Two OD 4 filters are not OD 8

The number

Optical density is the attenuation at one wavelength on a base-10 log scale, $\mathrm{OD} = -\log_{10} T$. OD 4 passes one part in 10 000. A blocking specification such as OD 4 at 532 nm is measured on one filter, alone, in a collimated beam.

Two OD 4 filters in series suggest OD 8. Two OD 4 reflective (interference) blockers mounted parallel give OD 4.30.

What it means

A reflective blocker rejects light by reflecting it. Between two parallel blockers the rejected light bounces: the second filter returns almost all of it to the first, which returns almost all of it again, and a small share leaks through on every pass. Summing the bounces incoherently, as a geometric series, gives the transmission of the pair:

Here $\rho_i$ is the share of the blocked light that filter $i$ reflects: close to 100 % for an interference blocker and about 4 % for absorbing glass, where only an uncoated surface reflects. For two OD 4 reflective blockers $1 - R_1 R_2 \approx 2 \times 10^{-4}$, so the pair transmits $10^{-8} / (2 \times 10^{-4}) = 5 \times 10^{-5}$, which is OD 4.30. A third filter combines with the pair the same way, using the pair's reflectance from the side of filter 2, $R_{12} = R_2 + T_2^2 R_1 / (1 - R_1 R_2)$. Three parallel OD 4 blockers give OD 4.48.

Compute it

Where it misleads

The sum of the ratings holds only when the layout removes the light reflected between filters. Three layouts do: tilt one filter a few degrees so its reflection leaves the beam into a beam dump; put an absorbing element between the reflective blockers; or space the filters and baffle the gap so reflected light leaves the aperture. An absorbing OD 4 element with 4 % surface reflection in front of a reflective OD 4 blocker reaches OD 7.98 even when parallel.

Tilting moves the passband of an interference filter to the blue; check the shift with the angle shift calculator. A quoted OD is also no better than the dynamic range of the instrument that measured it: a measurement that reads to OD 6 cannot confirm OD 8 on one filter or on a stack.

The optical density stack calculator adds a power budget to the same equations. The glossary defines optical density and stacked OD.

What to specify

• The measured OD curve of each filter at the wavelengths to block, not only a rated OD.
• The dynamic range of the measurement behind each OD figure.
• Whether each filter blocks by reflection, by absorption or both.
• The stack OD in the mounted layout, with the tilt, absorbing element or baffles that make the sum valid.

References

• Macleod, H. A. (2017). Thin-Film Optical Filters, 5th ed. CRC Press, Boca Raton.