Geometric and Wave Optics: Final Takeaways

Troubleshooting

Cheatsheet

=== GEOMETRIC OPTICS ===
Eikonal:      |∇S|² = n²(x)
Ray eq:       d/ds(n t̂) = ∇n
Fermat:       δ ∫ n ds = 0
Snell:        n₁ sinθ₁ = n₂ sinθ₂
Étendue:      n²A·Ω = const

=== PARAXIAL / GAUSSIAN BEAM ===
Paraxial wave eq:  2ik ∂_z u + ∇_⊥² u = 0
Beam waist:        w₀ = minimum 1/e radius
Rayleigh range:    z_R = π w₀²/λ
Beam radius:       w(z) = w₀√(1 + (z/z_R)²)
Wavefront R:       R(z) = z[1 + (z_R/z)²]
Far-field angle:   θ = λ/(π w₀)
Gouy phase:        η(z) = arctan(z/z_R)

ABCD: r' = (Ar + Br')/(Cr+Dr')
Free space:  [[1,d],[0,1]]
Thin lens:   [[1,0],[-1/f,1]]
Mirror R:    [[1,0],[-2/R,1]]
For Gaussian: 1/q = 1/R - iλ/(πw²); q' = (Aq+B)/(Cq+D)

=== DIFFRACTION ===
Helmholtz:      (∇² + k²)U = 0
Fresnel-Kirchhoff:  U(P) = (-i/λ) ∫ U₀ e^(ikr)/r K(θ) dA
Fresnel number: N_F = a²/(λz)
  N_F >> 1 → Fresnel (near field)
  N_F << 1 → Fraunhofer (far field) = FT of aperture

Single slit (width a):   I = I₀ sinc²(πa sinθ/λ)
Double slit (sep d):     I = single-slit · cos²(πd sinθ/λ)
N-slit grating:          I = single-slit · [sin(Nπd sinθ/λ)/sin(πd sinθ/λ)]²
Circular (Airy):         I = I₀ [2J₁(x)/x]², x = ka sinθ
  first null: sinθ = 1.22 λ/D

Resolving power (grating): R = λ/Δλ = mN
Free spectral range:       Δλ_FSR = λ/m
Rayleigh resolution:       θ_min = 1.22 λ/D

Fresnel zones: r_n = √(nλz)
Zone plate f = a²/λ for first-zone radius a

=== INTERFERENCE / COHERENCE ===
Visibility:    V = (I_max - I_min)/(I_max + I_min)
              = 2√(I₁I₂)/(I₁+I₂) × |γ₁₂|
Mutual coh:    Γ₁₂(τ) = ⟨E*(r₁,t)E(r₂,t+τ)⟩
γ₁₂ = Γ₁₂/√(Γ₁₁(0)Γ₂₂(0)),  |γ|≤1
Coh time:      τ_c ~ 1/Δν
Coh length:    L_c = c τ_c

Wiener-Khinchin: γ_11(τ) ↔ S(ω)  (FT pair)
Van Cittert-Zernike: γ₁₂(0) = FT of source brightness

Fabry-Perot:    I/I₀ = 1/[1 + F sin²(δ/2)],  F = 4R/(1-R)²
FSR:            Δν = c/(2L)
Finesse:        F̃ = π√R/(1-R)

HBT:            g²(0)=2 thermal, 1 coherent, 0 single-photon
Siegert:        g²(τ) = 1 + |g¹(τ)|²  (chaotic Gaussian)

Photon noise:
  Poisson (coherent): Δn = √n̄
  Bunched (thermal):  Δn = √(n̄ + n̄²)
  Squeezed:           Δn < √n̄ (beats SQL)

=== NONLINEAR OPTICS ===
P = ε₀ [χ⁽¹⁾E + χ⁽²⁾E² + χ⁽³⁾E³ + ...]
χ⁽²⁾ ≠ 0 only in non-centrosymmetric media
χ⁽³⁾ in all media

