Optics
Reflection, refraction, lenses, instruments, interference, diffraction and polarisation. 61 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Optics is the study of light. This area covers the nature of light, rectilinear propagation, reflection from plane and spherical mirrors, refraction and total internal reflection, lenses and optical instruments, and wave phenomena such as interference, diffraction and polarisation. It shows how the ray model and the wave model each explain different observations.
How the topics connect
Optics covers how light reflects, refracts and interferes. Ray optics handles mirrors and lenses, which is enough for cameras, glasses and telescopes, while wave optics explains interference, diffraction and polarisation, the effects that show light is a wave.
Real-world uses
Understand how converging and diverging lenses correct long-sightedness and short-sightedness.
Relate focal length, object distance and magnification to how a camera forms an image or a microscope enlarges a specimen.
Explain how total internal reflection keeps light trapped inside a fibre to carry internet and telephone signals over long distances.
Use thin-film interference and polarisation to reduce glare on lenses and water surfaces.
Opticians design glasses and contact lenses using the lens equations and the eye's focusing to correct myopia and hypermetropia.
Optical fibres carry internet traffic by total internal reflection, and microscopes and telescopes depend on lenses and mirrors.
Nature of light
Light behaves as both wave and particle.
Light behaves as a wave (interference, diffraction, polarisation) and as particles called photons (photoelectric effect). Which description to use depends on the situation. Light is electromagnetic radiation at 400–700 nm.
Double-slit pattern vs. single photon detection.
Light is neither a pure wave nor a tiny ball.
Rectilinear propagation
Light travels in straight lines in uniform media.
In a uniform medium light travels in straight lines, explaining shadows, eclipses and the pinhole camera. It breaks down when apertures are comparable to the wavelength.
Sharp shadow of a tree at noon.
Diffraction shows the limit of this idea.
Reflection of light
Angle of incidence equals angle of reflection.
θ_i = θ_rThe reflected ray lies in the plane of incidence and θᵢ = θᵣ, measured from the normal. Rough surfaces scatter light (diffuse reflection).
A ray at 30° to the normal reflects at 30°.
Angles are measured from the normal, not the surface.
Plane mirrors
Form virtual images as far behind as the object is in front.
A plane mirror gives a virtual, upright, same-size image as far behind the mirror as the object is in front, with left–right reversal. A person needs only a half-height mirror to see the whole body.
Rotating a mirror by θ turns the reflected ray by 2θ.
The image can't be projected on a screen.
Spherical mirrors
Concave and convex mirrors form real or virtual images.
Concave mirrors converge light and can form real images; convex mirrors diverge light and always give virtual, reduced, upright images (wide view in car mirrors).
Makeup mirrors are concave.
Real images are inverted.
Mirror equation
Relates object, image and focal distances.
1/v + 1/u = 1/f1/v + 1/u = 1/f (real-is-positive convention as written here). Magnification m = −v/u.
Concave mirror f = 10 cm with the object 30 cm away: the image forms 15 cm in front of the mirror, real and inverted, m = −0.5.
Set sign convention before plugging in numbers.
Magnification of a mirror
Ratio of image to object height.
m = −v/um = h'/h = −v/u. |m| > 1 enlarged, negative means inverted.
Object at 2f gives m = −1.
Real images are inverted, virtual are upright.
Focal length and radius
Focus is half the radius of curvature.
f = R/2For a spherical mirror f = R/2 for paraxial rays. Large aperture rays focus elsewhere (spherical aberration); parabolic mirrors avoid this.
R = 40 cm: f = 20 cm.
Sign: concave f is negative in Cartesian convention.
Refraction of light
Bending of light at a boundary.
n₁ sinθ₁ = n₂ sinθ₂At a boundary the speed changes and the ray bends toward the normal when entering a denser medium. The frequency stays constant; wavelength changes.
A straw looks bent in water.
Bending happens because of the speed change.
Refractive index
Ratio of light speed in vacuum to that in a medium.
n = c/vn = c/v is greater than 1: air 1.0003, water 1.33, glass 1.5, diamond 2.42. A higher n means slower light and more bending.
