This chapter discusses the phenomenon of diffraction, which describes the spreading of light waves as they propagate through apertures or around obstacles, leading to intensity patterns that deviate from the predictions of geometrical optics. It begins by explaining how light propagation in vacuum can be understood as an expansion in plane waves, where the transverse spatial frequencies determine the angular divergence of the beam. The uncertainty principle is invoked to show that a finite beam size inherently leads to a minimum angular divergence. The core of the chapter introduces the Huygens–Fresnel principle, which states that every point on a wavefront can be considered a source of secondary spherical wavelets, and the total field at any point is the superposition of these wavelets. This principle is mathematically formalized through the Huygens–Fresnel integral, providing a method to calculate diffraction patterns. The counter-intuitive Fresnel-Arago-Poisson spot (a bright spot in the center of a shadow cast by an opaque disk) is presented as a compelling experimental validation of the wave theory of light, alongside Babinet’s theorem, which relates the diffraction patterns of complementary obstacles. The chapter then introduces the Fresnel (paraxial) approximation, a simplification of the Huygens–Fresnel integral valid for small angles and distances much larger than the wavelength. This approximation leads to the paraxial wave equation, a fundamental equation for describing beam propagation in optics. The behavior of Gaussian beams, a common solution to this equation that describes laser beams, is analyzed in detail, including their waist, Rayleigh length, and divergence. A significant section explores diffraction by thin lenses, demonstrating how lenses perform a Fourier transform of the incident light field at their focal plane, a concept central to Fourier optics. This principle is applied to spatial filtering, where masks in the Fourier plane can selectively block or transmit spatial frequencies, enabling image processing techniques like edge enhancement. The chapter also discusses image formation and the diffraction limit, showing that the image of a point source formed by a lens is an Airy pattern, which defines the fundamental limit of resolution for optical imaging systems (the Rayleigh criterion and numerical aperture). Finally, the chapter distinguishes between Fresnel diffraction (near-field) and Fraunhofer diffraction (far-field), providing analytical solutions for common geometries like single and double slits. It concludes by demonstrating how diffraction gratings utilize the principles of interference and diffraction to function as spectrometers, spatially separating different wavelengths of light and enabling high-resolution spectral analysis, characterized by their resolving power.

Diffraction

章节
发表日期：

2025年

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Fabian Cadiz,
Arnaud Couairon

摘要

This chapter discusses the phenomenon of diffraction, which describes the spreading of light waves as they propagate through apertures or around obstacles, leading to intensity patterns that deviate from the predictions of geometrical optics. It begins by explaining how light propagation in vacuum can be understood as an expansion in plane waves, where the transverse spatial frequencies determine the angular divergence of the beam. The uncertainty principle is invoked to show that a finite beam size inherently leads to a minimum angular divergence. The core of the chapter introduces the Huygens–Fresnel principle, which states that every point on a wavefront can be considered a source of secondary spherical wavelets, and the total field at any point is the superposition of these wavelets. This principle is mathematically formalized through the Huygens–Fresnel integral, providing a method to calculate diffraction patterns. The counter-intuitive Fresnel-Arago-Poisson spot (a bright spot in the center of a shadow cast by an opaque disk) is presented as a compelling experimental validation of the wave theory of light, alongside Babinet’s theorem, which relates the diffraction patterns of complementary obstacles. The chapter then introduces the Fresnel (paraxial) approximation, a simplification of the Huygens–Fresnel integral valid for small angles and distances much larger than the wavelength. This approximation leads to the paraxial wave equation, a fundamental equation for describing beam propagation in optics. The behavior of Gaussian beams, a common solution to this equation that describes laser beams, is analyzed in detail, including their waist, Rayleigh length, and divergence. A significant section explores diffraction by thin lenses, demonstrating how lenses perform a Fourier transform of the incident light field at their focal plane, a concept central to Fourier optics. This principle is applied to spatial filtering, where masks in the Fourier plane can selectively block or transmit spatial frequencies, enabling image processing techniques like edge enhancement. The chapter also discusses image formation and the diffraction limit, showing that the image of a point source formed by a lens is an Airy pattern, which defines the fundamental limit of resolution for optical imaging systems (the Rayleigh criterion and numerical aperture). Finally, the chapter distinguishes between Fresnel diffraction (near-field) and Fraunhofer diffraction (far-field), providing analytical solutions for common geometries like single and double slits. It concludes by demonstrating how diffraction gratings utilize the principles of interference and diffraction to function as spectrometers, spatially separating different wavelengths of light and enabling high-resolution spectral analysis, characterized by their resolving power.