This work introduces a new class of PT-symmetry grating assisted devices for switching or modulation applications. Their operation is based on a four-wave interaction, thus marking a step forward in the development of PT-symmetry devices which currently are essentially based on two-wave interactions. A remarkable feature of the new structure is that all its properties also hold in the case of imperfect PT-symmetry operation, corresponding to the important practical case of fixed losses.

Using optical PT-symmetry for switching applications

This work introduces a new class of PT-symmetry grating assisted devices for switching or modulation applications. Their operation is based on a four-wave interaction, thus marking a step in the further development of PT-symmetry devices which currently are essentially based on two-wave interactions. A remarkable feature of the new device is that all their properties also hold for the case of imperfect PT-symmetry operation, corresponding to the important practical case of fixed losses.

Using PT-symmetry for switching applications

Two configurations of volume holographic grating couplers are rigorously analyzed by means of the finite-difference frequency-domain method (FDFD) for both TE and TM polarizations and for 0- and 45-deg output coupling. The two configurations depend on the position of the grating coupler, which can be placed either in the film or in the cover waveguide region. The FDFD results are compared with those obtained by the rigorous coupled-wave analysis in conjunction with the leaky-mode approach (RCWA-LM). Because the FDFD method is a rigorous solution of the Maxwell equations, it simulates the VHGC configuration and takes into account the waveguide-coupler discontinuity effects as well as the multimode excitation and interference effects, all of which are neglected by the traditional RCWA-LM.

Volume holographic grating couplers: rigorous analysis by use of the finite-difference frequency-domain method.

Comprehensive mathematical modelling of ultra-high Q grating-assisted ring resonators

A rigorous mathematical method to simulate an ultra-high Q-factor 1D PhC ring resonator (1D PhCRR) is proposed. The integration of a 1D PhC in a ring cavity resonator allows an enhancement of the Q-factor of a single ring resonator up to 109 or more, without enlarging the device footprint, with an improvement of at least two orders of magnitude in comparison with the values obtained without the grating. Rigorous modelling and simulations of such ultra high-Q resonant cavities are very challenging, because of the structure complexity, due to the large device footprint and the requirements for an accurate spectral analysis. Conventional numerical methods, such as Finite Element Method (FEM) and Finite Difference Time Domain (FDTD) require very long simulation time and huge computing resource. Therefore, a general mathematical method, able to reduce the computing time, assuring accurate solutions, and investigating the effect of the grating in the coupling region, is a real need. Several configurations of 1D PhCRRs have been investigated. An ultra-high Q-factor (> 109) for a 1D PhCRR in Si 3 N 4 technology with a footprint of 64 mm2 has been calculated. Similar performance makes the 1D PhC ring resonator suitable for several applications, such as optical gyroscopes for space environment and ultra-sensitive biosensors.

Rigorous model for the design of ultra-high Q-factor resonant cavities

Grating-assisted backward coupling between two parallel waveguides is analyzed with a set of four coupled equations. The equations were recently derived by a unified approach that is appropriate for other coupling problems as well. Examples include three different operation regions and provide for the field variation along the guides as well as for the reflectivity and transmission coefficients. It is shown that the reduction of the four-wave coupling problem to a set of two coupled equations is possible only for some specific cases and at the cost of reduced accuracy. The model may be especially useful for the design of optical components for various applications such as optical switching and multiplexing.

Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism

Characteristics of grating-assisted coupling between two parallel waveguides are analyzed. The influence of the grating parameters, such as groove depth, duty cycle, and refractive indices is considered. Chirped and parallel gratings as well as gratings with sinusoidal envelope periodicity are also addressed. The analysis is based on a unified coupled-mode formalism, with the transfer-matrix method as a general solution technique. It is shown how to modify the grating parameters to provide a specific spectral response (reflectivity and transmission coefficients). As an example, two parallel gratings are used to obtain a similar response to a single grating of double length. The location of the grating between the two waveguides as well as the light-wave injection direction are very important. The presented methods and effects may be useful for design and analysis in the fields of optical communications, sensing, and processing.

Characteristics of grating-assisted couplers

The problem of oblique incidence of plane waves and Gaussian beams on finite-aperture gratings (the number of grooves and their length and depth are all finite) in slab waveguides is analyzed by means of a four-wave two-dimensional coupled-mode theory (2D-CMT). This model considers the finite aperture of the gratings and the correct simultaneous interaction among all four relevant waves (TE+,TE-,TM+, and TM-) by means of Bragg diffraction at oblique incidence. The grating’s geometry and boundary conditions are properly accounted for, and the problem is solved numerically by a finite-difference method. Near-field and far-field distributions, as well as reflection and transmission (power) coefficients (as functions of the plane-wave incidence angle), are calculated. The model is compared with the degenerate case of two-wave coupling that considers interaction only between pairs (e.g., TE+↔TE-), and significant differences may be observed. Compatibility and differences between the 2D-CMT and the one-dimensional CMT (grooves with infinite length) are also presented, in addition to the influence of the beam width and the groove length on the emerging waves. The analysis is general and can be performed on many kinds of realistic beams, grating shapes, and applications.

Obliquely incident wave coupling on finite-aperture waveguide gratings

Diffraction of plane waves, obliquely incident on a (waveguide) grating with infinitely long grooves, is analyzed with the four-wave coupled-mode theory (CMT). Unlike the ordinary analysis that considers only coupling between pairs (i.e., TE+↔TE-,TE+↔TM-,TM+↔TE-, or TM+↔TM-), the simultaneous interaction among these waves is properly accounted for. The field variation along the grating and the impulse response representation of the grating (reflectivity and transmission as functions of the plane-wave incidence angle) are calculated. Significant differences between the ordinary (two-wave) CMT and the four-wave CMT are observed over a wide range of incident angles.

Four-wave coupled-mode theory of obliquely incident plane waves on waveguide diffraction gratings

Diffraction of obliquely incident Gaussian beams on waveguide gratings with infinitely long grooves is described. The analysis is based on the angular spectrum of the beam and on the grating response derived from the four-wave coupled-mode theory, which simultaneously considers all four (approximately) synchronous waves. Results concerning the shapes of the emerging beams and their directions, the power content, and the possibility of beam splitting could differ significantly from those obtained by means of two-wave coupling. The analysis method is general and can be performed on many kinds of realistic beam shapes and applications.

Nahum Izhaky

Papers

Analysis of grating-assisted backward coupling employing the unified coupled-mode formalism

Characteristics of grating-assisted couplers

Obliquely incident wave coupling on finite-aperture waveguide gratings

Four-wave coupled-mode theory of obliquely incident plane waves on waveguide diffraction gratings

Oblique incidence of gaussian beams on waveguide gratings with the four-wave coupled-mode theory.