Optical Ring Resonator Notch Filter 3D

The simplest optical ring resonator consists of a straight waveguide and a ring waveguide. The two waveguide cores are placed close to each other, so light couples from one waveguide to the other.

When the length of the ring waveguide is an integer number of wavelengths, the ring waveguide is resonant to the wavelength and the light power stored in the ring builds up.

The wave transmitted through the straight waveguide is the interference of the incident wave and the wave that couples over from the ring to the straight waveguide.

Schematically, you can think of the ring resonator as shown in Figure 1. A part of the incident wave Ei1 is transmitted in the straight waveguide, whereas a fraction of that field couples over to the ring. Similarly, some of the light in the ring couples over to the straight waveguide, whereas the rest of that wave continues around the ring waveguide.

Figure 1: Schematic of an optical ring resonator, showing the incident fields Ei1 and Ei2 and the transmitted/coupled fields Et1 and Et2. The transmission and coupling coefficients t and κ are also indicated, as well as the round-trip loss L.

The transmitted fields are related to the incident fields through the matrix-vector relation

(1)

The matrix elements defined above assure that the total input power equals the total output power,

(2)

by assuming that the transmission and coupling coefficients are related by

(3)

Furthermore, as the wave propagates around the ring waveguide, one gets the relation

(4)

where L is the loss coefficient for the propagation around the ring and

is the accumulated phase.

Combining Equation 1, Equation 3, and Equation 4, the transmitted field can be written

(5)

Here the transmission coefficient is separated into the transmission loss |t| and the corresponding phase

(6)

Notice that on resonance, when

is an integer multiple of 2π, and when |t| = L, the transmitted field is zero. The condition |t| = L is called critical coupling. Thus, when the coupler transmission loss balances the loss for the wave propagating around the ring waveguide, one gets the optimum condition for a bandstop filter, a notch filter.

The procedure to optimize the filter is as follows:

Calculate the transmittance |t|2 for some values of the distance between the straight and the ring waveguide. Here, your should just include half (or a part) of the ring.

Calculate the loss coefficient L for some values of the ring radius. In this case, define a geometry with a short piece of straight waveguide, followed by half of the ring, and, finally, another short piece of straight waveguide. The short pieces of straight waveguide help to launch and properly absorb the propagating wave.

Find the geometry parameters where the transmittance and the loss coefficient are equal, |t| = L.

Make a wavelength sweep over a couple of free spectral ranges to find the resonances.

If the exact resonance wavelength is important, fine tune the ring radius to obtain the target resonance wavelength.

However, before starting a full 3D design, it is often good to begin with a 2D model, as described in the Optical Ring Resonator Notch Filter model.

Model Definition

This application is set up using the Electromagnetic Waves, Beam Envelopes interface, to handle the propagation over distances that are many wavelengths long. Since the wave propagates in essentially one direction along the straight waveguide and along the waveguide ring, the unidirectional formulation is used. This assumes that the electric field for the wave can be written as

(7)

where E1 is a slowly varying field envelope function and

is an approximation of the propagation phase for the wave. The definitions used for the phase in the straight and ring waveguide are shown in Table 1, Table 2, and Table 3.

Table 1: phase definition in straight waveguide domains.
Name	expression	unit	description
phi	ewbe.beta_1*y	rad	Phase

Table 2: phase definition in ring waveguide - LEFT domain.
Name	expression	unit	description
phi	ewbe.beta_1r0atan2(y,-(x-r0-dx))	rad	Phase

Table 3: phase definition in ring waveguide - RIGHT domain.
Name	expression	unit	description
phi	ewbe.beta_1r0atan2(-y,(x-r0-dx))	rad	Phase

The parameters r0 and dx correspond, respectively, to the curvature radius of the ring waveguide and to the separation between the straight and ring waveguide cores. The phase approximation defined in the tables above is discontinuous at the boundary between the straight waveguide and the ring waveguide as well as at the boundary between the left and the right ring waveguide domains. To handle this phase discontinuity and thereby the discontinuity in the field envelope, E1, a Field Continuity boundary condition is used at the aforementioned boundaries. The Field Continuity boundary condition ensures that the tangential components of the electric and the magnetic fields are continuous at the boundary, despite the phase jump.

In this model, not only the guided wave needs to be resolved. There is also coupling to radiating modes that needs to be resolved. Thus, the mesh needs resolve the beating between these different waves. Instead of making a very fine mesh, cubic shape orders are used when solving for the electric field. However, when running this model on a Windows PC, approximately 24 GB of RAM is required.

