Ray Optics and Optical Instruments

Light is an electromagnetic wave, but when the objects it meets are very much larger than its wavelength (about 0.4 to 0.7 μm) we can forget the wave and follow straight-line rays. That is the whole idea of ray optics. A ray is the direction along which energy travels; a bundle of rays is a beam. The speed of light in vacuum, c = 3 × 10⁸ m s⁻¹, is the highest speed in nature.

The chapter uses two laws. The laws of reflection: the angle of reflection equals the angle of incidence, and the incident ray, the reflected ray and the normal at the point of incidence lie in one plane. The laws of refraction (Snell's law): the incident ray, the refracted ray and the normal lie in one plane, and sin i / sin r is a constant for a given pair of media and a given colour. Every result in the chapter, from the mirror equation to the telescope, is these two laws applied to a curved surface with a bit of geometry.

Keep the NCERT sequence in mind because the derivations build on each other: reflection at a spherical mirror gives the mirror equation; refraction at a plane surface gives Snell's law, apparent depth and total internal reflection; refraction at one spherical surface gives the lens maker's formula and the thin lens formula; two refractions in a prism give the prism formula; and lenses combined in the right way give the microscope and the telescope.

A spherical mirror is a piece of a sphere. Its centre P is the pole, the centre of the sphere is the centre of curvature C, the radius of the sphere is the radius of curvature R, and the line PC is the principal axis. Rays parallel to the axis and close to it (paraxial rays) meet the axis at the principal focus F after reflection from a concave mirror, or appear to come from F for a convex mirror. PF is the focal length f.

Cartesian sign convention (NCERT). All distances are measured from the pole (for a lens, from the optical centre). Distances measured in the direction of the incident light are positive; distances against it are negative. Heights above the principal axis are positive, below it negative. With light travelling from left to right, an object on the left has u negative. A concave mirror has f and R negative; a convex mirror has f and R positive. A convex lens has f positive, a concave lens f negative.

Derivation of f = R/2. Let a ray parallel to the axis strike a concave mirror at M. CM is the normal, so the ray makes angle θ with CM, and by the law of reflection the reflected ray also makes θ with CM. Hence ∠MCP = θ and ∠MFP = 2θ (exterior angle of triangle CMF). Drop the perpendicular MD on the axis. Then tan θ = MD/CD and tan 2θ = MD/FD. For paraxial rays the angles are small, so tan θ ≈ θ and tan 2θ ≈ 2θ, giving MD/FD = 2 · MD/CD, i.e. FD = CD/2. For a mirror of small aperture D is very close to P, so FD ≈ f and CD ≈ R. Therefore f = R/2. The same argument holds for a convex mirror.

Standard ray-diagram rays: (i) a ray parallel to the axis passes through F after reflection (or appears to diverge from F); (ii) a ray through C retraces its path; (iii) a ray through F emerges parallel to the axis; (iv) a ray hitting the pole is reflected symmetrically about the axis. Any two of these locate the image.

Derivation. Take a concave mirror with an object AB standing on the axis beyond C, and let its real inverted image be A'B'. Draw the ray from A parallel to the axis, striking the mirror at M and passing through F, and the ray from A through the pole P, which is reflected symmetrically. Because the mirror is of small aperture, M is essentially vertically above P, so MP = AB.

Triangles A'B'F and MPF are similar (right-angled, equal angles at F): B'A'/PM = B'F/FP, so B'A'/BA = B'F/FP = (B'P − FP)/FP. Triangles A'B'P and ABP are also similar (the angles at P are equal by the law of reflection): B'A'/BA = B'P/BP. Equating, (B'P − FP)/FP = B'P/BP.

Now put in signs: B'P = −v, FP = −f, BP = −u. Then (−v + f)/(−f) = (−v)/(−u), i.e. (v − f)/f = v/u. Dividing by v: 1/f − 1/v = 1/u, which gives the mirror equation 1/v + 1/u = 1/f. Though derived for one case, it holds for all positions of the object and both kinds of mirror with the sign convention.

Linear magnification m = h'/h, where h and h' are the heights of object and image. From the similar triangles A'B'P and ABP, h'/h with signs gives m = −v/u. A negative m means an inverted (real) image; a positive m means an erect (virtual) image. |m| > 1 means magnified.

Results the examiner expects you to know from the equation: a convex mirror always gives a virtual, erect, diminished image between P and F, which is why it is used as a rear-view mirror (wide field of view); a concave mirror gives a virtual, erect, magnified image only when the object is between P and F (shaving mirror); an object between F and C gives a real image beyond C; an object beyond C gives a real image between F and C.

When light crosses the boundary between two transparent media its direction changes. Snell's law: sin i / sin r = n₂₁, the refractive index of medium 2 with respect to medium 1. If n₂₁ > 1 the ray bends towards the normal (medium 2 is optically denser); if n₂₁ < 1 it bends away. Also n₂₁ = 1/n₁₂, and for three media n₃₂ = n₃₁ × n₁₂. The absolute refractive index of a medium is n = c/v, its value with respect to vacuum, so n₂₁ = n₂/n₁ = v₁/v₂. Optical density (refractive index) is not the same as mass density: turpentine is less dense than water but optically denser.

Rectangular glass slab. The ray refracts towards the normal at the first face and away from it at the second. Since the two faces are parallel the angle of emergence equals the angle of incidence, so the emergent ray is parallel to the incident ray but shifted sideways. This sideways displacement is the lateral shift; it grows with the thickness of the slab and with the angle of incidence.

Apparent depth. An object at the bottom of a tank of water looks raised because rays from it bend away from the normal on leaving the water and appear to come from a shallower point. For near-normal viewing, real depth / apparent depth = n, so a tank h deep appears h/n deep. This is why a coin in a glass of water looks nearer, and why a glass slab of thickness t raises an object under it by t(1 − 1/n), a shift that does not depend on where the slab is placed between the object and the eye.

Atmospheric refraction is an application: the sun is seen about two minutes before actual sunrise and after actual sunset because its light bends towards the normal as it enters denser and denser layers of air; the flattened look of the sun at the horizon has the same cause.

When light goes from a denser to a rarer medium (say glass to air) it bends away from the normal, so the angle of refraction is larger than the angle of incidence. As the angle of incidence increases, the refracted ray grazes the surface (r = 90°) at a particular angle of incidence called the critical angle i_c (C). For any larger angle of incidence there is no refracted ray at all; the whole of the light is reflected back into the denser medium. This is total internal reflection. The two conditions are: light must travel from a denser to a rarer medium, and the angle of incidence must exceed the critical angle. The reflection is 'total' because, unlike a mirror, nothing is absorbed or transmitted.

Relation with n. Apply Snell's law at the critical angle with the rarer medium as medium 1: n₁₂ = sin i_c / sin 90°, so sin i_c = n₁₂ = 1/n₂₁. For glass–air with n = 1.5, sin i_c = 2/3, i_c ≈ 42°. For water (n = 1.33), i_c ≈ 49°. For diamond (n = 2.42), i_c ≈ 24°; the small critical angle lets a cut diamond trap light by repeated total reflections, which is its sparkle (NCERT exercise 9.18(e)).

Applications the board asks for. (i) Totally reflecting prisms: a right-angled isosceles glass prism (45°, 45°, 90°) turns a ray through 90° when the ray enters one short face normally and hits the hypotenuse at 45°, greater than 42°; it turns a ray through 180° (erecting an inverted image) when the ray enters the hypotenuse normally and reflects off both short faces. Used in binoculars and periscopes; better than mirrors because there is no silvering to tarnish and no loss of intensity. (ii) Optical fibres: a thin glass or quartz core (higher n) coated with a cladding of lower n. Light entering one end strikes the core–cladding interface at an angle greater than the critical angle and is totally reflected thousands of times, following the fibre round bends with almost no loss. A bundle of fibres carries images in endoscopes, and single fibres carry telephone and internet signals as light pulses. The cladding, not air, is used so that the critical angle is well defined and the surface is protected.

This is the derivation that everything about lenses is built on, and the board asks for it either alone (3 marks) or as the first half of the lens maker's formula (5 marks).

Set-up. A convex spherical surface of radius R separates medium 1 (index n₁, on the left) from medium 2 (index n₂). P is the pole, C the centre of curvature, and a point object O lies on the axis in medium 1. A ray from O strikes the surface at N, refracts and meets the axis at the image I. Draw NM perpendicular to the axis. Assume the aperture is small, so all angles are small and tan of an angle equals the angle. Let i and r be the angles of incidence and refraction at N; CN is the normal.

Geometry. tan ∠NOM = MN/OM, tan ∠NCM = MN/MC, tan ∠NIM = MN/MI, and for small angles these equal the angles themselves. In triangle NOC, i is the exterior angle: i = ∠NOM + ∠NCM = MN/OM + MN/MC. In triangle NCI, r = ∠NCM − ∠NIM = MN/MC − MN/MI.

Snell's law for small angles. n₁ sin i = n₂ sin r becomes n₁ i = n₂ r. Substituting: n₁(MN/OM + MN/MC) = n₂(MN/MC − MN/MI). Cancel MN: n₁/OM + n₂/MI = (n₂ − n₁)/MC.

Signs. OM = −u, MI = +v, MC = +R (M is effectively at P). Hence n₂/v − n₁/u = (n₂ − n₁)/R. This single formula covers convex and concave surfaces and real and virtual images once the signs are put in correctly. NCERT Example 9.5 checks it: a point source in air 100 cm from a glass surface (n = 1.5, R = 20 cm) gives 1.5/v + 1/100 = 0.5/20, so v = 100 cm inside the glass.

Lens maker's formula (5 marks). A thin double convex lens of material n₂ is in a medium n₁. Its first surface ABC has radius R₁ and centre C₁; its second surface ADC has radius R₂ and centre C₂. A point object O on the axis sends a ray that refracts at the first surface. Had the second surface not been there, it would have formed an image I₁. Applying the spherical-surface result at the first surface: n₁/OB + n₂/BI₁ = (n₂ − n₁)/BC₁.

The image I₁ acts as a virtual object for the second surface, where light goes from n₂ to n₁ and the final image forms at I: −n₂/DI₁ + n₁/DI = (n₂ − n₁)/DC₂. For a thin lens BI₁ = DI₁. Adding the two equations: n₁/OB + n₁/DI = (n₂ − n₁)[1/BC₁ + 1/DC₂].

Put the object at infinity (OB → ∞); the image is then at the focus, DI = f: n₁/f = (n₂ − n₁)[1/BC₁ + 1/DC₂]. With the sign convention BC₁ = +R₁ and DC₂ = −R₂, and writing n₂₁ = n₂/n₁:

1/f = (n₂₁ − 1)(1/R₁ − 1/R₂). This is the lens maker's formula. It gives f from the radii and the refractive index, and it explains why f depends on the surrounding medium: in water n₂₁ is smaller, so a glass convex lens has a longer focal length in water than in air, and if n₂₁ = 1 the lens does nothing at all (1/f = 0). If the lens is put in a liquid of higher refractive index, n₂₁ < 1 and a convex lens becomes diverging.

Thin lens formula. Going back one step, with OB = −u, DI = v and n₁/f on the right: 1/v − 1/u = 1/f. Notice the minus sign, different from the mirror equation. Magnification for a lens: m = h'/h = v/u. A convex lens forms a real inverted image for an object beyond F (real, magnified if the object is between F and 2F, diminished if beyond 2F) and a virtual erect magnified image for an object inside F; a concave lens always forms a virtual, erect, diminished image between the lens and F.

Power. The power of a lens measures how strongly it converges or diverges a beam: P = 1/f with f in metres. The SI unit is the dioptre (D); 1 D is the power of a lens of focal length 1 m. A convex lens has positive power, a concave lens negative power. An optician prescribing +2.5 D means a converging lens of f = 0.4 m. Geometrically, P = tan δ / h, where δ is the angle through which a ray at height h from the axis is bent; for h = 1 m, tan δ = 1/f.

Two thin lenses in contact (derivation, 3 marks). Lenses A (f₁) and B (f₂) are in contact. For a point object O on the axis, lens A alone would form an image at I₁: 1/v₁ − 1/u = 1/f₁. I₁ is the object for lens B, which forms the final image at I: 1/v − 1/v₁ = 1/f₂. Adding: 1/v − 1/u = 1/f₁ + 1/f₂. If the combination is regarded as a single lens of focal length f, then 1/v − 1/u = 1/f, so 1/f = 1/f₁ + 1/f₂. For several lenses 1/f = 1/f₁ + 1/f₂ + 1/f₃ + …, and in terms of power P = P₁ + P₂ + P₃ + …. The total magnification is the product m = m₁m₂m₃….

Such combinations are used in camera lenses, microscopes and telescopes to get the required magnification, sharpness and field of view while reducing the defects of a single lens. Example: a convex lens of f = 30 cm in contact with a concave lens of f = −20 cm gives 1/f = 1/30 − 1/20 = −1/60, f = −60 cm, a diverging combination (NCERT exercise 9.10).

Note for questions where two lenses are separated by a distance: the answer depends on which side the light enters, so the idea of one effective focal length does not apply; you must trace the image lens by lens.

A prism has two refracting faces meeting at the refracting edge; the angle between them is the angle of the prism A. A ray PQ enters face AB at Q with angle of incidence i and angle of refraction r₁, travels to face AC, and leaves at R with internal angle of incidence r₂ and angle of emergence e. The emergent ray is bent towards the base; the angle δ between the incident and emergent directions is the angle of deviation.

Derivation of A + δ = i + e. The normals at Q and R meet at N. In the quadrilateral AQNR the angles at Q and R are right angles, so A + ∠QNR = 180°. In triangle QNR, r₁ + r₂ + ∠QNR = 180°. Comparing, r₁ + r₂ = A. The deviation at Q is (i − r₁) and at R is (e − r₂), so δ = (i − r₁) + (e − r₂) = i + e − A, i.e. A + δ = i + e.

