Oblique semi-guided waves: 2-D waveguide optics with negative effective permittivity
Semi-guided waves confined in dielectric slab waveguides are being considered, for oblique angles of propagation. If the waves encounter a linear discontinuity of quite arbitrary shape and extension, a variant of Snell's law applies, separately for each pair of incoming and outgoing modes. Depending on the effective indices involved, and on the angle of incidence, power transfer to specific outgoing waves can be allowed or forbidden. In particular, critical angles of incidence can be identified, beyond which any power transfer to non-guided waves is forbidden, i.e. all radiative losses are suppressed. In that case the input power is carried away from the discontinuity exclusively by reflected semi-guided waves in the input slab, or by semi-guided waves that are transmitted into other outgoing slab waveguides.
Vectorial equations on a 2-D cross sectional domain apply. These are formally identical to the equations that govern the eigenmodes of 3-D channel waveguides. Here, however, these need to be solved not as an eigenvalue problem, but as an inhomogeneous problem with a right-hand-side that is given by the incoming semi-guided wave, and subject to transparent boundary conditions. The equations resemble a standard 2-D Helmholtz problem, with an effective permittivity in place of the actual relative permittivity. Depending on the properties of the incoming wave, including the angle of incidence, this effective permittivity can become locally negative, causing the suppression of propagating outgoing waves. A semi-analytical solver based on a vectorial variant of the scheme for quadridirectional eigenmode propagation (QUEP) has been implemented.
The list of example structures investigated so far includes simple linear (abrupt, or tapered) discontinuities between slab waveguides of different thickness, spiral modes supported by bent slab waveguides, but also corner- (cf. figure) and step-like discontinuities that could permit the lossless power transfer between guiding layers at different levels in 3-D photonic circuits.
Hybrid analytical / numerical coupled mode theory
Computational tools are indispensable for concrete design tasks as well as for more fundamental investigations in integrated optics / photonics. Difficulties arise from the usually very limited range of applicability of purely analytical models, and from the frequently prohibitive effort required for rigorous numerical simulations of practical structures in 3-D. Hence we pursue an intermediate strategy. For specific classes of devices there exist relatively clear physical ideas about the optical behaviour, such that field templates can be written down which cover the major features of the light propagation. The templates are expressed in terms of certain known basis fields of lower dimensions. Typically these are guided modes supported by the local straight or bent optical channels, or resonances associated with the cavities in the device. One introduces unknowns in the form of varying amplitude functions, or in the form of coefficients for localized resonances.
What then remains is to determine these unknown parameters in the field template, i.e. to quantify the interactions between the basis fields. Here we use numerical techniques: upon discretization of the amplitude functions by finite elements (FE), variational procedures are applied to reduce the problems to systems of linear equations of small to moderate size.
Building on the physical/engineering notions underlying the field templates, and on the tools necessary to compute the basis fields, the FE/variational procedures can be implemented in a flexible, device independent way. Although restricted to configurations where these prerequisites are available, this Hybrid analytical / numerical variant of Coupled Mode Theory (HCMT) still constitutes a tool of some generality for accurate quantitative ab-initio simulations of real-world photonic devices.
Some of the programs related to a 2-D version of the approach are available online, included in the Metric subroutine collection (section HCMT). Our present range of examples covers circuits that involve straight waveguides, finite periodic corrugations (waveguide Bragg gratings), curved waveguides, square cavities, and circular micro-ring or -disk cavities described either in terms of bend modes or in terms of whispering gallery resonances. First results of a 3-D HCMT implementation concern models of intersections of straight channels at fairly arbitrary positioning and orientation.
Quadridirectional eigenmode propagation scheme
With their potential for high integration densities, devices based on optical microcavities or on photonic crystals attract considerable attention in photonics research at present. Specific obstacles for the modeling originate from the high permittivity contrasts which are exploited for effective guiding and trapping of light waves: Phenomena like wide angle propagation and pronounced reflections are essential for the interference effects that the new structures rely upon. Some of the most promising simulation tools, the class of bidirectional mode expansion or eigenmode propagation (BEP) techniques, are so far set up exclusively (for simulations in the optical regime) along one major axis of light propagation. This viewpoint is decidedly inadequate for the high contrast structures.
Our present approach removes that limitation, basically by means of superposition of eigenmode expansions along two perpendicular coordinate axes, where the (real) basis fields satisfy simple Dirichlet boundary conditions. The relevant Helmholtz wave equation is solved on a rectangular computational domain, formed by the overlap of the lateral windows of the two BEP sets, with fully transparent boundaries (exception: the corner points). Modeling simultaneous influx and outflux over all four boundaries is straightforward; the computational effort remains moderate. While preserving the advantages (accuracy, efficiency, interpretability) of the BEP tools, the proposed technique is capable to adequately capture the phenomena related to omnidirectional (2D) light propagation. Modes propagating in positive and negative direction along two coordinate axes play a role, hence this could be called a ``quadridirectional eigenmode propagation'' (QUEP) method.
