Bandpass filters: all-dielectric cavities

An all-dielectric bandpass filter is a Fabry–Pérot cavity built entirely from quarter-wave layers, so that nothing absorbs and the mirrors can be as good as the period count allows. Two Bragg mirrors of high- and low-index layers surround a spacer that is a whole number of half waves thick at the design wavelength $\lambda_0$:

where $p$ is the number of pairs per mirror and $m$ the cavity order. The notation is Macleod's: each letter a quarter wave at $\lambda_0$, read from the incident medium to the substrate.

Why it transmits

At $\lambda_0$ the spacer is a half-wave layer, which is absent in the transfer-matrix sense ($M = -I$). What is left is $(\mathrm{HL})^p(\mathrm{LH})^p$: the two mirrors face each other, and because the sequence is symmetric the admittance that the second mirror builds up from the substrate is exactly undone by the first, layer by layer, back to the substrate's own admittance $n_s$. The whole filter at $\lambda_0$ is therefore optically the bare substrate, with $R = \left(\frac{n_0 - n_s}{n_0 + n_s}\right)^2$, 4 % for glass, and $T \approx 96 \%$ (a single antireflecting layer on the back or an absentee-corrected design removes the rest). A little away from $\lambda_0$ the spacer is no longer absent, the mirrors are not undone, and since each reflects nearly everything the transmittance collapses: a line, inside a stop band.

What sets the width

The single-cavity line has the Airy shape of the etalon, which near its peak is a Lorentzian whose half-width Macleod derives in closed form. For a spacer of low index with $p$ pairs on each side and a substrate $n_s$,

and with a high-index spacer the roles of $n_H$ and $n_L$ in the prefactor exchange. Every design knob is visible in it:

• each extra pair on both mirrors multiplies the width by $(n_L/n_H)^2$, 0.4 for TiO₂/SiO₂: the width falls geometrically with $p$, as the mirrors' $1 - R$ does;
• the cavity order $m$ narrows the line in proportion, because the phase in a thicker spacer changes faster with wavelength; second- and third-order spacers are the usual compromise between narrowness and the tolerances of thicker layers;
• a high-index spacer is narrower than a low-index one with the same mirrors by $(n_H/n_L)^2$, and also shifts less with angle (below).

A $p = 4$, $m = 1$ TiO₂/SiO₂ filter has a half-width of about 1 % of $\lambda_0$, 5.5 nm at 550 nm; $p = 6$ gives 0.9 nm. The rejection outside the passband is the mirrors' own stop band: its width comes from the index contrast, so TiO₂/SiO₂ blocks about 490–630 nm around a 550 nm design, and a practical filter adds absorbing glass or further stacks as broadband blockers beyond that, where the mirrors are transparent again.

import numpy as np
from pystratify import planar
from pystratify.docs import material  # n + ik of a tabulated page (refractiveindex.info format)

n = lambda wl: [1.52, material('docs/data/tutorials/TiO2_Sarkar.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/TiO2_Sarkar.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/TiO2_Sarkar.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/TiO2_Sarkar.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/TiO2_Sarkar.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/TiO2_Sarkar.yml', wl), 1.0]  # incident medium, layers, substrate
d = [np.inf, 0.063524, 0.094184, 0.063524, 0.094184, 0.063524, 0.094184, 0.188, 0.094184, 0.063524, 0.094184, 0.063524, 0.094184, 0.063524, np.inf]  # thicknesses, µm
lam = (400 + 2 * np.arange(201)) * 1e-3  # wavelength, µm
th = np.radians(0)
R, T = {}, {}
for pol in ('s', 'p'):
    rows = [planar.coh_tmm(pol, n(wl), d, th, wl) for wl in lam]
    R[pol], T[pol] = np.array([r['R'] for r in rows]), np.array([r['T'] for r in rows])
R_avg, T_avg = (R['s'] + R['p']) / 2, (T['s'] + T['p']) / 2  # unpolarized

Moving the spacer away from a half wave shifts the passband through the stop band; it disappears into the edge once the cavity's resonance leaves the band. Absorption in the layers, ignored in the ideal design, is what limits the peak transmittance of very narrow filters: by the potential-transmittance argument of the etalon section, what counts is the absorption per pass relative to $1 - R$, and a filter of 0.5 nm width with layers that absorb $10^{-4}$ per pass already loses a tenth of its peak.

Multiple cavities: a square top

A single cavity has a Lorentzian line: sharp at the top but with slowly falling wings, so that a 1 nm filter still passes a thousandth of the light 10 nm away. Putting two or three cavities in series, coupled through a shared quarter-wave layer, $(\mathrm{HL})^p 2\mathrm{L} (\mathrm{LH})^p\, \mathrm{L}\, (\mathrm{HL})^p 2\mathrm{L}(\mathrm{LH})^p$, splits the resonance into as many coupled modes, which together make a flat top and steep sides; the shape is the optical analogue of a Butterworth or Chebyshev electrical filter, and the design rules carry over (Baumeister 1958). Nearly every commercial narrowband filter is a two- or three-cavity design, and the wavelength-division multiplexers of optical fibre links use up to a hundred cavities' worth of layers to pass one 0.4 nm channel and reject its neighbours by $10^{-4}$.

Angle: tuning and the converging-beam limit

Tilting the filter moves its passband to shorter wavelengths, as every interference feature moves, by

where $n_\mathrm{eff}$ is an effective index close to the spacer's (slightly modified by the mirrors). A few degrees of tilt is the standard way to fine-tune a filter after it has been made; it also means that a filter in a cone of half-angle $\alpha$ passes a band broadened by the same formula with $\theta_0 \to \alpha$, so a 1 nm filter in an f/2 beam no longer has a 1 nm band. High-index spacers, with their larger $n_\mathrm{eff}$, are preferred where the beam cannot be collimated, and a filter's data sheet always states the cone angle its curve was measured in.

The notch filter

Swap the roles and a stack rejects one band while passing the rest. A quarter-wave mirror with very weak index contrast (two materials differing by a few hundredths, or a mixture graded between them) has a stop band whose width, by the Bragg formula $\Delta g = \frac{4}{\pi}\arcsin\frac{n_H - n_L}{n_H + n_L}$, is as narrow as one likes, and needs correspondingly many periods to reach high reflectance. Southwell's rugate filters (1989) replace the discrete layers by a sinusoidal index profile, whose single Fourier component produces one stop band with no higher-order bands and no ripple; the laser-line blocking filters of Raman spectroscopy, which must reject the excitation by $10^{-6}$ a few nanometres from a band they transmit, are built this way.