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The Linear Canonical Transformation: Definition and Properties

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Linear Canonical Transforms

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

Abstract

In this chapter we introduce the class of linear canonical transformations, which includes as particular cases the Fourier transformation (and its generalization: the fractional Fourier transformation), the Fresnel transformation, and magnifier, rotation and shearing operations. The basic properties of these transformations—such as cascadability, scaling, shift, phase modulation, coordinate multiplication and differentiation—are considered. We demonstrate that any linear canonical transformation is associated with affine transformations in phase space, defined by time-frequency or position-momentum coordinates. The affine transformation is described by a symplectic matrix, which defines the parameters of the transformation kernel. This alternative matrix description of linear canonical transformations is widely used along the chapter and allows simplifying the classification of such transformations, their eigenfunction identification, the interpretation of the related Wigner distribution and ambiguity function transformations, among many other tasks. Special attention is paid to the consideration of one- and two-dimensional linear canonical transformations, which are more often used in signal processing, optics and mechanics. Analytic expressions for the transforms of some selected functions are provided.

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References

  1. E.G. Abramochkin, V.G. Volostnikov, Generalized Gaussian beams. J. Opt. A Pure Appl. Opt. 6, S157–S161 (2004)

    Article  ADS  Google Scholar 

  2. M. Abramowitz, I.A. Stegun (eds.), Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, 1984)

    Google Scholar 

  3. G.S. Agarwal, SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum. J. Opt. Soc. Am. A 16, 2914–2916 (1999)

    Article  ADS  Google Scholar 

  4. O. Akay, G.F. Boudreaux-Bartels, Fractional convolution and correlation via operator methods and an application to detection of linear FM signals. IEEE Trans. Signal Process. 49, 979–993 (2001)

    Article  ADS  Google Scholar 

  5. T. Alieva, M.J. Bastiaans, Alternative representation of the linear canonical integral transform. Opt. Lett. 30, 3302–3304 (2005)

    Article  ADS  Google Scholar 

  6. T. Alieva, M.J. Bastiaans, Mode mapping in paraxial lossless optics. Opt. Lett. 30, 1461–1463 (2005)

    Article  ADS  Google Scholar 

  7. T. Alieva, M.J. Bastiaans, Properties of the linear canonical integral transformation. J. Opt. Soc. Am. A 24, 3658–3665 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  8. T. Alieva, M.J. Bastiaans, Phase-space rotations and orbital Stokes parameters. Opt. Lett. 34, 410–412 (2009)

    Article  ADS  Google Scholar 

  9. T. Alieva, M.J. Bastiaans, M.L. Calvo, Fractional transforms in optical information processing. EURASIP J. Appl. Signal Process. 2005, 1498–1519 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero. Twisted Gaussian Schell-model beams: a superposition model. J. Mod. Opt. 41, 1391–1399 (1994)

    Article  ADS  Google Scholar 

  11. M.J. Bastiaans, The Wigner distribution function applied to optical signals and systems. Opt. Commun. 25, 26–30 (1978)

    Article  ADS  Google Scholar 

  12. M.J. Bastiaans, Wigner distribution function and its application to first-order optics. J. Opt. Soc. Am. 69, 1710–1716 (1979)

    Article  ADS  Google Scholar 

  13. M.J. Bastiaans, Application of the Wigner distribution function to partially coherent light. J. Opt. Soc. Am. A 3, 1227–1238 (1986)

    Article  ADS  Google Scholar 

  14. M.J. Bastiaans, Second-order moments of the Wigner distribution function in first-order optical systems. Optik 88, 163–168 (1991)

    Google Scholar 

  15. M.J. Bastiaans, ABCD law for partially coherent Gaussian light, propagating through first-order optical systems. Opt. Quant. Electron. 24, 1011–1019 (1992)

    Article  Google Scholar 

  16. M.J. Bastiaans, Application of the Wigner distribution function in optics, in The Wigner Distribution: Theory and Applications in Signal Processing, ed. by W. Mecklenbräuker, F. Hlawatsch (Elsevier Science, Amsterdam, 1997), pp. 375–426

