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Development of Linear Canonical Transforms: A Historical Sketch

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Linear Canonical Transforms

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 198))

Abstract

Linear canonical transformations (LCTs) were introduced almost simultaneously during the early 1970s by Stuart A. Collins Jr. in paraxial optics, and independently by Marcos Moshinsky and Christiane Quesne in quantum mechanics, to understand the conservation of information and of uncertainty under linear maps of phase space. Only in the 1990s did both sources begin to be referred jointly in the growing literature, which has expanded into a field common to applied optics, mathematical physics, and analogic and digital signal analysis. In this introductory chapter we recapitulate the construction of the LCT integral transforms, detailing their Lie-algebraic relation with second-order differential operators, which is the origin of the metaplectic phase. Radial and hyperbolic LCTs are reviewed as unitary integral representations of the two-dimensional symplectic group, with complex extension to a semigroup for systems with loss or gain. Some of the more recent developments on discrete and finite analogues of LCTs are commented with their concomitant problems, whose solutions and alternatives are contained the body of this book.

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Notes

  1. 1.

    All lens centers are assumed to be on a common straight optical axis with their planes orthogonal to it; the “center” of cylindrical lenses is a line that should also intersect this axis. The consideration of displacement and (paraxial) tilt can be made using 2 + 2 more parameters for inhomogeneous LCTs, which are not explicitly considered here. See [18].

  2. 2.

    The paper by Collins uses momenta in the form n i p i with \(\vert \mathbf{p}_{i}\vert =\sin \theta _{i}\), and orders the 4-vector components as \((x_{1},p_{1},x_{2},p_{2})^{\top }\).

  3. 3.

    I thank Dr. George Nemeş for the remark that when dimensions are respected, \(\mathbf{F}\neq \boldsymbol{\Omega }\) because the parameters b and 1∕c have units of momentum/position, while a and d have no units. In our presentation of the kernel (1.17) we assume that momentum p bears no units (as in optics), and that a unit of distance has been agreed for position so that x is its numerical multiple.

  4. 4.

    Once I said in front of a large student audience that I had devoted much work to understand 2 × 2 matrices, the giggles in the hall were sobering.

  5. 5.

    For γ < 0 there is a doubling of the Hilbert space that requires some extra analytical finesse [57], which stems from a separation in hyperbolic coordinates such as that seen in the previous subsection.

  6. 6.

    The generators present a one-parameter family of self-adjoint extensions with non-equally spaced spectra [58] and also harbor the \(\mathcal{E}_{k}\) exceptional (or supplementary) representation series [59].

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Acknowledgements

I must thank the inspiring interaction I had with Professor Marcos Moshinsky for many years, the continuing interest of my colleagues, and especially that of the Editors of this volume. I have also incorporated some remarks graciously offered by Professor Stuart A. Collins Jr. Support for this research has been provided by the Óptica Matemática projects of UNAM (papiit in101115) and by the National Council for Science and Technology (sep-conacyt 79899).

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  1. Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Morelos, 62251, México

    Kurt Bernardo Wolf

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Editors and Affiliations

  1. School of Electrical and Electronic Engineering, University College Dublin, Dublin, Ireland

    John J. Healy

  2. The Scientific and Technological Research Council of Turkey, Ankara, Ankara, Turkey

    M. Alper Kutay

  3. Department of Electrical Engineering, Bilkent University, Ankara, Ankara, Turkey

    Haldun M. Ozaktas

  4. School of Electrical and Electronic Engineering, University College Dublin, Dublin, Dublin, Ireland

    John T. Sheridan

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Wolf, K.B. (2016). Development of Linear Canonical Transforms: A Historical Sketch. In: Healy, J., Alper Kutay, M., Ozaktas, H., Sheridan, J. (eds) Linear Canonical Transforms. Springer Series in Optical Sciences, vol 198. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3028-9_1

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