Optical Waveguides: Basic Geometries-2

Optical Waveguides: Basic Geometries-2

As the two former examples have shown, a first waveguide classification can be

made by looking at the number of dimensions in which the light is confined (Table 3.1).

Figure 3.3 shows the three basic types of waveguides depending on their number of

dimensions for light confinement: while a planar waveguide (or 1D waveguide) confines

the radiation in one dimension (Figure 3.3a), channel waveguides (or 2D waveguides)

confine the light in two dimensions (Figure 3.3b).

There also exist structures that confine light in the three dimensions. These constitute

a very special case of optical waveguides: since the radiation is confined in

all directions, it cannot propagate. Therefore, these structures in fact form light traps,

and are often called photonic crystals (Figure 3.3c). The light confinement in this case

obviously cannot be based on total internal reflection; instead, photonic crystals are

fabricated by means of tridimensional periodical structures, in which the light confinement

is based on Bragg reflection. Photonic crystals have very interesting properties,

and their use in several devices or applications has been proposed, such as miniaturised

lasers with virtually no threshold power, waveguide bends with very small curvature

radii and dimensions, or narrow-band filters [2].

Up to now, we have not imposed any restriction on the size of the guiding structures.

In fact, the LSC and the light tubes can be called macroscopic waveguides. If we

start from a planar waveguide, and progressively reduce the thickness of the guiding

film, when sizes of the order of the wavelength of the radiation are reached, a new

phenomenon occurs: we found that due to the interference produced by the reflected

wave coming from total internal reflection at the upper boundary and the reflected

wave from the lower interface, now the light propagation is only allowed for a discrete

set of angles. For each permitted angle of propagation, the transversal structure of

the electromagnetic field associated with the radiation is maintained as the light beam

propagates along the film; these characteristics form a propagation mode. Thus, a

propagation mode is the result of combining total internal reflection and constructive

interference.

The description that we have given to illustrate the concept of propagation modes

was based on considering the light as plane waves, or in other words, that the direction

of the light propagation within the waveguide can be described by using rays. In fact,

although the ray optics treatment can give some interesting results, for a complete

understanding and description of light propagation within guiding structures, it is necessary

to consider light as electromagnetic waves, and use the formalism developed in

the previous chapter.

Now we will describe the typical geometries found in optical waveguides, including

planar waveguides, channel waveguides and optical fibres, and will discuss some of

their basic characteristics.

Planar waveguides Planar waveguides are optical structures than confine optical

radiation in a single dimension. Considering the refractive index distribution in the planar

structure, planar waveguides can be classified as step-index waveguides or graded

index waveguides.

The step-index planar waveguide is the simplest structure for light confinement, and

is formed by a uniform planar film with a constant refractive index (homogeneous film,

nf = constant), surrounded by two dielectric media of lower refractive indices [3].

The homogeneous upper medium, or cover, has a refractive index of nc, and the lower

medium, with refractive index ns , is often called substrate. Usually, it is assumed

that the refractive index of the cover is less than or equal to the refractive index

of the substrate, nc ≤ ns , and in this way we have nf < ns ≤ nc. In fact, in many

cases the cover medium is air, and therefore nc = 1, which fulfils the assumption

previously mentioned.

If the upper and the lower media are the same (equal optical constants), the structure

forms a symmetric planar waveguide. On the other hand, in integrated photonics

the upper and lower media are different, and in this case we are dealing with an

asymmetric planar waveguide (Figure 3.4). Asymmetric step-index planar waveguides

are fabricated by depositing a high-index film on top of a lower index substrate, by

means of physical methods (thermal evaporation, molecular beam epitaxy, sputtering,

etc.) or chemical methods (chemical vapour deposition, metal-organic chemical vapour

deposition, etc.).

If the high index film is not homogeneous, but its refractive index is depth dependent

(along the x-axis in Figure 3.5), the structure is called a graded index planar

waveguide [4]. Usually the refractive index is maximum at the top surface, and its

value decreases with depth until it reaches the value corresponding to the refractive

index of the substrate (Figure 3.5). This kind of structure is present in waveguide fabrication

methods based on the surface modification of a substrate, whether by physical

processes (ion implantation, metal diffusion, etc.), or by chemical modification of the

substrate (ionic exchange methods).

Channel waveguides In planar waveguides the light confinement is restricted to a

single dimension (the x-axis in Figures 3.4 and 3.5), and if the light propagates along

a given direction (z-axis), the light can spread out in a perpendicular direction (y-axis)

due to diffraction. When we want to avoid this effect and keep the light beam well

confined, it is necessary for total internal reflection to take place not only at the upper

and lower interfaces, but also at the lateral boundaries. This confinement is attained

in channel waveguides, or 2D waveguides, in which the core region (where the radiation

is concentrated) has a refractive index greater than any of the surrounding media

(Figure 3.3b). The classification made for planar waveguides, in terms of symmetric/

asymmetric or step index/graded index, is also valid for channel waveguides, but

with the difference that we are now dealing with the extra dimension which characterises

the waveguide structure.

Although many types of channel waveguides have been proposed, three are the

most common basic structures used. The easiest way to build a channel waveguide is

to deposit a stripe made of a high refractive index material on top of a lower refractive

index substrate. This kind of channel waveguide is called stripe waveguide, and

can be made by either depositing the stripe directly onto the substrate, or simply by

conveniently etching a previously deposited film (Figure 3.6a) [5]. If the etch process

is not complete and does not reach the substrate, a channel waveguide is also

produced, providing that the thickness and height of the structure are conveniently

tailored; this waveguide geometry is called rib waveguide (Figure 3.6b). Another type

of channel waveguide common in integrated photonic is the buried channel waveguide

Optical fibres A special type of channel waveguides, from the point of view of their

geometry and manufacturing methods as well as their applications, is called optical

fibres (Figure 3.7) [7]. Optical fibres have cylindrical geometry, and are constituted

by a cylindrical core of radius a and refractive index n1, surrounded by a cladding

of slightly lower refractive index n2. As we did with planar waveguides, we can

classify the optical fibres as step-index fibres, with homogeneous core (n1 = constant,

Figure 3.8a), or graded index fibres, in which the refractive index of the core varies

as a function of the radial distance (n1 = n1(r), Figure 3.8b). This last type of optical

fibres is the best choice when high transmission bandwidth is required in long-distance

optical communications [7].