Center Wavelength Isn't the Average

The λ0 Formula That Surprises Most Engineers

The Intuition Trap

"Center wavelength" sounds like it should be the midpoint between the band edges. That intuition is so strong that many people never question it—until a bandpass filter doesn't behave the way they expected in a real system.

Here's the truth: center wavelength (λ0) for bandpass filters is defined using the half-power points via this formula: λ0 = 2 × λ1 × λ2 ÷ (λ1 + λ2). That's the harmonic mean of the two half-power wavelengths—not the simple arithmetic average.

How Much Does It Matter?

The difference can be small for narrow bands, but it grows as the band widens.

Example 1: If your half-power points are at 500 nm and 600 nm, the arithmetic mean is 550 nm. But the actual center wavelength by the formula is about 545 nm—a 5 nm difference. Small, but real.

Example 2: If your half-power points are at 400 nm and 700 nm (a wide band), the arithmetic mean is 550 nm. But the actual center wavelength is about 509 nm—a 41 nm difference. That's big enough to wreck a "close enough" illumination match.

The deeper reason for this formula: many filter behaviors are more symmetric in frequency (or wavenumber) space than in wavelength space. So the "center" isn't always where your wavelength-based intuition points.

Practical Implications

This matters whenever you're matching an LED wavelength to a bandpass filter and trying to maximize signal while crushing ambient light.

Use the manufacturer's definition. Verify whether they define center wavelength by peak transmission, by HPP points, or by a specific calculation. Don't assume.

Sanity-check the overlap. Look at your emitter spectrum and camera sensitivity curve together with the filter passband. Make sure you're actually getting the light you expect through to the sensor.

Watch for the double-whammy. Remember that physically inserting or removing filters can also shift working distance. A "center wavelength" spec mistake can compound with a focus shift—giving you less light and an unexpected mechanical problem at the same time.

The Bottom Line

Center wavelength uses the harmonic mean formula, not the arithmetic average. For wide-band filters, the difference can be significant. Always verify the manufacturer's definition and check the actual overlap with your LED and sensor. Your illumination budget depends on it.