Bragg filter, coupled mode theory, onedimensional photonic crystal. A few books summarily describe the dielectric guides. This chapter emphasizes the nature of optical modes and their group velocities in stepindex and gradedindex fibers. A set of equations describing pulse propagation in multimode optical fibers in the presence of an intensitydependent refractive index is derived by taking advantage of the coupledmode theory usually employed for describing the influence of fiber imperfections on linear propagation. Use is made of the statistical ensemble of similar fibers introduced by marcuse for studying mode coupling in optical waveguides. Weakly guiding optical fibers are discussed along with coupled mode theory and coupled power theory. V coupled mode equations is derived to describe mode propagation in uniform and slightly nonuniform cylindrical optical fiber systems. The general theory is applied to a simplified description of guided and radia tion modes of. It is customary in the theory of optical fibers to make the weakly guiding approximation n. Theory of imperfect nonconventional singlemode optical fibers. A set of equations describing pulse propagation in multimode optical fibers in the presence of an intensitydependent refractive index is derived by taking advantage of the coupled mode theory usually employed for describing the influence of fiber imperfections on linear propagation. Pdf coupledmode theory approach to nonlinear pulse. The paper begins with the derivation of a general coupled wave theory based on the modes of the ideal fiber. All the preceding methods have advantages and disad vantages.
The software rp fiber power can be used for simulating mode coupling effects. Multipole method for microstructured optical fibers. Chan, alan pak tao lau, and hwayaw tam department of electrical engineering, the hong kong polytechnic university, hong kong, china fym. Coupledmode theory of multipath interference in quasi. The coupling between fibers of an array made up of n identical fibers each at the vertex of a polygon and one at the center, which is not necessarily the same as its n neighbors, is determined. 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. This paper presents a theoretical framework for analyzing, in a general way, the imperfect nonconventional optical fibers met with in practice. We use the power coupledmode theory to study the interplay between multi path interference mpi and differential mode attenuation dma in quasisingle mode qsm fibers. Coupled mode theory is particularly useful when a slight perturbation has a large effect on the distribution of modal power, e.
Coupledmode theory of nonlinear propagation in multimode. Aug 28, 2008 raman amplification in weakly guiding optical fibers is investigated for the two cases of weak and strong mode coupling. Theory of dielectric optical waveguides 2e 2nd edition. The paper begins with the derivation of a general coupled. Optical fibers based communication networks are part and parcel of todays knowledge based society. Fullvectorial coupled mode theory for the evaluation of macro. Rp photonics encyclopedia mode coupling, coupledmode. Once the parameters of one of the cores are specified, the roots of the resulting. Coupled mode and coupled power theories are described for multicore fiber design and analysis. Ankiewicz, fibre couplers composed of unequal cores, electon. A set of coupledmode equations is derived to describe mode. Optical fiber interferometers and their applications 5 polarization aligned with one of the principal axes of the birefrigent fiber, the light propagates without any disturbance in its pola rization state. We take advantage of the perturbed coupledmode equations to discuss the form of coupledmode equations for optical fiber when the term of twostage approximation has been reserved.
The fiber grating bending sensing technology is based on its axial strain sensing characteristics. We have arranged them in order but feel free to navigate around as you like. Pdf coupled mode theory for optical waveguides jianwei mu. Nevertheless, following the applications of these guides in the form of optical fibers, it has become essential for a course on applied electromagnetism to cover this theory and emphasize on the dispersion minimisation which allows an extreme bandwidth. We discuss the twostage approximation of coupled mode equations and apply it to allfiber acoustooptic modulator. We derive an analytical solution to the add and drop spectra arising from such interactions in the limit of linear dispersion.
A systematic approach for designing zerodgd coupled multi. Coupledmode theory of multipath interference in quasisingle. The light emmited by a source of area asource can be butt coupled into an optical fiber which is larger the the source area. Propagating beam theory of optical fiber cross coupling. Coupled mode theory based modeling and analysis of circular. The perturbed coupled mode theory is used to discuss the solutions of coupled mode equations for optical fiber when there are multiple parts being perturbed, and the multiplechannel modulation is. We use the power coupled mode theory to study the interplay between multipath interference mpi and differential mode attenuation dma in quasisingle mode qsm fibers. The nonorthogonality of modes as emphasized by hardy and streifer is part of this formalism as well. Nonlinear interactions in a waveguide can also couple the modes as calculated for low light intensities to each other. We discuss the twostage approximation of coupledmode equations and apply it to allfiber acoustooptic modulator.
Abstract bragg gratings in optical fibers in multimode propagation are investigated theoretically by using the coupled mode theory and considering two modes coupling. Coupled mode theory is a method to analyze the light propagation in perturbed or weakly coupled waveguides. Jun 12, 2001 extensions of coupled mode theory for splice of different optical fiber with arbitrary refractive index profiles is considered. In the preceding two chapters, the transmission characteristics of independent planar optical waveguides and optical fibers have been investigated. Coupled mode theory of round optical fibers wiley online library. The formalism of coupledmode theory is adopted for describing nonlinear pulse propagation in optical fibers, the coupling being induced by the nonlinear part of the refractive index. The formalism of coupled mode theory is adopted for describing nonlinear pulse propagation in optical fibers, the coupling being induced by the nonlinear part of the refractive index.
