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 a 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 continuous 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, assures that the total input power equals the total output power,

(2)

by assuming that coupler’s transmission and coupling coefficients are related by

(3)

Furthermore, as the wave propagates around the ring waveguide, you get 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 number times 2π, and when |t| = L, the transmitted field is zero. The condition that |t| = L is called critical coupling. Thus, when the coupler’s transmission loss balances the loss for the wave propagating around the ring waveguide you get the optimum condition for a bandstop filter, a notch filter.

Model Definition

This application is setup 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 and Table 2.

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 domains.
Name	expression	unit	description
phi	ewbe.beta_1r0atan2(-y,x)	rad	Phase

Notice in Figure 2 that the phase approximation defined in Table 1 and in Table 2 is discontinuous at the boundary between the straight waveguide and the ring waveguide. To handle this phase discontinuity and thereby the discontinuity in the field envelope, E1, a Field Continuity boundary condition is used at the boundary between the straight waveguide and the ring waveguide. This 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 that the phase jump at y = 0 in the cladding of the left part of the ring waveguide is neglected, as the light is mainly confined to the waveguide core.

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 is out of phase, when they interfere in the coupler. Thereby the outgoing field in the straight waveguide is almost zero.