PHALIN THAKKAR
Why the square matters
Why does projection give cosθ while intensity gives cos²θ?
Note 1 · Explanatory note
Analytical result · Hypothetical examples are not physical measurements.
For an ideal linearly polarized monochromatic wave, write the incoming field as E0 cos(ωt). Projection onto the analyzer multiplies its amplitude by cosθ. A detector responding to average energy flow measures a quantity proportional to the time average of the squared field.
The incoming time average contains the same factor ½, so the intensity ratio is cos²θ.
At 60°, the amplitude is ½ of its original value and the intensity is ¼. At 120° the projected field has a negative amplitude factor, but the same squared intensity. An intensity detector does not recover that field sign.
Extension: angular uncertainty matters through the slope. For the normalized intensity J = cos²θ:
Angles must be in radians in this derivative. At 30°, a small 1° standard uncertainty gives σJ ≈ 0.0151, with the reference intensity treated as known.
At alignment or extinction the first derivative vanishes. This does not make every uncertainty zero; higher order effects and other uncertainty sources remain.