| A rigorous classical analytic model of confined optical wave propagation along 2-D bent slab waveguides and curved dielectric interfaces is investigated, based on a piecewise frequency domain ansatz for bend mode profiles in terms of Bessel and Hankel functions. This approach provides a clear picture of the behaviour of bend modes, concerning their decay for large radial arguments or effects of varying bend radius. Fast and accurate routines are required to evaluate Bessel functions with large complex orders and large arguments. Our implementation enabled detailed studies of bent waveguide properties, including higher oder bend modes and whispering gallery modes, their interference patterns, and issues related to bend mode normalization and orthogonality properties. Also a perturbational expression is derived for the shift in the propagation constant due to changes in the core refractive index. |
Parts of this chapter are adapted from:
K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Ctyroký.
Analytic approach to dielectric optical bent slab waveguides. Optical
and Quantum Electronics, 37(1-3):37-61, January 2005.
Bent dielectric waveguides play an important role in photonic integrated circuits. Accurate evaluation of mode profiles, phase propagation constants, and of optical losses associated with the leaky wave propagation is the central task for theoretical modeling of the curved structures. The present work on this - rather old - topic is motivated by the recent interest in circular optical microresonator devices as building blocks for large-scale integrated optics [22,31]. During our participation in a related European project [32], we experienced that certain notions about the properties of bend modes deserved clarification. This concerns e.g. the behaviour of the mode profiles for large radial coordinates, profile integrability, mode orthogonality, or a clear picture of propagation and interference of the bend modes.2.1
A sound modal analysis of bent slabs becomes particularly relevant if the mode profiles are to be employed as basis fields for a description of integrated optical microresonators with circular, ring- or disk-shaped cavities. In a framework of coupled mode theory [33,70,44], an as far as possible analytic representation of the basic field profiles on a radially unbounded domain must be regarded as highly advantageous. This is provided by the approach of this chapter. Preliminary promising studies on CMT modeling of circular resonators are contained in [34,71,72]; further details follow in Chapters 3, 4.
This chapter presents an analytical model of (2-D) bent waveguides. Using the uniform asymptotic expansions of Bessel/Hankel functions as provided in Ref. [73,74], we found that with present standard computers it is not a problem to carry out the rigorous analytic evaluation of the problem. Details on the implementation of Bessel and Hankel functions are given in Section 2.3. See Refs. [67,68,75], for steps towards a 3-D generalization of the present 2-D model.
In Section 2.1 various methods for modeling of bent waveguides are briefly reviewed. Section 2.2 introduces the bend mode ansatz and outlines the analytic steps towards a solution. Remarks on bend mode normalization, on orthogonality properties of bend modes are added in Sections 2.2.1, 2.2.2. Section 2.4 summarizes the results of the analytic model for a series of bend configurations including higher order modes, and modes which are ``effectively'' guided by just one dielectric interface. Wherever benchmark results are available, the present analytical results are compared with them. In Section 2.5 a perturbational analysis of the effect of changes in the core refractive index on the propagation constants is carried out.