I am using Siegman's formalism, but I think I am missing something. I want to build a matrix formalism in which x' = M delta_x, where these are vectors. When I write ray tracing, I seem to see that the algebra forms are incompatible.
You can add an addition dimension to your ABCD matrix to shift all rays in the opposite direction of lens-decentration before the transmission through the lens and shift it back after the transmission.
Your lens-matrix will be like
A B 0
C D 0
0 0 1
and you have a displacement-matrix with before the lens:
0 0 -dx
0 0 0
0 0 1
and after the lens you apply it again with switched sign of dx.
your ray vector than has to be
alpha
x
1
You cannot represent the full effects of lens de-centering through this formalism, because you are using Gaussian approximation. However, this might be sufficient for your application.
I have used this approach to study the effect of a decentered lens inside the eye (DOI: 10.1364/JOSAA.35.000561) and it corresponded well with the results from numerical ray tracing.
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