    Google Scholar 

  17. M.J. Bastiaans, Wigner distribution function applied to twisted Gaussian light propagating in first-order optical systems. J. Opt. Soc. Am. A 17, 2475–2480 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  18. M.J. Bastiaans, Applications of the Wigner distribution to partially coherent light beams, in Advances in Information Optics and Photonics, ed. by A.T. Friberg, R. Dändliker (SPIE – The International Society for Optical Engineering, Bellingham, WA, 2008), pp. 27–56

    Chapter  Google Scholar 

  19. M.J. Bastiaans, Wigner distribution in optics, in Phase-Space Optics: Fundamentals and Applications, ed. by M.E. Testorf, B.M. Hennelly, J. Ojeda-Castañeda (McGraw-Hill, New York, 2009), pp. 1–44

    Google Scholar 

  20. M.J. Bastiaans, T. Alieva, Generating function for Hermite-Gaussian modes propagating through first-order optical systems. J. Phys. A Math. Gen. 38, L73–L78 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. M.J. Bastiaans, T. Alieva, Propagation law for the generating function of Hermite-Gaussian-type modes in first-order optical systems. Opt. Express 13, 1107–1112 (2005)

    Article  ADS  Google Scholar 

  22. M.J. Bastiaans, T. Alieva, First-order optical systems with unimodular eigenvalues. J. Opt. Soc. Am. A 23, 1875–1883 (2006)

    Article  ADS  Google Scholar 

  23. M.J. Bastiaans, T. Alieva, Classification of lossless first-order optical systems and the linear canonical transformation. J. Opt. Soc. Am. A 24, 1053–1062 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  24. M.J. Bastiaans, T. Alieva, First-order optical systems with real eigenvalues. Opt. Commun. 272, 52–55 (2007)

    Article  ADS  Google Scholar 

  25. M.J. Bastiaans, T. Alieva, Signal representation on the angular Poincaré sphere, based on second-order moments. J. Opt. Soc. Am. A 27, 918–927 (2010)

    Article  ADS  Google Scholar 

  26. S.A. Collins Jr., Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60, 1168–1177 (1970)

    Article  ADS  Google Scholar 

  27. L.S. Dolin, Beam description of weakly-inhomogeneous wave fields. Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964)

    Google Scholar 

  28. D. Dragoman, The Wigner distribution function in optics and optoelectronics. Prog. Opt. 37, 1–56 (1997)

    Article  Google Scholar 

  29. A.T. Friberg, B. Tervonen, J. Turunen, Interpretation and experimental demonstration of twisted Gaussian Schell-model beams. J. Opt. Soc. Am. A 11, 1818–1826 (1994)

    Article  ADS  Google Scholar 

  30. F.R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1974–1977)

    Google Scholar 

  31. J. García, D. Mendlovic, Z. Zalevsky, A.W. Lohmann, Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier transform filters. Appl. Opt. 35, 3945–3952 (1996)

    Article  ADS  Google Scholar 

  32. G. García-Calderón, M. Moshinsky, Wigner distribution functions and the representations of canonical transformations in quantum mechanics. J. Phys. A 13, L185–L188 (1980)

    Article  ADS  Google Scholar 

  33. J.W. Goodman, Introduction to Fourier Optics, 2nd edn. (McGraw-Hill, New York, 1996)

    Google Scholar 

  34. I.S. Gradshteyn, I.M. Ryzhik, A. Jeffrey (eds.), Table of Integrals, Series, and Products, 5th edn. (Academic, San Diego, CA, 1994)

    MATH  Google Scholar 

  35. A.J.E.M. Janssen, Positivity of weighted Wigner distributions. J. Math. Anal. 12, 752–758 (1981)

    MATH  Google Scholar 

  36. G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (McGraw-Hill, London, 1968)

    MATH  Google Scholar 

  37. W.W. Lin, V. Mehrmann, H. Xu, Canonical forms for Hamiltonian and symplectic matrices and pencils. Linear Algebra Appl. 302–303, 469–533 (1999)

    Article  MathSciNet  Google Scholar 

  38. A.W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform. J. Opt. Soc. Am. A 10, 2181–2186 (1993)

    Article  ADS  Google Scholar 

  39. A.W. Lohmann, Z. Zalevsky, D. Mendlovic, Synthesis of pattern recognition filters for fractional Fourier processing. Opt. Commun. 128, 199–204 (1996)

