Optics — formula sheet
Every boxed result from the 16 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Optical Wave and Helmholtz Equations from Maxwell
The optical field is a wave travelling at \(v=1/\sqrt{\mu\varepsilon}\). Time-harmonic light replaces the two derivatives in the wave equation with a single spatial eigenvalue problem: \(k^2\) is the eigenvalue of \(-\nabla^2\) that a spatial field must carry to oscillate at frequency \(\omega\). The magnetic field \(\vec{B}\) satisfies the identical equations, so both fields propagate together with the same speed and wavenumber.
The Eikonal Equation as the Short-Wavelength Limit
The magnitude of the phase gradient equals the local refractive index everywhere. Geometrically, the wavefronts \(S=\text{const}\) are surfaces of constant optical path from the source, and light energy streams along the rays — the orthogonal trajectories that thread these surfaces. Where \(n\) is uniform the rays are straight; where \(n\) varies they bend toward higher index. The accompanying transport equation \(\nabla\cdot(A^2\nabla S)=0\) conserves power in each ray tube, fixing how brightness changes as rays converge or diverge.
Fermat's Principle from the Eikonal Equation
The left form is a local force-like law: the "optical momentum" \( n\,\hat{\mathbf{t}} \) changes along the ray at a rate set by the gradient of the index. A ray bends toward the region of higher \( n \), because \( \nabla n \) supplies the curving push. The right form is the global statement — Fermat's principle — that the actual ray makes the optical path length (equivalently the travel time \( \int n\, ds / c \)) stationary between its endpoints. The two are the same physics: the ray equation is the Euler–Lagrange equation of the optical path integral.
Laws of Reflection and Refraction from Fermat
The stationary-path requirement at an interface says nothing more nor less than: the quantity \(n\sin\theta\) — the transverse component of the optical wavevector, up to \(2\pi/\lambda_0\) — is conserved across the boundary. Reflection is the special case of equal indices; refraction bends the ray toward the normal on entering a denser medium (\(n_2>n_1\Rightarrow\theta_2<\theta_1\)). All three rays share the plane of incidence.
Paraxial Ray-Transfer (ABCD) Matrices
A ray's transverse state \((y,\,n\theta)\) transforms linearly through any paraxial system. Propagation is a shear that tilts height without changing reduced slope; refraction and the thin lens are focusing kicks that change slope in proportion to height (with strength the power \(P=1/f\)) without moving the ray. The generic entries \(\left(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right)\) mean: \(A\) is transverse (angular) magnification bookkeeping, \(B\) has the dimension of a reduced length, \(C=-P\) is minus the system power, and \(D\) is angular magnification bookkeeping. The constraint \(AD-BC=1\) is the conserved phase-space area.
Lensmaker's Formula and the Imaging Equation
The focal length is set entirely by the index contrast \((n-n_0)/n_0\) and the curvature difference \(1/R_1 - 1/R_2\); a lens in a medium of matched index (\(n=n_0\)) has \(1/f=0\) and does nothing. The imaging relation then says that once \(f\) is fixed, object and image distances are reciprocally linked: pushing the object toward the front focal point (\(s\to -f\)) sends the image to infinity, and vice versa.
Brewster's Angle, Total Internal Reflection, and Evanescent Waves
At \(\theta_B\) the reflected beam is purely \(s\)-polarised because the \(p\)-component reflects with zero amplitude — the microscopic dipoles in medium 2 oscillate along the would-be reflected \(p\)-ray and cannot radiate along their own axis. At \(\theta_c\) the refracted ray lies flat in the surface; for any \(\theta_i>\theta_c\) refraction is forbidden, \(|r|=1\) for both polarisations, and 100% of the time-averaged power is reflected. Yet the fields do not stop at the boundary: an evanescent wave leaks a distance \(d=1/\kappa\) (the penetration depth) into the rarer medium, carrying no net energy across but able to couple to a third body placed within \(d\).
Polarization States and the Jones Calculus
Every state of fully polarized monochromatic light is a two-component complex Jones vector, fixed by an amplitude ratio and a single relative phase \(\delta\). A polarizer is a rank-one projector (\(\det=0\), lossy, idempotent); a retarder is a unitary diagonal matrix in its eigenbasis (\(|\det|=1\), lossless). An optical train is the ordered product of these matrices, last element on the left. The relative phase \(\delta\) and amplitude ratio \(\tan\psi\) sort the state into linear, circular, or elliptical.