Simulation results for several model systems (Gaussian beams, waveguide crossings, a square resonator cavity with perpendicular ports, a bent photonic crystal waveguide, a chain of nine coupled square dielectric cavities, finite high-contast dielectric gratings, and a periodic dielectric waveguide) illustrate the performance of the approach. According to the semianalytical ansatz, the technique is intended for structures with piecewise constant, rectangular, and (so far) lossless permittivity distributions, with preferably only few segments in the slice/layer decomposition. The example of a circular microdisk-resonator shows that the approach can handle also non-rectangular shapes in terms of suitable staircase approximations, unfortunately at the cost of a significantly increased computational effort. For predominant rectangular configurations, however, the efficient simulations allow to play with quite large composite structures. Exciting TET!
Some of the related programs are available online, included in the Metric subroutine collection.
Rectangular integrated optical microresonators
Compact integrated optical microresonator devices are currently discussed as one of the most promising concepts for applications in optical wavelength division multiplexing. While the majority of proposals deals with cylindrical or spherical resonators, coupling two port waveguides by small square or rectangular cavities can also lead to resonator devices that show the desired filter functionality.
Apart from proper performance and extreme compactness, a square resonator unit has the nice property that it can be simulated quite efficiently by well established mode expansion techniques (bidirectional eigenmode propagation, BEP). We consider the structure as being composed of three longitudinally homogeneous waveguide segments (here 'longitudinally' refers to the direction parallel to the waveguides). For fixed frequency, the electromagnetic field is then expanded into the local modes of the segments, including propagating and evanescent, forwards and backwards directed terms. Matching adjacent fields at the segment boundaries yields the transmission characteristics of the resonator.
The technique enables quite rigorous investigations of the spectral response of the microresonators to changes in the various geometrical and material parameters, in particular with respect to the shape of the cavity, and it allows for a straightforward extension to the modeling of a sequence of cavities, as it is required for the realization of add-drop-filters. Finally, the simulations constitute an efficient numerical complement to a general theory of ''localized states'' in the framework of conventional waveguides.
Once a suitable resonator specification has been identified for a given target wavelength and material system, that resonance can be used to construct composite structures of quite arbitray shapes, e.g. by evanescently coupling a sequence of individual squares, or by enforcing a hard continuation through specific dielectric spacers. The resulting configurations have no dominant axial direction, hence they are handled best by an omnidirectional propagation tool (QUEP).
Cross strip interferometer concept
While searching for technologically simplified integrated optical device concepts, we have discovered the promising properties of a specific class of waveguide interferometers. A typical structure consists of nothing more than a single strip, etched from a bimodal dielectric film. Guided light is launched perpendicularly to the strip. Provided that specific conditions for the film thickness and etching depth are observed, the strip acts as an interferometer of surprisingly good quality. Avoiding the necessity for bend or laterally precisely dimensioned waveguides, the geometry offers an attractive alternative to common integrated interferometric devices like Mach-Zehnder structures or directional couplers.
Understanding its behaviour starts with a division into three homogenous waveguide sections. The light enters from the thinner single mode waveguide outside the strip into the thick strip segment, which is configured to support two guided modes. At the end of the strip, the amount of power that is guided in the following lower output region depends on the local phases of these two fields. For a properly adjusted geometry there are two extreme scenarios. In the first one, the superposition of the two strip modes matches the output profile well. Most of the input power passes the device; the modes interfere constructively. In the opposite scenario the field that arrives at the output junction is orthogonal to the mode of the subsequent segment. The power is scattered into the substrate and cover regions; consequently we call this destructive interference. In both cases the achievable levels of transmission respectively suppression are remarkable: Our numerical calculations predict insertion losses below 0.1 dB and extinction ratios well above 30 dB for realistic device designs.
Based mainly on semianalytic mode expansion simulations, our modeling and design activities have led to concrete proposals for polarizing strips, and for experiments on magnetooptic isolator devices. Realizations were considered in both cases, in the Integrated Optical Microsystems Group, University of Twente, Enschede, The Netherlands, , and in the Applied Magnetooptics Group, Department of Physics, University of Osnabrück, Germany, respectively.
Mode expansion modeling of integrated optical devices
Several interesting structures in integrated optics consist of a sequence of longitudinally homogeneous waveguide segments. Examples are the rectangular microresonators and the cross strip devices as described above, a waveguide end facet, and certainly also isolated waveguides and simple coupler segments.
