Memory tricks for light and lenses

Both angles measured from the normal (perpendicular to the surface). Flat mirror: angle in = angle out. This is how billiard ball bounces are predicted.

n = refractive index. Higher n = slower light = more bending. Light going from air to glass bends toward the normal. From glass to air it bends away.

Convex (thicker in middle): magnifying glasses, cameras, eyes. Concave (thinner in middle): corrects nearsightedness, flashlights. Memory trick: conCAVE is caved in at the middle.

When light hits a boundary at an angle greater than the critical angle, it reflects entirely back inside. No refraction out. Critical angle = arcsin(n₂/n₁). Fiber optics, diamonds, and mirages all use this.

f = focal length (positive for converging, negative for diverging). do = object distance. di = image distance (positive = real image on other side; negative = virtual image on same side as object). Magnification m = -di/do. |m| > 1 = enlarged. |m| < 1 = reduced.

Same equation as thin lens but for mirrors. Concave (converging) mirror: f is positive. Used in telescopes, flashlights, makeup mirrors. Convex (diverging) mirror: f is negative. Always produces virtual, upright, reduced images. Used in car side mirrors and security mirrors — wider field of view.

Different wavelengths of light travel at slightly different speeds in glass — different refractive indices. Violet light has the highest refractive index → bends most. Red light has the lowest → bends least. Rainbows are caused by dispersion and internal reflection inside water droplets.

n = c/v where c = speed of light in vacuum, v = speed of light in medium. n is always ≥ 1. Higher n: light travels more slowly and bends more when entering from air. Vacuum: n=1.000. Air: n=1.0003. Water: n=1.33. Glass: n~1.5. Diamond: n=2.42.

Grating equation: d sinθ = mλ (m = order number: 0, ±1, ±2...). d = slit spacing. More slits → sharper, brighter maxima. Used in spectrometers to measure wavelengths of light. CDs and DVDs work as reflection diffraction gratings — different wavelengths reflect at different angles → rainbow colors.

Light enters fiber at a shallow angle → hits the boundary at angle greater than critical angle → total internal reflection → light bounces along the fiber. Core has higher refractive index than cladding. Single-mode fiber: one light path, used in telecommunications. Multi-mode: multiple paths, shorter distances.

The eye is a converging optical system. Cornea does most focusing; lens fine-tunes. Image formed on retina is real and inverted — brain flips it. Nearsighted (myopia): eyeball too long or lens too strong → image forms in front of retina → corrected by diverging (concave) lens. Farsighted: opposite — corrected by converging (convex) lens.

Constructive interference: path difference = mλ (m = 0, 1, 2...) — bright fringes. Destructive interference: path difference = (m + ½)λ — dark fringes. Double-slit fringe spacing: y = mλL/d where L = screen distance, d = slit separation. Thin film interference: accounts for phase shift on reflection (light going from low to high n flips phase by π — like reflection off fixed end). Newton's rings: circular interference pattern from spherical lens on flat glass.

Unpolarized light: electric field oscillates in all transverse directions. Linear polarizer: passes only one plane of oscillation. Unpolarized → polarizer: intensity halved (I = I₀/2). Polarizer → analyzer at angle θ: I = I₀cos²θ (Malus's Law). At θ = 90° (crossed polarizers): I = 0 — no light passes. Brewster's angle: angle of incidence where reflected light is completely polarized: tan(θ_B) = n₂/n₁. Applications: sunglasses (reduce glare), LCD screens, 3D movies, photography.

Compound microscope: objective (short focal length) forms enlarged real image of nearby object. Eyepiece acts as magnifying glass — views that image. Total magnification = m_obj × m_eye. Resolution limit: d_min = 0.61λ/NA where NA = n sin θ. Refracting telescope: objective (long focal length) forms real image of distant object at its focal plane. Eyepiece magnifies that image. Angular magnification = f_obj / f_eye. Large f_obj → higher magnification. Reflecting telescope (Newton): uses concave mirror instead of objective lens — avoids chromatic aberration.

Chromatic aberration: different wavelengths (colors) refract at different angles → lens focuses red and blue at different points → color fringing. Fixed by achromatic doublet (two lenses of different glass). Spherical aberration: rays hitting edge of lens focus at different point than rays through center → blurry image. Fixed by aspherical lens or stop. Coma: off-axis point sources appear comet-shaped. Astigmatism: different focal lengths in vertical vs horizontal planes. Modern cameras use 7-10+ lens elements to minimize all aberrations.