Light in glass travels at 2×10⁸ m/s.
n depends on wavelength.
Snell's law
Law of refraction.
refractionn₁ sinθ₁ = n₂ sinθ₂ relates the angles in two media, measured from the normal. It follows from Fermat's principle.
Air to glass at 40°: θ₂ = 25.4°.
Check the angles in the correct order.
Apparent depth
Objects in water look shallower.
d' = d/nAn object under water seems shallower: apparent depth = real depth/n (viewed from above). A pool of depth 2 m appears 1.5 m deep.
Spearfishers aim lower.
The formula holds for near-normal viewing.
Critical angle
Angle beyond which total internal reflection occurs.
sinθ_c = n₂/n₁When light goes from a denser to a rarer medium, the refracted ray bends away; at the critical angle θ_c = sin⁻¹(n₂/n₁) it travels along the boundary. Beyond it there's total internal reflection.
Glass to air: 41.8°; water to air: 48.8°.
Needs n₁ > n₂.
Total internal reflection
Complete reflection inside a denser medium.
refractionIf the incidence angle exceeds θ_c, all light is reflected back with no loss. It makes diamonds sparkle and optical fibres work.
Binoculars use prisms for TIR.
No energy is transmitted (apart from an evanescent field).
Applications of TIR
Optical fibres, prisms and diamonds.
Optical fibres, endoscopes, prism binoculars, reflectors on bicycles and cat's-eyes on roads, and the sparkle of cut diamonds.
A fibre loses only ~4% per km.
Fibre cladding has a lower n than the core.
Prism and deviation
A prism bends light and disperses colours.
n = sin((A+δ)/2)/sin(A/2)A prism deviates light twice and disperses colours. n = sin((A + δ_min)/2)/sin(A/2) at minimum deviation.
A = 60°, δ_min = 37°: n = 1.5.
Violet bends the most.
Dispersion
Different colours have different refractive indices.
n varies with wavelength (higher for blue), so white light splits into colours when refracted. It causes rainbows and chromatic aberration.
Red n_glass = 1.51, violet 1.53.
Different from diffraction.
Rainbow formation
Refraction, reflection and dispersion in raindrops.
Sunlight is refracted into a raindrop, internally reflected once and refracted out, with deviation minimum at about 138°, so the bow appears at 42° from the antisolar point. Red on the outside.
Secondary bow at 51° with reversed colours.
The observer sees different drops for each colour.
Scattering of light
Rayleigh scattering makes the sky blue.
I ∝ 1/λ⁴Molecules scatter light with intensity ∝ 1/λ⁴ (Rayleigh): blue sky and red sunsets. Larger particles scatter all colours (white clouds).
Blue scatters 5.5 times more than red.
Scattering is not the same as reflection.
Mirage
Light bends in layers of air at different temperatures.
Hot air near the ground has lower n, so light bends upward gradually giving an inverted image of the sky that looks like water.
Seen on hot roads.
It's refraction, not reflection from a surface.
Refraction at a spherical surface
Relation between object and image distance.
n₂/v − n₁/u = (n₂−n₁)/RFor a single surface: n₂/v − n₁/u = (n₂ − n₁)/R. Basis for the lens maker's formula.
n₁ = 1, n₂ = 1.5, R = 10 cm, u = −30: v = 45 cm.
Signs: R positive if the centre is on the outgoing side.
Thin lens
Converging and diverging lenses.
Converging (convex) lenses bring parallel rays to a focus; diverging (concave) lenses spread them. Their power is measured in dioptres.
A magnifying glass is a convex lens.
Thick lenses need more care.
Lens formula
Thin lens equation.
1/v − 1/u = 1/f1/v − 1/u = 1/f in the Cartesian convention (u negative for real objects). Magnification m = v/u.
f = 10 cm, u = −30 cm: v = 15 cm, m = −0.5.
Keep the sign convention consistent.
Lens maker's formula
Focal length from curvature and index.
1/f = (n−1)(1/R₁−1/R₂)1/f = (n − 1)(1/R₁ − 1/R₂) gives focal length from refractive index and curvatures.
Biconvex lens with R₁ = 20 cm, R₂ = −20 cm, n = 1.5: f = 20 cm.
