Coupled ring resonator filters

We will consider how INTERCONNECT can be used to simulate parallel and serial-coupled ring resonator optical filters.

Problem definition: More details

Bus coupled optical ring resonators are versatile components for wavelength filtering, multiplexing, switching, and modulating in photonic integrated circuits. A number of methods [1] have been described to obtain wide passband, large free spectral range (FSR) and high finesse. Some of these methods apply to series-coupled, others to parallel-coupled configurations. The following figure depicts a circuit configuration of an N stage parallel-coupled ring resonator optical filter:

Where L_r is the circumference of the ring, L_c is the arm or connecting waveguide length, and K_q represents power coupling coefficients for ring q (1 to N). The INTERCONNECT schematic diagram for a second order (N=2) parallel-coupled ring resonator optical filter is:

The following figure depicts a circuit configuration of serial-coupled resonator filter [2] and the correspondent INTERCONNECT schematic diagram:

Discussion and results

Filter design begins with the specifications for the free spectral range (FSR)  and for the 1 dB bandwidth (B) of the filter [2]. The FSR is the spacing between adjacent modes of the ring resonator, and is:

$$
F S R=\frac{c}{n_{g} L_{r}}
$$ 

Where c is the speed of light in a vacuum and n_g is the group index of the waveguide, which is the same for the rings and for connecting waveguides. The power coupling coefficients is calculated from the Q factor of each resonator [1]:

$$
K_{q}=\frac{\pi^{2}}{2 Q_{q}^{2}}\left[\left(1+\frac{4 Q_{q}^{2}}{\pi^{2}}\right)^{1 / 2}-1\right]
$$ 

$$
Q_{q}=\frac{F S R}{B g_{q}}
$$ 

where:

$$
g_{q}=\sqrt{2} \sin \left(\frac{2 q-1}{2 N} \pi\right)
$$ 

The main filter periodicity arises from the signal circulating around the rings and is given by the FSR. A secondary periodicity is caused by the phase delay in the connecting waveguides 2L_c. By selecting 2L_c=L_r/n_r, where n_r is an integer, the secondary periodicity will be a multiple of the FSR and given by:

$$
\Delta f=\frac{c}{n_{g} 2 L_{c}}
$$ 

The length of the connecting waveguide L_c should be as nearly equal as possible to an odd multiple (n) of a quarter-wavelength over the operating frequency band of the filter, or more generally, 2L_c should be a small odd multiple of a half-wavelength [2]:

$$
2 \mathrm{L}_{\mathrm{c}}=\left(n+\frac{1}{2}\right) \frac{\lambda_{0}}{n_{g}}
$$ 

Where λ₀ if the wavelength at the design frequency of the filter. When the condition above is satisfied the contributions for all the rings add in phase at the resonant frequencies and the signal is dropped at the output port.

The figure bellow shows the frequency response of the filters for orders from 2 to 5, over two FSRs. Circuit parameters are FSR/B = 10 and L_c = (L_r/100)_λ/4, where the subscript means the odd multiple of a quarter-wavelength closer to (L_r/100):

The periodicity of the frequency response for a fourth-order filter with FSR = 100 GHz, B = 20 GHz and L_c = (L_r/100)_λ/4 is depicted bellow. The smaller the L_c, the higher the number of usable FSRs:

The next figure illustrates the influence of the length of the connecting waveguide on the frequency response for a third order filter with FSR/B = 10 and three values of L_c. When the distance L_c is increased, the transmission nulls move toward the dropped band and the rejection decreases [1].

Finally, we will consider a filter where the length of the combined connecting waveguides has the same length of the ring for a fourth-order filter with FRS/B = 30. The figure bellow shows the frequency response for L_c = L_r/2 and L_c = (L_r/2)_λ/4. For  L_c = (L_r/2)_λ/4the filter has poor rejection, if L_c = L_r/2 the rejection values are better, but the dropped band widens considerably:

The free spectral range is inversely proportional to the length of the resonator and, therefore, bending losses can become non-negligible. A viable alternative to increase the free spectral range is to use the Vernier effect [3]. The Vernier effect intends to suppress interstitial microring resonances to create an extended, virtual, FSR without decreasing the ring circumference. Its operation has been described both in connection with parallel-coupled as well as series-coupled ring resonator filters as the one illustrated in the next figure:

The extended FSR is related to the FSR of each resonator by [3]:

 $$
F S R_{e x}=m_{1} F S R_{1}=m_{2} F S R_{2}
$$

Where m₁ and m₂ are co-prime integers and subscript index 1 and 2 are defined by ring lengths L₁ and L₂ respectively. The length of each ring resonator is determined using

$$
\frac{m_{2}}{m_{1}}=\frac{L_{2}}{L_{1}}
$$ 

The extended FSR can also be defined as

$$
F S R_{e x}=\frac{\left(m_{2}-m_{1}\right) F S R_{1} F S R_{2}}{\left(F S R_{1}-F S R_{2}\right)}
$$ 

For example, if m₂-m₁=1, from (1):

$$
F S R_{e x}=\frac{c}{n_{e f f}\left(L_{2}-L_{1}\right)}
$$ 

The figure bellow shows the response of an un-optimized second order series-coupled ring resonator filter, illustrating the twin peaks and extended FSR. The twin peaks occur because m₁ and m₂ were chosen to be 9 and 7, respectively by [3]:

By choosing m₂ to be equal to m₁-1 we can remove the occurrences of the twin peaks. The figure bellow depicts the frequency response after optimizing the device, and setting  m₁ and m₂  to be 3 and 2, respectively [3]:

Simulation setup

The table bellow shows a list of projects and script files, and the correspondent circuit parameters from  [1] and [3]. For each project, a script file is used to set element properties calculated from the circuit parameters.

Related publications

[1]  Andrea Melloni, "Synthesis of a parallel-coupled ring-resonator filter," Opt. Lett. 26, 917-919 (2001)

[2] O. Schwelb and I. Frigyes, “A design for a high finesse parallel-coupled microring resonator filter”, Microwave Opt Technol Lett. 38, 125–129 (2003)

[3] Robi Boeck, Nicolas A. Jaeger, Nicolas Rouger, and Lukas Chrostowski, "Series-coupled silicon racetrack resonators and the Vernier effect: theory and measurement," Opt. Express 18, 25151-25157 (2010)

See also

Optical Filters