We derive a coupled mode theory for the interaction of an optical cavity with a waveguide that includes waveguide dispersion. It can solve various modal conversion and scattering problems in elastic waveguides with small changes of cross sections and stress free boundaries. Reformulation of the coupled mode theory of multiwaveguide systemsr. In this chapter we will describe how optical energy couples. Consequently, the cmt formulism is far less computationally demanding than the maxwells equations. Chapter 2 is focused on artificial transmission lines based on periodic structures including nonuniform transmission lines and reactivelyloaded lines, and provides a comprehensive analysis of the coupled mode theory. We apply the concepts of nonlinear guidedwave optics to a boseeinstein condensate bec trapped in an external potential.
Resonant coupling of light beams via highindex media or gratings to planar waveguiding. Pdf we present an extension of the canonical coupled mode theory of. The coupled mode theory cmt is the easiest and in many cases the only possible means for analytical investigations of these new systems of modern optics. The superposition principle says that eigenmodes of linear systems are independent of each other. To be of use in any particular problem, some method of deriving the coefficients must be available. In the term mode coupling, as used in physics and electrical engineering, the word mode refers to eigenmodes of an idealized, unperturbed, linear system. Coupledmode theory of multipath interference in quasi. Coupledmode theory for general freespace resonant scattering of waves ra. Pdf coupledmode theory for electromagnetic pulse propagation.
Coupled mode theory and its applications on computational. Joannopoulos, and marin soljacic center for materials science and engineering and research laboratory of electronics, massachusetts institute of technology, cambridge, massachusetts 029, usa. Temporal coupledmode theory for the fano resonance in. The theory can be applied to photonic crystal cavity waveguide structures. It has found extensive applications in the theory of fiber arrays 1, 2, discrete optical systems 3, and high power fiber lasers 4, 5, 6. Distributed feedback structures and waveguide gratings, coupled mode theory for first order bragg gratings, dbr lasers, multimode and single mode operation, threshold gain, photon lifetime, and longitudinal mode frequencies of dbr lasers. Winick contradirectional propagation through active, firstorder, almostperiodic, corrugated waveguide grat. A coupledmode formulation is described in which the radiation fields are represented in terms of discrete complex modes. Coupledmode theory of field enhancement in complex. Our theory thus assumes the same regime of validity as the standard temporal coupled mode theory. Mode coupling is ubiquitous in physics, where different modes belonging to the same system or to different systems, can exchange energy through a coupling perturbation. As an example, we consider a parabolic doublewell potential and derive coupled mode equations for the complex amplitudes of the bec macroscopic collective modes. Coupled mode theory for resonant excitation of waveguiding structures.
Nanodisks cavities and a plasmonic waveguide mohamed nady abstracta metalinsulatormetal mim plasmonic waveguide coupled with two nanodisks as a resonator has been examined and numerically simulated with the. Summary we present a universal coupled mode theory treatment of freespace scattering of waves from resonant objects. The reflected signature of an optical fiber bragg grating is analyzed using the transfer function method. Finally, we show that the coupled mode theory can be applied even when more than two modes are involved in the heat transfer. Coupled mode and coupled power theories are described for multicore fiber design and analysis. Implementation of a fully nonlinear hamiltonian coupled. Skorobogatiy,2 mihai ibanescu,1 elefterios lidorikis,1 and j. The cross power is necessary for the selfconsistency. Threedimensional coupledmode theory of freeelectron lasers in the collective regime. Coupled mode theory a rigorous approach to investigate an electromagnetic wavepropagation problem is to solve the maxwell equations in differential or integral form for a given device, along with the material interface conditions, and relevant initial and boundary conditions. In the matrix transfer function, the bragg grating is simulated as.
Calculation of the spatial derivatives of the vertical basis functions. The effects of direct and indirect couplings, if coexisting in one system, can offset each other such that no mode splitting. Coupled mode theory allows a wide range of devices and systems to be modeled as one or more coupled resonators. Effectiveindex method and coupled mode theory for almostperiodic waveguide gratings. Similar to the coupled mode theory in optical waveguides, the analysis is presented by the evolution of modal amplitudes. The range of applicability of the presented approach is fairly. Coupled mode and parametric electronics louisell, william h on. Here we investigate a frequency domain spatial coupled mode theory based model of 2d circular integrated optical microresonators. Mode splitting based on the coupling between modes of two. This lecture steps the student through coupledmode theory using perturbation analysis. The coupling coefficients agree with those of hardy and streifer derived for te modes of loss free guides. Artificial transmission lines for rf and microwave.
Coupled mode analysis for 3d stressfree elastic acoustic. Conclusions the coupled mode theory as developed here is useful in many areas of research. The overwhelming majority of these formalisms relates to. The crosspower terms appear logically as a result of modal nonorthogonality. Adiabatic theorem and continuous coupledmode theory for. The complex coupledmode theory is applied to bragg grating in slab waveguides and validated by rigorous modematching method. The method of analysis is known as coupled mode theory, or zdependent perturbation theory.