Placed in a medium, replace n by n_lens/n_medium.
Power of a lens
Reciprocal of focal length in metres, in dioptres.
P = 1/fP = 1/f with f in metres, in dioptres (D). Converging lenses have positive power.
f = 25 cm: 4 D.
Glasses prescriptions use dioptres.
Combination of lenses
Powers add for thin lenses in contact.
P = P₁+P₂Thin lenses in contact: P = P₁ + P₂. For separated lenses, the image from the first is the object for the second.
+3 D and −1 D: +2 D.
Don't just add focal lengths.
Lens aberrations
Spherical and chromatic aberrations.
Spherical aberration (edge rays focus differently) and chromatic aberration (colours focus at different points) blur images. Cured by combining lenses (achromatic doublets) or aspheric surfaces.
Cheap lenses show colour fringes.
Mirrors don't suffer chromatic aberration.
Human eye
Lens focuses light onto the retina.
Cornea and lens focus light onto the retina; the ciliary muscle changes the lens shape (accommodation). Near point is 25 cm and far point infinity for normal vision.
Rods work in dim light and cones for colour.
The retinal image is inverted.
Defects of vision
Myopia and hypermetropia corrected by lenses.
Myopia (short-sightedness): image forms in front of the retina; corrected by a diverging lens. Hypermetropia: behind the retina; corrected by a converging lens. Presbyopia: with age.
A myopic person with far point 2 m needs −0.5 D.
Astigmatism needs cylindrical lenses.
Simple microscope
Magnifying glass angular magnification.
M = 1 + D/fA convex lens used as a magnifier with the object inside the focal length. Maximum angular magnification M = 1 + D/f, with D = 25 cm.
f = 5 cm: M = 6.
Image is virtual.
Compound microscope
Two lenses give high magnification.
Objective forms a real magnified image that the eyepiece magnifies again: M ≈ (L/f_o)(D/f_e). Used up to ~1500×.
f_o = 0.5 cm, f_e = 2.5 cm, L = 16 cm: M = 320.
Resolution limited by diffraction.
Astronomical telescope
Refractor with large objective.
M = f_o/f_eObjective lens (large f) collects light; eyepiece magnifies. In normal adjustment M = f_o/f_e and the tube length is f_o + f_e.
f_o = 100 cm, f_e = 2 cm: M = 50.
Image is inverted.
Reflecting telescope
Uses a concave mirror.
Uses a large concave mirror as objective, avoiding chromatic aberration and allowing very large apertures with a lightweight support. The Hubble and Webb use mirrors.
Webb's mirror is 6.5 m across.
Needs a secondary mirror to bring the image out.
Camera and f-number
Aperture controls exposure and depth of field.
The f-number N = f/D controls how much light enters; each stop (f/2.8, 4, 5.6...) halves the light. Smaller aperture increases depth of field.
f/2 lets in 4 times as much light as f/4.
Exposure is a trade-off with shutter time.
Wave theory of light
Huygens' principle explains reflection and refraction.
Huygens and Young treated light as waves; Maxwell showed they are electromagnetic. Wave theory explains interference, diffraction and polarisation.
Fringe patterns in thin films.
Particle behaviour appears in photon detection.
Huygens' principle
Each point on a wavefront is a source of secondary wavelets.
Every point on a wavefront acts as a source of secondary spherical wavelets; the new wavefront is their envelope. It explains reflection, refraction and diffraction.
Plane waves hitting a boundary and bending.
Ignores the backward wave (needs Kirchhoff).
Coherent sources
Constant phase difference needed for interference.
Sources with a constant phase difference produce stable interference patterns. Light from two ordinary bulbs isn't coherent, but two slits lit by the same source are.
Lasers are highly coherent.
Same frequency is necessary but not sufficient.
Young's double slit
Bright and dark fringes from two slits.
β = λD/dLight from two narrow slits forms bright and dark fringes on a screen. Bright when path difference = nλ: d sinθ = nλ.
d = 0.2 mm, D = 1.5 m, λ = 550 nm: β = 4.1 mm.
Small angles allow sinθ ≈ y/D.