    Article  ADS  Google Scholar 

  40. R.K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, CA, 1966)

    Google Scholar 

  41. A.A. Malyutin, Use of the fractional Fourier transform in π∕2 converters of laser modes. Quantum Electron. 34, 165–171 (2004)

    Article  ADS  Google Scholar 

  42. A.C. McBride, F.H. Kerr, On Namias fractional Fourier transforms. IMA J. Appl. Math. 39, 159–175 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  43. W. Mecklenbräuker, F. Hlawatsch (eds.), The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier Science, Amsterdam, 1997)

    MATH  Google Scholar 

  44. M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations. J. Math. Phys. 12, 1772–1780 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. V. Namias, The fractional order Fourier transform and its applications to quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Nazarathy, J. Shamir, First-order optics—a canonical operator representation: lossless systems. J. Opt. Soc. Am. 72, 356–364 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  47. H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001)

    Google Scholar 

  48. M.J. Padgett, J. Courtial, Poincaré-sphere equivalent for light beams containing orbital angular momentum. Opt. Lett. 24, 430–432 (1999)

    Article  ADS  Google Scholar 

  49. S.C. Pei, J.J. Ding, Eigenfunctions of linear canonical transform. IEEE Trans. Signal Process. 50, 11–26 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  50. A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev (eds.), Elementary Functions. Integrals and Series, vol. 1 (Gordon and Breach, New York, 1986)

    Google Scholar 

  51. C. Quesne, M. Moshinsky, Canonical transformations and matrix elements. J. Math. Phys 12, 1780–1783 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. J. Serna, R. Martínez-Herrero, P.M. Mejías, Parametric characterization of general partially coherent beams propagating through ABCD optical systems. J. Opt. Soc. Am. A 8, 1094–1098 (1991)

    Article  ADS  Google Scholar 

  53. R. Simon, G.S. Agarwal, Wigner representation of Laguerre-Gaussian beams. Opt. Lett. 25, 1313–1315 (2000)

    Article  ADS  Google Scholar 

  54. R. Simon, N. Mukunda, Twisted Gaussian Schell-model beams. J. Opt. Soc. Am. A 10, 95–109 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  55. R. Simon, N. Mukunda, Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams. J. Opt. Soc. Am. A 15, 2146–2155 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  56. R. Simon, N. Mukunda, Twist phase in Gaussian-beam optics. J. Opt. Soc. Am. A 15, 2373–2382 (1998)

    Article  ADS  Google Scholar 

  57. R. Simon, K.B. Wolf, Fractional Fourier transforms in two dimensions. J. Opt. Soc. Am. A 17, 2368–2381 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  58. R. Simon, K.B. Wolf, Structure of the set of paraxial optical systems. J. Opt. Soc. Am. A 17, 342–355 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  59. R. Simon, N. Mukunda, E.C.G. Sudarshan, Partially coherent beams and a generalized ABCD-law. Opt. Commun. 65, 322–328 (1988)

    Article  ADS  Google Scholar 

  60. R. Simon, K. Sundar, N. Mukunda, Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum. J. Opt. Soc. Am. A 10, 2008–2016 (1993)

    Google Scholar 

  61. R. Simon, A.T. Friberg, E. Wolf, Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems. J. Eur. Opt. Soc. A Pure Appl. Opt 5, 331–343 (1996)

    Article  ADS  Google Scholar 

  62. K. Sundar, N. Mukunda, R. Simon, Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams. J. Opt. Soc. Am. A 12, 560–569 (1995)

    Article  ADS  Google Scholar 

  63. K. Sundar, R. Simon, N. Mukunda, Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics. J. Opt. Soc. Am. A 10, 2017–2023 (1993)

    Google Scholar 

  64. M.E. Testorf, J. Ojeda-Castañeda, A.W. Lohmann, (eds.), Selected Papers on Phase-Space Optics, SPIE Milestone Series, vol. MS 181 (SPIE, Bellingham, WA, 2006)

