Channel Waveguides
This is a continuation from the previous tutorial - graded-index planar waveguides.
So far we have discussed the characteristics of planar waveguides. In practice, most waveguides used in device applications are nonplanar waveguides. For a nonplanar waveguide, the index profile n(x, y) is a function of both transverse coordinates x and y. There are many different types of nonplanar waveguides that are differentiated by the distinctive features of their index profiles. One very unique group is the circular optical fibers discussed in the next tutorial. Another important group of nonplanar waveguides is the channel waveguides, which include the buried channel waveguides, the strip-loaded waveguides, the ridge waveguides, the rib waveguides, and the diffused waveguides, shown in figure 14 below.
A buried channel waveguide is formed with a high-index waveguiding core buried in a low-index surrounding medium. The waveguiding core can have any cross-sectional geometry though it is often intended to have a rectangular shape, as shown in figure 14(a).
A strip-loaded waveguide is formed by loading a planar waveguide, which already provides optical confinement in the x direction, with a dielectric strip of index n3 < n1 or a metal strip to facilitate optical confinement in the y direction, as shown in figure 14(b). The waveguiding core of a strip waveguide is the n1 region under the loading strip, with its thickness d determined by the thickness of the n1 layer and its width w defined by the width of the loading strip.
A ridge waveguide, shown in figure 14(c), has a structure that looks like a strip waveguide, but the strip, or the ridge, on top of its planar structure has a high index and is actually the waveguiding core. A ridge waveguide has strong optical confinement because it is surrounded on three sides by low-index air.
A rib waveguide, shown in figure 14(d), has a structure similar to that of a strip or ridge waveguide, but the strip has the same index as the high-index planar layer beneath it and is part of the waveguiding core.
These four types of waveguides are usually classified as rectangular waveguides with a thickness of d in the x direction and a width w in the y direction, though their shapes are normally not exactly rectangular.
A diffused waveguide, shown in figure 14(e), is formed by creating a high-index region in a substrate through diffusion of dopants, such as a LiNbO3 waveguide with a core formed by Ti diffusion. Because of the diffusion process, the core boundaries in the substrate are not sharply defined. However, a diffused waveguide also has thickness d defined by the diffusion depth of the dopant in the x direction an a width w defined by the distribution of the dopant in the y direction.
One distinctive property of nonplanar dielectric waveguides versus planar waveguide is that a nonplanar waveguide supports hybrid modes in addition to TE and TM modes, whereas a planar waveguide supports only TE and TM modes.
Except for those few exhibiting special geometric structures, such as circular optical fibers, nonplanar dielectric waveguide generally do not have analytical solutions for their guided mode characteristics. Numerical methods, such as the beam propagation method, exist for analyzing such waveguides. Here we are interested in obtaining approximate solutions that give the mode characteristics without full-blown numerical analysis. One of the methods for this purpose is the effective index method discussed below.
Effective Index Method
The basic concept of the effective index method, illustrated in figure 15 below, is to convert the problem of a channel waveguide into that of two planar waveguides. The effective index method is a good approximation if the waveguide satisfies the following two conditions:
(1) the waveguide width is larger than its thickness, w > d
(2) waveguiding in the y direction across its width is not stronger than that in the x direction across its thickness
Many useful waveguides satisfy these conditions. The effective index method applies to both step-index and graded-index channel waveguides, including all of those shown in figure 14, as long as these two conditions are satisfied.
When these two conditions are satisfied, the characteristics of the guided modes are primarily determined by the layered structure perpendicular to the x direction, much like a planar waveguide of thickness d, but are modified by a lateral structure of width w. The planar structure defines TE and TM polarizations, but the lateral structure distorts them. Therefore, a mode with its electric field mostly in the y direction parallel to the planar layers is called a TE-like mode, and one with its magnetic field mostly in this direction is called a TM-like mode.
The procedure of applying the effective index method is straightforward. Because an effective index is mode dependent, we first decide on the specific mode, either TEmn or TMmn, with specific mode indices m and n, to be analyzed. The waveguide is then divided into three structures for the three vertical regions, I, II, and III, shown in figure 15. The structure associated with each region is then treated as a planar waveguide to find the propagation constant βm for the mode m. The x dependence of the y component of the mode field, \(\mathcal{E}_{m, y}(x)\) in the case of a TE-like mode or \(\mathcal{H}_{m, y}(x)\) in the case of a TM-like mode, for central waveguide region I is also found through the same procedure. The propagation constants for the three regions are used to determine the effective indices, \(n_I=c\beta_m^I/\omega=\lambda\beta_m^I/2\pi,\,n_{II}=c\beta_m^{II}/\omega=\lambda\beta_m^{II}/2\pi,\,\text{and }n_{III}=c\beta_m^{III}/\omega=\lambda\beta_m^{III}/2\pi,\) for a vertical planar waveguide of core width w. This structure is then treated as a planar slab waveguide to solve for the propagation constant βmn of the desired mode and for the y dependence of the y component of the mode field, \(\mathcal{E}_{n, y}(y)\) in the case of a TE-like mode or \(\mathcal{H}_{n, y}(y)\) in the case of a TM-like mode.
