Part II: Optics

Published

September 22, 2025

Work in Progress

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Last updated: September 22, 2025

This part of the book describes the fundamental principles of image formation. We transition from the abstract concept of the light field to the practical physics of how a lens gathers light and forms an image. The chapters build progressively, starting with the intuitive ray-tracing model of geometric optics, introducing the essential formulas for lenses, and then exploring the inevitable imperfections (aberrations) of real-world systems. The final chapters introduce a more powerful linear systems framework, using concepts like the point spread function and Fourier transforms, which is essential for understanding modern image quality analysis and computational optics.

Optics topics

Geometric Optics

This chapter introduces the foundational models of light propagation. It begins with the simplest imaging device, the pinhole camera, and immediately confronts the limitations of a pure ray-based model by introducing the phenomenon of diffraction.

  • Explains image formation with a pinhole camera.
  • Introduces diffraction and the failure of the simple ray model.
  • Describes Huygens’ wave model as an alternative.
  • Recounts the historical Fresnel-Poisson-Arago experiment, which provided definitive proof for the wave nature of light.

Lenses

Building on geometric optics, this chapter derives the fundamental principles of how lenses work. It covers the key formulas that govern ideal thin lenses and introduces essential photographic concepts related to aperture and focus.

  • Derives Snell’s Law for refraction at a surface.
  • Develops the thin lens and lensmaker’s equations.
  • Defines and explains f-number, aperture, and depth of field.
  • Introduces basic concepts of radiometry, including radiance and irradiance.

Aberrations

This chapter addresses the ways in which real lenses deviate from the idealized thin lens model. It categorizes and provides intuitive explanations for the primary optical aberrations that limit image quality.

  • Distinguishes between chromatic (wavelength-dependent) and monochromatic (geometric) aberrations.
  • Explains longitudinal and transverse chromatic aberration (LCA, TCA).
  • Describes the five primary monochromatic aberrations: spherical, coma, astigmatism, field curvature, and distortion.

Advanced Lenses and Matrices

Here, we move beyond single lenses to analyze realistic, multi-element systems. The chapter introduces the formalism for thick lenses and presents the powerful ray transfer matrix (ABCD) method for systematically analyzing complex optical paths.

  • Discusses historical multi-element lens designs (Tessar, Petzval).
  • Introduces the concept of thick lenses and their cardinal points (principal planes, focal points).
  • Provides a step-by-step derivation of the ABCD ray transfer matrix method.
  • Shows how to use the system matrix to calculate properties like effective focal length.

The Spatial Domain

This chapter reframes optical systems as linear signal processors. It introduces the point spread function (PSF) as the fundamental descriptor of a system’s performance in the spatial domain.

  • Models an optical system as a linear, shift-invariant (LSI) system.
  • Defines the point spread function (PSF) as the system’s impulse response.
  • Describes common analytical PSFs used in simulation (Airy, Gaussian, Pillbox).
  • Explores how chromatic aberrations affect the PSF.
  • Introduces PSF engineering with the example of a coronagraph for exoplanet imaging.

The Transform Domain

The final chapter in this part analyzes optical systems in the frequency domain. It explains why harmonic functions are the “eigenfunctions” of LSI systems and introduces the Fourier tools used to characterize system performance.

  • Introduces the Optical Transfer Function (OTF) as the Fourier transform of the PSF.
  • Defines the Modulation Transfer Function (MTF) and Phase Transfer Function (PTF).
  • Explains how the MTF describes a system’s ability to reproduce contrast at different spatial frequencies.
  • Discusses the application of 2D Fourier transforms for analyzing and filtering images.