Standard Coupled-mode Theory Research Articles

Cross-axis Bragg gratings in few-mode fibers inscribed with a femtosecond laser point-by-point technique

Point-by-point (PbP) cross-axis Bragg gratings in a step-index two-mode fiber are fabricated by tightly focused femtosecond laser pulses. The grating is formed by inscribing a line of periodic damage points crossing the fiber axis at a certain tilting angle. We experimentally show that the cross-axis structure can introduce multiple mainband resonances associated with the higher-order modes, with wavelength spacing proportional to the tilting angle. We also develop a numerical model for the PbP-inscribed gratings based on pulse-wave approximation and the standard coupled-mode theory, the resulting simulation exhibits a good agreement with the experimental findings. This work suggests the higher-order fiber modes, synergizing with the PbP technique, has the potential to add a new dimension to designing gratings with desired spectral spacing; such an inverse problem of PbP gratings can be assisted by the numerical modal we developed.

Unified Coupled-Mode Theory for Geometric and Material Perturbations in Optical Waveguides

Coupled-mode theory is a powerful tool to understand and control the effects of deployment and fabrication imperfections on optical waveguides. Although it provides many advantages compared to the finite element method, it still lacks the ability to treat geometric and material perturbations when they act simultaneously on the waveguide. This work fills this gap, providing a novel framework for a unified treatment of geometric and material perturbations in the coupled-mode analysis. The proposed approach consists of, first, applying the theory of transformation optics to convert geometric deformation into material perturbations and, second, studying the obtained waveguide by using a custom-developed coupled-mode theory able to deal with perturbations of both the permittivity and the permeability tensor. The framework is applied to three examples: a solid-core fiber affected by intrinsic perturbations, a bent solid-core fiber, and an elliptical hollow-core fiber. Results are validated against simulations based on the finite element method and compared with the standard coupled-mode theory most suitable for each specific example; they show that the proposed unified coupled-mode theory performs consistently better than standard theories, confirming it as a general and accurate tool for the design and analysis of optical waveguides.

Enhanced light-extraction efficiency and emission directivity in compact photonic-crystal based AlGaInP thin-films for color conversion applications.

We investigated the use of photonic crystals with different opto-geometrical parameters for light extraction from AlGaInP/InGaP MQW color converters. Blue-to-red and green-to-red color conversions were demonstrated using room-temperature photoluminescence with excitation wavelengths at 405nm and 514nm. Complete, compact and highly directional light extraction was demonstrated. 3D-FDTD and a herein-developed phenomenological model derived from the standard coupled-mode theory were used to analyze the results. The highest light extraction gains were ∼8 times better than unpatterned reference structures, which were paired with short extraction lengths (between 2µm and 6µm depending on the acceptance angle) and directional light emission for square lattice of nanopillars with a lattice period of 400nm. The design guidelines set in this work could pave the way for the use of inorganic MQW epi-layer color converters to achieve full color microdisplays on a single wafer.

Application of the standard coupled-mode formalism to the analysis of holey photonic crystals

ABSTRACTPresented below is a new, Standard Coupled-mode Theory (CMT) based approach for analysis of optical characteristics of holey photonic crystals (i.e., photonic devices, built on periodicity of holes in dielectric media). This class of devices encompasses the majority of photonic crystal fibres and several kinds of modern thresholdless lasers. Naturally, holey photonic crystals were considered as a sequence of holes, surrounded by dielectric media. This model made it impossible to utilise CMT for analysing their characteristics. The underlying idea of our approach is a different physical model, considering holey photonic devices as a sequence of coupled dielectric spots (waveguides), surrounded by air. This model can be combined with the Standard Coupled-mode formalism. The latter combination allows fast (on a timescale of several minutes to tens of minutes) and accurate analysis of holey photonic devices. Moreover, it gives a deeper insight into the behaviour of EM fields in holey photonic crystals.

Modeling of On-Chip Optical Nonreciprocity with an Active Microcavity

On-chip nonreciprocal light transport holds a great impact on optical information processing and communications based upon integrated photonic devices. By harvesting gain-saturation nonlinearity, we recently demonstrated on-chip optical asymmetric transmission at telecommunication bands with superior nonreciprocal performances using only one active whispering-gallery-mode microtoroid resonator, beyond the commonly adopted magneto-optical (Faraday) effect. Here, detailed theoretical analysis is presented with respect to the reported scheme. Despite the fact that our model is simply the standard coupled-mode theory, it agrees well with the experiment and describes the essential one-way light transport in this nonreciprocal device. Further discussions, including the connection with the second law of thermodynamics and Fano resonance, are also briefly made in the end.

