Optics Formulas Cheat Sheet: 30+ Key Equations in One Place

What is the optics formulas cheat sheet? It is a single-page reference of every key equation in optics — from Snell's law and the thin lens equation to interference conditions and polarisation — with each variable defined and a link to a full explanation. Use it to quickly find the formula you need and see which variables go where.

Reflection

Law of Reflection

θᵢ = θᵣ

The angle of incidence equals the angle of reflection. Both angles are measured from the normal (the line perpendicular to the surface). This holds for all smooth reflecting surfaces. See the reflection physics guide for examples with diagrams.

Mirror Formula

1/f = 1/v + 1/u

Where f is focal length, v is image distance from the mirror, and u is object distance from the mirror. Sign conventions differ between concave and convex mirrors — the concave mirror guide and convex mirror guide show how to apply them correctly.

Magnification for Mirrors

m = −v/u = hᵢ / hₒ

A negative m means the image is inverted. |m| > 1 means enlarged, |m| < 1 means reduced. For a plane mirror, m = +1 (same size, upright).

Focal Length and Radius of Curvature

f = R/2

The focal length of a spherical mirror is half its radius of curvature. For a concave mirror, f and R are positive; for a convex mirror, they are negative.

Refraction

Snell's Law

n₁ sin θ₁ = n₂ sin θ₂

The fundamental law of refraction. n₁ and n₂ are the refractive indices of the two media, θ₁ is the angle of incidence, and θ₂ is the angle of refraction. The Snell's law guide has worked examples with air, water, and glass.

Speed of Light in a Medium

v = c / n

The speed of light in any transparent medium equals the vacuum speed (299,792,458 m/s) divided by the medium's refractive index. In water (n = 1.33), light travels at roughly 225,000 km/s.

Critical Angle

sin θc = n₂ / n₁ (for n₁ > n₂)

If the angle of incidence exceeds θc, all light reflects internally — total internal reflection. The critical angle formula guide works through examples for water, glass, and diamond.

Brewster's Angle

tan θB = n₂ / n₁

At Brewster's angle, reflected light is perfectly s-polarised. For an air-to-glass interface (n₁ = 1.00, n₂ = 1.50), θB ≈ 56°. See the s and p polarisation guide for details.

Prism Deviation

δ = i + i' − A

The total deviation δ of a ray passing through a prism equals the sum of the two incidence angles minus the apex angle A. At minimum deviation, i = i' and n = sin[(A + δm)/2] / sin(A/2). The refraction through a prism post has the derivation.

Apparent Depth

n = real depth / apparent depth

When you look straight down into water, objects appear shallower. If the real depth is 3.0 m and n = 1.33, the apparent depth is about 2.26 m.

Lenses

Lensmaker's Equation (Thin Lens)

1/f = (n − 1)(1/R₁ − 1/R₂)

The focal length f of a thin lens depends on the refractive index n of the material and the radii of curvature R₁ and R₂ of its two surfaces. The lens maker formula guide has a detailed step-by-step walkthrough with sign conventions.

Thin Lens Equation

1/f = 1/v − 1/u

Where f is focal length, v is image distance, and u is object distance. A positive f means a converging lens; negative f means a diverging lens. Worked examples are in the convex lens guide and concave lens guide.

Lens Magnification

M = v / u = hᵢ / hₒ

For lenses, a positive M means upright, negative means inverted. |M| > 1 means enlarged.

Optical Power

P = 1 / f (f in metres, P in dioptres)

Common spectacle powers range from −6 D (strong myopia) to +4 D (strong hyperopia). A +2 D lens has f = 0.5 m.

Combined Focal Length (Two Thin Lenses in Contact)

1/F = 1/f₁ + 1/f₂

For two thin lenses separated by distance d: 1/F = 1/f₁ + 1/f₂ − d/(f₁f₂)

Wave Optics — Interference and Diffraction

Double Slit — Bright Fringes

d sin θ = mλ (m = 0, ±1, ±2, …)

d is the slit separation, θ is the angular position of the bright fringe, m is the order number, and λ is the wavelength. The double slit experiment guide shows how this pattern forms.