Three-wave (χ⁽²⁾): ω₃ = ω₁ + ω₂, k₃ = k₁+k₂
SHG: 2ω from ω, η ∝ d_eff² L² I_ω sinc²(ΔkL/2)
Parametric amp: A_s(z) ∝ cosh(γz), γ ∝ √I_p
SPDC: ω_p → ω_s + ω_i (entangled)
Quasi-phase matching: PPLN with period 2π/Δk

Kerr (χ⁽³⁾): n = n₀ + n₂ I
Self-focusing critical power: P_cr ~ λ²/(n₀n₂)
Self-phase mod: δφ(t) = n₂ I(t) ωL/c
NLSE soliton:  A = A₀ sech(t/T₀)

=== USEFUL NUMBERS ===
λ_vis: 400-700 nm
1 photon @ 1eV: λ ≈ 1240 nm
σ_SB = 5.67e-8 W/m²K⁴
NA_oil_max ≈ 1.4
Atmospheric seeing: θ ~ 1" (good site)
LIGO arm: 4 km, F ~ 450

Glossary

• ABCD matrix — 2×2 transfer matrix for paraxial ray tracing.
• Aberration — Departure from ideal imaging (spherical, coma, astigmatism, chromatic).
• Airy disk — Diffraction pattern of circular aperture; central spot + concentric rings.
• Antibunching — $g^{(2)}(0) < 1$; non-classical photon statistics.
• Apodization — Smoothly tapering aperture transmission to reduce sidelobes.
• Babinet's principle — Complementary apertures sum to unobstructed wave.
• Beam waist — Location of minimum radius in Gaussian beam.
• Birefringence — Polarization-dependent $n$; enables phase matching.
• Bunching — $g^{(2)}(0) > 1$; thermal/chaotic light photon clumping.
• Caustic — Locus where rays converge; geometric-optics singularity.
• Coherence area — Transverse area over which $|\gamma_{12}| > $ threshold.
• Coherence length ($L_c$) — Distance over which temporal coherence persists.
• Coherent state — Laser-like quantum state; Poisson photon statistics, $g^{(2)} = 1$.
• Eikonal — Scalar function whose gradient is local wavevector / $k_0$.
• Étendue — Phase-space area $A\Omega n^2$; conserved by lossless optics.
• Fabry-Perot — Multi-beam interferometer formed by parallel mirrors.
• Fermat's principle — Light takes path of extremal optical length.
• Finesse ($\mathcal{F}$) — FSR/FWHM of Fabry-Perot peaks.
• FSR (free spectral range) — Spacing between modes / interference orders.
• Fraunhofer diffraction — Far-field; FT of aperture.
• Fresnel diffraction — Near-field; quadratic-phase integral.
• Fresnel number ($N_F$) — $a^2/(\lambda z)$; selects regime.
• Fresnel zones — Annuli on aperture contributing alternately ± to axis field.
• Frequency comb — Equally spaced laser spectrum: $f_n = n f_r + f_0$.
• Gaussian beam — Lowest-order paraxial mode; Gaussian transverse profile.
• Geometric optics — Limit $\lambda \to 0$; rays propagate by eikonal.
• Gouy phase — Extra $\pi/2$ accumulated by Gaussian beam through focus.
• Hanbury Brown-Twiss (HBT) — Intensity correlation interferometry.
• Helmholtz equation — Monochromatic wave: $(\nabla^2 + k^2)U = 0$.
• Holography — Recording interference of object + reference wave; restores 3-D.
• Huygens-Fresnel principle — Wavefront = superposed secondary sources.
• Kerr effect — Intensity-dependent $n$ (from $\chi^{(3)}$).
• Kirchhoff integral — Boundary integral solution of Helmholtz.
• Mandel $Q$ — Photon-statistic parameter; negative ⇒ non-classical.
• Michelson interferometer — Two-arm amplitude division device.
• Mutual coherence function — $\Gamma_{12}(\tau) = \langle E^*_1 E_2(\tau)\rangle$.
• Numerical aperture (NA) — $n\sin\theta_{\max}$; sets resolution.
• Optical path length (OPL) — $\int n\, ds$; phase = $k_0 \cdot $ OPL.
• Paraxial — Small-angle ray / wave limit.
• Phase matching — $\sum \mathbf{k} = 0$ for nonlinear process; necessary for efficient conversion.
• Pockels effect — Linear electro-optic ($\chi^{(2)}$).
• Polarization — Direction of $\mathbf{E}$; transverse for plane wave.
• Quasi-phase matching (QPM) — Periodically poled $\chi^{(2)}$ to compensate momentum mismatch.
• Rayleigh criterion — Two sources resolved when separation ≥ Airy radius.
• Rayleigh range ($z_R$) — Distance for Gaussian beam to expand by $\sqrt 2$.
• Resolving power (R) — $\lambda/\Delta\lambda$ for spectrometers / gratings.
• Self-phase modulation — Pulse Kerr-induced spectral broadening.
• SHG — Second-harmonic generation; $\omega\to 2\omega$.
• Soliton — Self-preserving wavepacket from nonlinear-dispersive balance.
• Spatial coherence — $|\gamma_{12}(0)|$; transverse phase correlation.
• SPDC — Spontaneous parametric down-conversion; entangled photon pair source.
• Squeezed light — Quantum state with sub-Poisson noise in one quadrature.
• Temporal coherence — $|\gamma_{11}(\tau)|$; correlation with delayed self.
• Van Cittert-Zernike theorem — Spatial coherence = FT of source brightness.
• Visibility ($V$) — Normalized fringe contrast.
• Wave optics — Full Helmholtz / Maxwell treatment; includes diffraction & interference.
• Wiener-Khinchin theorem — Autocorrelation ↔︎ power spectrum FT.
• WKB approximation — Semiclassical wave expansion; equivalent to eikonal.

