OSA cheat sheet

Snell's law: n₁sinθ₁ = n₂sinθ₂; total internal reflection occurs when θ₁ exceeds the critical angle θc = arcsin(n₂/n₁) for n₁ > n₂.

Thin-lens equation 1/f = 1/s + 1/s′ with transverse magnification m = −s′/s; sign conventions must be stated explicitly before use.

Gaussian beam waist radius w(z) = w₀√(1 + (z/z_R)²), with Rayleigh range z_R = πw₀²/λ; divergence half-angle θ ≈ λ/(πw₀) for TEM₀₀.

Diffraction limit: Airy disk first zero at radius 1.22λ·f/#; Rayleigh criterion resolution = 0.61λ/NA for coherent-free imaging.

Fresnel equations give amplitude reflection/transmission at a dielectric interface; Brewster angle θB = arctan(n₂/n₁) yields zero p-polarized reflection.

Numerical aperture NA = n sinθ_max; for a step-index fiber NA = √(n_core² − n_clad²).

Interference condition for thin films: constructive when optical path difference 2nt·cosθ = (m + ½)λ for a single phase flip at one surface.

Abbe sine condition h = f·sinθ relates image height to marginal ray angle; violation produces coma in the presence of spherical correction.

Polarization states are described by Stokes parameters S₀–S₃ or Jones vectors; a quarter-wave plate converts linear to circular when its axis is at 45° to the input.

Optical glass is specified by nd (587.6 nm), vd = (nd − 1)/(nF − nC), and internal transmittance; low vd indicates high dispersion.

Common substrates: N-BK7 (nd ≈ 1.517, vd ≈ 64), fused silica (nd ≈ 1.459, vd ≈ 68, deep-UV to IR transmission), and sapphire (hard, IR to ~4.5 µm).

Anti-reflection coatings rely on destructive interference; a single-layer λ/4 MgF₂ coating gives minimum reflection at the design wavelength only.

Dielectric high-reflectance stacks alternate high- and low-index quarter-wave layers; reflectance increases with the number of layer pairs.

Beam-splitter types: cube (cemented, polarization-sensitive if a dielectric coating is used) versus plate (introduces astigmatism in converging beams).

Grating equation d(sinα ± sinβ) = mλ; the blaze angle maximizes efficiency in a chosen order and wavelength.

Waveplate retardance is temperature- and wavelength-dependent; a true zero-order retarder is far less sensitive to angle and wavelength than a multi-order one.

Optical cements and adhesives must match thermal expansion of substrates; index-matched epoxies reduce interface reflections but limit power handling.

Start from first-order (paraxial) layout: choose focal lengths and spacings to satisfy EFL, back focal distance, and magnification before optimizing aberrations.

Seidel aberrations: spherical, coma, astigmatism, field curvature (Petzval), distortion - each has a characteristic field or aperture dependence.

Stop and pupil logic: the aperture stop limits on-axis bundle; entrance pupil is its image through preceding optics, exit pupil through following optics.

Telecentric design requires the aperture stop at the front focal point (object-space telecentric) or rear focal point (image-space telecentric).

Chromatic correction: achromat pairs a positive low-dispersion element with a negative high-dispersion element; apochromats correct secondary spectrum.

Tolerance analysis assigns compensators, then Monte Carlo or sensitivity tables convert element tolerances into RMS wavefront or spot-size degradation.

MTF targets are set from detector pixel pitch and sampling: a common rule is to place the diffraction cutoff (1/(λ·f/#)) near the Nyquist frequency.

Stray light control uses baffles, blackened surfaces, and careful placement of the field stop; ghost analysis traces double reflections explicitly.

Laser threshold requires round-trip gain ≥ round-trip loss: 2g₀L ≥ 2αL + ln(1/(R₁R₂)) for a linear cavity.

Population inversion needs a three- or four-level scheme; four-level systems reach threshold at far lower pump power.