Double Slit — Fringe Spacing

Δy = λD / d

The distance Δy between adjacent bright fringes on a screen at distance D from the slits. For green light (550 nm), slit spacing 0.5 mm, and D = 1.5 m, Δy ≈ 1.65 mm.

Double Slit — Dark Fringes

d sin θ = (m + ½)λ (m = 0, ±1, ±2, …)

Half-integer path differences produce destructive interference. The constructive vs destructive interference guide compares both conditions side by side.

Single Slit — Dark Fringes

a sin θ = mλ (m = ±1, ±2, ±3, …)

a is the slit width. Note: the central maximum is twice as wide as the other fringes. The single slit diffraction equation guide has the full derivation.

Diffraction Grating

d sin θ = nλ (n = 0, ±1, ±2, …)

d is the grating spacing (d = 1/N where N is the number of lines per metre). A 600 lines/mm grating has d = 1667 nm. See the diffraction grating guide for how spectrometers use this.

Bragg's Law

nλ = 2d sin θ

d is the distance between crystal lattice planes, θ is the glancing angle. Used in X-ray diffraction to determine crystal structure — covered in the XRD guide.

Rayleigh Criterion

θ = 1.22 λ / D

The smallest angular separation two point sources can have and still be resolved, where D is the aperture diameter. For a 10 cm telescope at 550 nm, θ ≈ 1.34 × 10⁻⁵ radians (about 2.8 arcseconds).

Abbe Diffraction Limit

d = λ / (2 × NA)

The minimum resolvable feature in a microscope. NA is the numerical aperture. A good oil-immersion objective with NA = 1.4 and λ = 550 nm can resolve features down to about 196 nm.

Thin Film Interference

Constructive: 2nt = (m + ½)λ | Destructive: 2nt = mλ

t is the film thickness, n is the refractive index of the film. For an oil slick (n ≈ 1.4) on water, the colours you see come from different thicknesses satisfying this condition for different wavelengths.

Polarisation

Malus's Law

I = I₀ cos²θ

I is the transmitted intensity, I₀ is the incident intensity, θ is the angle between the light's polarisation direction and the filter axis. At θ = 0°, all light passes; at θ = 90°, none passes. The polarised light guide shows real examples from sunglasses to LCD screens.

Fresnel Equations (Reflection Coefficients)

s-polarised: rₛ = (n₁ cos θᵢ − n₂ cos θₜ) / (n₁ cos θᵢ + n₂ cos θₜ)

p-polarised: rₚ = (n₂ cos θᵢ − n₁ cos θₜ) / (n₂ cos θᵢ + n₁ cos θₜ)

These give the fraction of light reflected at an interface for each polarisation state. At Brewster's angle, rₚ = 0. The s and p polarisation guide explains the full set.

Optical Instruments

Compound Microscope — Total Magnification

M_total = M_objective × M_eyepiece

A typical lab microscope with a 10× eyepiece and a 40× objective gives 400× total magnification. The microscope guide explains how both lens systems contribute.

Astronomical Telescope — Angular Magnification

M = −f_objective / f_eyepiece

The negative sign means the image is inverted (which does not matter for astronomy). A telescope with a 1000 mm objective and a 10 mm eyepiece gives 100× magnification.

Simple Magnifier

M ≈ 250 / f (f in millimetres)

A 50 mm focal length lens gives about 5× magnification. The magnifying glass guide explains the near-point formula.

Photon Energy

E = hf = hc / λ

h is Planck's constant (6.626 × 10⁻³⁴ J·s), f is the frequency in Hz. Red light (700 nm) carries about 1.77 eV per photon; violet light (400 nm) carries about 3.10 eV. The light energy guide connects this to how we perceive brightness.

Quick-Reference Table

External References

The Physics Classroom optics tutorial is a well-structured introduction to how each formula is derived. The RP Photonics Encyclopedia provides advanced coverage of laser optics and nonlinear effects. For historical context, the Wikipedia list of optics equations includes derivations and variations of every formula here.