Neutral-density filter

A neutral-density (ND) filter reduces the power of a beam by a fixed factor that is approximately independent of wavelength over its design range, so the spectrum passes through unchanged in shape. Its strength is given as an optical density, OD $= -\log_{10}T$, where $T$ is the transmittance: ND 0.3 transmits 50%, ND 1 transmits 10%, ND 2 transmits 1% and ND 4 transmits 0.01%. Laboratory sets usually run from OD 0.1 to OD 4, and camera ND filters use the same scale or a stop count, where one stop halves the light and corresponds to OD 0.3.

Absorptive and reflective types

Absorptive ND filters are glass bodies containing absorbing dopants that give a nearly flat response across the visible. The absorbed power becomes heat. Their OD usually changes toward the ultraviolet and near infrared, often falling in the near infrared, so a filter labeled ND 2 in the visible may transmit noticeably more than 1% at 1064 nm; the datasheet curve at the working wavelength is the one that counts.

Reflective ND filters carry a thin metallic coating, often a nickel-chromium alloy, on a glass or fused silica substrate. They reflect a large part of the light they do not transmit and absorb the rest, and their response is flat over a wider range, commonly from the ultraviolet into the near infrared. The reflected beam can carry much of the input power and has to be dumped safely.

Variable ND filters have a coating whose OD changes continuously with position, around a circular wheel or along a linear slide, giving adjustable attenuation over a range such as OD 0–2 or 0–4. The OD varies across a large beam, so they are best used with small beams. In fiber systems the same role is filled by a variable optical attenuator.

Stacking and the OD arithmetic

Because OD is logarithmic, the ODs of filters in series add while their transmittances multiply:

$$T = 10^{-(\text{OD}_1+\text{OD}_2)}.$$

An ND 2 filter followed by an ND 0.5 gives OD 2.5 and $T = 0.316\%$. A 50 mW laser beam through an OD 3 stack leaves 50 µW, a level suited to most silicon photodiodes; a camera usually needs further attenuation. The attenuation in decibels is ten times the OD, so OD 3 is 30 dB.

Addition holds only when light that reflects between the filters is not sent back through them. Two reflective ND 1 filters placed parallel form a low-finesse cavity with the etalon geometry: if each transmits 10% and reflects 50%, the multiply reflected light raises the combined transmittance, averaged over interference fringes, to

$$T = \frac{T_1T_2}{1-R_1R_2} = \frac{0.01}{1-0.25} = 1.3\%,$$

an effective OD of 1.88 instead of 2. Tilting one filter by a few degrees walks the multiple reflections out of the beam and restores the expected value.

Measurement

The OD at a given wavelength is measured by recording the power with an optical power meter with and without the filter, in the same beam and with the detector's range held fixed where possible: $\text{OD} = \log_{10}(P_0/P)$. For OD up to about 3 this is straightforward with a laser; higher ODs approach the detector noise floor and the stray-light level, and are measured by chaining calibrated filters or by changing the source power, as described in measuring optical density beyond OD 4.

Pitfalls with lasers

An absorptive filter in a 1 W beam at ND 1 absorbs 0.9 W. With a millimeter-size beam, this heats a small spot of glass, which can distort the transmitted wavefront by thermal lensing, crack the filter or, in pulsed beams, damage its surface. A common arrangement for high power is to put a reflective filter or a wedge first and absorptive filters after it, where the power is lower.

The front reflection of any ND filter, and the whole rejected beam of a reflective one, returns toward the source if the filter is normal to the beam. A few degrees of tilt keeps it out of the laser, where it could cause instability, and away from the input beam path.

ND filters are also not a substitute for laser safety eyewear or solar filters, which are specified for particular wavelengths and damage conditions. The label value applies at the calibration wavelength and angle; for accurate work the filter is calibrated at the wavelength in use.

Common questions

What does ND 8 or ND 64 mean on a camera filter?

Photographic ND filters are often labeled by attenuation factor. ND 8 reduces the light eightfold, three stops, equivalent to OD 0.9; ND 64 is six stops, OD 1.8.

What is the difference between OD and ND number?

For laboratory filters they are the same number: an ND 2 filter has OD 2. Photographic labeling sometimes uses the attenuation factor instead, as above.

Should an ND filter be absorptive or reflective for a laser?

Reflective filters handle higher power because they absorb little, and their spectrum is flatter, but they create a strong reflected beam. Absorptive filters have no strong reflection and are insensitive to angle, but heat up.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 4 and 9; H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).