Normal-Mode Analysis of Switching Dynamics in Nonlinear Directional Couplers

The dynamic properties of a nonlinear directional coupler made of Kerr materials are investigated using a normal-mode theory based on even and odd supermodes. This analysis provides a good platform to study the pulse propagation in nonlinear multimode waveguides, since the device is simple and can be regarded as a bimodal waveguide. Although the nonlinear directional coupler has already been well investigated using the standard coupled-mode theory, this paper presents new knowledge concerning the device operation from the standpoint of supermodes. Under the normal-mode model, power exchange between the two supermodes is performed through non-phase-matched four-wave mixing (FWM) and new frequency components are created on propagation, leading to spectral broadening of optical pulses. It is found that the group-delay mismatch between the two modes induces instability in collaboration to the nonlinearity when the duration of input pulses is of the order of sub-picoseconds. The dynamic properties of the nonlinear directional coupler are quite similar to those of the nonlinear MMI coupler that were more recently investigated by the present authors.

Light-propagation management in coupled waveguide arrays: Quantitative experimental and theoretical assessment from band structures to functional patterns

We assess the band structure of arrays of coupled optical waveguides both by ab initio calculations and by experiments, with an excellent quantitative agreement without any adjustable physical parameter. The band structures we obtain can deviate strongly from the expectations of the standard coupled mode theory approximation, but we describe them efficiently by a few parameters within an extended coupled mode theory. We also demonstrate that this description is in turn a firm and simple basis for accurate beam management in functional patterns of coupled waveguides, in full accordance with their design.

Coupled-resonator optical waveguides: Q-factor influence on slow-light propagation and the maximal group delay

Coupled-resonator optical waveguides hold potential for slow-light propagation of optical pulses. The dispersion properties may adequately be analyzed within the framework of coupled-mode theory. We extend the standard coupled-mode theory for such structures to also include complex-valued parameters which allows us to analyze the dispersion properties also in presence of finite Q factors for the coupled resonator states. Near the band-edge the group velocity saturates at a finite value v_g/c \propto \sqrt{1/Q} while in the band center, the group velocity is unaffected by a finite Q factor as compared to ideal resonators without any damping. However, the maximal group delay that can be envisioned is a balance between having a low group velocity while not jeopardizing the propagation length. We find that the maximal group delay remains roughly constant over the entire bandwidth, being given by the photon life time of the individual resonators.

Coupled-resonator optical waveguides: Q-factor influence on slow-light propagation and the maximal group delay

Coupled-resonator optical waveguides hold potential for slow-light propagation of optical pulses. The dispersion properties may adequately be analyzed within the framework of coupled-mode theory. We extend the standard coupled-mode theory for such structures to also include complex-valued parameters which allows us to analyze the dispersion properties also in presence of finite Q factors for the coupled resonator states. Near the band-edge the group velocity saturates at a finite value υg/c∝1/Q while in the band center, the group velocity is unaffected by a finite Q factor as compared to ideal resonators without any damping. However, the maximal group delay that can be envisioned is a balance between having a low group velocity while not jeopardizing the propagation length. We find that the maximal group delay remains roughly constant over the entire bandwidth, being given by the photon life time τp = Q/Ω of the individual resonators.

Water-wave gap solitons: An approximate theory and numerical solutions of the exact equations of motion

It is demonstrated that a standard coupled-mode theory can successfully describe weakly nonlinear gravity water waves in Bragg resonance with a periodic one-dimensional topography. Analytical solutions for gap solitons provided by this theory are in reasonable agreement with numerical simulations of the exact equations of motion for ideal planar potential free-surface flows, even for strongly nonlinear waves. In numerical experiments, self-localized groups of nearly standing water waves can exist up to hundreds of wave periods. Generalizations of the model to the three-dimensional case are also derived.

Time-domain approach to parametric conversion: A comparative study on the impact of numerical dispersion

Numerical integration schemes of time-domain Maxwell equations in optical dispersive and nonlinear media are considered. In this context, we study quantitatively the impact of numerical dispersion on a typical nonlinear parametric conversion process, comparing the widely used finite difference (FDTD) approach and pseudo-spectral (PSTD) methods. Our results show that, unless using very dense grid, only fourth-order PSTD gives results in good quantitative agreement with standard coupled-mode theory.

Bandpass filters based on phase-shifted photonic crystal waveguide gratings

In this paper, a bandpass transmission filter realized in phase-shifted waveguide gratings based on photonic crystals (PCs) is proposed. Phase-shift regions each composed of one period of photonic crystal (PC) waveguide are incorporated into PC waveguide gratings. The magnitudes of the phase-shifts are modified by involving small changes in the size of the border rods in the phase-shift regions. Using standard coupled-mode theory and finite-difference time-domain (FDTD) method, we show that by properly choosing the magnitudes of phase-shifts and the lengths of waveguide gratings, a flat-top and sharp roll-off response with a narrow bandwidth is theoretically and numerically achieved by the designed filter. A further analysis shows that the center frequency of the transmission band can be changed by altering the magnitude of the phase-shift and the response performance exhibits relaxed sensitivity to the phase-shift variation. As a specific application, we theoretically demonstrate a third-order Chebyshev bandpass filter based on compound phase-shifted PC waveguide gratings. The filter performance is suitable for dense wavelength-division-multiplexed (DWDM) optical communication systems with a channel spacing of 100-GHz.