Fringe width
Spacing between adjacent bright fringes.
β = λD/dβ = λD/d: fringes are wider for longer wavelengths, larger screen distances and closer slits. In water β reduces by n.
Red fringes are wider than blue.
Measure over several fringes to reduce error.
Intensity in interference
Intensity distribution across the fringes.
I = 4I₀cos²(φ/2)I = 4I₀cos²(φ/2) for two equal sources, varying between 0 and 4I₀. Average intensity is 2I₀, so energy is redistributed, not created.
Bright fringes are 4 times the single-slit intensity.
Intensity ∝ amplitude squared.
Thin film interference
Colours in soap bubbles and oil films.
2nt = mλLight reflected from the two surfaces of a film interferes; the condition depends on thickness and phase changes on reflection. Soap bubbles, oil slicks and anti-reflective coatings.
A coating of thickness λ/4n on glass cancels reflection.
Count the π phase changes.
Newton's rings
Circular fringes from a lens on a flat surface.
A convex lens on a flat plate produces concentric rings from an air-wedge interference; the radius of the nth dark ring is r_n = √(nλR). Used to test lens surfaces.
R = 1 m, λ = 589 nm: r₁₀ = 2.4 mm.
Centre is dark due to the phase change.
Michelson interferometer
Precise length measurements through interference.
A beam splitter sends light along two perpendicular arms; recombining shows fringes that shift by one for each λ/2 mirror move. Used to measure tiny lengths and in LIGO.
Michelson–Morley experiment.
Needs stability to nanometres.
Single slit diffraction
Central maximum with decreasing side peaks.
a sinθ = mλLight through a slit of width a gives a central maximum with width 2λD/a and minima at a sinθ = mλ. A narrower slit gives a broader pattern.
a = 0.1 mm, λ = 600 nm, D = 1 m: central width 12 mm.
Bright fringes outside the central are much dimmer.
Diffraction grating
Many slits give sharp spectra.
d sinθ = mλMany equally spaced slits give sharp maxima at d sinθ = mλ. More lines produce narrower and brighter peaks; used in spectrometers.
600 lines/mm, λ = 500 nm: first order at 17.5°.
Different from the single slit pattern.
Resolving power
Ability to distinguish nearby objects; Rayleigh criterion.
θ = 1.22λ/DRayleigh criterion: two point sources are just resolved when the first minimum of one coincides with the maximum of the other: θ = 1.22λ/D. Bigger apertures resolve more.
Human eye (D = 3 mm): about 1 arc minute.
Diffraction limits all instruments.
Polarisation of light
Light with E field in a single plane.
Unpolarised light has random E orientations; a polariser transmits only one direction. Light can be polarised by absorption, reflection, scattering or birefringence.
Polaroid sunglasses reduce glare.
Shows light is a transverse wave.
Malus's law
Intensity through an analyser.
I = I₀cos²θIntensity through an analyser is I = I₀cos²θ. Two crossed polarisers block all light, but a third at 45° between them lets a fraction through.
Unpolarised light through one polariser halves; then cos²θ.
Initial halving occurs only for unpolarised input.
Brewster's angle
Reflected light is fully polarised at this angle.
tanθ_B = nAt the incidence angle with tanθ_B = n₂/n₁ the reflected light is completely polarised (parallel to the surface). For glass n = 1.5 it is 56.3°.
Reflected and refracted rays are perpendicular.
Used in lasers and glare filters.
Polarisers and LCDs
Liquid crystal displays use polarisers.
Liquid-crystal displays place a liquid crystal between crossed polarisers; voltage controls how much it rotates the polarisation, switching pixels on or off.
Phone screens.
Polarised sunglasses can make LCD screens look dark at some angles.
Birefringence
Different indices for different polarisations.
Some crystals (calcite, quartz) have different refractive indices for different polarisations, splitting a beam into ordinary and extraordinary rays. Used in wave plates.
Double image through calcite.
Depends on the crystal axis.
Optical activity
Some substances rotate the plane of polarisation.
Certain substances (sugar solutions, quartz) rotate the plane of polarisation. The rotation is proportional to path length and concentration, measured in polarimeters.
