In this example, we will demonstrate how INTERCONNECT can be used to simulate finiteimpulse response (FIR) and infiniteimpulse response (IIR) optical filters.
Problem definition: More details
FIR filters consist simply of feedforward waveguides, and their impulse responses are limited in finite time. IIR filters include feedback loops such as ring waveguides, and their impulse responses continue for infinite time. The following figure depicts a circuit configuration for a FIR optical halfband filter [1] using a lattice form configuration, composed of cascaded MachZehnder interferometers (MZI’s):
Where the L is the unit path length difference, φ is a phase shift, θ is the angle representing power coupling, and N is the number of phase shifts. The INTERCONNECT schematic diagram for N=4 is depicted bellow:
The following figure depicts a circuit configuration for an IIR optical halfband filter [1] . The circuit configurations consist of an MZI whose two arms are connected with a number of ring waveguides:
Where M is the number of upper rings, and N is the number of lower rings. INTERCONNECT schematic diagram for M=1 and N=1 is depicted bellow:
Discussion and Results
The free spectral range (FSR) is the spacing between adjacent modes of the filter, and is:
$$
\Delta v=\frac{c}{n_{g} \Delta L}
$$
Where n_{g} is the group index for the waveguide and c is the speed of light in a vacuum. The straight waveguide model allows for the definition of the loss, effective index, group index and dispersion at a given center frequency f_{0}. These propagation parameters defined the transfer function of the waveguide. Without including losses or dispersion, the transfer function of the straight waveguide is:
$$
\mathrm{H}(\mathrm{f})=e^{i\left(\frac{2 \pi \mathrm{f}_{0}}{\mathrm{c}} \mathrm{n}_{\mathrm{eff}}+\frac{2 \pi}{\mathrm{c}} \mathrm{ng}\left(\mathrm{f}\mathrm{f}_{0}\right)\right) \cdot \Delta \mathrm{L}}
$$
The phase shift φ is:
$$
\varphi=\frac{2 \pi f_{0}}{c} \Delta L\left(n_{e f f}n_{g}\right)
$$
The power coupling coefficient is:
$$
c=\sin (\theta)^{2}
$$
For a FIR filter with a given FSR and group index, we can then calculate the properties for the cascaded MZI’s from the required phase shifts and coupling circuit parameters. For IIR filters, we can calculate the properties for the ring waveguides. The schematic diagram for each circuit from table I and the corresponding calculated transmission spectra is depicted bellow:
FIR maximallyflat halfband filter (N=2) 


Project file 
JLT_vol18_N_2_Feb2000_252_Fig6_N2.icp 
Script file 
JLT_vol18_N_2_Feb2000_252_TableI_N2.lsf 
Schematic:

Result:

FIR maximallyflat halfband filter (N=3) 


Project file 
JLT_vol18_N_2_Feb2000_252_Fig6_N3.icp 
Script file 
JLT_vol18_N_2_Feb2000_252_TableI_N3.lsf 
Schematic:

Result:

FIR maximallyflat halfband filter (N=4) 


Project file 

Script file 

Schematic:

Result:

FIR Chebyshev halfband filter (N=4) 


Project file 

Script file 

Schematic:

Result:

IIR elliptic halfband filter (M=1, N=1) 


Project file 

Script file 

Schematic:

Result:

Related publications
[1] K. Jinguji and M. Oguma, "Optical HalfBand Filters," J. Lightwave Technol. 18, 252 (2000)