Final Takeaways

1. Three regimes, three formalisms. Eikonal for rays, Fresnel-Kirchhoff for diffraction, $\chi^{(n)}$ for nonlinear. Always check which one applies before computing.
2. Light's geometric limit is mechanics' classical limit. Eikonal ↔︎ Hamilton-Jacobi makes ray tracing a special case of Hamiltonian dynamics.
3. The Fourier transform is the heart of wave optics. Fraunhofer pattern = FT of aperture. Lens = FT machine. Spatial filtering = applying $H(\mathbf{k})$ in FT plane.
4. Diffraction sets the resolution floor unless you exploit nonlinearity, near-field, or statistics. $\theta = 1.22\lambda/D$ for a circular aperture is the canonical bound.
5. Coherence is the bridge from waves to photons. Mutual coherence $\Gamma_{12}$ encodes all classical interference and connects to photon statistics through HBT-Siegert.
6. Van Cittert-Zernike is to space what Wiener-Khinchin is to time. Both reveal: correlations and brightness distributions are Fourier duals.
7. Phase matching is non-negotiable in nonlinear optics. Without it, conversion efficiency drops by orders of magnitude. QPM made nonlinear optics ubiquitous.
8. Quantum optics begins where classical visibility ends. $g^{(2)}(0) < 1$ has no classical-wave analog; squeezing beats the SQL.
9. Holography stores phase, which is most of the information. Same idea reappears in modern AdS/CFT and quantum gravity.
10. Frequency combs link microwave to optical clocks. A self-referenced comb makes every optical frequency directly measurable; built modern atomic clock accuracy ($10^{-19}$).
11. Solitons are universal. They live in fibers, plasmas, water, BECs — wherever dispersion and nonlinearity balance.
12. The eyes / brain / instruments are all matched filters — they decode light by inner-product against templates. Optics provides the templates (impulse responses, transfer functions).
13. Everything in this Part directly enables later Parts: wave propagation in plasma (Part VI), gravitational waves (Part VII), and fluid acoustics (Part V) all reuse eikonal, dispersion, and coherence machinery.

Next: Part IV — Elasticity (stress-strain, elastic constants, elastostatic equilibrium, elastodynamics & seismic waves). The first deep continuum-mechanics chapter; builds directly on the stress tensor from Part I.