Ray Optics and Optical Instruments

When a ray of light passes from one transparent medium into another, it changes direction at the boundary because light travels at different speeds in different media. This bending is called refraction, and it is governed by two laws: the incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane, and the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media. This second law is Snell's law. The absolute refractive index of a medium, n = c/v, compares the speed of light in vacuum, c, to its speed v in that medium; a medium with a higher refractive index is called optically denser.

Because refraction bends light towards the normal on entering a denser medium, objects viewed through a denser medium appear closer to the surface than they really are. For near-normal viewing, the real depth and apparent depth of an object are related to the refractive index of the medium in which the object lies. This is why a coin at the bottom of a pool of water looks shallower than it really is, and why a swimming pool always looks less deep than its true depth.

Total internal reflection (TIR) is a special case of refraction that happens when light travels from an optically denser medium towards a rarer medium. As the angle of incidence inside the denser medium increases, the refracted ray bends further away from the normal. At one particular angle of incidence, called the critical angle θc, the refracted ray grazes along the boundary at 90°. For any angle of incidence larger than θc, no light is refracted out at all — the entire ray is reflected back into the denser medium. This reflection is remarkably efficient, which is why TIR is used wherever light must be redirected with minimal loss, such as in totally reflecting prisms used in binoculars and periscopes, and in optical fibres that guide light along a curved path by repeated total internal reflection at the fibre wall. TIR also explains the shimmering "mirage" seen above hot road surfaces, and the brilliant sparkle of a cut diamond, whose very high refractive index (about 2.42) gives it an unusually small critical angle, so light entering it undergoes multiple total internal reflections before emerging.

Worked example (Snell's law): A ray of light travelling in air is incident on a glass slab (n = 1.5) at 45° to the normal. Find the angle of refraction.
Using n₁ sin θ₁ = n₂ sin θ₂ with n₁ = 1, θ₁ = 45°, n₂ = 1.5:
sin θ₂ = (1 × sin 45°)/1.5 = 0.7071/1.5 = 0.4714
θ₂ = sin⁻¹(0.4714) ≈ 28.1°. The ray bends towards the normal on entering the denser glass, as expected.

Worked example (critical angle): Find the critical angle for light travelling from glass (n = 1.5) into water (n = 1.33).
At the critical angle the refracted ray grazes the boundary (angle of refraction = 90°): n_glass sin θc = n_water sin 90° = n_water.
sin θc = 1.33/1.5 = 0.8867
θc = sin⁻¹(0.8867) ≈ 62.5°. Any ray inside the glass striking the glass–water surface at more than about 62.5° is totally internally reflected.

When light refracts at a single spherical surface separating two media of refractive indices n₁ (object side) and n₂ (image side), the object distance u, image distance v and radius of curvature R (all measured from the pole of the surface, using the Cartesian sign convention where distances measured in the direction of the incident light are positive) are related by a single formula. Applying this relation at both curved surfaces of a thin lens gives the lens maker's formula, which connects the focal length of a lens to the refractive index of its material and the radii of curvature of its two faces.

For a thin lens in air, once f is known, image formation is described compactly by the thin lens formula. The same sign convention applies: distances are measured from the optical centre, a convex (converging) lens has positive focal length, and a concave (diverging) lens has negative focal length. The linear magnification produced by a lens is the ratio of image height to object height, equal to v/u.

The ability of a lens to converge or diverge light is expressed by its power, the reciprocal of its focal length in metres; the unit is the dioptre (D). A converging lens has positive power, a diverging lens negative power. When two or more thin lenses are placed in contact on a common axis, the power of the combination is simply the sum of the individual powers — this is why an optician's prescription is written directly in dioptres.

Worked example (lens maker's formula + image formation): A thin biconvex lens is made of glass of refractive index 1.5, with R₁ = +15 cm and R₂ = −30 cm.
1/f = (n₂₁ − 1)(1/R₁ − 1/R₂) = 0.5 × [1/15 − (−1/30)] = 0.5 × (2/30 + 1/30) = 0.5 × 0.1 = 0.05 cm⁻¹, so f = 20 cm (converging).
Now place an object 50 cm in front (u = −50 cm). 1/v = 1/f + 1/u = 1/20 − 1/50 = (5 − 2)/100 = 3/100, so v ≈ 33.3 cm.
Magnification m = v/u ≈ 33.3/(−50) ≈ −0.67: the image is real, inverted, and about two-thirds the object's height.

Worked example (combination of lenses): A converging lens of focal length 25 cm is placed in contact with a diverging lens of focal length 40 cm.
P₁ = 1/0.25 = +4 D, P₂ = 1/(−0.40) = −2.5 D
P = P₁ + P₂ = +1.5 D
F = 1/1.5 ≈ 0.667 m = 66.7 cm. The combination is converging, with focal length about 66.7 cm.

A prism is bounded by two plane refracting surfaces inclined to each other at the refracting angle A (the angle of the prism). A ray entering one face is refracted twice — on entry and on exit — and bends towards the thicker part (the base) of the prism. The total bending between the incident and emergent rays is the angle of deviation, δ. The geometry of the two refractions gives two simple relations: the sum of the two internal refraction angles at the two faces equals the prism angle, and the sum of the angle of incidence and the angle of emergence equals the prism angle plus the deviation.

As the angle of incidence is varied, the deviation first decreases, reaches a single minimum, and then increases again. At this angle of minimum deviation, Dm, the ray inside the prism travels parallel to the base, and the angle of incidence equals the angle of emergence (i = e). This symmetric condition gives a convenient formula for the refractive index of the prism material in terms of A and Dm — the standard experimental method for measuring the refractive index of a transparent solid.