Minimum deviation. As i increases from a small value, δ first decreases, reaches a minimum D_m, then increases again; the graph of δ against i is a U-shaped curve, and each value of δ (other than D_m) is produced by two angles of incidence, i and e interchanged. At minimum deviation the ray inside the prism runs parallel to the base and by symmetry i = e and r₁ = r₂ = r. Then 2r = A, r = A/2, and A + D_m = 2i, i = (A + D_m)/2. Snell's law at the first face gives the prism formula: n₂₁ = sin i / sin r = sin[(A + D_m)/2] / sin(A/2). This is how the refractive index of a prism material is measured in the laboratory.

Thin prism. For small A, sin[(A + D_m)/2] ≈ (A + D_m)/2 and sin(A/2) ≈ A/2, so n₂₁ = (A + D_m)/A, giving D_m = (n₂₁ − 1)A. A thin prism does not deviate the light much. Since n is slightly larger for violet than for red, violet deviates more; replacing violet light by red reduces the angle of minimum deviation.

The unaided eye sees an object clearly and comfortably no nearer than the near point, D = 25 cm for a normal eye. Optical instruments enlarge the angle the object subtends at the eye.

Simple microscope (magnifying glass). A single convex lens with the object held inside its focal length gives an erect, magnified, virtual image. If the image is formed at the near point D, the magnification is m = 1 + D/f. If the object is placed exactly at the focus the image is at infinity and the eye is most relaxed; then m = D/f. The magnifying power is the ratio of the angle subtended by the image to the angle subtended by the object when the object is at D. A single lens is rarely used beyond a magnifying power of about 6 (f = 5 cm gives m = 6 for the image at D) before aberrations spoil the image, so for more we use two lenses.

Compound microscope. The objective is a convex lens of very short focal length f_o and small aperture; the eyepiece has a slightly larger focal length f_e and larger aperture. The object is placed just beyond f_o, so the objective forms a real, inverted, magnified image A'B' near the focus of the eyepiece. The eyepiece works as a simple microscope on this image and gives a final virtual image, inverted with respect to the object and greatly enlarged, either at D or at infinity.

Magnifying power. The objective magnifies m_o = h'/h = v_o/u_o. Because the object is close to the focus and the intermediate image close to the eyepiece, u_o ≈ f_o and v_o ≈ L, the tube length (distance between the second focal point of the objective and the first focal point of the eyepiece), so m_o ≈ L/f_o. The eyepiece gives m_e = 1 + D/f_e (image at D) or D/f_e (image at infinity). The total magnifying power is m = m_om_e = (L/f_o)(D/f_e) for normal adjustment. So both focal lengths must be small, with f_o smaller than f_e. In the ray diagram, mark f_o on both sides of the objective, f_e on the eyepiece, the intermediate image A'B' between the eyepiece and its focus, and the final image A''B'' on the same side as the object.

Refracting (astronomical) telescope. Two convex lenses: an objective of large focal length f_o and large aperture, and an eyepiece of short focal length f_e and small aperture. Parallel rays from a distant object form a real, inverted, diminished image A'B' at the focal plane of the objective. The eyepiece magnifies this image. In normal adjustment the intermediate image lies at the focus of the eyepiece, so the final image is at infinity, and the length of the tube is L = f_o + f_e.

Magnifying power derivation. Magnifying power is the ratio of the angle β subtended at the eye by the final image to the angle α subtended at the lens (or eye) by the object. Since the object is far away, α is the angle A'B' subtends at the objective: tan α ≈ α = h/f_o. The same image subtends β at the eyepiece: β = h/f_e. Hence m = β/α = f_o/f_e. If instead the final image is formed at D, m = (f_o/f_e)(1 + f_e/D). For large magnifying power f_o must be large and f_e small; a large aperture of the objective gathers more light (brighter image) and gives higher resolving power. A telescope with f_o = 100 cm and f_e = 1 cm magnifies 100 times. The final image is inverted, which does not matter for astronomy; terrestrial telescopes add an erecting lens or an inverting prism pair.

Reflecting telescope. The objective is a concave mirror (paraboloidal) instead of a lens. In the Cassegrain arrangement a large concave primary mirror with a hole at its centre reflects light to a small convex secondary mirror, which sends it back through the hole to the eyepiece. Advantages over a refracting telescope, each a separate marking point: (i) no chromatic aberration, since a mirror reflects all colours alike; (ii) spherical aberration is reduced by using a parabolic mirror; (iii) a mirror can be made much larger than a lens because it is supported from behind and does not need a flawless large block of glass, so it gathers more light and resolves better; (iv) a mirror is lighter and cheaper than a lens of the same aperture; (v) the Cassegrain design makes the tube short. The largest telescopes, such as the 10 m Keck and India's 2.34 m Kavalur telescope, are reflecting telescopes.