We consider the Maxwell equations in the frequency domain. The interesting region is enclosed by artificial boundaries, which are oriented parallel to the dominant direction of light propagation, and located such that all guided fields in the structure are negligible at the boundary positions. This leads to a discretization of the mode spectra on the waveguide segments; the mode sets become numerically manageable. Separately on each segment, the electromagnetic field is written as a mode superposition, where depending on the investigated structure forward and backward traveling, propagating and evanescent terms are included. Bidirectional projection of the adjacent fields at the junctions allows then to establish a linear system of equations for the coefficients in the mode superpositions.
For these structures, the mode expansion ansatz can replace modeling techniques of a more numerical nature, like finite difference or finite element based beam propagation or time-domain calculations, as a reference simulation tool, at least in a twodimensional setting. Usually the quasianalytical calculations are only moderately demanding in terms of computational resources, and they yield continuous representations of the electromagnetic field, with the solutions of the Maxwell equations being directly accessible in terms of guided mode profiles and mode amplitudes. The related programs are available online, included in the Metric subroutine collection.
Nonreciprocal integrated optical devices
Transparent materials with specific optical properties are of particular interest for general as well as for integrated optics. Our special emphasis lies on waveguides made of epitaxially grown rare earth iron garnet films. Along with low losses in the relevant wavelength regions around 1.3 µm and 1.5 µm, a window of transparency of optical fibers, these materials show comparably large magnetooptic effects of first order (Faraday effect) and second order (Cotton-Mouton-effect).
Garnet films are thus well qualified for integrated optics, where their pronounced Faraday effect enables the design of nonreciprocal devices. These are intended to distinguish between optical waves propagating in opposite directions. In the current framework, a device with two ports A and B that establishes a connection from A to B, but blocks all power transmission from B to A, is called an `isolator'. A more sophisticated `circulator' comprises more ports, three at least, which are connected in a circular manner.
The coefficients of coupled mode theory for the magnetooptic permittivity contribution allow a classification of the influences of gyrotropy on guided wave propagation. For mirror symmetric waveguides with well polarized fundamental modes, one identifies the dominant effects of TE phase shift, TM phase shift, and TE/TM polarization conversion, for polar, equatorial, and longitudinal magnetooptic configurations, respectively. In the framework of project D10 of the SFB 225 at the Department of Physics, University of Osnabrück, Germany, we have investigated applications of all three geometries for isolator and circulator devices. Among these are well established concepts such as the common polarizer / 45o Faraday rotator / polarizer setting for an isolator, several interferometric device proposals such as Mach-Zehnder structures, directional couplers, or multimode interference (MMI) devices, and more exotic proposals, e.g. a polarization independent circulator based on unidirectional polarization converters between radiatively coupled waveguides. The theoretical research in the Integrated Optics Modeling Group was closely related to the experiments in the Applied Magnetooptics Group in Osnabrück.
Wave-matching mode analysis of dielectric waveguides with rectangular cross sections
For dielectric integrated optical waveguides with piecewise constant, rectangular permittivity profiles, a simple local expansion into factorizing exponential or harmonic trial functions forms a promising ansatz for the calculation of guided modes. This function space turns out to be large enough to adequately approximate the unknown mode fields. Inside the homogeneous rectangles the trial functions solve the Maxwell equations exactly. A least squares expression for the mismatch in the continuity conditions at dielectric boundaries connects the fields on neighbouring regions. Minimization of this error allows to identify propagation constants and to compute the mode fields.
The procedure has been implemented both for semivectorial simulations and for fully vectorial mode analysis, where each of the common formulations for the vectorial mode problem may be employed. The vectorial Wave Matching Method (WMM) performs truly vectorial calculations in the sense that both transverse directions are treated alike. The piecewise defined trial fields are well suited to deal with field discontinuities or discontinuous derivatives. At the corners of dielectric waveguides, the method yields correct qualitative features of the divergent field behaviour.
For widely established benchmark problems we observed reassuring agreement between the WMM and several previously published methods. While in principle the current implementation allows to simulate cross sections with an arbitrary rectangular decomposition, the method turns out to be effective especially for structures described by few rectangles and boundary lines only. This class includes single rib and raised strip waveguides as well as coupler geometries constructed from these waveguides. For the composite structures investigated so far, the results are reasonable and accurate.
Unlike methods based on finite differences or finite elements, the WMM yields semianalytical mode field representations which are defined on the entire plane of the waveguide cross section, including the dielectric discontinuities explicitely and accurately. The fields are therefore perfectly suited for further processing, for instance in the framework of propagating mode analysis, coupled mode theory, or for the evaluation of perturbation theory integrals. These techniques are implemented in the WMM subroutine library, illustrated by several example program files.
Next generation active integrated optical subsystems
Online content related to our contributions to the project: