PU-206 · Optics
This unit develops optics as a disciplined sequence of controlled approximations to Maxwell's equations: the short-wavelength eikonal limit yields rays, Fermat's principle, and the paraxial matrix formalism, after which the full wave nature of light is restored through Fresnel's boundary-value analysis, polarization, and dispersion in matter. It culminates in scalar diffraction theory and Fourier optics, showing how coherence, resolution, imaging, and spatial filtering all descend from a single diffraction integral.
Lectures
| L01 | What Is Light? Scope and Hierarchy of Optical Models — |
| L02 | From Maxwell's Equations to the Optical Wave Equation |
| L03 | The Eikonal Limit and the Birth of Rays |
| L04 | Fermat's Principle and the Ray Equation |
| L05 | Reflection and Refraction at Interfaces |
| L06 | Paraxial Optics and the ABCD Matrix Method |
| L07 | Lenses, Mirrors, and the Imaging Equation |
| L08 | Optical Instruments and Aberrations |
| L09 | The Fresnel Equations |
| L10 | Brewster's Angle, Total Internal Reflection, and Evanescent Fields |
| L11 | Energy Flow, Reflectance, and the Stokes Relations |
| L12 | Polarization and the Jones Calculus |
| L13 | Dispersion: The Lorentz Oscillator Model of Matter |
| L14 | Phase Velocity, Group Velocity, and Pulse Propagation |
| L15 | Scalar Diffraction Theory: The Kirchhoff Integral |
| L16 | Fraunhofer Diffraction as a Fourier Transform |
| L17 | The Single Slit and the Diffraction Grating |
| L18 | Circular Apertures and the Resolution Limit |
| L19 | Two-Beam Interference and Interferometry — |
| L20 | Fresnel Diffraction and the Near Field |
| L21 | Gaussian Beams and Laser Resonators |
| L22 | Optical Coherence and the van Cittert-Zernike Theorem |
| L23 | Fourier Optics: The Abbe Theory of Image Formation |
| L24 | Spatial Filtering, the Microscope, and Frontiers |
Derivations homed in this unit
Optical Wave and Helmholtz Equations from Maxwell
Derive the vector and scalar wave equations for the optical field in a linear medium and reduce them to the Helmholtz equation for a monochromatic field.
The Eikonal Equation as the Short-Wavelength Limit
Obtain the eikonal equation |grad S|^2 = n^2 by taking the lambda -> 0 asymptotic limit of the Helmholtz equation, defining rays as trajectories orthogonal to wavefronts.
Fermat's Principle from the Eikonal Equation
Show that ray paths make the optical path length integral n ds stationary, deriving the ray equation d/ds(n dr/ds) = grad n.
Laws of Reflection and Refraction from Fermat
Derive the law of reflection and Snell's law n1 sin theta1 = n2 sin theta2 as stationary-path conditions at an interface.
Paraxial Ray-Transfer (ABCD) Matrices
Construct the 2x2 ray-transfer matrices for free propagation, refraction at a spherical surface, and thin lenses, and show system matrices compose by multiplication with unit determinant.
Lensmaker's Formula and the Imaging Equation
Derive the lensmaker's equation and the thin-lens imaging relation 1/s' - 1/s = 1/f from refraction at two spherical surfaces via the ABCD formalism.
Fresnel Equations from Electromagnetic Boundary Conditions
Derive the s- and p-polarization amplitude reflection and transmission coefficients, and the reflectance/transmittance energy relations, by matching tangential E and H at a planar interface.
Brewster's Angle, Total Internal Reflection, and Evanescent Waves
From the Fresnel coefficients derive the Brewster angle of zero p-reflection, the critical angle for total internal reflection, and the exponentially decaying evanescent field beyond it.
Polarization States and the Jones Calculus
Represent polarization by two-component complex Jones vectors and optical elements by 2x2 Jones matrices, deriving the action of polarizers and wave plates and the classification of polarization states.
Dispersion and Absorption from the Lorentz Oscillator
Derive the complex refractive index n(omega) of a dielectric from the driven damped electron oscillator, obtaining normal and anomalous dispersion and the connection between absorption and the imaginary index.
Phase Velocity, Group Velocity, and Pulse Spreading
From the dispersion relation derive phase velocity omega/k and group velocity domega/dk, and show how group-velocity dispersion broadens a Fourier wave packet.
The Helmholtz-Kirchhoff Diffraction Integral
Derive the Kirchhoff (and Rayleigh-Sommerfeld) integral expressing the diffracted field at a point as a surface integral over an aperture, using Green's theorem and the outgoing spherical Green's function.
Fraunhofer Diffraction as a Fourier Transform
Reduce the Kirchhoff integral in the far field to a two-dimensional Fourier transform of the aperture transmission function over spatial frequencies set by the diffraction angles.
Diffraction Grating: Grating Equation and Resolving Power
From the Fourier transform of N equally spaced slits derive the grating equation d sin theta = m lambda, the sharpening of principal maxima, and the resolving power R = mN.
Airy Pattern and the Rayleigh Resolution Limit
Derive the Airy disk intensity from the Fourier transform of a circular aperture and obtain the Rayleigh criterion theta = 1.22 lambda/D for angular resolution.
Gaussian Beams from the Paraxial Wave Equation
Solve the paraxial Helmholtz equation for the fundamental Gaussian mode, deriving the complex beam parameter q, waist, Rayleigh range, and Gouy phase, and the ABCD law for its propagation.
The van Cittert-Zernike Coherence Theorem
Show that the complex degree of spatial coherence of light from an incoherent extended source equals the normalized Fourier transform of its intensity distribution.
Abbe Theory of Imaging and Spatial Filtering
Derive image formation as a double Fourier transform through the lens's back focal plane, giving the diffraction-limited transfer function and the basis of spatial filtering.