Geometrical optics

In short

Geometrical optics treats light as rays travelling in straight lines and bending at surfaces, which is valid whenever the apertures and objects are far larger than the wavelength. From this single approximation come the mirror and lens formulas, prism dispersion, total internal reflection, image defects (aberrations) and the design of every microscope, telescope, camera and spectrometer. It is a guaranteed scoring block in B.Sc Optics papers, appears in GATE (PH) under "ray optics, matrix methods, lens aberrations, optical instruments", and is tested in JAM, CSIR-UGC NET and IIT/IISER entrance interviews. Employers using it directly include optical-instrument makers and labs in India — Bharat Electronics, Instruments Research & Development Establishment (IRDE, Dehradun), Centre for Advanced Technology (RRCAT, Indore), Holmarc, Essilor and Lenskart lens plants, and ISRO/DRDO payload optics groups.

Explained

Foundations: rays, Fermat's principle and sign convention

• Ray - the normal to a wavefront; the direction of energy flow. Geometrical optics is the λ → 0 limit of wave optics, so it cannot explain interference or the diffraction-limited spot.
• Laws of reflection: incident ray, reflected ray and normal are coplanar; angle of incidence = angle of reflection.
• Laws of refraction (Snell): n₁ sin i = n₂ sin r; the ray bends towards the normal on entering a denser medium.
• Optical path length = ∫n ds. Fermat's principle: light travels a path for which the optical path length is stationary (minimum, maximum or constant). Both laws of reflection and refraction follow from it by minimisation.
• Cartesian sign convention (used in Indian syllabi): all distances measured from the pole/optical centre; distances along the incident light are positive, against it negative; heights above the axis positive. Hence a real object has u negative in the convention where light travels left to right.
• Paraxial approximation: rays make small angles with the axis, so sin θ ≈ tan θ ≈ θ. All simple formulas are paraxial; departures from it are aberrations.
• Real vs virtual: a real image is formed by actually converging rays and can be caught on a screen; a virtual image is the apparent intersection of diverging rays.

Reflection at plane and spherical surfaces

• Plane mirror: image is virtual, erect, same size, laterally inverted, as far behind the mirror as the object is in front. Rotating the mirror by θ rotates the reflected ray by 2θ.
• Two inclined mirrors at angle θ give n = (360/θ) − 1 images (360/θ even); the basis of the kaleidoscope.
• Spherical mirrors: concave (converging) and convex (diverging). Pole - centre of the reflecting surface; centre of curvature C; radius R; focus F at R/2 paraxially.
• Mirror formula 1/v + 1/u = 1/f with f = R/2; magnification m = −v/u = h′/h.
• Image table for a concave mirror: object beyond C → real, inverted, diminished, between F and C; at C → real, inverted, same size; between F and C → real, inverted, magnified beyond C; at F → at infinity; inside F → virtual, erect, magnified. A convex mirror always gives a virtual, erect, diminished image between P and F.
• Uses: convex mirrors as road and vehicle rear-view mirrors (wide field); concave as shaving, dental and headlight reflectors; parabolic mirrors in reflecting telescopes and solar concentrators (no spherical aberration on axis).

Refraction: plane surfaces, slabs, prisms, dispersion

• Refractive index n = c/v (absolute); relative index n₂₁ = n₂/n₁ = v₁/v₂ = λ₁/λ₂ (frequency unchanged). Typical values: water 1.33, crown glass 1.52, flint glass ≈ 1.62, diamond 2.42.
• Apparent depth: real depth/apparent depth = n. Normal shift for a slab of thickness t: shift = t(1 − 1/n); lateral shift = t sin(i − r)/cos r.
• Total internal reflection (TIR): for light going from denser to rarer, if i > critical angle C where sin C = n₂/n₁, all light is reflected. C ≈ 48.8° for water-air, ≈ 41.1° for crown glass-air, ≈ 24.4° for diamond-air (hence the brilliance of a cut diamond). TIR is lossless — superior to metal mirrors — and underlies porro prisms in binoculars, right-angle prisms, mirages and optical fibres.
• Prism: for refraction through a prism of apex angle A, r₁ + r₂ = A and deviation δ = i₁ + i₂ − A. δ is minimum when the ray passes symmetrically (i₁ = i₂, r₁ = r₂ = A/2), giving the prism formula n = sin((A + δm)/2)/sin(A/2) — the standard spectrometer experiment for measuring n.
• Thin prism: δ = (n − 1)A. Dispersion: n decreases with wavelength (Cauchy n = A + B/λ²), so violet deviates most, red least. Angular dispersion = (nV − nR)A; dispersive power ω = (nV − nR)/(nY − 1).
• Achromatic combination of two thin prisms: ω₁A₁(n₁−1) ... practically, (nV−nR)₁A₁ = (nV−nR)₂A₂ with the prisms opposed — zero net dispersion, some net deviation (direct-vision prism does the opposite).