All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures

We analyze the resonant linear and nonlinear transmission through a photonic crystal waveguide side-coupled to a Kerr-nonlinear photonic crystal resonator. First, we extend the standard coupled-mode theory analysis to photonic crystal structures and obtain explicit analytical expressions for the bistability thresholds and transmission coefficients which provide the basis for a detailed understanding of the possibilities associated with these structures. Next, we discuss limitations of standard coupled-mode theory and present an alternative analytical approach based on the effective discrete equations derived using a Green's function method. We find that the discrete nature of the photonic crystal waveguides allows a geometry-driven enhancement of nonlinear effects by shifting the resonator location relative to the waveguide, thus providing an additional control of resonant waveguide transmission and Fano resonances. We further demonstrate that this enhancement may result in the lowering of the bistability threshold and switching power of nonlinear devices by several orders of magnitude. Finally, we show that employing such enhancements is of paramount importance for the design of all-optical devices based on slow-light photonic crystal waveguides.

Transfer matrix and full Maxwell time domain analysis of nonlinear gratings

In this paper we show that the transfer matrix method and the finite difference time domain technique applied to the full time-dependent Maxwell equations (in their one-dimensional form) permit to improve the description of nonlinear distributed feedback structures such as Bragg gratings. In fact, no simplifying hypothesis are introduced in the electromagnetic problem by these two numerical approaches. The two techniques validate qualitatively the results based on the standard coupled-mode theory for weakly nonlinear and shallow gratings, while significant quantitative deviations can appear in other cases.

Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals.

We prove that an adiabatic theorem generally holds for slow tapers in photonic crystals and other strongly grated waveguides with arbitrary index modulation, exactly as in conventional waveguides. This provides a guaranteed pathway to efficient and broad-bandwidth couplers with, e.g., uniform waveguides. We show that adiabatic transmission can only occur, however, if the operating mode is propagating (nonevanescent) and guided at every point in the taper. Moreover, we demonstrate how straightforward taper designs in photonic crystals can violate these conditions, but that adiabaticity is restored by simple design principles involving only the independent band structures of the intermediate gratings. For these and other analyses, we develop a generalization of the standard coupled-mode theory to handle arbitrary nonuniform gratings via an instantaneous Bloch-mode basis, yielding a continuous set of differential equations for the basis coefficients. We show how one can thereby compute semianalytical reflection and transmission through crystal tapers of almost any length, using only a single pair of modes in the unit cells of uniform gratings. Unlike other numerical methods, our technique becomes more accurate as the taper becomes more gradual, with no significant increase in the computation time or memory. We also include numerical examples comparing to a well-established scattering-matrix method in two dimensions.

Dual-core single-mode polymer fiber coupler

We have observed light coupling between dual-core dye-doped polymer optical fibers. The experimentally observed coupling length is consistent with the standard coupled-mode theory for single-mode–single-mode coupling between step-index waveguides [ P. D. McIntyre A. W. Snyder , J. Opt. Soc. Am.63, 1518–1527 (1973)]. We also show that a graded refractive-index profile of such dual-core fibers can be made by controlled dye diffusion. The concentration, as measured by optical absorption, shows a graded profile that matches the refractive-index profile, confirming that dye concentration determines the refractive index.

Phase-shifted Bragg-grating filters with improved transmission characteristics

Standard Bragg filters (dielectric waveguides with a grating overlay) show a stopband only in reflection. Filters with quarter-wave-shifted gratings have an extremely narrow transmission peak the center of the stopband but its shape (a sharp triangle with a broad bottom) is not suitable for system design. Using standard coupled-mode theory together with transfer-matrix formalism, we show that by introducing several phase-shift regions and properly choosing their locations and magnitudes the transmission spectrum can be tailored into a nearly rectangular shape. For InP-InGaAsP based Bragg-grating filters estimates are made of achievable channel spacings in wavelength-division multiplexing systems.

A coupled-mode theory of heisenberg paramagnets

The inadequacy of the standard coupled-mode theory of one dimensional paramagnets is resolved by renormalisation of the exchange coupling and time scale, in such a way that the refined theory is consistent with some static spin cumulants. With the proposed refinements the coupled-mode theory gives the exact spin dynamics in the limit of vanishing temperatures. For low temperatures the theory is studied numerically to achieve comparisons with computer simulation data that vindicate the renormalisation scheme.

Critical spin dynamics of EuO: comparison of theory and experiment

The standard coupled-mode theory of spin dynamics is evaluated for an isotropic, Heisenberg model of EuO at the critical temperature. The predicted long-wavelength limit of the spin-relaxation function is approximately a gaussian function of time in the time interval probed in recent neutron spin-echo measurements. This functional form is distinctly different from the observed, almost exponential, decay.