Note that \(\mathcal{E}_{n, y}(y)\) for a TE-like mode of the original channel waveguide is obtained from the \(\mathcal{E}_y\) component of the TMn field of the effective vertical planar waveguide, whereas \(\mathcal{H}_{n, y}(y)\) for a TM-like mode of the original waveguide is obtained from the \(\mathcal{H}_y\) component of the TEn field of the effective vertical planar waveguide. Finally, the y component of the total mode field for the original channel waveguide is \(\mathcal{E}_{mn, y}(x, y)=\mathcal{E}_{m, y}(x)\mathcal{E}_{n, y}(y),\) in the case when the TEmn mode is considered. Other significant field components are found by using (32) and (33) for a TE-like mode and by using (35) and (36) for a TM-like mode [refer to the wave equations for optical waveguides tutorial]. The propagation constant is simply βmn found from the effective vertical planar waveguide.
Example
A strip-loaded waveguide with a silica strip on top of the polymer layer, as shown in figure 16 below. The silica loading strip has a width of w = 5 μm and a thickness of t = 2 μm. We are interested in the TM-like modes that have fundamental-mode characteristics in the x direction. How many such modes exist at λ = 1 μm? Find their characteristics using the effective index method.
To apply the effective index method to this problem, the waveguide is divided into three regions as shown in figure 16. The structure in region I can be treated as a symmetric waveguide if t is sufficiently large so that the evanescent wave in the x direction does not reach the air above the strip.
We find that γ2 = 5.84 μm-1 for the TM0 mode of interest here. Because exp(-γ2t) ≈ 8 x 10-6 for t = 2 μm, we can safely say that the evanescent field is completely confined within the strip. Therefore, the structure in region I is simply the symmetric waveguide with \(\beta_0^I=10.8208\,\mu{m}^{-1}\) for the TM0 mode. The structures in regions II and III are just the asymmetric waveguide solved in the first example in the step-index planar waveguides tutorial, with \(\beta_0^{II}=\beta_0^{III}=10.7800\,\mu{m}^{-1}\) for the TM0 mode. The vertical planar waveguide is thus a symmetric waveguide that has a width of w = 5 μm and effective indices of nI = 1.7222 and nII = nIII = 1.7157. The V number of this effective planar waveguide at λ = 1 μm is
\[V = \frac{2\pi}{\lambda}w\sqrt{n_I^2-n_{II}^2}=4.696\]
It has two TE modes and two TM modes because 2π > V > π.
Because the TM-like modes of the strip waveguide are polarized in the x direction, we have to consider the TE modes of the effective vertical planar waveguides. Because there are two such TE modes, we have two TM-like modes, TM00 and TM01, which are associated with the TM0 mode of the horizontal planar waveguide and the TE0 and TE1 modes, respectively, of the effective vertical planar waveguide. The characteristics of these TE0 and TE1 modes can be solved in the same manner as that describe in the example in the symmetric slab waveguides tutorial, because the effective vertical waveguide is symmetric. The results are summarized below.
In this table, wn is the effective width of the TM0n mode in the y direction similar to the effective thickness dM in the x direction for the TM0 mode.
Except for those in the last two columns, the parameters listed above are for the y dependence of the modes. Both mode fields have the same x dependence described by \(\mathcal{H}_{0, y}(x)\), which has the characteristics of the TM0 mode listed in the example in the symmetric slab waveguides tutorial. The y dependence is found by using the parameters listed above to obtain \(\mathcal{E}_{0, x}(y)\) and \(\mathcal{E}_{1, x}(y)\) of the TE0 and TE1 modes of the effective planar waveguide, respectively. Using \(\mathcal{H}_y=-\beta\mathcal{E}_x/\omega\mu_0\) after exchanging the x and y coordinates for (32) [refer to the wave equations for optical waveguides tutorial], we then find \(\mathcal{H}_{0,y}(y)\) and \(\mathcal{H}_{1,y}(y)\). The y component of the total mode field for the TM00 mode is then \(\mathcal{H}_{00,y}(x, y)=\mathcal{H}_{0,y}(x)\mathcal{H}_{0,y}(y)\) and that for the TM01 mode is \(\mathcal{H}_{01,y}(x, y)=\mathcal{H}_{0,y}(x)\mathcal{H}_{1,y}(y)\). The propagation constants are simply those found from solving for the effective vertical waveguide: β00 = β0 = 10.8120 μm-1 and β01 = β1 = 1.7892 μm-1. The effective mode confinement factor Γmn, defined as its fractional power in the d x w two-dimensional guiding core, can be found by multiplying its two confinement factors in the x and y dimensions. Thus, we have Γ00 = 0.967 x 0.930 = 0.899 for the TM00 mode and Γ01 = 0.967 x 0.634 = 0.613 for the TM01 mode.
The next part continues with the Step-Index Fibers tutorial.