Dielectric Slab Waveguide

A planar dielectric slab waveguide demonstrates the principles behind any kind of dielectric waveguide such as a ridge waveguide or a step index fiber, and has a known analytic solution. This model solves for the effective index of a dielectric slab waveguide as well as for the fields, and compares to analytic results.

Figure 1: The guided modes in a dielectric slab waveguide have a known analytic solution.

Model Definition

A dielectric slab of thickness hslab = 1 μm and refractive index ncore = 1.5 forms the core of the waveguide, and sits in free space with ncladding = 1. Light polarized out of the plane of propagation, of wavelength λ = 1550 nm, is perfectly guided along the axis of the waveguide structure, as shown in Figure 1. Here, only the TE0 mode can propagate. The structure varies only in the y direction, and it is infinite and invariant in the other two directions.

The analytic solution is found by assuming that the electric field along the direction of propagation varies as Ez = E(y)exp(-ikxx), where E(y) = C1cos(kyy) inside the dielectric slab, and E(y) = C0exp(−α(|y| − (hslab/2)) in the cladding. Because the electric and magnetic fields must be continuous at the interface, the guidance condition is

where ky and α satisfy

with kcore = 2πncore/λ and kcladding = 2πncladding/λ. It is possible to find the solution to the above two equations via the Newton–Raphson method, which is used whenever COMSOL Multiphysics detects a system of nonlinear equations, the only requirement being that of an adequate initial guess.

This model considers a section of a dielectric slab waveguide that is finite in the x and y directions. Because the fields drop off exponentially outside the waveguide, the fields can be assumed to be zero at some distance away. This is convenient as it makes the boundary conditions in the y direction irrelevant, assuming that they are imposed sufficiently far away.

Use Numeric Port boundary conditions in the x direction to model the guided wave propagating in the positive x direction. These boundary conditions require first solving an eigenvalue problem that solves for the fields and propagation constants at the boundaries.

Results and Discussion

Figure 2 shows the results. The numeric port boundary condition at the left side excites a mode that propagates in the x direction and is perfectly absorbed by the numeric port on the right side. The analytic and numerically computed propagation constants agree.

Figure 2: The electric field in a dielectric slab waveguide.

Wave_Optics_Module/Verification_Examples/dielectric_slab_waveguide

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
lda0	1550[nm]	1.55E-6 m	Wavelength
n_core	1.5	1.5	Refractive index, core
n_cladding	1	1	Refractive index, cladding
h_core	1[um]	1E-6 m	Thickness, core
h_cladding	7[um]	7E-6 m	Thickness, cladding
w_slab	5[um]	5E-6 m	Slab width
k_core	2pi[rad]n_core/lda0	6.0805E6 rad/m	Wave number, core
k_cladding	2pi[rad]n_cladding/lda0	4.0537E6 rad/m	Wave number, cladding
f0	c_const/lda0	1.9341E14 1/s	Frequency

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type w_slab.

In the text field, type h_core.

Locate the section. From the list, choose .

Click

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type w_slab.

In the text field, type h_cladding.

Locate the section. From the list, choose .

Click

Click the

button in the toolbar.

Electromagnetic Waves, Frequency Domain (ewfd)

Only solve for the out-of-plane electric field component, since we are only interested in solving for a transverse electric (TE) mode.

In the window, under click .

In the window for , locate the section.

From the list, choose .

The wave is excited at the port on the left side.

Port 1

In the toolbar, click

and choose .

Select Boundaries 1, 3, and 5 only.

In the window for , locate the section.

From the list, choose .

For the first port, wave excitation is by default.

Now, add the exit port.

Port 2

In the toolbar, click

and choose .

Select Boundaries 8–10 only (the boundaries on the right side).

In the window for , locate the section.

From the list, choose .

Materials

Cladding

In the window, under right-click and choose .

In the window for , type Cladding in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n_cladding	1	Refractive index

By default, the first material you add applies on all domains. Add a core material.

Core

Right-click and choose .

In the window for , type Core in the text field.

Select Domain 2 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n_core	1	Refractive index

Study 1

Boundary Mode Analysis

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type n_core. This value should be in the vicinity of the value that you expect the fundamental mode to have.

In the text field, type f0.

Boundary Mode Analysis 2

Add another boundary mode analysis, for the second port.

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type n_core.

In the text field, type 2.

In the text field, type f0.

Frequency Domain

Finally, add the study step for the propagating wave in the waveguide.

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type f0.

In the toolbar, click

Results

Electric Field (ewfd)

The default plot shows the norm of the electric field. Modify the plot to shows the z component (compare with Figure 2).

Surface 1

In the window, expand the node, then click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Click

In the dialog box, select in the tree.

Click .

In the toolbar, click

Now inspect the mode field plot and the effective mode index resulting from the boundary mode analysis performed for each port.

Electric Mode Field, Port 1 (ewfd)

Electric Mode Field, Port 2 (ewfd)

Finish by comparing the simulation results to the analytic solution. To compute the latter, add a Global ODEs and DAEs interface and then set up and solve the relevant equations.

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

In the toolbar, click

to close the window.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the table, clear the check box for .

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Global ODEs and DAEs (ge)

Global Equations 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	f(u,ut,utt,t) (1)	Initial value (u_0) (1)	Initial value (u_t0) (1/s)	Description
alpha	alpha-k_ytan(k_yh_core/2)	k_core/2	0
k_y	k_y^2-(k_core^2-k_cladding^2-alpha^2)	k_core/2	0