Week 18 — Electrodynamics and Optics: Fields, Waves, and Imaging

Overview

The final week ties the course together: the vector calculus of this week’s fields is the multivariable extension of Weeks 1–2, Maxwell’s equations produce electromagnetic waves, and those waves — through geometrical optics, diffraction, and Fourier optics — are how cameras and sensors form images. Fourier optics in particular closes the loop with the signal processing of Weeks 12–13: an imaging system is a linear, (shift-invariant) system whose transfer function lives in spatial frequency. This is the physics under camera and sensor simulation.

Readings

Griffiths Ch 1 — Vector analysis. Gradient, divergence, curl, the divergence and Stokes theorems, the Dirac delta, curvilinear coordinates (Appendix A).
Griffiths §5.1 — The Lorentz force law.
Griffiths Ch 7 — Electrodynamics. Faraday’s law, Maxwell’s correction, Maxwell’s equations.
Griffiths Ch 9 — Electromagnetic waves. Waves in vacuum and matter, reflection/transmission.
Hecht Ch 2 — Wave motion; Ch 5–6 — geometrical optics; Ch 10 — diffraction; Ch 11 — Fourier optics.

Key Concepts

Vector calculus of fields

Scalar and vector fields are differentiated by the gradient, divergence, and curl. The fundamental theorems relate a field’s derivative on a region to its values on the boundary — the divergence theorem and Stokes’ theorem:

\[\int_V (\nabla\!\cdot\!\mathbf{F})\,dV = \oint_{\partial V} \mathbf{F}\cdot d\mathbf{A}, \qquad \int_S (\nabla\times\mathbf{F})\cdot d\mathbf{A} = \oint_{\partial S} \mathbf{F}\cdot d\boldsymbol{\ell}.\]

These are the higher-dimensional fundamental theorem of calculus, and they convert Maxwell’s integral and differential forms into each other.

Maxwell’s equations and the Lorentz force

The four Maxwell equations (in vacuum, differential form) are

\[\nabla\!\cdot\!\mathbf{E} = \rho/\epsilon_0,\quad \nabla\!\cdot\!\mathbf{B} = 0,\quad \nabla\times\mathbf{E} = -\partial_t\mathbf{B},\quad \nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\,\partial_t\mathbf{E},\]

and the Lorentz force \(\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})\) says how fields act on charge. Faraday’s law and Maxwell’s displacement-current correction are what make electrodynamics dynamic — and what produce waves.

Electromagnetic waves

In source-free regions Maxwell’s equations combine into the wave equation

\[\nabla^2 \mathbf{E} = \frac{1}{c^2}\,\partial_t^2 \mathbf{E}, \qquad c = 1/\sqrt{\mu_0\epsilon_0},\]

with plane-wave solutions \(\mathbf{E} = \mathbf{E}_0\,e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}\). Light is an EM wave; its speed in matter sets the refractive index, and boundary conditions give the reflection/transmission (Fresnel) relations. This is the continuum analogue of the oscillator (Week 16).

Wave motion and geometrical optics

Hecht’s wave motion covers harmonic waves, phase, and superposition — the same Fourier vocabulary as Week 12. In the short-wavelength limit, geometrical optics treats light as rays: reflection, Snell’s law of refraction, and imaging by lenses/mirrors via the thin-lens equation

\[\frac{1}{s_o} + \frac{1}{s_i} = \frac{1}{f}.\]

Stops, apertures, and aberrations (Ch 6) describe real optical systems and cameras.

Diffraction and Fourier optics

When wavelength is not negligible, diffraction spreads light through apertures. In the Fraunhofer (far-field) regime the diffraction pattern is the Fourier transform of the aperture function — the same transform as Week 13, now over spatial coordinates. Fourier optics formalizes this: an imaging system acts as a linear shift-invariant system on the image, characterized by a spatial-frequency transfer function (the OTF/MTF). Image formation is therefore convolution with a point-spread function — directly the LTI/convolution theory of Week 10 applied to optics.

Connections

Backward: the wave equation extends the oscillator (Week 16); Fourier optics is the Fourier/LTI theory of Weeks 12–13 in two spatial dimensions; the field calculus extends the linear algebra of Weeks 1–2.
Across courses: camera/sensor image formation, depth and rendering (Course 2), the optics and signal front end behind computer vision (Course 1).