayashi, Asai, and Yamakita represents a significant advancement in the understanding of electromagnetic wave interactions with composite dielectric gratings embedded with conducting strips. Their work has introduced scattering factors as an important tool for analyzing these complex structures and has unveiled the reciprocity of diffraction efficiencies and scattering factors in both propagating and evanescent wave incidences.
Figure : Numerical analysis of scattering fields by an asymmetric triple plane grating using shadow theory. “[Adapted with permission from JOSA A, Vol.40, 305 (2023) Copyright Optica.]”
Wakabayashi H, Asai M, Yamakita J. Numerical analysis of scattering fields by a multiple plane grating using shadow theory. Journal of the Optical Society of America A: Optics and Image Science, and Vision 2023 ;40(2):305-315. doi: 10.1364/JOSAA.475015.