Bi-directional Mode Expansion Research Articles

Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range

In this paper we investigate the band structure of guided modes in a segmented planar waveguide for the terahertz range. The dispersion curves are obtained using two different methods. The first approach is based on a close relationship between the band structure and narrow resonances in transmission spectra: transmission spectra are therefore calculated using a transfer matrix method and compared to the experimental ones. The second technique involves a modal analysis approach of diffraction gratings and it is used for direct calculation of the band structure. Propagation of THz guided waves along the grating considered as a segmented dielectric waveguide is also investigated using a bi-directional mode expansion and propagation method. All approaches are compared to each other and discussed.

Dual resonance in a waveguide-coupled ring microresonator

Eigenmodes of an isolated circular ring or disk microresonator are degenerate because of its rotational symmetry. Coupling of the microresonator to a straight bus guide breaks the rotational symmetry of the device. As a result, two slightly different resonant frequencies occur. While this effect is known in microring lasers, it is usually neglected in microring-based filters and add-drop de/multiplexers for telecom applications. Resonant frequency splitting does not follow from the usual ‘optical circuit’ analysis of the microring devices in which the coupling between the bus guide and ring is described by the 2 × 2 coupling matrix. It is shown here that the full 4 × 4 scattering matrix that takes into account also the back-reflections in the coupling region has to be used in order to include this effect into consideration. The magnitude of the resonant frequency splitting due to coupling to the bus guides in small high-contrast structures is estimated by two-dimensional numerical modelling based on bi-directional mode expansion and FDTD algorithms.

A Simple Bi-directional Mode Expansion Propagation Algorithm Based on Modes of a Parallel-plate Waveguide

The bi-directional mode expansion propagation algorithm (BEP) is known to be an accurate and efficient method for modelling field distribution in high-index contrast waveguide structures with strong back-reflections like Bragg gratings and photonic crystals. The main difficulty of this method is that for lossy structures, the propagation constants of modes are to be searched in the complex plane. To speed-up this procedure, a two-step algorithm for eigenmode calculation based on the expansion into the modes of an empty metallic waveguide has recently been proposed. Proper truncation rules possessing good convergence of the expansion method for both TE and TM modes have also been recently published. In this contribution, both these approaches are combined in the development of an extremely simple version of the two-dimensional BEP method that makes use of the field expansion into the eigenmodes of a parallel-plate waveguide. The method is strictly reciprocal and appeared to be computationally reliable also for strongly lossy structures. High numerical stability is ensured using the scattering matrix formalism, and an efficient method of calculating Bloch modes for symmetric as well as asymmetric periodic waveguide structures is adopted. A wide range of applicability of the method is demonstrated by a few examples.

A Comparative Study of Higher Order Bragg Gratings: Coupled-Mode Theory Versus Mode Expansion Modeling

The modal reflectivity and loss of lamellar diffraction gratings to be used in distributed-feedback and distributed-Bragg reflector lasers were computed in dependence of wavelength, duty cycle and Bragg order. Different methods based on the approximate coupled-mode theory and and the exact bidirectional mode expansion modeling were compared and a good mutual agreement was found. The slab Green's function needed to compute the coupling coefficients can be approximated by that of a homogeneous unbounded medium with sufficient accuracy.

Analysis of a long-period grating inscribed in the core of a fibre with inverse parabolic-index cladding

The operation of a long-period grating inscribed into the core of a single-mode fibre with the cladding having an inverse parabolic-index profile is theoretically investigated. Two-dimensional bidirectional mode expansion and propagation method is used to simulate mutual interaction of rotationally symmetric LP modes in the fibre. Numerical analysis has shown that the inverse parabolic-index profile of the cladding helps increase the sensitivity of a fibre as a refractometric sensor.

Modeling of fully etched waveguiding photonic bandgap structures

Modeling of a 1-D finite-height, finite-length fully etched waveguiding photonic bandgap structures, based on the leaky mode propagation method, is proposed for the first time. So far only infinitely long gratings have been modeled by this approach. Finite extension structures having deep grooves, high refractive index contrast, and an arbitrary profile of the etched region can be modeled in very short computer time, starting from the infinitely-long photonic bandgap structure. Useful analytical and closed-form expressions for the reflection and transmission coefficients and out-of-plane losses are derived, which are valid for any operating conditions. One of the most important applications of the model relevant to 1-D photonic bandgap devices is to determine the losses occurring also in 2-D devices. Comparisons of results in terms of transmittance, losses, bandgap position and complex propagation constant with those obtained by the bi-directional mode expansion and propagation method and an exact vectorial method show an excellent agreement together with a strongly reduced CPU time for our method. Full investigations of three different etching profiles (i.e., rectangular, triangular and saw-tooth) are carried out. Particular attention is paid to the physical behavior around the first and second Bragg interaction regions. We demonstrate that the rectangular shape exhibits the highest losses and the widest bandgap, while the saw-tooth grating exhibits the lowest losses and the narrowest bandgap. Quick and accurate determination of the out-of-plane losses in a large variety of photonic bandgap devices is also demonstrated.

Theory and modelling of optical waveguide sensors utilising surface plasmon resonance

A theoretical analysis of the phenomenon of excitation of surface plasma waves in integrated-optical waveguide structures is carried out. Rigorous approach to analysis of light propagation through a waveguide structure with a thin metal overlayer supporting surface plasma waves is formulated using a bi-directional mode expansion and propagation method. It is demonstrated that because the back-reflections in the structure are very weak and most of optical power is transmitted by only a limited number of modes of the sensing structure, the method can be considerably simplified.

Modelling of the surface plasmon resonance waveguide sensor with Bragg grating

The operation of a novel device – a waveguide surface plasmon resonance sensor with a UV-written Bragg grating – is theoretically analysed using two methods. In the simple perturbation approach, the metal/dielectric layer system supporting the resonance excitation of the surface plasma wave is considered to be a perturbation of the original dielectric waveguide with Bragg grating that is analysed using a coupled-mode theory. The second approach consists of the rigorous method of bi-directional mode expansion and propagation using the Floquet mode formalism developed recently for the analysis of waveguide grating structures. The results of both approaches are mutually compared, and the operation characteristics of this novel sensing device are briefly described.

Analysis of a deep waveguide Bragg grating

Spectral properties of a very deep Bragg grating operating in a first diffraction order on a single-mode planar waveguide have been studied theoretically. It is shown that the scattering loss can be low (a few percent), the reflectivity very high (over 90%), the reflection band is shifted against the ‘Bragg’ wavelength toward the shorter wavelengths, and its spectral shape is very different from that of a shallow grating. Inside a reflection band, a part of the input optical power penetrates through the grating even if it is infinitely long. These properties are predicted by modelling using two independent computer codes based on different modelling methods, namely the bi-directional mode expansion and propagation method (BEP), and a method of lines (MoL). The first method is discussed in some detail here. The work has been performed within the framework of European Action COST 240.

The bidirectional mode expansion method for two-dimensional waveguides: the TM case

The mode expansion propagation method is a modelling technique for a large variety of waveguide components. Until now only the case with TE modes has been reported; we give details on the mode expansion propagation method for TM modes. Instead of one type of overlap integral in the TE case, a TM analysis requires two kinds of overlap integrals. The inclusion of radiation modes in this method is discussed. The modifications in the algorithm to include waveguides with gain or loss structures are also considered.