We will consider how INTERCONNECT can be used to simulate parallel and serialcoupled 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 seriescoupled, others to parallelcoupled configurations. The following figure depicts a circuit configuration of an N stage parallelcoupled 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) parallelcoupled ring resonator optical filter is:
The following figure depicts a circuit configuration of serialcoupled 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:
$$
g_{q}=\sqrt{2} \sin \left(\frac{2 q1}{2 N} \pi\right)
$$
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 quarterwavelength over the operating frequency band of the filter, or more generally, 2L_{c} should be a small odd multiple of a halfwavelength [2]:
$$
2 \mathrm{L}_{\mathrm{c}}=\left(n+\frac{1}{2}\right) \frac{\lambda_{0}}{n_{g}}
$$
Where λ_{0} 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 quarterwavelength closer to (L_{r}/100):
The periodicity of the frequency response for a fourthorder 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 fourthorder 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/4}the 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 nonnegligible. 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 parallelcoupled as well as seriescoupled 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 m_{1} and m_{2} are coprime integers and subscript index 1 and 2 are defined by ring lengths L_{1} and L_{2} 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 m_{2}m_{1}=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 unoptimized second order seriescoupled ring resonator filter, illustrating the twin peaks and extended FSR. The twin peaks occur because m_{1} and m_{2} were chosen to be 9 and 7, respectively by [3]:
By choosing m_{2} to be equal to m_{1}1 we can remove the occurrences of the twin peaks. The figure bellow depicts the frequency response after optimizing the device, and setting m_{1} and m_{2} 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.
Parallelcoupled ringresonator filter [ N=2, FSR/B = 10, L_{c} = (L_{r}/100) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N2.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 10e9;#bandwidth N = 2; #order of the filter nr = 100;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=3, FSR/B = 10, L_{c} = (L_{r}/100) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N3.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 10e9;#bandwidth N = 3; #order of the filter nr = 100;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=4, FSR/B = 10, L_{c} = (L_{r}/100) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N4.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 10e9;#bandwidth N = 4; #order of the filter nr = 100;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=5, FSR/B = 10, L_{c} = (L_{r}/100) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N5.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 10e9;#bandwidth N = 5; #order of the filter nr = 100;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=4, FSR/B = 5, L_{c} = (L_{r}/100) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N4.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 20e9;#bandwidth N = 4; #order of the filter nr = 100;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 2e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=3, FSR/B = 10, L_{c} = (L_{r}/π) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N3.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 10e9;#bandwidth N = 3; #order of the filter nr = pi;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=3, FSR/B = 10, L_{c} = (L_{r}/10) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N3.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 10e9;#bandwidth N = 3; #order of the filter nr = 10;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=4, FSR/B = 30, L_{c} = (L_{r}/2) _{λ/4} ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N4.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 3.333e9;#bandwidth N = 4; #order of the filter nr = 2;#integer multiple of FSR (second periodicity) oddm = 1;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Parallelcoupled ringresonator filter [ N=4, FSR/B = 30, L_{c} = L_{r}/2 ] 


Project file 
optics_letters_vol26_n_12_jun2001_917_fig2_N4.icp 
Script file 
optics_letters_vol26_n_12_jun2001_917.lsf 
Circuit parameters (script) 
#filter parameters FSR = 100e9;#free spectral range B = 3.333e9;#bandwidth N = 4; #order of the filter nr = 2;#integer multiple of FSR (second periodicity) oddm = 0;#odd multiple of a quarterwavelength (1=true, 0=false) range = 0.5e12;#frequency range analysis 
Schematic diagram 

Serialcoupled ringresonator filter [ N=2 ] 


Project file 
optics_express_v_18_i_24_pp2151_2010.icp 
Script files 
optics_express_v_18_i_24_pp2151_2010_Fig1_2.lsf optics_express_v_18_i_24_pp2151_2010_circuit.lsf 
Circuit parameters (script) 
ng = 4.306; neff = ng; loss = 3 / 1e2; # 3 dB/cm #figure 1 k1 = 0.35;#power coupling coeff 1 k2 = 0.1;#power coupling coeff 2 k3 = 0.35;#power coupling coeff 3 l1 = 127.91e6; #ring 1 length l2 = 99.487e6;#ring 2 length #figure 2 k1 = 0.015;#power coupling coeff 1 k2 = 0.00005;#power coupling coeff 2 k3 = 0.015;#power coupling coeff 3 l1 = 42.637e6; #ring 1 length l2 = 28.425e6;#ring 2 length 
Schematic diagram 

Related publications
[1] Andrea Melloni, "Synthesis of a parallelcoupled ringresonator filter," Opt. Lett. 26, 917919 (2001)
[2] O. Schwelb and I. Frigyes, “A design for a high finesse parallelcoupled microring resonator filter”, Microwave Opt Technol Lett. 38, 125–129 (2003)
[3] Robi Boeck, Nicolas A. Jaeger, Nicolas Rouger, and Lukas Chrostowski, "Seriescoupled silicon racetrack resonators and the Vernier effect: theory and measurement," Opt. Express 18, 2515125157 (2010)