Refraction at spherical surfaces, thin lenses and combinations

• Single refracting surface: n₂/v − n₁/u = (n₂ − n₁)/R.
• Lens maker's formula (thin lens in air): 1/f = (n − 1)(1/R₁ − 1/R₂). In a medium of index nm, replace (n − 1) by (n/nm − 1) — a glass convex lens becomes diverging in a liquid of higher index.
• Thin lens formula 1/v − 1/u = 1/f; magnification m = v/u. Power P = 1/f (metres) in dioptres; converging lens P positive.
• Lenses in contact: P = P₁ + P₂ + …; separated by distance d: 1/F = 1/f₁ + 1/f₂ − d/(f₁f₂), equivalently P = P₁ + P₂ − dP₁P₂.
• Equivalent focal length and cardinal points of a thick lens or system: two focal points, two principal points with their principal planes (unit transverse magnification), two nodal points (unit angular magnification; coincide with principal points when the end media are the same).
• Newton's formula: x₁x₂ = f₁f₂ with distances measured from the focal points.
• Longitudinal (axial) magnification = m² for small objects; Lagrange–Helmholtz invariant n·h·u = constant limits how much magnification a system can give.
• Matrix (ABCD) method: each element is a 2×2 matrix acting on (height, n·angle). Translation [[1, d/n],[0, 1]]; refraction at a surface [[1, 0],[−P, 1]] with P = (n₂−n₁)/R; a system matrix is the ordered product. The matrix is unimodular (determinant 1) and its elements give the system's focal length (f = −1/C) and cardinal points. This is the method GATE and CBCS papers set for thick lenses and lens systems.

Aberrations: why real images are imperfect

• Monochromatic (Seidel) aberrations — arise because sin θ ≠ θ:
  1. Spherical aberration - marginal rays focus nearer than paraxial rays; longitudinal and lateral SA. Reduced by stopping down, using a stop at the right place, aplanatic surfaces, a plano-convex lens with the curved side to the parallel beam, crossed lenses, or an aspheric/parabolic surface. Two thin lenses separated by d = f₁ − f₂ minimise SA (Huygens eyepiece condition).
  2. Coma - off-axis point images as a comet-shaped flare; removed when the Abbe sine condition n h sin u = n′ h′ sin u′ holds (aplanatic system).
  3. Astigmatism - off-axis beams focus into two mutually perpendicular line images (tangential and sagittal).
  4. Curvature of field (Petzval) - image surface is curved; flattened by making Σ(1/nf) = 0.
  5. Distortion - magnification varies with field angle: barrel (stop in front of lens) and pincushion (stop behind).
• Chromatic aberration - f depends on λ. Longitudinal CA = fR − fV; lateral CA = size difference.
• Achromatic doublet: two lenses in contact of crown (low ω) and flint (high ω) glass satisfying ω₁/f₁ + ω₂/f₂ = 0 — the standard camera and telescope objective. For separated lenses, the condition is ω₁/f₁ + ω₂/f₂ = d(ω₁ + ω₂)/(f₁f₂), i.e. d = (ω₁f₂ + ω₂f₁)/(ω₁ + ω₂), which reduces to d = (f₁ + f₂)/2 when the glasses are the same (eyepiece achromatism).
• Limitation to remember: even an aberration-free lens cannot focus to a point — diffraction sets the limit (Airy pattern), which belongs to wave optics.

