Lens formulas, types, camera optics, refractive indices, telescope types.
Fundamental Lens Formulas
| Formula | Variables | Description |
|---|---|---|
1/f = 1/dₒ + 1/dᵢ | f=focal length, dₒ=object dist, dᵢ=image dist | Thin lens equation |
m = −dᵢ / dₒ = hᵢ / hₒ | m=magnification, h=height | Linear magnification |
P = 1/f (m⁻¹) | P=optical power in diopters | Lens power (diopters) |
n₁sinθ₁ = n₂sinθ₂ | n=refractive index, θ=angle | Snell's Law |
θ_c = arcsin(n₂/n₁) | Critical angle for total internal reflection | TIR condition |
f/# = f / D | D=aperture diameter | f-number (aperture) |
NA = n·sin(θ_max) | Numerical Aperture, n=medium RI | Microscope / fiber NA |
Lens Types
| Type | Profile | Focal Length | Use Case |
|---|---|---|---|
| Convex (converging) | Thick center | Positive (real focus) | Magnifying glass, camera, eye |
| Concave (diverging) | Thin center | Negative (virtual focus) | Myopia correction, beam expander |
| Plano-convex | One flat, one convex | Positive | Collimating point sources |
| Plano-concave | One flat, one concave | Negative | Beam divergence |
| Biconvex | Both sides convex | Positive | Imaging, relay lens |
| Biconcave | Both sides concave | Negative | Beam expansion |
| Achromat (doublet) | Two cemented lenses | Positive | Minimizes chromatic aberration |
| Aspheric | Non-spherical surface | Positive or negative | Reduces spherical aberration |
| Fresnel | Concentric rings (thin) | Positive | Lighthouse, overhead projector, VR |
| GRIN (gradient-index) | Refractive index gradient | Positive | Fiber coupling, endoscopes |
Camera Optics
| Concept | Formula | Description |
|---|---|---|
| Depth of Field | DoF ∝ f²/(D·f) | Larger f/# → greater DoF |
| Circle of Confusion | c = f/N | Max acceptable blur spot |
| Hyperfocal distance | H = f²/(Nc) | Focus at H → DoF from H/2 to ∞ |
| Magnification (macro) | m = f/(dₒ−f) | At dₒ=2f → m=1 (life size) |
| Field of View | FoV = 2·arctan(d/2f) | d=sensor dimension |
| Crop factor | CF = 43.27mm / √(w²+h²) | 43.27=diagonal of 35mm full frame |
Refractive Indices (λ=589nm)
| Material | n | Notes |
|---|---|---|
| Vacuum | 1.0000 (exact) | Reference |
| Air (STP) | 1.0003 | ≈ 1 for most calculations |
| Water (20°C) | 1.333 | — |
| Glass (crown) | 1.52 | BK7 optical glass |
| Glass (flint) | 1.62 | SF11 dense flint |
| Diamond | 2.417 | High dispersion, total internal reflection |
| Silicon (IR) | 3.48 | Used for IR optics |
| Fused silica | 1.458 | UV-transparent glass |
Telescope Types
| Type | Principle | Advantage | Example |
|---|---|---|---|
| Refractor | Objective lens | Sharp image, low maintenance | Galileo scope |
| Newtonian reflector | Parabolic primary mirror | Low cost, large aperture | Dobsonian |
| Cassegrain | Primary + secondary mirror | Compact, long focal length | Most observatory scopes |
| Schmidt-Cassegrain | Corrector plate | Compact, versatile | Meade, Celestron |
| Ritchey-Chrétien (RC) | Hyperbolic mirrors | Wide field, no coma | Hubble Space Telescope |
