The properties of first-order optical systems are described paraxially by a ray transfer matrix, also called the ABCD matrix. Here we consider the inverse problem: an ABCD matrix is given, and we look for the minimal optical system that consists of only lenses and pieces of free-space propagation. Similar decompositions have been studied before but without the restriction to these two element types or without an attempt at minimalization. As the main results of this paper, we found that general lossless one- dimensional optical systems can be synthesized with a maximum of four elements and two-dimensional optical systems can be synthesized with six elements at most.
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Here scale S is a symplectic diagonal matrix with only three parameters, and Ψ is symplectic phase space rotation with only two parameters. The submatrix in the QR, QL, LQ, and RQ decompositions is a symmetric matrix. In the last decomposition, the parameters , , , and denote that the ABCD matrix is first transformed to a matrix with symmetric submatrix or using a rotation matrix.
Tables (4)
Table 1
Various Matrix Decompositions Using the Operators: Lens L, Free Space P, Scale S and Rotation Ψa
Here scale S is a symplectic diagonal matrix with only three parameters, and Ψ is symplectic phase space rotation with only two parameters. The submatrix in the QR, QL, LQ, and RQ decompositions is a symmetric matrix. In the last decomposition, the parameters , , , and denote that the ABCD matrix is first transformed to a matrix with symmetric submatrix or using a rotation matrix.