What is a Band-Pass filter?

A band pass filter lets a wavelentgh band to pass while not letting wavelenghts around it. It is normally made of a cavity/spacer layer sandwiched between bragg mirrors. Bragg mirrors are stack of alternating high and low refractive index dielectric layers, that as the name suggest provide excellent reflection (and less absorbtion in comparison to metallic mirrors). Bragg mirrors have the dielectric layers at quarter wave thickness (at a specific design wavelength).

Within this excellent reflectance spectral region, by introducing a spacer in betwen two bragg mirrors, a small wavelength range can be tranmitted. This happens as within this cavity (also known here as fabry-perot cavity), resonance happens.

Bragg Mirror to a Band Pass Filter

First, a bragg mirror with alternating stacks of TiO2 and SiO2 are modelled using Transfer Matrix Method. The layers are at quarter wave thickness at the deisgn wavelength of 1550 nm. The Reflectance spectra is shown below with the effect of number of layers. The result shows that as the number of layers increase, the reflectance improves,

Let us stick with N = 8 from the above figure and now introduce a spacer material of half wave thickness. Here SiO2 is used (although there is no reason for it to be lower refractive index or either of the two use materials). To highlight the effect on reflectance due to this cavity, for the followign figure N = 4 has been used.

As is visible in the figure above, the R dips at around the deisgn wavelength of 1550 nm and the T goes up. Next figure shows the Transmissvitiy peak in detail. For number of layers in each bragg mirror being 8, the FWHM for this peak is 1.05 nm and peak transmission is 96.6%.

This response is affected by the number of layers, layers thicknessses, as well as the cavity thickness.

Effect of number of layers in the bragg mirror

Below, the figure shows what happens to the peak when the number of layers in bragg mirror are changed. As the number of layers decrease, the peak broadens. If one of the perfomance criteria for this band pass filter is peak line-width or the quality factor, then the number of layers of the bragg mirror have to high enough.

Effect of Cavity Thickness: Changin cavity thickness changes (meaning that even though cavity thickness was half wave at a deisgn wavelenght, due to some manufacturing errors that was not so) the optical phase accumulated by the waves inside the cavity, thereby changing the wavelength where the phase is correspodnign to constrcutive interference for transmission. The figure below shows that the wavelenght at which the transmission peaks changes as the cavity thickness changes.

Effect of asymetry of bragg pairs:

What happens if the number of bragg pairs on either side of the cavity are not same. Asymmetry in bragg mirrors effect the peak maximum transmissivity as well as the line widht of the peak. The followign figure shows the effect. The Transmittance at peak maximum drops down away from symmetry.

Thus, if the concern is linewidth of the tranmitted wavelength , adding more bragg pairs might help; If the concern is the peak position, the optical cavity thickness correction would help; If the concern is the peak transmission, the reflectances of the two mirros and the symmetry between them as well as any absorption losses will help.

Tolerance Analysis:

For the Monte Carlo fabrication-tolerance analysis, each layer was independently given a thickness error drawn from a zero-mean Gaussian distribution with a specified standard deviation (σ). A 0.1% deposition accuracy means that the gaussian distribution from which the errors are randomly picked have 0.1% (0.001 * the nominal thickness) sigma value. For each simulated filter, the transmission and reflection spectra was calculated and evaluated with respect to the following criteria:

1. Center wavelength: The transmission peak must lie within ±0.5 nm of 1550 nm.
2. Bandwidth: The full width at half maximum (FWHM) must be ≤2 nm.
3. Peak transmission: The maximum transmission at the resonance wavelength must be ≥95%.
4. Out-of-band reflectance: Reflectance must be ≥99% throughout both blocking bands, 1500–1530 nm and 1570–1600 nm.

The manufacturing yield was calculated as the percentage of simulated filters satisfying all four criteria simultaneously. 1,000 Monte Carlo trials for each thickness-error standard deviation, ranging from 0.02% to 0.5% were performed in total. The figure below show hwo the yield (number of samples that pass the criteria) change as the range of fabrication error increases.

Assumptions:

Semi-infinite substrate: The fused-silica substrate is assumed to be semi-infinite. Therefore, the calculated transmittance represents power transmitted into the glass substrate, rather than power emerging into air from the rear surface. Internal reflections from the substrate–air interface are not considered.

Wavelength-independent refractive indices: The refractive indices of Ta₂O₅ and SiO₂ are assumed constant over the investigated wavelength range. Material dispersion is neglected.

Lossless materials: All dielectric layers are assumed to have purely real refractive indices. Material absorption is neglected.

MonteCarlo Tolerance Analysis: Independent errors are assumed on the thickness of the layers.