PHALIN THAKKAR
More intermediate filters, more transmission?
What changes if several ideal projections share the rotation between crossed axes?
Note 12 · Research extension
Analytical result · Hypothetical examples are not physical measurements.
Let N be the number of angular steps after a fixed first 0° filter and before the final 90° filter. There are N − 1 intermediate filters.
For monotonic axes with gaps δj between 0 and π/2:
Because ln cosδ has second derivative −sec²δ, equal gaps maximize the product:
With one intermediate filter, N = 2 and Tmax = ¼. With two, N = 3: use 30° and 60°, giving Tmax = (3/4)³ = 27/64 ≈ 0.421875.
The fraction is relative to the intensity after the first filter. Add a factor ½ if the reference is an unpolarized source before that filter.
In the ideal limit, Tmax approaches 1 as N increases. Actual aligned transmission losses change this. If each of the N later filters additionally transmits an intensity fraction τ:
For the illustrative τ = 0.95, evaluating integer N gives a maximum near N = 7, with T ≈ 0.4894. Adding filters indefinitely eventually loses more intensity.