Optical Ring Resonator Notch Filter

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 below. 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.

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.

Figure 2: Plot of the predefined phase approximation. Notice the phase jump at the boundary between the left and right part of the ring waveguide. The discontinuity at the boundary between the straight and the ring waveguide is not visible at this scale.

Results and Discussion

Figure 3 below shows the transmittance spectrum for the optical ring resonator.

Figure 3: Transmittance spectrum for the optical ring resonator.

and Figure 4 shows a field plot for a resonant wavelength. Notice that the field in the straight waveguide and the field incoming from the ring are out of phase when they interfere in the coupler. Thereby the outgoing field in the straight waveguide is almost zero.

Figure 4: The out-of-plane component of the electric field for the resonant wavelength.

Notes About the COMSOL Implementation

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

Modeling Instructions

First add the physics interface and the study sequence.

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

The geometry for the optical ring resonator is straightforward to set up. Load the Slab Waveguide Straight-to-Ring Coupler geometry part from the COMSOL Part Libraries and then modify the input parameters in order to build the desired geometry.

Part Libraries

In the toolbar, click

and choose .

In the window, select in the tree.

Click

Geometry 1

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

In the window, under click .

In the window for , click

Global Definitions

Start by loading a few more parameters required for building the physics and defining the materials.

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_parameters.txt.

Geometry 1

Slab 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	2E-7 m	Core width
cladding_width	w_clad	2E-6 m	Cladding width
element_length	2*r0+w_clad	1.44E-5 m	Element length
coupler_core_separation	dx	7.1666E-7 m	Core separation in coupler region
ring_radius	r0	6.2E-6 m	Ring radius

and leave the rest of the input parameters unchanged.

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

Click

Click the

button in the toolbar.

Choose to keep those domain and boundary selections that will be useful later when adding materials, boundary conditions and the mesh sequence.

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


Name	Keep	Physics	Contribute to
All		√	None
Core	√	√	None
Cladding	√	√	None
Ring domain 1	√	√	None
Ring domain 2	√	√	None
Straight 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 cladding		√	None
Port 2	√	√	None
Port 2 core		√	None
Port 2 cladding		√	None
Transverse perimeter	√	√	None
Edge mesh	√	√	None
Field continuity	√	√	None