    Google Scholar 

  65. M.E. Testorf, B.M. Hennelly, J. Ojeda-Castañeda (eds.), Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, New York, 2009)

    Google Scholar 

  66. A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, Amsterdam, 2005)

    Google Scholar 

  67. A. Walther, Radiometry and coherence. J. Opt. Soc. Am. 58, 1256–1259 (1968)

    Article  ADS  Google Scholar 

  68. A. Walther, Propagation of the generalized radiance through lenses. J. Opt. Soc. Am. 68, 1606–1610 (1978)

    Article  ADS  Google Scholar 

  69. E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)

    Article  ADS  Google Scholar 

  70. J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58, 141–163 (1936)

    Article  Google Scholar 

  71. K.B. Wolf, Canonical transforms. I. Complex linear transforms. J. Math. Phys. 15, 1295–1301 (1974)

    MATH  Google Scholar 

  72. K.B. Wolf, Canonical transforms. II. Complex radial transforms. J. Math. Phys. 15, 2102–2111 (1974)

    MATH  Google Scholar 

  73. K.B. Wolf, Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations. J. Math. Phys. 17, 601–613 (1976)

    Article  ADS  MATH  Google Scholar 

  74. K.B. Wolf, Integral Transforms in Science and Engineering, Chap. 9 (Plenum Press, New York, 1979)

    Google Scholar 

  75. K.B. Wolf, Geometric Optics on Phase Space (Springer, Berlin, 2004)

    MATH  Google Scholar 

  76. P.M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, London, 1953)

    MATH  Google Scholar 

  77. B. Zhu, S. Liu, Optical image encryption based on the generalized fractional convolution operation. Opt. Commun. 195, 371–381 (2001)

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Eindhoven University of Technology, 513, 5600 MB, Eindhoven, Netherlands

    Martin J. Bastiaans

  2. Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Ciudad Universitaria s/n, Madrid, 28040, Spain

    Tatiana Alieva

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  1. Martin J. Bastiaans
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  2. Tatiana Alieva
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Correspondence to Martin J. Bastiaans .

Editor information

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

Appendices

Appendix

Derivation of the Phase-Space Relation (2.5)

We start with (2.4) and substitute from (2.1):

$$\displaystyle\begin{array}{rcl} W_{o}(\mathbf{r}_{o},\mathbf{q}_{o})& =& \int f\left (\mathbf{r}_{o} + \tfrac{1} {2}\mathbf{r}_{o}'\right )\,f^{{\ast}}\left (\mathbf{r}_{ o} -\tfrac{1} {2}\mathbf{r}_{o}'\right )\,\exp [-\mathrm{i\,}2\pi \,\mathbf{q}_{o}^{\,t}\mathbf{r}_{ o}']\,\mathrm{d}\mathbf{r}_{o}' {}\\ & =& \vert \det \mathbf{L}_{io}\vert \iiint \exp [\mathrm{i\,}\pi (E_{1} - E_{2} - 2\mathbf{q}_{o}^{\,t}\mathbf{r}_{ o}')]\,f_{i}(\mathbf{r}_{1})\,f_{i}^{{\ast}}(\mathbf{r}_{ 2})\,\mathrm{d}\mathbf{r}_{1}\,\mathrm{d}\mathbf{r}_{2}\,\mathrm{d}\mathbf{r}_{o}' {}\\ \end{array}$$
$$\displaystyle{ \mbox{ with}\quad \left \{\begin{array}{l} E_{1} = \left (\mathbf{r}_{o} + \frac{1} {2}\mathbf{r}_{o}'\right )^{t}\mathbf{L}_{ oo}\left (\mathbf{r}_{o} + \frac{1} {2}\mathbf{r}_{o}'\right ) - 2\mathbf{r}_{1}^{\,t}\mathbf{L}_{ io}\left (\mathbf{r}_{o} + \frac{1} {2}\mathbf{r}_{o}'\right ) + \mathbf{r}_{1}^{\,t}\mathbf{L}_{ ii}\mathbf{r}_{1}\;, \\ E_{2} = \left (\mathbf{r}_{o} -\frac{1} {2}\mathbf{r}_{o}'\right )^{t}\mathbf{L}_{ oo}\left (\mathbf{r}_{o} -\frac{1} {2}\mathbf{r}_{o}'\right ) - 2\mathbf{r}_{2}^{\,t}\mathbf{L}_{ io}\left (\mathbf{r}_{o} -\frac{1} {2}\mathbf{r}_{o}'\right ) + \mathbf{r}_{2}^{\,t}\mathbf{L}_{ ii}\mathbf{r}_{2}\;.\end{array} \right. }$$