Optical instruments

• Human eye: cornea + aqueous + crystalline lens focus on the retina; accommodation changes lens power; least distance of distinct vision D ≈ 25 cm, far point infinity. Defects: myopia (corrected by a concave lens of P = −1/far point in m), hypermetropia (convex lens), presbyopia (bifocals), astigmatism (cylindrical lens).
• Simple microscope (magnifier): M = 1 + D/f (image at D), M = D/f (image at infinity).
• Compound microscope: M = mo × Me = (vo/uo)(1 + D/fe); short fo and fe, object just beyond Fo. Resolving limit (Abbe, wave optics) = 1.22λ/(2 n sin θ); numerical aperture NA = n sin θ — oil immersion raises NA.
• Astronomical (refracting) telescope: M = fo/fe (normal adjustment, length fo + fe), M = (fo/fe)(1 + fe/D) for image at D; large fo and aperture. Terrestrial telescope adds an erecting lens; Galilean uses a concave eyepiece (erect image, short tube).
• Reflecting telescopes (Newtonian, Cassegrain) - no chromatic aberration, large aperture possible, cheaper per unit aperture; used in Indian facilities such as the 2 m Himalayan Chandra Telescope (Hanle), the 3.6 m Devasthal Optical Telescope (ARIES, Nainital) and the 2.34 m Vainu Bappu Telescope (Kavalur).
• Eyepieces: Ramsden (two plano-convex lenses, convex faces facing, separation 2f/3) - has a real external focal plane, so cross-wires can be placed; used in spectrometers and travelling microscopes. Huygens (field lens 3f, eye lens f, separation 2f) - better achromatism and lower SA, but focal plane is internal, so it cannot take cross-wires.
• Camera lens: f-number N = f/D controls illumination (∝ 1/N²) and depth of field; India's spectrometers, autocollimators and theodolites all use the same paraxial theory.

Key points

• Geometrical optics - the ray treatment of light, valid when the aperture ≫ λ; the λ → 0 limit of wave optics.
• Fermat's principle - light follows a stationary optical-path route; both reflection and refraction laws follow from it.
• Snell's law - n₁ sin i = n₂ sin r; frequency is unchanged on refraction, wavelength and speed change.
• Critical angle - sin C = n_rarer/n_denser; beyond it, total internal reflection with no loss.
• Mirror formula - 1/v + 1/u = 1/f, f = R/2; the single relation for both mirror types with sign convention.
• Lens maker's formula - 1/f = (n − 1)(1/R₁ − 1/R₂); f changes if the surrounding medium changes.
• Power in dioptres - P = 1/f(m); in contact, powers add; separated lenses lose dP₁P₂.
• Prism minimum deviation - n = sin((A + δm)/2)/sin(A/2); the standard lab method for refractive index.
• Dispersive power - ω = (nV − nR)/(nY − 1); dimensionless, a property of the glass only.
• Achromatic doublet - ω₁/f₁ + ω₂/f₂ = 0, crown + flint in contact.
• Spherical aberration - marginal rays focus closer than paraxial; cured by aspherics, stops, or lens splitting.
• Five Seidel aberrations - spherical, coma, astigmatism, field curvature, distortion.
• Cardinal points - two focal, two principal, two nodal points fully describe a thick system.
• ABCD matrices - unimodular 2×2 matrices multiply in reverse order of travel; f = −1/C.
• Magnifying power - microscope (vo/uo)(1 + D/fe); telescope fo/fe in normal adjustment.
• Ramsden vs Huygens eyepiece - external vs internal focal plane; cross-wires possible only with Ramsden.
• Lagrange invariant - n h u is conserved; magnification and field cannot both be increased freely.
• Diffraction limit - geometrical optics predicts a point image; the real limit is 1.22λ/D, a wave-optics result.

Formulas and examples

Example 1 — Concave mirror, image position and size. An object 2 cm tall stands 30 cm in front of a concave mirror of radius 40 cm.

