Part II: Optics

Published

September 29, 2026

Work in Progress

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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 using the ray approximation, focusing on image formation with a pinhole camera.

  • Explains image formation with a pinhole camera using ray tracing.
  • Discusses on-axis and off-axis points and collimated light.
  • Covers pinhole models in computer graphics and historical camera obscuras.

The Wave Theory of Light

This chapter explores why light must be modeled as a wave and how wave optics triumphed over the corpuscular theory.

  • Confronts the failure of the simple ray model near small apertures.
  • Introduces Huygens’ wave model and secondary wavelets.
  • Recounts Young’s double-slit experiment and Fresnel’s mathematics confirmed by Poisson’s spot.
  • Explains polarization as evidence of transverse electromagnetic waves.

Diffraction and the PSF

This chapter analyzes the physical limits of diffraction and introduces the point spread function (PSF).

  • Calculates the trade-off between geometric pinhole blur and wave diffraction (the Airy disk).
  • Introduces the point spread function (PSF) as the fundamental building block of optical imaging.
  • Explores diffraction patterns from rectangular and polygonal apertures.
  • Examines the PSF and optical aberrations of the human eye.

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.

Optics resources

  • Glossary: Optics: A reference table and guide clarifying core terminology—including lens surface geometry (radius of curvature \(R\) and curvature \(C\)) and optical distances (focal length \(f\), conjugate image distance \(d_i\), and physical sensor plane \(d_s\)).
  • The Ideal Thin Lens: Conditions and surface profiles (aspheric hyperboloids and Coddington best-form spherical singlets) for achieving diffraction-limited focus.
  • Multi-Element Lens Calculations: The ABCD Matrix: Ray transfer matrix analysis for multi-element and thick lens systems.
  • The Poisson Spot and Wave Theory: Historical context and experimental demonstrations of Huygens’ wave theory.
  • Diffraction and Wave Propagation Notes: Further mathematical derivations and notes on wave optics and diffraction.