We reorder the exponent E 1 − E 2 to get

$$\displaystyle\begin{array}{rcl} E_{1} - E_{2}& =& (\mathbf{r}_{o}'^{\,t}\mathbf{L}_{ oo}\mathbf{r}_{o} + \mathbf{r}_{o}^{\,t}\mathbf{L}_{ oo}\mathbf{r}_{o}') - 2(\mathbf{r}_{1} -\mathbf{r}_{2})^{t}\mathbf{L}_{ io}\mathbf{r}_{o} - (\mathbf{r}_{1} + \mathbf{r}_{2})^{t}\mathbf{L}_{ io}\mathbf{r}_{o}' {}\\ & & +(\mathbf{r}_{1}^{\,t}\mathbf{L}_{ ii}\mathbf{r}_{1} -\mathbf{r}_{2}^{\,t}\mathbf{L}_{ ii}\mathbf{r}_{2})\;. {}\\ \end{array}$$

We substitute \(\mathbf{r}_{1} = \mathbf{r}_{i} + \frac{1} {2}\mathbf{r}_{i}'\) and \(\mathbf{r}_{2} = \mathbf{r}_{i} -\frac{1} {2}\mathbf{r}_{i}'\) and get

$$\displaystyle\begin{array}{rcl} W_{o}(\mathbf{r}_{o},\mathbf{q}_{o})& =& \vert \det \mathbf{L}_{io}\vert \iiint \exp [\mathrm{i\,}\pi (E_{1} - E_{2} - 2\mathbf{q}_{o}^{\,t}\mathbf{r}_{ o}')] {}\\ & & \times f_{i}\left (\mathbf{r}_{i} + \tfrac{1} {2}\mathbf{r}_{i}'\right )\,f_{i}^{{\ast}}\left (\mathbf{r}_{ i} -\tfrac{1} {2}\mathbf{r}_{i}'\right )\,\mathrm{d}\mathbf{r}_{i}\,\mathrm{d}\mathbf{r}_{i}'\,\mathrm{d}\mathbf{r}_{o}' {}\\ \end{array}$$

with \(E_{1} -E_{2} = (\mathbf{r}_{o}'^{\,t}\mathbf{L}_{oo}\mathbf{r}_{o} + \mathbf{r}_{o}^{\,t}\mathbf{L}_{oo}\mathbf{r}_{o}') -\mathbf{r}_{i}'^{\,t}\mathbf{L}_{io}\mathbf{r}_{o} - 2\mathbf{r}_{i}^{\,t}\mathbf{L}_{io}\mathbf{r}_{o}' + (\mathbf{r}_{i}'^{\,t}\mathbf{L}_{ii}\mathbf{r}_{i} + \mathbf{r}_{i}^{\,t}\mathbf{L}_{ii}\mathbf{r}_{i}')\;.\)

We substitute \(f_{i}\left (\mathbf{r}_{i} + \frac{1} {2}\mathbf{r}_{i}'\right )\,f_{i}^{{\ast}}\left (\mathbf{r}_{ i} -\frac{1} {2}\mathbf{r}_{i}'\right ) =\int W_{i}(\mathbf{r}_{i},\mathbf{q}_{i})\,\exp [\mathrm{i\,}2\pi \,\mathbf{r}_{i}'^{\,t}\mathbf{q}_{ i}]\,\mathrm{d}\mathbf{q}_{i}\) and get

