A relay system is built from a converging thin lens

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Optics & Photonics · Geometrical optics and imaging

What an Instrument Can See, Ch The paraxial system as an operator

Problem

A relay system is built from a converging thin lens f1=+50 mmf_1=+50~\text{mm}, an air gap d=30 mmd=30~\text{mm}, then a diverging thin lens f2=−75 mmf_2=-75~\text{mm}, in that order along the axis (light travelling left to right). Someone computes the system matrix as Mwrong=M1 T M2=(1.430 mm−0.01467 mm−10.4),\mathbf{M}_{\text{wrong}}=\mathbf{M}_1\,\mathbf{T}\,\mathbf{M}_2=\begin{pmatrix}1.4 & 30~\text{mm}\\ -0.01467~\text{mm}^{-1} & 0.4\end{pmatrix}, where M1\mathbf{M}_1, M2\mathbf{M}_2 are the refraction matrices of the two lenses (in the (y,θ)(y,\theta) column, air throughout) and T\mathbf{T} is the 30 mm30~\text{mm} translation matrix, and checks det⁡Mwrong=1\det\mathbf{M}_{\text{wrong}}=1 as confirmation that the matrix is correct. ((a)) Find the error in this working. ((b)) Compute the correct system matrix, and state whether the effective focal length changes. ((c)) Compute the front principal-plane position (from the front vertex) and back principal-plane position (from the back vertex) under the correct matrix and under the faulty one, and compare.

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