Reading the curve

Question

How does the transmitted intensity depend on the angle between two ideal polarizers?

A linear polarizer selects the component of an incoming electric field parallel to its transmission axis. Let the incoming field have amplitude E0 and let the second axis make angle θ with it. The field after the analyzer is the vector projection E = E0 cos θ. For a monochromatic wave, average intensity is proportional to the square of field amplitude, so I/I0 = (E/E0)² = cos² θ. At θ = 0° the ideal analyzer transmits all modeled light; at 90° it transmits none.

Model and variables

I0 is the intensity before the analyzer, I is transmitted intensity, θ is the relative angle, and E is electric field amplitude. Assumptions: monochromatic light, ideal polarizers, no scattering, a detector with a linear response, and a beam already linearly polarized along the first axis.

Derivation

Analytical result

Numerical example. For this illustrative calculation, I0 = 120 intensity units before the analyzer. At 30°, the field amplitude ratio is ≈ 0.8660, and I = 120 cos²30° ≈ 90 units. At 60°, I ≈ 30 units. Calculations use full precision, not the rounded amplitude shown here.

Reproducible method

I calculate the predicted intensity as 120 × cos² θ, using full precision before rounding the displayed values. The predicted curve below evaluates angles from 0° to 180° in 1° steps. For future physical measurements, I would fix the source and detector, mark the zero axis, rotate only the analyzer, take repeated readings and subtract a dark reading. I would compare those measurements, with their repeat spread, against the analytical prediction.

Reading the plot

The graph rises to a maximum at 0° and 180°, falls to zero at 90°, and has mirror symmetry about those axes. A residual plot, measured minus predicted, should show noise around zero if the model and alignment are adequate. A smooth curve alone cannot establish that the light is perfectly polarized.

Analytical result

Proposed physical validation

I would establish the zero axis, check that the detector is linear and unsaturated, and record a dark reading with a stable aligned reference. I would repeat readings and check source drift. Physical uncertainty must be reported separately from the one degree sampling used for this predicted curve.

Uncertainty and limits

Real filters leak, rotate imperfectly, and absorb light even at alignment. Ambient light adds an offset. A detector can saturate. Elliptical or unpolarized input needs a richer model. A good agreement in simulated data is only a check of the code because the data were generated from the same equation.

Further question

Field square and angular sensitivity

Time averaging the squared field produces the intensity ratio. Near alignment or extinction a vanishing first derivative does not eliminate higher order uncertainty.

Next question

How does an intermediate filter change the prediction when the first and last axes are crossed? In The third filter, I compare the predicted stage intensities using the ideal product of projections. Comparing that prediction with physical measurements remains future work.