Linear retardance formula

I found this formula in https://doi.org/10.1364/BOE.426653 (NLM), a paper titled: Stokes polarization imaging applied for monitoring dynamic tissue optical clearing.

The formula is for calculating the linear retardance of a sample image using a Stokes polarization camera and a right circular polarizing filter over a source light. I already have several other parameters calculated for a camera I'm making and this seems like an interesting one to include.

Unfortunately, I'm not familiar with the notation the researchers used to write the formula. My best guess is the normalized vectors multiplied together for in and out divided by the absolute value of the normalized vectors. But that doesn't make any sense because normalized vectors are already positive.

  • $\begingroup$ Hint. $\endgroup$
    – Kurt G.
    Sep 23 at 10:35
  • 3
    $\begingroup$ normalized vectors are already positive This statement doesn’t make sense. Vectors aren’t positive or negative. Individual components of a vector in a basis can be positive or negative. The magnitude of any nonzero vector is positive, regardless of whether it is normalized or not. $\endgroup$
    – Ghoster
    Sep 23 at 17:27

1 Answer 1


This is just the rearrangement of one definition of the dot product, $$ \vec{a}\cdot \vec{b} = |\vec{a}||\vec{b}|\cos (\theta)$$ where $\theta$ is the angle the two vectors make with one another. They are solving for the angle using the arc cosine in this case, $$\theta = \cos^{-1} \left( \frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|} \right)$$

  • $\begingroup$ So the elongated ‘^’ signs in the question denote merely vectors — and not normalised vectors (as OP may have been thinking, and as I would have otherwise assumed)? $\endgroup$
    – gidds
    Sep 23 at 14:51
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    $\begingroup$ The only other use of $\hat{\cdot}$ that I'm aware of is to denote an estimated quantity in statistics. Are these $\hat{S}$ vectors coming from some kind of statistical model? $\endgroup$ Sep 23 at 14:59
  • 1
    $\begingroup$ Okay, on reading the paper, it seems that that's exactly the sense in which the hat is being used here. $\endgroup$ Sep 23 at 15:01

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