The Helmholtz-Kirchhoff Diffraction Integral
The field at \(P_0\) is a coherent sum of secondary spherical wavelets \(e^{ikr_{01}}/r_{01}\), one from every aperture point, each carrying the incident amplitude and phase \(A\,e^{ikr_{21}}/r_{21}\), weighted by the directional factor \(K(\chi)\), and advanced in phase by \(90^\circ\) and scaled by \(1/\lambda\) through the \(1/(i\lambda)\) prefactor. This is Huygens' principle made quantitative — the wavelet strength, phase lead, and forward bias are all derived, not assumed. The inclination factor \(K=\tfrac12(1+\cos\chi)=1\) forward and \(=0\) backward, so no wavelet radiates back toward the source.
Fraunhofer Diffraction as a Fourier Transform
The complex amplitude on a distant screen is the Fourier transform of the aperture, sampled at the spatial frequency \((x/\lambda z,\,y/\lambda z)\). A point at screen position \((x,y)\) reports how much of the spatial frequency \(f_X=x/\lambda z\) is contained in the aperture. The prefactor \(e^{ikR}/(i\lambda z)\) is a pure phase and \(1/z\) amplitude spreading; it drops out of the measured intensity, which is the squared modulus of the transform. Larger apertures (more high-frequency content) produce narrower patterns — the reciprocal-width property of Fourier pairs.
Diffraction Grating: Grating Equation and Resolving Power
Each spectral order \(m\) is diffracted to the angle where the path difference between adjacent slits, \(d\sin\theta\), is a whole number of wavelengths. Adding slits does not move the maxima — it makes them taller (\(\propto N^2\)) and narrower (\(\propto 1/N\)), which is why resolving power grows linearly with \(N\). Going to higher order \(m\) also improves resolution proportionally, at the cost of overlapping orders and reduced free spectral range.
Airy Pattern and the Rayleigh Resolution Limit
The far field of a round aperture is a bright central Airy disk carrying about \(83.8\%\) of the energy, ringed by faint concentric maxima. The disk’s angular radius, to the first dark ring, is \(1.22\,\lambda/D\). Two point objects closer together than this on the sky merge into a single blur: their combined intensity profile loses its central dip. Resolution therefore improves with a bigger aperture (\(D\!\uparrow\)) or a shorter wavelength (\(\lambda\!\downarrow\)) — the two levers every optical designer pulls.
Gaussian Beams from the Paraxial Wave Equation
A single complex number \(q\) carries everything: its imaginary part fixes the beam width through \(w\), its real part fixes the wavefront curvature through \(R\). At the waist \(q=iz_R\) is purely imaginary (flat wavefront, minimum spot \(w_0\)). Moving a distance \(z\) just adds \(z\) to \(q\); a lens subtracts \(1/f\) from \(1/q\); any paraxial system acts on \(q\) through its ray ABCD matrix by the same fractional-linear map that would move a ray. The beam expands hyperbolically, has far-field half-angle \(\theta=\lambda/(\pi w_0)\), and accumulates an extra \(\pi\) of Gouy phase as it passes through focus.
The van Cittert-Zernike Coherence Theorem
The correlation between the light at two points is set entirely by the source's shape on the sky, not its brightness. Sweeping the baseline \(\Delta\vec{r}\) samples different spatial frequencies of the source; measuring \(\mu_{12}\) at many baselines and inverting the transform reconstructs \(I(\hat{\mathbf{s}})\). A point source (\(I\propto\delta\)) gives \(\mu_{12}=1\) for all baselines — perfect coherence — while a broad source decorrelates quickly, so the "coherence area" at the observation plane is inversely proportional to the source's angular size.
Abbe Theory of Imaging and Spatial Filtering
Imaging is a Fourier transform to the back focal plane followed by an inverse transform to the image plane. In between, the object's spatial spectrum lies exposed and separated by frequency, and the lens pupil (or any inserted mask) acts as a multiplicative transfer function \(H\). Whatever \(H\) leaves out never appears in the image; the finite pupil low-passes the object with cutoff \(f_c=\mathrm{NA}/\lambda\), blurring it by the amplitude point-spread function \(h\) (an Airy pattern for a circular aperture). Coherent imaging is linear and shift-invariant in complex amplitude, not intensity.
Young's Double-Slit Interference
The intensity varies as the square of the cosine of half the phase difference. Bright fringes (maxima) occur when the argument is \(m\pi\), i.e., \(d\sin\theta = m\lambda\). Dark fringes (minima) occur when the argument is \((m+\tfrac{1}{2})\pi\), i.e., \(d\sin\theta = (m+\tfrac{1}{2})\lambda\).