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I am looking at this paper (Multicoated gratings, J. Opt. Soc. Am., 1981) and I am getting confused around equation 22. I do not completely understand where he comes up with the equation

$$\xi^j_q=b_q^j(R^{-1}V_q^j)\qquad \text{(22)}$$

And then what is the meaning of the $b^{j+1}$. I initially thought they were the eigenvalues of the T matrix he defines, but all he says is that it is a vector of components $b^q_j$.

This includes the $\bf b^0$ and $\bf b^{Q+1}$ which I do not see how they are vectors?

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It is said (just before $(22)$) that the $\xi^j_q$ are proportional to the eigenvectors $R^{-1}V_q^j$ So $b_q^j$ is just the ratio (which is different for each $j$ and $q$), between the two terms ,that is : $$\xi^j_q=b_q^j(R^{-1}V_q^j)$$ $\bf b^j$ is the vector with coordinates $b_q^j$, $q = 1,2,....+\infty$ (see just after $(23)$ –  Trimok Jul 17 '13 at 17:45
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The formal structure shows, that the matrix in brackets is of same dimension as the matrix $\xi^j_q$. Therefore $b_q^j$ is just a proportionality constant. The coatings/media of the optical structure are labeled by index $j$. $$\xi^j_q=b_q^j(R^{-1}V_q^j)\qquad \text{(22)}$$

And then what is the meaning of the $\bf b^{j+1}$?

The set of equations (22) is expressed in equation (23). By by introducing a new vector $$\bf{b^j}= \begin{pmatrix} b^j_1\\ b^j_2\\ \dots\\ b^j_\infty \end{pmatrix} $$ you get rid of index q. Where the index $j$ labels the medium of the coating. So you got some vector description of each coating layer $j$. Citing the paper

The problem reduces now to the search for vectors $\bf b^j$ in each medium.

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Any idea on how to solve for the $\mathbf b^j$ –  yankeefan11 Jul 18 '13 at 17:41
    
Linear algebra, since equation (23) is of form $$\bf{\xi}=M\quad\bf{b^j}$$Take inverse $M^{-1}$ –  Stefan Bischof Jul 18 '13 at 17:48
    
Equation 24, has $b^j$ in both sides of the equation. Do I want equation 23 and actually solve for the $\xi$ –  yankeefan11 Jul 18 '13 at 17:49
    
You are correct. I had to write (23). This was my first thought. –  Stefan Bischof Jul 18 '13 at 17:51
    
Not a problem. Any chance you understand what the author says in 4A about the matrix sizes that reduces equation 28 to 4N+2 equations with that many unknowns? –  yankeefan11 Jul 18 '13 at 17:52
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