Black and white work frequently calls for two filters at once: a contrast filter to separate tones, plus a polarizer to deepen a sky or kill reflections, or a neutral density filter to reach a longer shutter speed. Combining them is straightforward in principle, but the exposure arithmetic trips up anyone who treats it as addition, and the optical cost of extra glass is real.

Factors Multiply, Stops Add

Every filter carries a filter factor: the multiplicative amount by which it reduces light reaching the film. A factor of 2 halves the light and costs one stop; a factor of 4 costs two stops; a factor of 8 costs three. The relationship is logarithmic, stops equal the base-2 logarithm of the factor, which is why factors and stops behave differently when filters are combined.

When filters are stacked, their factors multiply, while their stop values add: the combined factor of two filters is the product of the individual factors. A Kodak Wratten No. 25 red filter has a factor of 8 (three stops). A linear polarizer used at a non-polarizing orientation behaves much like a neutral density filter, with a factor of roughly 2.5, about one and a third stops. Stacked, the factor is 8 × 2.5 = 20, not 10.5. Twenty corresponds to roughly 4.3 stops, the sum of three and one and a third. Adding the published stop values reaches the same answer more safely than multiplying factors in the head.

Polarizer and ND Caveats

A polarizer complicates the figure because its effect depends on orientation. The nominal 2.5 factor applies near the non-polarizing angle; rotated to maximum effect against a clear sky, the apparent loss shifts, and through-the-lens metering captures the actual value more reliably than a fixed factor. Neutral density filters are simpler, their density is deliberately constant across the visible spectrum, so a 3-stop ND stacked on a 3-stop red filter simply totals six stops.

The Optical Penalty of Extra Glass

Each filter adds two air-to-glass surfaces, and each surface can reflect light. Stacked filters increase internal reflections between elements, raising the risk of ghosting and veiling flare, particularly with a bright source in or near the frame. Multi-coating reduces but does not eliminate this. The second mechanical penalty is vignetting: two stacked rings extend the filter assembly forward, and on a wide-angle lens the front ring can intrude into the image circle, darkening the corners. Slim or low-profile mounts mitigate the obstruction, but the cleanest images on wide lenses come from limiting the stack to a single filter where the result allows.