Results and Discussion

Figure 2 shows the mode field at the launch port. As the height of the waveguide core is slightly larger than the width of the core, the lowest order mode is polarized in the z direction.

Figure 2: The mode field norm and polarization at the launch port.

Figure 3 shows that the transmittance at the resonance wavelength, 1.56 mm, is very small (below 5%), as the device was designed to approximately match the transmittance through the coupler with the loss coefficient in the ring (see the discussion in the Introduction).

Figure 3: The transmittance and loss spectra.

Figure 4 shows the z-component of the electric field at the resonance wavelength. Notice that the field in the straight waveguide is very weak after the coupler region, due to the destructive interference between the light passing straight through the coupler region and the light coupled back in from the ring. Furthermore, it is clear that there is a noticeable loss when the wave propagates around the ring.

Figure 4: The z-component of the electric field at the resonance wavelength 1.56 μm.

Notes About the COMSOL Implementation

This model geometry is easily set up by importing a geometry part from the COMSOL Part Libraries. The rectangular waveguide coupling between a straight and a ring waveguide section, with the core embedded in a cladding domain, is available in the Wave Optics Module Part Library under Rectangular Waveguides.

Predefined geometry parts can be quickly modified by changing the default input parameters. Moreover, geometry parts provide targeted selections of domains and boundaries that greatly simplify the model building. As demonstrated in this model, these built-in selections are useful when adding materials, physics features and mesh sequences.

Wave_Optics_Module/Waveguides_and_Couplers/optical_ring_resonator_3d

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.

Click

Browse to the model’s Application Libraries folder and double-click the file optical_ring_resonator_3d_parameters.txt.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Part Libraries

In the toolbar, click

and choose .

In the window, select in the tree.

Click

Geometry 1

Rectangular Waveguide Straight-to-Ring Coupler 1 (pi1)

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
core_width	w_core	0.31 µm	Core width
core_height	h_core	0.372 µm	Core height
cladding_width	w_clad	3.1 µm	Cladding width
cladding_height	w_clad	3.1 µm	Cladding height
element_length	2*r0+w_clad	22.32 µm	Element length
coupler_core_separation	dx	0.713 µm	Core separation in coupler region
ring_radius	r0	9.61 µm	Ring radius

Locate the section. Find the subsection. In the text field, type -r0-w_clad/2.

Click to expand the section. Click the

button in the toolbar, to make it easier to see the selections.

In the table, enter the following settings:


Name	Keep	Physics	Contribute to
All		√	None
Substrate		√	None
Superstrate		√	None
Core	√	√	None
Embedding		√	None
Cladding	√	√	None
Straight domain	√	√	None
Ring domain 1	√	√	None
Mesh source domain	√	√	None
Ring domain 2	√	√	None
Mesh destination domain	√	√	None

Click to expand the section. In the table, enter the following settings:


Name	Keep	Physics	Contribute to
Exterior		√	None
Port 1	√	√	None
Port 1 core		√	None
Port 1 substrate		√	None
Port 1 embedding		√	None
Port 1 superstrate		√	None
Port 1 cladding		√	None
Port 2	√	√	None
Port 2 core		√	None
Port 2 substrate		√	None
Port 2 embedding		√	None
Port 2 superstrate		√	None
Port 2 cladding		√	None
Transverse perimeter	√	√	None
Triangular mesh	√	√	None
Field continuity	√	√	None

Click

Definitions

First add a few selections that will be useful when defining the mesh.

Core Boundaries

In the toolbar, click

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

Locate the section. Under , click

In the dialog box, select in the list.

Click .

In the window for , locate the section.

Select the check box.

Triangular Mesh Core Boundaries

In the toolbar, click

In the window for , type Triangular Mesh Core Boundaries in the text field.

Locate the section. From the list, choose .

Locate the section. Under , click

In the dialog box, in the list, choose and .

Click .

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_clad	1	Refractive index

Core

Right-click and choose .

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

Locate the section. From the list, choose .

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

Definitions

Before setting up the physics, first add the definition of the phase variable that will be used by the Electromagnetic Waves, Beam Envelopes interface.

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

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


Name	Expression	Unit	Description
phi	ewbe.beta_1*y

Here, ewbe.beta_1 is the propagation constant for the first port. This port will be defined when the physics is set up in the next steps. As the variable not yet exists, COMSOL warns about this condition by displaying the expression in orange.

Variables 2

Right-click and choose .