Finding the hidden direction

Question

Can intensity readings at known analyzer angles identify an unknown linear polarization direction?

The forward problem predicts a reading from a direction. The reconstruction problem reverses that process. Because intensity is unchanged when the field direction is rotated by 180°, the direction is an axis, not an arrow. If I0 is known, one reading narrows the possibilities but leaves a cosine ambiguity. Measurements at several nonparallel axes constrain the same φ from different views.

Model and variables

φ is the unknown direction, a is a known analyzer angle, I0 is the unfiltered scale, and Iₐ is a reading. Here the pure model fixes I0 = 100 and fits only φ. A separate background model, Iₐ = b + I0 cos²(φ − a), has three free parameters if background, scale and direction are all fitted. I assume linear polarization and stable source output.

Derivation

Analytical result

Numerical example. Illustrative model data for φ = 37° and I0 = 100 intensity units: at 0°, I ≈ 63.8; at 30°, I ≈ 98.5; at 60°, I ≈ 84.7; at 90°, I ≈ 36.2. A 0.1° grid search uses the full precision readings, not these rounded labels, and recovers 37.0° with SSE = 0 within floating point tolerance. Directions 37° and 217° describe the same linear polarization axis.

Reproducible method

I generated four intensity readings with I0 fixed at 100 and compared them with predictions for candidate directions from 0° to 179.9° in 0.1° steps. I selected the direction that minimised the sum of squared errors, using full precision readings rather than the rounded labels. For future physical measurements, I would calibrate the reference intensity, repeat readings at each axis and randomise their order if the source drifts. I would examine residuals and repeat spread before reporting a recovered direction. An uncertainty interval would also require a stated noise model and estimation method; the grid spacing or an arbitrary width around the minimum is not a confidence interval.

Reading the plot

The fit curve shows how one hidden direction explains all readings. The SSE versus φ graph has repeated minima 180° apart. A broad minimum means the data do not resolve direction precisely. If the four points cannot sit near one curve, drift, background, ellipticity, or a wrong scale assumption may be responsible.

Computational illustration

Four readings, one fitted axis

These readings are generated from φ = 37°, not detector measurements. I compare the full precision readings with the pure known intensity model using a 0.1° grid from 0° through 179.9°.

Recovered axis: 37.0° modulo 180°. SSE: 0.00e+0 intensity units², zero within floating point tolerance.