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Fractional-order Fourier transformations as pseudo-differential operators. Applications to diffraction and imaging by optical systems

Pierre Pellat-Finet

TL;DR

The paper recasts circular and hyperbolic fractional-order Fourier transformations as Weyl pseudo-differential operators with explicit kernels and symbols, enabling Weyl-calculus-based composition rules. It shows that optical field transfers by diffraction can be represented by these transforms and classifies diffraction into three classes via geometrical criteria. A central result is that imaging through a refracting spherical cap corresponds to the composition of two fractional-order transforms of the same kind, yielding fundamental imaging laws such as double conjugation and radius/longitudinal magnification that align with classical geometrical optics. The framework unifies diffraction theory and geometrical imaging under Weyl calculus, providing rigorous tools to analyze field transfers, resonator stability, and centered optical systems. Overall, the work links fractional Fourier optics to a solid pseudo-differential operator theory with clear implications for diffraction-based imaging and optical design.

Abstract

Circular and hyperbolic fractional-order Fourier transformations are actually Weyl pseudo-differential operators. Their associated kernels and symbols are written explicitly. Products of fractional-order Fourier transformations are obtained by composing their kernels or symbols, in accordance with the Weyl calculus. On the other hand, optical field transfers by diffraction, from spherical emitters to receivers, are mathematically expressed using fractional-order Fourier transforms and are classified into three categories, for which we provide geometrical characterizations. Respecting the Huygens-Fresnel principle requires that the associated transformations be adequately composed, which is achieved through the composition rules of pseudo-differential operators. By applying the Weyl calculus, we prove that, in general, imaging through a refracting spherical cap can be described as the composition of two fractional-order Fourier transformations, but only if those transformations are of the same kind. Basic laws of coherent geometrical imaging are thus deduced from the composition of two circular fractional-order Fourier transformations. We also prove that the arrangement of a sequence of points along the axis of a centered optical system is preserved by imaging, up to a circular permutation. This recovers a classical result from geometrical optics, but here it is derived within the framework of diffraction theory.

Fractional-order Fourier transformations as pseudo-differential operators. Applications to diffraction and imaging by optical systems

TL;DR

The paper recasts circular and hyperbolic fractional-order Fourier transformations as Weyl pseudo-differential operators with explicit kernels and symbols, enabling Weyl-calculus-based composition rules. It shows that optical field transfers by diffraction can be represented by these transforms and classifies diffraction into three classes via geometrical criteria. A central result is that imaging through a refracting spherical cap corresponds to the composition of two fractional-order transforms of the same kind, yielding fundamental imaging laws such as double conjugation and radius/longitudinal magnification that align with classical geometrical optics. The framework unifies diffraction theory and geometrical imaging under Weyl calculus, providing rigorous tools to analyze field transfers, resonator stability, and centered optical systems. Overall, the work links fractional Fourier optics to a solid pseudo-differential operator theory with clear implications for diffraction-based imaging and optical design.

Abstract

Circular and hyperbolic fractional-order Fourier transformations are actually Weyl pseudo-differential operators. Their associated kernels and symbols are written explicitly. Products of fractional-order Fourier transformations are obtained by composing their kernels or symbols, in accordance with the Weyl calculus. On the other hand, optical field transfers by diffraction, from spherical emitters to receivers, are mathematically expressed using fractional-order Fourier transforms and are classified into three categories, for which we provide geometrical characterizations. Respecting the Huygens-Fresnel principle requires that the associated transformations be adequately composed, which is achieved through the composition rules of pseudo-differential operators. By applying the Weyl calculus, we prove that, in general, imaging through a refracting spherical cap can be described as the composition of two fractional-order Fourier transformations, but only if those transformations are of the same kind. Basic laws of coherent geometrical imaging are thus deduced from the composition of two circular fractional-order Fourier transformations. We also prove that the arrangement of a sequence of points along the axis of a centered optical system is preserved by imaging, up to a circular permutation. This recovers a classical result from geometrical optics, but here it is derived within the framework of diffraction theory.

Paper Structure

This paper contains 78 sections, 19 theorems, 272 equations, 9 figures, 10 tables.

Table of Contents

  1. Weyl calculus EgoHor3UntNou
  2. Two-dimensional Fourier transformation
  3. Pseudo-differential operators
  4. Definition
  5. Composition of kernels
  6. Weyl pseudo-differential operators
  7. Definition
  8. Composition of symbols
  9. Symbols in ${\mathcal{S}'}({\mathbb R}^2\times {\mathbb R}^2)$ Hor3Nou
  10. The parity operator
  11. Definition
  12. Kernel and symbol
  13. Fractional-order Fourier transformations as Weyl pseudo-differential operators
  14. Three classes of fractional-order Fourier transformations
  15. Circular fractional-order Fourier transformations
  16. ...and 63 more sections

Key Result

Proposition 1

When $\alpha\in ]-\pi ,\pi [$, the circular fractional-order Fourier transformation ${\mathcal{F}}_\alpha$ is a Weyl pseudo-differential operator whose symbol is

Figures (9)

  • Figure 1: Elements for representing diffraction from a spherical emitter ${\mathcal{A}}$ to a spherical receiver ${\mathcal{B}}$ at a distance $D$. Light propagates from left to right. Algebraic measures $D=\overline{V_AV_B}$, $R_A=\overline{V_AC_A}$, etc. are positive if taken along light propagation. For example above: $R_B=\overline{V_BC_B}>0$, whereas $R_A<0$.
  • Figure 2: Fraunhofer diffraction phenomenon. Spherical caps ${\mathcal{A}}$ and ${\mathcal{F}}$ are confocal: the vertex of one is the curvature center of the other ($R_A=D=-R_F$). The field transfer from ${\mathcal{A}}$ to its Fourier sphere ${\mathcal{F}}$ is expressed by an "optical" Fourier transform---see Eq. (\ref{['eq2.1b']}).
  • Figure 3: A sequence of spherical caps with increasing curvatures: $\frak{C}_{A_1}<\frak{C}_{A_2}<0<\frak{C}_{A_3}<\frak{C}_{A_4}<\frak{C}_{A_5}$.
  • Figure 4: Two configurations for which $J>0$. The receiver ${\mathcal{B}}$ is located between ${\mathcal{S}}$ and ${\mathcal{T}}$. The vertices $V_A$ and $V_B$ and the centers of curvature $C_A$ and $C_B$ are arranged along the axis according to $(V_A C_B C_A V_B)$, on the left diagram, and according to $(C_BV_AV_BC_A)$, on the right one.
  • Figure 5: Two configurations for which $J<-1$. The receiver ${\mathcal{B}}$ should not lie between ${\mathcal{S}}$ and ${\mathcal{T}}$. The vertices $V_A$ and $V_B$ and the centers $C_A$ and $C_B$ are ordered along the axis according to $(V_A C_A C_B V_B)$ and to $(V_AV_BC_BC_A)$.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Remark 1.1
  • Proposition 1
  • Proposition 2
  • Remark 2.1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Remark 2.2
  • ...and 17 more