Used in the sugar industry.
Different enantiomers rotate in opposite directions.
Coherence and lasers
Stimulated emission produces coherent beams.
Lasers emit light that is nearly monochromatic, coherent and highly directional through stimulated emission in a resonant cavity.
Laser pointers, surgery and barcode readers.
Not necessarily powerful.
Holography
Records interference patterns to reproduce 3D images.
A hologram records both the amplitude and phase of light through interference with a reference beam; illuminating it reconstructs a 3D image.
Security holograms on bank cards.
Needs coherent light for recording.
Fermat's principle
Light takes the path of stationary travel time.
Light travels along the path of stationary (usually minimum) travel time between two points. Both reflection and refraction laws follow.
A lifeguard problem: run along the beach and swim.
Stationary doesn't always mean minimum.
Fresnel and Fraunhofer diffraction
Near-field and far-field diffraction regimes.
Fraunhofer (far-field) diffraction has parallel waves and simple patterns; Fresnel (near-field) has curved wavefronts. The Fresnel number a²/λL tells which applies.
Single slit with a screen far away is Fraunhofer.
Poisson spot is a Fresnel effect.
Fibre optics numerical aperture
Acceptance angle of a fibre.
NA = √(n₁²−n₂²)NA = √(n₁² − n₂²) = sinθ_max for the acceptance cone. Light entering inside the cone is trapped by TIR.
n₁ = 1.48, n₂ = 1.46: NA = 0.24, θ = 14°.
Larger NA collects more light but increases modal dispersion.
Colour and the eye
Cones in the retina detect red, green, blue.
Three cone types (red, green, blue sensitive) give colour vision; all other colours are combinations of their responses. Colour blindness arises from missing or shifted cones.
Yellow light stimulates red and green cones.
Colour is a perception, not a property of the light alone.
Additive and subtractive colour mixing
Light adds to white; pigments subtract.
Mixing lights adds (red + green = yellow; all three = white). Mixing pigments subtracts (cyan + magenta + yellow gives dark) as each absorbs colours.
Screens use RGB; printers use CMYK.
Different primaries for each.
Photometry
Luminous intensity, flux and illuminance.
Measures light as perceived by the human eye: luminous flux (lumen), illuminance (lux = lm/m²) and luminous intensity (candela). Weighted by the eye's sensitivity (peak at 555 nm).
A 800 lm bulb over 1 m² gives 800 lux.
Radiometry measures energy not perception.
Geometrical vs wave optics
Ray optics is valid when objects are much larger than λ.
Ray optics is valid when objects and apertures are much larger than the wavelength. When sizes approach λ, diffraction and interference appear.
Lens design uses rays; resolution limits use waves.
The two pictures are consistent in their domains.
Frequently asked questions
Why does a straw look bent in a glass of water?
Light changes speed and direction when it passes from water into air, a process called refraction. Your brain traces the light rays back in straight lines, so the submerged part appears displaced.
What is total internal reflection?
It happens when light travelling in a denser medium hits the boundary with a less dense medium at an angle greater than the critical angle. All the light is reflected back into the denser medium.
What is the difference between a real and a virtual image?
A real image is formed where light rays actually converge and can be projected onto a screen. A virtual image is where the rays appear to come from, and it cannot be projected.
Why does a lens make an image upside down?
Light from the top of the object crosses the axis after passing the lens, so the real image is inverted. Your eye still sees it correctly because your brain interprets the picture.
Why do rainbows form?
Raindrops refract, reflect and refract sunlight again, and each colour leaves at a slightly different angle, spreading the light into a spectrum.
Other areas
Measurement & Math Tools · Kinematics · Newton's Laws & Forces · Work, Energy & Power · Momentum & Collisions · Rotational Motion · Gravitation & Orbits · Oscillations & SHM · Waves & Sound · Fluids & Elasticity · Heat & Thermodynamics · Kinetic Theory & Statistical Physics · Electrostatics · Current Electricity & Circuits · Magnetism & Induction · AC Circuits & EM Waves · Quantum Mechanics · Atomic & Nuclear Physics · Relativity · Particle Physics & Cosmology