In this condition, the alternating field amplitude in the primary inductor is constant, a 2 t a 2 e. Coupled mode theory is a perturbational approach for analyzing the coupling of vibrational systems in space or in time. Barybin, 9781589490079, available at book depository with free. We have studied such systems theoretically using both coupled mode theory cmt 34,35 and fullwave simulations. Mutual coupling between optical modes is essential in the design of integrated optic devices. A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation. For tm modes the coupling coefficients differ slightly for. Wireless power transfer via strongly coupled magnetic. It also offers an approach of grouptheoretical analysis to the mode coupling problems. We take the limit l to re cover the case of a single slit with a continuum of plane wave channels.
Temporal coupled mode theory linking to surfacewave. In optics, such systems include laser cavities, photonic crystal slabs, metamaterials, and ring resonators. Pdf complex coupledmode theory for optical waveguides. Temporal coupledmode theory for resonant apertures stanford.
Theory of dielectric optical waveguides 2e 2nd edition. The range of applicability of the presented approach is fairly broad. Timedomain simulations of nonlinear interaction in. Improved coupledmode theory for codirectionally and. Adiabatic theorem and continuous coupled mode theory for ef. We then add on driving and damping forces and apply some results from chapter 1. Coupled mode theory analysis of modesplitting in coupled. The rigorous cmt is established on the basis of a linear superposition of the modes for individual waveguides. Coupled mode theory describes this energy exchange, and serves as the primary tool for designing optical couplers, switches, and filters. The circuit theory behind coupledmode magnetic resonance.
An experimental demonstration of the theory is also carried out in a system. Coupled mode theory of field enhancement in complex metal nanostructures gregory sun1 department of physics, university of massachusetts boston boston, massachusetts 02125 jacob b. All covid19 related research articles, book chapters and information is free to access and available for anyone to read. Dfb lasers, threshold gain, photon lifetime, and longitudinal. A similar approach has been used previously to analyze bistability in the single mode cavity coupled to a waveguide 27. The conventional coupledmode formulations are set up in a simple, intuitive way. The cmt results agree well with rigorous fdtd simulations. Coupled mode theory is particularly useful when a slight perturbation has a large effect on the distribution of modal power, e.
Analytic solutions exist for rather a few problems only. A coupledmode theory is derived from the scalar wave equation for the interaction between forward and. The effects of direct and indirect couplings, if coexisting in one system, can offset each other such that no mode splitting occurs and the. Arbitrary order exceptional point induced by photonic spin. Joannopoulos1 1department of physics, massachusetts institute of technology, cambridge, massachusetts 029 2omniguide communications, one kendall. The theory has been used by letton 5 to find the transmissivity of an inhomogeneous plasma sheet when electroacoustic waves are included in the analysis. The primary objective, however, is to illustrate and explain coupled mode theory from a picturequalitative. In steady state analysis, which applies to the conventional inductive power transmission links, f s t in is a sinusoidal signal described as a s e. In reality, simple mechanisms can lead to significant energy exchange among the various modes of a structure. Coupled mode theory in periodic media, two approaches are generally used. First, in order to satisfy the law of power conservation, mode coupling coefficients are redefined and then, closedform powercoupling coefficients are derived based on exponential, gaussian, and triangular autocorrelation functions. The complex modes are obtained from a waveguide model facilitated by the combination of perfectly matched boundary pml and perfectly reflecting boundary prb condition. Mode splitting is observed for both directly cavitycavity and indirectly cavitywaveguidecavity coupled cavity systems.
Coupled mode theory based modeling and analysis of. All integrated optical components and devices make use of waveguides, where light is confined by total internal reflection. Course paperwork syllabus course assignments course material course resources animated visualization of a grating vector short course on generating spatiallyvariant lattices download. This thesis is based on our contributions to the project nais concerning the mod eling and simulation of microresonators as tunable wavelength lters. The elements in such photonic chip are interconnected through waveguides, and also the integrated optics components themselves are fabricated using waveguide configuration, such as couplers, switches, modulators. There are many different formalisms of coupled mode theory in optics, describing a plethora of cases 16. It was shown in the coupled mode analysis of the fel amplifier carried out in refs. This text presents an electrodynamics theory that provides a selfconsistent approach to describe mode excitations and mode couplings, including the optical guidedwave interactions. We derive an analytical solution to the add and drop spectra arising from such interactions in the limit of linear dispersion. Effectiveindex method and coupledmode theory for almost. First, in order to satisfy the law of power conservation, modecoupling coefficients are redefined and then, closedform powercoupling coefficients are derived based on exponential, gaussian, and triangular autocorrelation functions.
Spatiotemporal coupledmode theory in dispersive media. Coupledmode theory of optical waveguides ieee xplore. Pdf a coupledmode formulation is described in which the radiation. Using coupled mode theory to describe this physical system 11, we obtain the following set of linear equations. Here, we develop a general theoretical framework for modeling metasurface based on the coupled mode theory cmt, which fully describes the interaction between the metaunits and light by a simple set of coupled mode equations. The analysis using the above model shows that only a few most relevant basis fields are required to construct approximate solutions to the scattering problems that are sufficient for purposes of practical resonator design. The complex modes are obtained from a waveguide model facilitated by the combination of perfectly matched boundary pml.
Bloch wave formalism and coupledmode theory the coupledmode theory. Ieee photonics journal coupledmode theory of mpi in qsm fibers denote quantities referring to the fundamental mode and the subscript 1 to denote quantities refer ring to the higher order mode. Coupledmode theory for general freespace resonant scattering of waves. Coupledmode and coupledpower theories are described for multicore fiber design and analysis.
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