The authors give a brief historic perspective of the coupled mode theory. The development and applications of the theory in microwaves in early years and in optoelectronics and fiber optics in recent years are described. They then consider lossless coupling of two modes in time. Two coupled resonance circuits, or two coupled microwave or optical resonators, are the physical examples. The start-up of a parametric oscillator is another example. Then they look at the formal derivation of coupled mode theory and consider the more general case when the modes are not energy-orthogonal and the energies are not necessarily positive. A more detailed account of the nonorthogonal coupled mode theory developed in the last five years for optical waveguides is given. >

Coupled-mode theory

The coupled-mode theory (CMT) for optical waveguides is reviewed, with emphasis on the analysis of coupled optical waveguides. A brief account of the recent development of the CMT for coupled optical waveguides is given. Issues raised in the debates of the 1980’s on the merits and shortcomings of the conventional as well as the improved coupled-mode formulations are discussed. The conventional coupled-mode formulations are set up in a simple, intuitive way. The rigorous CMT is established on the basis of a linear superposition of the modes for individual waveguides. The cross-power terms appear logically as a result of modal nonorthogonality. The cross power is necessary for the self-consistency of the CMT for dissimilar waveguides. The nonorthogonal CMT, though more complicated, yields more-accurate results than the conventional orthogonal CMT for most practical applications. It also leads to the prediction of cross talk in directional couplers. The conventional orthogonal CMT is, however, reliably accurate for describing the power coupling between two weakly coupled, nearly identical waveguides. For dissimilar waveguides, a self-consistent orthogonal CMT can be derived by a redefinition of the coupling coefficients, and it predicts the coupling length and therefore the power exchange between the waveguides accurately if the two waveguides are far apart. Three typical coupler configurations—the uniform, the grating-assisted, and the tapered—are examined in detail. The accuracy, scope of validity, limitations, and extensions of the coupled-mode formulations are discussed in conjunction with each configuration. To verify the arguments in the discussions, comparisons with the exact analytical solutions and the rigorous numerical simulations are made.

Coupled-mode theory for optical waveguides: an overview

Since the first deployments of fiber-optic com- munication systems three decades ago, the capacity carried by a single-mode optical fiber has increased by a staggering 10 000 times. Most of the growth occurred in the first two decades with growth slowing to ten times in the last decade. Over the same three decades, network traffic has increased by a much smaller factor of 100, but with most of the growth occurring in the last few years, when data started dominating network traffic. At the current growth rate, the next factor of 100 in network traffic growth will occur within a decade. The large difference in growth rates between the delivered fiber capacity and the traffic demand is expected to create a capacity shortage within a decade. The first part of the paper recounts the history of traffic and capacity growth and extrapolations for the future. The second part looks into the technological chal- lenges of growing the capacity of single-mode fibers by pre- senting a capacity limit estimate of standard and advanced single-mode optical fibers. The third part presents elementary capacity considerations for transmission over multiple trans- mission modes and how it compares to a single-mode trans- mission. Finally, the last part of the paper discusses fibers supporting multiple spatial modes, including multimode and multicore fibers, and the role of digital processing techniques. Spatial multiplexing in fibers is expected to enable system capacity growth to match traffic growth in the next decades.

https://www.icg.isy.liu.se/courses/optical/Capacity_Trends.pdf

Capacity Trends and Limits of Optical Communication Networks Discussed in this paper are: optical communication network traffic evolution trends, required capacity and challenges to reaching it, and basic limits to capacity.

Since the first deployments of fiber-optic communication systems three decades ago, the capacity carried by a single-mode optical fiber has increased by a staggering 10 000 times. Most of the growth occurred in the first two decades with growth slowing to ten times in the last decade. Over the same three decades, network traffic has increased by a much smaller factor of 100, but with most of the growth occurring in the last few years, when data started dominating network traffic. At the current growth rate, the next factor of 100 in network traffic growth will occur within a decade. The large difference in growth rates between the delivered fiber capacity and the traffic demand is expected to create a capacity shortage within a decade. The first part of the paper recounts the history of traffic and capacity growth and extrapolations for the future. The second part looks into the technological challenges of growing the capacity of single-mode fibers by presenting a capacity limit estimate of standard and advanced single-mode optical fibers. The third part presents elementary capacity considerations for transmission over multiple transmission modes and how it compares to a single-mode transmission. Finally, the last part of the paper discusses fibers supporting multiple spatial modes, including multimode and multicore fibers, and the role of digital processing techniques. Spatial multiplexing in fibers is expected to enable system capacity growth to match traffic growth in the next decades.

Capacity Trends and Limits of Optical Communication Networks

The coupled-mode theory of parallel waveguides is derived from a variational principle for the propagation constant of the waveguide-wave solution using a superposition of the uncoupled modes as a trial field. The nonorthogonality of modes as emphasized by Hardy and Streifer is part of this formalism as well. The coupling Coefficients agree with those of Hardy and Streifer derived for TE modes of loss-free guides. For TM modes the coupling coefficients differ slightly for the simpler trial solution and agree exactly for a different trial solution. The simpler trial solution gives results closer to the exact solution. Conventional coupled-mode theory emerges from orthonormalization.

Coupled-mode theory of optical waveguides

A new coupled mode formulation for parallel dielectric waveguides is described. The results apply to any guided modes (TE, TM, or hybrid) in waveguides of arbitrary cross-section, dissimilar index, and nonidentical shape. Additional index perturbations not included within the waveguides are encompassed by the theory. Propagation constants and mode patterns for the coupled modes computed according to this theory are shown to agree very well with numerical solutions for the system modes when the latter can be determined. Moreover, the new results are more accurate than those obtained from prior coupled mode formulations. It is shown that even for Iossless guides the coupling coefficients from waveguide "b" to "a" and from "a" to "b," described by k ab and k ba respectively, are not related by their complex conjugates if the guides are not identical.

Coupled mode theory of parallel waveguides

A new theory describing coupled mode propagation in an array of arbitrary optical waveguides is developed. The formulation unambiguously accounts for real and/or imaginary index perturbations and its solutions yield supermode patterns and propagation constants. Several examples of coupled TE slab waveguides are evaluated and the results are shown to compare very favorably with exact solutions. The calculated propagation constants and mode patterns differ quantitatively and qualitatively from those derived from the prior theory. In particular, the supermode gains derived according to the new theory agree very well with exact solutions and predict different supermode ordering and splitting than those derived by the prior theory.

W. Streifer

Papers

Coupled mode theory of parallel waveguides

Coupled modes of multiwaveguide systems and phased arrays