$$\displaystyle\begin{array}{rcl} W_{o}(\mathbf{r}_{o},\mathbf{q}_{o})\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& & {}\\ & =& \vert \det \mathbf{L}_{io}\vert \mathop{\iint \iint }\nolimits W_{i}(\mathbf{r}_{i},\mathbf{q}_{i})\exp [\mathrm{i\,}\pi (E_{1} - E_{2} - 2\mathbf{q}_{o}^{\,t}\mathbf{r}_{ o}'] {}\\ & & \times \,\exp [\mathrm{i\,}2\pi \,\mathbf{r}_{i}'^{\,t}\mathbf{q}_{ i}]\,\mathrm{d}\mathbf{r}_{i}'\,\mathrm{d}\mathbf{r}_{o}'\,\mathrm{d}\mathbf{r}_{i}\,\mathrm{d}\mathbf{q}_{i} {}\\ & =& \vert \det \mathbf{L}_{io}\vert \iint W_{i}(\mathbf{r}_{i},\mathbf{q}_{i})\left (\int \exp [\mathrm{i\,}2\pi (\mathbf{L}_{oo}\mathbf{r}_{o} -\mathbf{L}_{io}^{t}\mathbf{r}_{ i} -\mathbf{q}_{o})^{t}\mathbf{r}_{ o}']\,\mathrm{d}\mathbf{r}_{o}'\right ) {}\\ & & \times \left (\int \exp [\mathrm{i\,}2\pi (\mathbf{L}_{ii}\mathbf{r}_{i} -\mathbf{L}_{io}\mathbf{r}_{o} + \mathbf{q}_{i})^{t}\mathbf{r}_{ i}']\,\mathrm{d}\mathbf{r}_{i}'\right )\,\mathrm{d}\mathbf{r}_{i}\,\mathrm{d}\mathbf{q}_{i} {}\\ & =& \vert \det \mathbf{L}_{io}\vert \iint W_{i}(\mathbf{r}_{i},\mathbf{q}_{i})\,\delta (\mathbf{L}_{oo}\mathbf{r}_{o} -\mathbf{L}_{io}^{t}\mathbf{r}_{ i} -\mathbf{q}_{o})\,\delta (\mathbf{L}_{ii}\mathbf{r}_{i} -\mathbf{L}_{io}\mathbf{r}_{o} + \mathbf{q}_{i})\,\mathrm{d}\mathbf{r}_{i}\,\mathrm{d}\mathbf{q}_{i} {}\\ & =& \vert \det \mathbf{L}_{io}\vert \int W_{i}(\mathbf{r}_{i},\mathbf{L}_{io}\mathbf{r}_{o} -\mathbf{L}_{ii}\mathbf{r}_{i})\,\delta (\mathbf{L}_{oo}\mathbf{r}_{o} -\mathbf{L}_{io}^{t}\mathbf{r}_{ i} -\mathbf{q}_{o})\,\mathrm{d}\mathbf{r}_{i} {}\\ & =& W_{i}\left (\mathbf{L}_{io}^{t-1}\mathbf{L}_{ oo}\mathbf{r}_{o} -\mathbf{L}_{io}^{t-1}\mathbf{q}_{ o},\mathbf{L}_{io}\mathbf{r}_{o} -\mathbf{L}_{ii}\mathbf{L}_{io}^{t-1}\mathbf{L}_{ oo}\mathbf{r}_{o} + \mathbf{L}_{ii}\mathbf{L}_{io}^{t-1}\mathbf{q}_{ o}\right )\;.{}\\ \end{array}$$

After substituting from (2.7), we finally get

$$\displaystyle{ W_{o}(\mathbf{r}_{o},\mathbf{q}_{o}) = W_{i}(\mathbf{D}^{\,t}\mathbf{r}_{ o} -\mathbf{B}^{\,t}\mathbf{q}_{ o},-\mathbf{C}^{\,t}\mathbf{r}_{ o} + \mathbf{A}^{t}\mathbf{q}_{ o}) }$$

and hence \(W_{o}(\mathbf{A}\mathbf{r} + \mathbf{B}\mathbf{q},\mathbf{C}\mathbf{r} + \mathbf{D}\mathbf{q}) = W_{i}(\mathbf{r},\mathbf{q})\), which is identical to (2.5).

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Bastiaans, M.J., Alieva, T. (2016). The Linear Canonical Transformation: Definition and Properties. 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_2

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