• f = R/2 = 20 cm. Using magnitudes with 1/v + 1/u = 1/f for a real object: 1/v = 1/20 − 1/30 = (3 − 2)/60 = 1/60 → v = 60 cm in front of the mirror (real image).
• m = −v/u = −60/30 = −2 → image is 4 cm tall, inverted. It lies beyond C (40 cm), as the table predicts for an object between F and C.

Example 2 — Prism at minimum deviation. A glass prism of apex angle A = 60° gives δm = 40°.

• n = sin((60 + 40)/2)/sin(60/2) = sin 50°/sin 30° = 0.7660/0.5 = 1.532 (crown glass).
• Critical angle: sin C = 1/1.532 = 0.6527 → C = 40.7°.
• At minimum deviation r = A/2 = 30°, so i = (A + δm)/2 = 50°, confirming the symmetric passage.

Example 3 — Lens maker and change of medium. An equiconvex lens of glass n = 1.5 has R = 20 cm on each face (R₁ = +20, R₂ = −20 cm).

• In air: 1/f = (1.5 − 1)(1/20 + 1/20) = 0.5 × 0.1 = 0.05 → f = +20 cm, P = +5 D.
• Immersed in water (nm = 1.33): 1/f = (1.5/1.33 − 1)(0.1) = (0.1278)(0.1) = 0.01278 → f ≈ +78.2 cm, i.e. P ≈ +1.28 D — much weaker.
• Immersed in carbon disulphide (n = 1.62): (1.5/1.62 − 1) = −0.0741 → f ≈ −135 cm: the convex lens now diverges.

Example 4 — Separated lens combination by formula and by matrix. Two thin convex lenses, f₁ = 10 cm and f₂ = 20 cm, are 5 cm apart in air.

• 1/F = 1/10 + 1/20 − 5/(10 × 20) = 0.1 + 0.05 − 0.025 = 0.125 → F = 8 cm (P = 12.5 D).
• Matrix check (P₁ = 0.1 cm⁻¹, P₂ = 0.05 cm⁻¹): M = [[1,0],[−0.05,1]] · [[1,5],[0,1]] · [[1,0],[−0.1,1]] = [[1,0],[−0.05,1]] · [[0.5, 5],[−0.1, 1]] = [[0.5, 5],[−0.125, 0.75]].
• det = 0.5(0.75) − 5(−0.125) = 0.375 + 0.625 = 1 ✓. C = −0.125 → f = −1/C = 8 cm ✓.

Example 5 — Achromatic doublet and a compound microscope. (a) A crown lens (ω₁ = 0.018) and a flint lens are cemented to give F = +50 cm with flint ω₂ = 0.036.

• Achromatism: ω₁/f₁ + ω₂/f₂ = 0 → f₂ = −(ω₂/ω₁) f₁ = −2 f₁.
• Power: 1/f₁ + 1/f₂ = 1/50 → 1/f₁ − 1/(2f₁) = 1/(2f₁) = 1/50 → f₁ = +25 cm (crown), f₂ = −50 cm (flint).

(b) A compound microscope has fo = 1 cm, fe = 5 cm, object 1.2 cm from the objective, D = 25 cm.

• Objective: 1/vo = 1/1 − 1/1.2 = (1.2 − 1)/1.2 = 0.1667 → vo = 6 cm; mo = 6/1.2 = 5.
• Eyepiece (image at D): Me = 1 + 25/5 = 6. M = 5 × 6 = 30×; tube length ≈ vo + ue = 6 + 4.17 ≈ 10.2 cm.

Where it is used

• Defence and space optics: IRDE Dehradun (sights, periscopes, thermal optics), BEL, ISRO/SAC payload telescopes — lens design, ray tracing and tolerancing in Zemax or Code V are everyday work.
• Ophthalmic and spectacle industry: Essilor, Lenskart and Titan Eyeplus lens labs — dioptre prescriptions, cylindrical corrections, surfacing and coating; optometry roles apply the eye-defect formulas directly.
• Scientific instruments: spectrometers, travelling microscopes, goniometers and autocollimators made by Holmarc, Scientech and Toshniwal; RRCAT and IUCAA build beamline and telescope optics.
• Astronomy: ARIES (3.6 m Devasthal), IIA (Kavalur, Hanle) and the India-TMT programme — mirror figuring, collimation and aberration budgets.
• Consumer and industrial imaging: smartphone camera modules, projectors, barcode scanners, machine-vision lenses, LED luminaire reflectors and optics (photometric design under IS/CIE illumination practice).
• Roles: optical design engineer, metrology and NDT engineer, physics lecturer, R&D assistant at DRDO/CSIR labs (GATE PH or JEST score is the usual entry route).

