Filter Stacks: Why Multiply Factors, Add Densities, and Stop Losing Light by Accident

The Filter Stacking Trap

Real machine vision systems rarely use just one filter. A typical build might include an ND filter to tame brightness, a bandpass filter to isolate your target wavelength, and maybe a polarizer to fight specular glare. Each filter solves a specific problem, and stacking them seems straightforward enough.

Here's where things go wrong: human intuition wants to add percentages. "I have a 50% filter and a 25% filter, so I should get... 75% transmission? No wait, 37.5%?"

Neither. The actual transmission is 12.5%, and this misunderstanding is how you accidentally design a system that's hopelessly starved for light.

The Physics of Stacked Filters

Think about what happens when light passes through multiple filters in sequence. The first filter takes its bite, say, passing 50% and absorbing the rest. The surviving photons then hit the second filter, which doesn't know or care what happened upstream. It takes its own bite from whatever light arrives.

If that second filter has 25% transmission, it passes 25% of what reached it (which was already only 50% of the original). The result: 50% × 25% = 12.5% of the original light makes it through both filters.

Stacked filters multiply their transmissions. They don't add them.

Two Safe Math Modes

To avoid filter stacking errors, use one of these two calculation methods:

Method A: Multiply Filter Factors

Remember that filter factor (F) equals 1/Transmission. When you multiply filter factors, you get the combined factor directly.

Take our example: a 50% filter has factor 2, and a 25% filter has factor 4. Multiply them: 2 × 4 = 8. Your combined stack has filter factor 8, meaning you need 8× the exposure compared to no filters.

Converting back to transmission: 1/8 = 0.125 = 12.5%. This matches our earlier calculation, confirming we did the math right.

Method B: Add Optical Densities

Because optical density is logarithmic, densities add linearly when you stack filters. This is often the fastest approach when working with filter spec sheets.

A 50% transmission filter has OD ≈ 0.3. A 25% transmission filter has OD ≈ 0.6. Add them: 0.3 + 0.6 = 0.9 total OD.

Converting OD 0.9 back to transmission: 10^(-0.9) ≈ 0.126, or about 12.5%. Same answer.

In stop terms: the 50% filter costs 1 stop, the 25% filter costs 2 stops, and combined they cost 3 stops—which corresponds to 8× light loss (2³ = 8).

A Worked Example Worth Memorizing

Stack a 50% ND and a 25% bandpass filter. What's the total system transmission?

Using factors: Factor 2 × Factor 4 = Factor 8 → need 8× more light
Using densities: OD 0.3 + OD 0.6 = OD 0.9 → 3 stops of loss
Final transmission: 12.5% (or roughly ⅛ of your original light)

Burn this example into memory. When you're in the lab estimating whether a filter stack is viable, this 50%/25% → 12.5% relationship gives you an instant calibration point.

Stacking Also Stacks Side Effects

The multiplicative light loss is the obvious consequence of stacking filters, but there are subtler effects that accumulate too.

More Surfaces Mean More Reflections

Every glass-to-air interface reflects some light—even a few percent for uncoated surfaces. Two filters means four surfaces; three filters means six. Each reflection loses light and can create ghost images or reduce contrast.

This is why quality filters use anti-reflective (AR) coatings. When stacking multiple filters, the cumulative benefit of AR coatings becomes significant. A 4% reflection per surface doesn't sound like much until you multiply it across six surfaces.

Optical Path Length Increases

Each filter adds to the total glass thickness in your optical path. In some systems—particularly those with tight working distances or precise focus calibration—this can shift the effective focal plane.

If you're designing a system where filters will be added or removed, account for how each combination affects focus. You may need to recalibrate or build in adjustment range.

Tilt Effects Compound

A single tilted filter introduces some aberration and possibly a small beam offset. Multiple tilted filters can compound these effects in complex ways. In systems with telecentric lenses or high-resolution sensors, maintaining perpendicular filter mounting becomes more critical as you add elements.

The Clean Workflow for Filter Stack Design

When you're planning a multi-filter system, follow this sequence:

First, do the multiplicative light math. Calculate your total transmission by multiplying individual transmissions (or adding optical densities). Determine whether your lighting system can compensate for the total loss. If you're looking at factor 20 or higher, seriously evaluate whether you have enough lighting headroom.

Second, sanity-check the optical consequences. Consider total glass thickness and its effect on focus. Assess how many surfaces you're introducing and whether AR coatings are adequate. Think about mechanical mounting and how to maintain perpendicularity.

Third, consider the spectral interactions. If you're stacking wavelength-selective filters, think about whether their passbands overlap appropriately. A bandpass combined with a poorly-matched longpass might block your target wavelength entirely.

The goal is to avoid the "just one more filter" trap, where you keep adding elements to solve problems without tracking the cumulative costs to your light budget and optical quality.

This is part of KUPO's educational series on optical filters for machine vision. Understanding how filter effects compound helps you design systems that work reliably in practice, not just on paper.