PHALIN THAKKAR / PORTFOLIO

PHALIN THAKKAR

Which analyzer settings carry more information?

Is the brightest reading always the most informative about an unknown angle?

Note 10 · Research extension

Analytical result · Hypothetical examples are not physical measurements.

For the known intensity model:

I(a;ϕ)=I0cos⁡2(ϕ−a),I(a;\phi)=I_0\cos^2(\phi-a),
∂I∂ϕ=−I0sin⁡2(ϕ−a).\frac{\partial I}{\partial\phi}=-I_0\sin2(\phi-a).

If each reading has independent Gaussian noise with the same known standard deviation σ, its local information about φ is:

Ia=I02σ2sin⁡2[2(ϕ−a)].\mathcal I_a=\frac{I_0^2}{\sigma^2}\sin^2[2(\phi-a)].

In that model the strongest sensitivity occurs 45° from the unknown axis. Alignment gives maximum intensity but a zero first derivative.

Four independent settings 0°, 45°, 90° and 135° satisfy:

∑asin⁡2[2(ϕ−a)]=2.\sum_a\sin^2[2(\phi-a)]=2.

Their combined local information is 2I0²/σ², independent of φ. The corresponding information bound on the standard deviation of a regular locally unbiased estimate is:

σϕ≥σI02.\sigma_\phi\ge\frac{\sigma}{I_0\sqrt2}.

For I0 = 100 and σ = 1 this is approximately 0.4051°.

This is a local bound under a specified model, not a measured uncertainty or a confidence interval. Estimating an unknown background or scale, including angle errors, or using intensity dependent counting noise changes the analysis.

Continue the reasoning

Finding the hidden direction

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