Notch filter

A notch filter, also called a band-stop or band-reject filter, is the complement of a bandpass: it transmits across a wide range and blocks one narrow band. In optics the blocked band is almost always a laser line. The canonical case is Raman spectroscopy, where the elastically scattered laser light is many orders of magnitude stronger than the Raman-shifted light a few nanometers away, and the notch removes the laser before the spectrometer sees it. The same part rejects the excitation in laser-induced fluorescence, strips a pump wavelength out of a fiber system, and protects a detector from one specific line.

Three numbers specify a notch. The center wavelength, which for laser-line filters is the laser wavelength (532, 633, 785 nm and so on). The blocking depth, quoted as optical density, $\mathrm{OD} = -\log_{10} T$, so OD 6 means a transmission of $10^{-6}$, or $-60$ dB. And the bandwidth, quoted as the full width at half transmission and often also as the narrower width over which the OD exceeds a stated value. Outside the notch a good thin-film filter transmits above 90%. Two further specifications matter on the bench, the angle-of-incidence tolerance and the polarization dependence, and both follow from how the notch is made.

Nearly every reflective notch is a Bragg reflector: a periodic modulation of refractive index with period $\lambda/2n$ reflects one wavelength and passes the rest (Bragg condition). A thin-film interference notch, typically a rugate coating whose index varies continuously through a layer tens of micrometers thick, is one realization; a volume Bragg grating written in photo-thermo-refractive glass is another; a fiber Bragg grating used in transmission is the same physics inside a fiber; and an all-pass ring resonator makes a notch by resonant coupling instead of by a grating. Coupled-mode theory gives the depth of a uniform index modulation of amplitude $\delta n$ and length $L$ in one line. With the coupling coefficient $\kappa = \pi\,\delta n/\lambda$, the transmission at the center of the notch is $T = \mathrm{sech}^2(\kappa L)$. OD 6 needs $\kappa L = 7.6$, and each further unit of $\kappa L$ deepens the notch by another 8.7 dB. The full stopband width is $\Delta\lambda = \lambda\,\delta n/n$. Those two relations set the design trade: depth comes from the product $\delta n \cdot L$, width from the ratio $\delta n/n$. A thin-film notch with $\delta n$ of a few hundredths reaches OD 6 in a few tens of micrometers and is tens of nanometers wide; a volume Bragg grating with $\delta n$ near $3 \times 10^{-4}$ needs about 3 mm of glass for OD 4 and is correspondingly narrow, about 0.1 nm at 532 nm.

Raman shifts are quoted in wavenumbers, and the conversion $\Delta\lambda = \lambda^2\,\Delta\tilde{\nu}$ makes a notch's bandwidth depend on the laser wavelength in a way that matters. At 532 nm, 1 cm$^{-1}$ is 0.028 nm; at 785 nm it is 0.062 nm. A 17 nm notch at 532 nm therefore blocks about 600 cm$^{-1}$ of spectrum in total, roughly 300 cm$^{-1}$ on the Stokes side, which is fine for the 520 cm$^{-1}$ silicon line and hopeless for anything below 300 cm$^{-1}$. Low-frequency Raman, the lattice and layer-breathing modes below 100 cm$^{-1}$ (2.8 nm at 532 nm), is what volume Bragg grating notches exist for: bandwidths of 5 to 10 cm$^{-1}$, OD 3 to 4 per filter, stacked two or three deep to reach OD 8 or more within a few wavenumbers of the laser.

Because the notch is an interference effect it shifts with angle of incidence. For a thin-film filter with effective index $n_\mathrm{eff}$ the center moves as $\lambda(\theta) = \lambda_0\sqrt{1 - \sin^2\theta/n_\mathrm{eff}^2}$: with $n_\mathrm{eff} = 2$, a 532 nm notch sits at 531.5 nm at 5° and 530.0 nm at 10°, a 2 nm shift that a 17 nm notch tolerates and a 0.1 nm volume-grating notch does not. Volume gratings are angle-tuned deliberately for the same reason. Away from normal incidence the s and p polarizations see different effective indices and the notch splits into two slightly offset notches, so specifications are given at normal incidence or over a stated cone half-angle. Thin-film notches are used at 0°; a reflective notch at 45° is a different design with a wider, polarization-split band, closer to a dichroic mirror.

Measuring a deep notch is harder than buying one. A grating spectrometer's stray-light floor is typically OD 3 to 4 in a single stage, so a spectrometer scan of an OD 6 filter shows the instrument's floor, not the filter's. The honest measurement is a laser and a calibrated power meter with 60 dB or more of dynamic range, or a double monochromator, with the out-of-band transmission measured separately with a broadband source; the procedure, with its error budget and the stacked-filter floor test, is in Measuring optical density beyond the stray-light floor. Two related traps: fluorescence from the filter substrate under intense laser illumination fills in the notch during the measurement, and a filter specified at OD 6 over its central band may be only OD 2 to 3 at the edges of its quoted bandwidth, which is where a laser's wavelength drift will eventually put the line.

In integrated photonics the all-pass ring is the notch. At resonance the through-port transmission is $T = \left(\frac{a - t}{1 - a t}\right)^2$ for round-trip amplitude transmission $a$ and coupling $t$, zero at critical coupling ($a = t$) and only a few dB deep when the two are mismatched, with a width set by the ring's Q rather than by an index modulation. The ring resonator explorer shows the through-port notch deepen and widen as the coupling is swept through critical.

References: T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277 (1997), for the coupled-mode results quoted here; H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010), on rugate and notch designs; O. M. Efimov, L. B. Glebov, L. N. Glebova, K. C. Richardson and V. I. Smirnov, "High-efficiency Bragg gratings in photothermorefractive glass," Appl. Opt. 38, 619 (1999); manufacturer data for laser-line notch filters (thin-film and volume Bragg grating types) for the typical bandwidth and OD figures quoted.