Inner layer core of radius aand refractive index n 1. Introduction one of the ubiquitous phenomenon in the field of photonics is resonance, in which resonant mode interacts with the propagated waves, the analyzing dynamics of the resonant system via coupled mode theory match with rigorous numerical simulations1234. Any reader interested in designing or actively working with optical devices must have a firm grasp of the principles of lightwave propagation. Coupled mode theory based modeling and analysis of. Fiber acts as an open optical waveguide may be analyzed using simple ray theory geometric optics not sufficient when considering all types of optical fibers electromagnetic mode theory for complete picture. Optical fibers are therefore classified as stepindex or gradedindex, and multimode or single mode, as illustrated in fig.
This approach effectively reduces the problem to a system of n1 algebraic equations involving the associated coupling coefficients and propagation constants as obtained from coupled mode theory. Coupled mode theory of round optical fibers nokia bell labs. Necessary to consider the optical waveguiding of a cylindrical glass fiber. We take advantage of the perturbed coupled mode equations to discuss the form of coupled mode equations for optical fiber when the term of twostage approximation has been reserved. Bragg gratings in multimode fibers mmfbgs show multiple reflection peaks due to the coupling between all bounded modes.
The preferred field of the electromagnetic theory of waveguides is that of single mode waveguides, or of guides supporting few modes. This approach describes in a natural way the influence of the waveguide, and in principle allows of the possibility of investigating soliton propagation when more. Examples of this array are two fibers, three fibers in a row. The analysis using the above model shows that only a few most relevant basis. We use the power coupledmode theory to study the interplay between multipath interference mpi and differential mode attenuation dma in quasisingle mode qsm fibers. Fundamentals of optical waveguides is an essential resource for any researcher, professional or student involved in optics and communications engineering. Optical fiber interferometers and their applications. Mode coupling dynamics and communication strategies for multicore fiber systems florence y. Using the coupled mode and coupled power theories, impacts of.
The general theory is applied to a simplified description of guided and radiation modes of the fiber that is valid for small core. The birefringence parameter of the fiber is defined by the difference between the two refractive indices corresponding to the. Coupledmode theory application for splice of different. Pier online coupledmode theory for optical waveguides. It is therefore anticipated that a range of devices could be modeled in a similar way. See demo cases for mode coupling in a fiber bragg grating or in a fiber coupler definition. Then, a linear combination of these modes is used as a trial solution to maxwells. In this chapter derivation of coupled mode equations based on perturbation theory is first presented and then concrete methods calculating the coupling coefficients for several practically important devices are explained in detail. This approach is often technically and conceptually much more convenient than, e. Light rays will be confined inside the fiber core if it is inputcoupled. Optical fiber theory series in optics and photonics.
The perturbed coupledmode theory is used to discuss the solutions of coupledmode equations for optical fiber when there are multiple parts being perturbed, and the multiplechannel modulation is. The analytical expressions derived assuming two mode propagation in qsm fibers show that mpi scales differently as a function of the span length for low and high dma. First, in order to satisfy the law of power conservation, mode coupling coefficients are redefined and then, closedform power coupling coefficients are derived based on exponential, gaussian, and triangular autocorrelation functions. Coupled mode theory of round optical fibers marcuse 1973.
Pdf a coupledmode formulation is described in which the. The basic idea of coupled mode theory is to decompose all propagating light into the known modes of the undisturbed device, and then to calculate how these modes are coupled with each other by some additional influence. The coupledmode theory of parallel waveguides is derived from a variational principle for the propagation constant of the waveguidewave solution using a superposition of the uncoupled modes as a trial field. 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. Applications of the coupled mode theory are also considered, taking into account a slab. Though lossless operation is assumed, it is simple to incorporate loss or gain, and no major modifications to the equations are incurred in the process. Spun and twisted fibers with various imperfections are taken as specific examples to illustrate the general theory. Dynamic coupledmode theory of nonlinear propagation in. The theory of optical fibres these few pages will explain the concepts behind optical fibres and how they work. The twostage approximation of coupledmode theory for.
Marcuse, coupled mode theory of round optical fibers, j bell syst tech 52, 817843 1973. 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. Coupled mode theory for the leaky modes of a bent fiber. 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.
Optical fiber a single solid dielectric of two concentric layers. Mode coupling dynamics and communication strategies for multi. The method of analysis is known as coupled mode theory, or zdependent perturbation theory. Mode coupling on a lossy fiber is investigated and a simple. Pdf complex coupledmode theory for optical waveguides. Coupled mode theory of round optical fibers marcuse. Coupledmode theory of nonlinear propagation in multimode and. This paper presents a comprehensive theory of mode coupling in optical fibers with imperfections. Koch, lowloss hollowcore silicaair photonic bandgap fibre, nature 424, 657 2003.
Using the coupledmode and coupledpower theories, impacts of. Coupledmode theory of optical waveguides ieee journals. The cmt results agree well with rigorous fdtd simulations. The theory can be applied to photonic crystal cavity waveguide structures. Coupledmode and coupledpower theories are described for multicore fiber design and analysis.
The basic idea of the coupled mode theory method is that the modes of the unperturbed or uncoupled structures are defined and solved first. We derive a coupled mode theory for the interaction of an optical cavity with a waveguide that includes waveguide dispersion. A set of coupledmode equations is derived to describe mode propagation in uniform and slightly nonuniform cylindrical opticalfiber systems. It simplifies the characteristic equation for the modes. The light incident outside the core area will not be guided by the fiber. Coupled mode theory as an important widely used methods for analyzing coupling between discrete guided modes plays a major. Coupledmode theory for allfiber acoustooptic modulator. Coupled mode theory for photonic crystal cavitywaveguide.