Regular integrated optical waveguides are fundamental components of integrated optics and waveguide optoelectronics. An example is the thin-film generalized waveguide Luneburg lens, a radially symmetrical structure that can focus light equally well from all directions (see Figure 1).^{1–5} For this reason, Luneburg lenses are of special interest for the aircraft industry, for example, as optical antennas or RF spectrum analyzers. In a regular multilayer waveguide, the boundaries between layers are smooth, and their physical structure is homogeneous (i.e., the refractive index of each layer is constant). However, the guided mode undergoes perturbations—changes in the amplitude-phase characteristics—caused by ‘smooth irregularities’ in the refractive index of the layers or their thickness. Several important problems in integrated optics require taking into account the vectorial character (related to the orientation of an electrical vector in space) of the electromagnetic fields propagating through these devices.^{1–5} For example, in realizing energy transfer through a variety of ‘conjugation’ elements (lenses, prisms, and so on), the efficiency depends on the compatibility of the electromagnetic fields at the entrance and exit of each element.

**Figure 1.** The cross section of a researched integrated optical structure. (1) The framing medium or covering layer (air) with an index of refraction n_{c}. (2) The first waveguide layer with an index of refraction n_{f}. (3) The substrate with an index of refraction n_{s}. (4) The thin-film generalized waveguide Luneburg lens (irregular four-layered part of the structure) with an index of refraction n_{l}. R: Radius of aperture of a thin-film lens. d: Thickness of a regular part of an integrated waveguide structure. h(y, z): Boundary of the waveguide layer (4) and covering layer (1). TE: Transverse electric.

Currently, all practical waveguiding structures and integrated-optical processors are statistically irregular or inhomogeneous to some extent owing both to their physical nature and to the technologies used to make them. For example, the substrate of a basic integrated-optical circuit ‘regular waveguide’ might have a rough surface because the technologies used to fabricate and polish it are imperfect. For similar reasons the thin film (waveguide layer) subsequently applied to the substrate might also be statistically heterogeneous, with an uneven external surface.

Consequently, much effort has been devoted to analyzing the propagation of planar monochromatic electromagnetic waves in regular multilayer integrated optical waveguides. A number of methods^{1–11} are available for evaluating the transformation of waveguide and quasi-transverse electric/quasi-transverse magnetic (TE/TM) modes, accompanied by an exchange of energy between the modes themselves and with the environment. However, these solutions replace real ‘inclined’ tangent boundary conditions with their projections on the horizontal plane. In other words, the obtained solution is 2D, which is not suited to investigating actual 3D waveguides (a critical requirement for on-chip optical links).

Equations that describe the vectorial character of an electromagnetic field in a smoothly irregular 3D segment of a multilayer integrated-optical multimode waveguide involve terms proportional to the gradient of dielectric permeability. These terms make the equations difficult to solve.^{4, 5} If one neglects polarizing effects, however, this reduces the problem of 3D propagation, transformation, and scattering of the wave to the solution of a known 2D wave equation.^{1–8}

**Figure 2. **Amplitude of the field of a guided TE_{5}-mode for h≈0:5. E_{y}: Amplitude of the transverse component of the electric field. h: Maximal thickness of a waveguide layer with an index of lens. arb. u.: Arbitrary units.

**Figure 3.** Distribution of a refractive index (n) for the generalized waveguide Luneburg lens with the various normalized focal lengths s. r: Lens radius.

Figure 1 shows in schematic form the propagation features of eigenmodes (characteristic vectors) in the smoothly irregular segment of an asymmetrical thin-film integrated optical waveguide. Shown are the framing medium or covering layer (air) with an index of refraction *n*_{c}, the first waveguide layer with an index of refraction *n*_{f}, the substrate with an index of refraction *n*_{s}, and the thin-film generalized waveguide Luneburg lens (irregular four-layered part of the structure) with an index of refraction *n*_{l}. *R* is the radius of aperture of a thin-film lens, *d* is the thickness of a regular part of an integrated waveguide structure, and *h*(*y, z*) is the boundary of waveguide layer and covering layer.

We previously proposed^{4,5} an asymptotic approach to solving a set of Maxwell's equations for a smoothly irregular dielectric waveguide. Here, we have preserved two terms from our prior work that circumscribe a so-called adiabatic approximation.^{4, 5,11} In the case of a statistically irregular dielectric waveguide the problem is defined as that of propagation of TE and TM eigenmodes in an asymmetrical thin-film integrated optical waveguide with stochastic (loosely defined as random) irregularities. Directed mode scattering in such a waveguide can be solved with the help of perturbation theory.^{1, 4–10}

We considered a number of structures. The parameters of our first example, a 3D smoothly irregular waveguide, are the following (for laser wavelength λ=0:9μm): *n*_{s} (silicon dioxide)=1:470, *n*_{f} (glass, Corning 7059)=1:565, *n*_{l} (tantalum pentoxide) with varying thickness *h(y; z)*=2:100, *n*_{c} (air) *n*_{c}=1:000. Figure 2 shows an example of the amplitude of the field in arbitrary units of the guided TE_{5} mode (where 5 indicates the number of the mode) for maximal thickness h(y; z)=0:5 (in the units of the given wavelength λ). Figure 3 shows the solution to the ‘synthesis’ problem^{11} of the 3D thin-film generalized waveguide Luneburg lens.

**Figure 4.** Amplitude of the field of a scattered radiation in the plane xy.

As a second waveguide structure we considered a three-layer polystyrene planar-integrated optical device where *h*=0. The parameters are the following (for laser wavelength λ=0:63μm): *n*_{s} (glass) =1:510, *n*_{f} (polystyrene) =1:590, and *n*_{c} (air)=1:000. Figure 4 shows an example of the calculated amplitude of the field of a radiated mode Es(y) near a waveguide layer (x=2μm) based on the size of a heterogeneity: 4×100×100μm. The waveguide parameters are β=1:58828 (TE_{0} mode) and d=3:98955μm.

The results of our numerical simulation show very good agreement of our least accurate solution with those of approaches using comparable regular waveguides.^{3–5, 10,12,13} Our findings also illustrate the advantages of our method: a more rigorous solution to the problem (i.e., that the multiple components of the waveguide—each with its own irregularities—make it difficult to predict the overall behavior of the device), more comprehensive consideration of its physical peculiarities, and more accurate calculations. The proposed method of perturbation theory is applicable to analysis of similar dielectric, magnetic, and metamaterial structures. In future, we plan to study the energetic contribution of radiation propagating into the surrounding space (e.g., the air) of the waveguide.

*The authors acknowledge Anton Sevastyanov, Konstantin Lovetskiy, and Aleksey Stavtsev for their productive cooperation.*

Alexandre A. Egorov

General Physics Institute of the Russian Academy of Sciences

Moscow, Russia

Alexandre Egorov is a professor and head research fellow. He has published around 190 scientific and technical papers. His main interests are integrated optics, coherent optics, laser physics, statistical optics, computer modeling, and optoelectronics.

Leonid A. Sevastyanov

Peoples' Friendship University of Russia

Moscow, Russia

Leonid Sevastyanov is a professor. He has published some 150 scientific and technical papers. His main interests are mathematical modeling, integrated optics, theoretical physics, and computer modeling.

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