Longitudinal mode spacing for a cavity of optical length L is Δν = c/(2L); mode-locking produces pulses of duration ~1/Δν_gain.

Q-switching stores inversion then dumps it, giving nanosecond pulses; mode-locking gives picosecond to femtosecond pulses.

Gaussian beam propagation through a lens follows ABCD matrices: q₂ = (Aq₁ + B)/(Cq₁ + D) with 1/q = 1/R − iλ/(πw²).

Semiconductor diode lasers: emission wavelength set by bandgap; catastrophic optical damage limits facet power density.

Erbium-doped fiber amplifiers operate near 1550 nm; gain bandwidth and noise figure (quantum limit ~3 dB) drive link design.

Nonlinear effects: second-harmonic generation requires a non-centrosymmetric crystal and phase matching; self-phase modulation scales with n₂·I·L_eff.

Detector sampling: pixel pitch p and focal length f give instantaneous field of view IFOV = p/f; Nyquist sampling requires at least 2 pixels per resolvable element.

Sensor metrics: quantum efficiency, dark current (doubles roughly every 6–8 °C for silicon), read noise, and full-well capacity set SNR and dynamic range.

Radiometric chain: source radiance → optics transmission → detector irradiance → digital counts via gain and offset calibration.

Time-of-flight and structured-light depth sensors trade range resolution against ambiguity range and ambient-light rejection.

Lidar range equation: received power ∝ (P_t·ρ·A_r·η)/(π·R²) for an extended target, giving R ∝ √P for fixed SNR.

Hyperspectral sensing uses a slit spectrometer or tunable filter; spectral resolution and spatial resolution compete for etendue.

Adaptive optics corrects wavefront error using a wavefront sensor plus deformable mirror; the Greenwood frequency sets required loop bandwidth.

Image quality metrics for sensing: MTF, SNR, and NETD (noise-equivalent temperature difference) for thermal imagers.

Interferometric testing reports wavefront error in waves at a stated wavelength; fringe density must stay below detector Nyquist to avoid aliasing.

Fizeau interferometry compares a test surface with a reference flat; the measured map includes reference and mount errors unless calibrated.

Surface figure versus finish: figure is low-spatial-frequency deviation (PV, RMS); finish is roughness measured by profilometry or white-light interferometry.

MTF is measured with slanted-edge, knife-edge, or sinusoidal targets; the slanted-edge method yields an edge spread function whose derivative is the line spread function.

Centering and radius measurement use a lens bench or autocollimator; wedge and decenter appear as coma or lateral color.

Power spectral density of surface error links roughness to scatter via the Rayleigh-Rice or Harvey models.

Calibration traceability requires unbroken chain to a national standard; uncertainty must be reported per GUM methods.

Environmental control matters: thermal gradients and air turbulence shift fringes and bias long-path interferometric measurements.

Laser classes per IEC 60825-1 / ANSI Z136.1: Class 1 safe under normal use, Class 2 visible ≤1 mW, Class 3R/3B restricted, Class 4 hazardous and fire-capable.

Mpe and nominal ocular hazard distance (NOHD) depend on wavelength, exposure duration, and pulse repetition; use the applicable standard's tables or calculator.

Laser controlled area requires warning signs, beam enclosures, eyewear rated at the specific wavelength and optical density, and a laser safety officer.

Optical radiation hazards: UV (photokeratitis, erythema), visible (blue-light photochemical), IR-A (retinal burn), IR-B/C (corneal burn, cataract).

Standards commonly cited: ISO 10110 for optical drawing tolerances, ISO 13696 for scatter, MIL-PRF-13830 for surface defects, ISO 9001 for quality systems.

Optical density OD = log₁₀(1/T); required OD is derived from the maximum permissible exposure and expected beam irradiance.

Professional practice: document design inputs, assumptions, and tolerance rationale; peer review of safety calculations before hardware build.

Chemical and material safety: many optical fabrication fluids and coating precursors require SDS review, ventilation, and waste handling procedures.