Interview questions

• Q: When does geometrical optics fail? A: When apertures or features approach the wavelength — then diffraction and interference dominate and the wave treatment is needed; it is the λ → 0 limit of wave optics.
• Q: Derive Snell's law from Fermat's principle in one line. A: Minimising the optical path n₁√(x² + a²) + n₂√((d−x)² + b²) with respect to x gives n₁ sin i = n₂ sin r.
• Q: Why does a glass convex lens diverge in carbon disulphide? A: 1/f ∝ (n_lens/n_medium − 1); when the medium is denser than the lens this factor turns negative, reversing the lens action.
• Q: Why is total internal reflection preferred over metallic mirrors in prisms? A: TIR reflects essentially 100% of the light with no metallic absorption, and it needs no coating that can tarnish.
• Q: Why can cross-wires be used in a Ramsden eyepiece but not a Huygens one? A: The Ramsden's focal plane lies outside, in front of the field lens, so a graticule placed there is imaged with the object; the Huygens focal plane is between its two lenses.
• Q: What is the difference between spherical and chromatic aberration? A: Spherical aberration is a monochromatic defect from non-paraxial rays at a curved surface; chromatic aberration comes from the refractive index varying with wavelength, so mirrors show the former but never the latter.
• Q: State the achromatism condition for two cemented lenses. A: ω₁/f₁ + ω₂/f₂ = 0, so one lens must be converging and the other diverging, made of glasses with different dispersive powers (crown and flint).
• Q: What are the cardinal points of an optical system? A: Two focal, two principal and two nodal points; the principal planes have unit transverse magnification and the nodal points unit angular magnification, and they coincide when the object and image media are the same.
• Q: Why is the determinant of a system (ABCD) matrix equal to 1 in air? A: Each translation and refraction matrix is unimodular when the reduced angle n·θ is used, and the determinant of a product of unimodular matrices is 1 — a built-in check on any ray-matrix calculation.
• Q: Can magnifying power be increased indefinitely by shortening the eyepiece focal length? A: No — aberrations grow and the diffraction-limited resolving power, set by the objective aperture, is unchanged, so beyond about 1,000×NA you only get "empty magnification".
• Q: Why is oil immersion used in a microscope? A: Immersion oil of index ≈ 1.52 raises the numerical aperture n sin θ, lowering the resolvable distance 1.22λ/(2NA) and letting more of the cone of light into the objective.
• Q: Why does a prism disperse light but a glass slab does not? A: A slab's two faces are parallel, so the deviations of each colour cancel and only a lateral shift remains; a prism's inclined faces make the deviation (n − 1)A different for each wavelength.

Common mistakes

• Mixing sign conventions mid-problem — using magnitudes in the mirror formula and Cartesian signs in the lens formula, or forgetting R₂ is negative for an equiconvex lens.
• Writing 1/v + 1/u = 1/f for a lens (it is 1/v − 1/u = 1/f) and the reverse for a mirror.
• Treating dispersive power ω as having units, or confusing it with angular dispersion (nV − nR)A, which depends on the prism angle.
• Assuming minimum deviation means grazing incidence; it occurs at symmetric passage with r₁ = r₂ = A/2.
• Taking f of a lens as fixed when the surrounding medium changes, or forgetting that mirrors are unaffected by medium.
• Adding powers of separated lenses as P₁ + P₂ and dropping the −dP₁P₂ term; also using d in cm while powers are in dioptres.
• Claiming a perfect lens gives a point image — diffraction limits it — and confusing resolving power (wave) with magnification (ray).

Related topics

Wave optics: interference, diffraction and polarisation · Lasers and fibre optics · Electromagnetic waves and Maxwell's equations · Photometry and radiometry · Optical instrument design and metrology · Spectroscopy