Optical filters

INTERCONNECT Photonic Integrated Circuits - Passive

In this example, we will demonstrate how INTERCONNECT can be used to simulate finite-impulse response (FIR) and infinite-impulse response (IIR) optical filters.

Problem definition: More details

FIR filters consist simply of feed-forward 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 half-band filter [1] using a lattice form configuration, composed of cascaded Mach-Zehnder 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 half-band 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 f₀. 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 maximally-flat half-band 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: