Optical Density Calculator

This tool allows you to calculate the optical density from the incident light intensity and transmitted light intensity.

Absorbance (optical density)2−log₁₀ of 0.01 transmittance

Transmitted intensity750 W/m²I₀cos²30° = 75% of the incident

Transmitted fraction0.75note the SQUARE — intensity goes as cos², while the amplitude goes as cos

Amplitude fraction0.866025

At 45°50%exactly half, since cos²45° = ½

At 90°0%crossed polarisers block everything — and inserting a THIRD polariser between them at 45° lets light through again, which is genuinely counter-intuitive

Crossed pair with a 45° polariser inserted25% of the incidentcos²45° twice — a quarter gets through where none did before, because each polariser reprojects the polarisation rather than merely filtering it

Same total rotation over 2 polarisers75% transmitted30° per stage — spreading the rotation over more stages passes MORE light, because cos² of a small angle is close to 1

Percent transmittance1%

Absorbance of 1, 2 and 310%, 1%, 0.1%each unit of absorbance is another factor of ten, so the scale is logarithmic and absorbances of stacked filters simply add

The formula

I = I₀cos²θ (Malus); A = −log₁₀(T)

Polarisers reproject, they do not just filter

Malus's law gives the transmitted intensity as I₀cos²θ. The square matters: at 45° exactly half the intensity passes, not 71% — that figure is the amplitude, and intensity goes as amplitude squared.

The three-polariser result is the one worth sitting with. Two crossed polarisers pass nothing. Insert a third between them at 45° and a quarter of the light gets through. Nothing was added — so the middle polariser cannot be merely blocking. What it does is reproject the polarisation onto a new axis, and light polarised at 45° to the final filter can then pass it. This is a genuinely quantum-mechanical behaviour showing up in a classroom demonstration.

The same reprojection is why spreading a rotation over more polarisers transmits more light: many small cos² factors near 1 multiply to more than one large one. With enough stages you can rotate polarisation through 90° with almost no loss.

Absorbance is a base-ten logarithmic scale, so absorbance 1 transmits 10%, absorbance 2 transmits 1%, and the absorbances of stacked filters add rather than multiply — which is exactly why the scale is defined that way.