The refractive index of a real material is slightly higher for violet light than for red light, a property called dispersion. When white light passes through a prism, each colour is deviated by a slightly different amount, with violet deviating most and red least, spreading the beam into a continuous band of colours (violet, indigo, blue, green, yellow, orange, red) called a spectrum. This differs from a rectangular glass slab, whose two parallel faces refract each colour back to nearly its original direction, producing only a small lateral shift and no visible spectrum.

Worked example: An equilateral prism (A = 60°) is made of glass of refractive index 1.52. Find its angle of minimum deviation.
n = sin[(A + Dm)/2] / sin(A/2)
sin[(60° + Dm)/2] = 1.52 × sin 30° = 1.52 × 0.5 = 0.76
(60° + Dm)/2 = sin⁻¹(0.76) ≈ 49.5°
60° + Dm ≈ 99.0°, so Dm ≈ 39.0°.

The human eye behaves like a camera: light entering through the transparent cornea and the variable-aperture iris/pupil is focused by the flexible crystalline eye lens to form a real, inverted, diminished image on the light-sensitive retina. The brain interprets this inverted image the right way up. Ciliary muscles change the eye lens's curvature (and hence its focal length) so that objects at different distances can all be focused sharply, a process called accommodation. The closest distance the eye can focus on comfortably is the near point (about 25 cm for a normal young adult, the standard "least distance of distinct vision", D), and the farthest is the far point (infinity, for a normal eye).

When the eyeball's shape or the lens's accommodation range does not match this ideal, refractive defects occur:

• Myopia (near-sightedness): the eyeball is too long, or the lens too strongly converging, so images of distant objects form in front of the retina; the far point is nearer than infinity. Corrected with a concave (diverging) lens of suitable power.
• Hypermetropia (far-sightedness): the eyeball is too short, or the lens too weakly converging, so images of nearby objects would form behind the retina; the near point is farther than 25 cm. Corrected with a convex (converging) lens.
• Presbyopia: with age, the ciliary muscles weaken and the lens loses flexibility, reducing the accommodation range (the near point recedes); often needs bifocal lenses combining both corrections.
• Astigmatism: the cornea (or lens) is not perfectly spherical, so horizontal and vertical lines cannot both be focused sharply; corrected with a cylindrical lens.

Worked example (myopia): A person's far point is 5 m instead of infinity. Find the power of the corrective lens needed.
The lens must form a virtual image of an object at infinity exactly at the far point: u = −∞, v = −5 m.
1/f = 1/v − 1/u = 1/(−5) − 0 = −0.2 m⁻¹, so f = −5 m, P = 1/f = −0.2 D (concave lens).

Worked example (hypermetropia): A person's near point is 1 m instead of the normal 25 cm. Find the power of lens needed to read at the normal near point.
The lens must form a virtual image, of an object held at 25 cm, at the person's actual near point (1 m): u = −0.25 m, v = −1 m.
1/f = 1/v − 1/u = 1/(−1) − 1/(−0.25) = −1 + 4 = 3 m⁻¹, so f ≈ 0.33 m, P = +3 D (convex lens).

The apparent size of an object depends on the visual angle it subtends at the eye, not its actual size — bringing an object closer makes it look bigger, up to the near point D, beyond which the eye cannot focus. A simple microscope (magnifying glass) is a single convex lens of short focal length that lets an object be brought closer than D while still forming a sharp, enlarged, virtual, erect image, because the lens increases the visual angle further. Its magnifying power compares the visual angle with the lens to the visual angle when the object is viewed unaided at the near point.

A single lens cannot usefully magnify beyond about 10–20 times without distortion, so much higher magnification uses a compound microscope: two converging lenses on a common axis — the objective, of very short focal length, close to the object, and the eyepiece, of somewhat larger focal length, close to the eye. The object is placed just outside the objective's focal point, so the objective forms a real, inverted, magnified image inside the tube. This image acts as the "object" for the eyepiece, used as a simple magnifier, producing a final image that is virtual, further magnified, and inverted relative to the original object. The overall magnifying power is the product of the objective's linear magnification and the eyepiece's angular magnification.

Worked example (simple microscope): A magnifying glass has focal length 5 cm. Find its magnifying power with the final image at the near point (D = 25 cm).
M = 1 + D/f = 1 + 25/5 = 6×.

Worked example (compound microscope, normal adjustment): An objective of focal length 2 cm and an eyepiece of focal length 6 cm are mounted 18 cm apart. Find the magnifying power with the final image at infinity.
The intermediate image must lie at the eyepiece's focal point: v₀ = 18 − 6 = 12 cm.
For the objective: 1/v₀ − 1/u₀ = 1/f₀ → 1/12 − 1/u₀ = 1/2 → 1/u₀ = 1/12 − 1/2 = −5/12 → u₀ = −2.4 cm.
m₀ = v₀/u₀ = 12/(−2.4) = −5. Eyepiece angular magnification (normal adjustment) = D/f_e = 25/6 ≈ 4.17.
M = m₀ × (D/f_e) = (−5) × 4.17 ≈ −20.8, i.e. about 20.8× (final image inverted).

Unlike a microscope, a telescope is used to view large but very distant objects — the extra visual angle comes from its optics, not from bringing the object closer, which is impossible. An astronomical (refracting) telescope uses two converging lenses: an objective of large focal length and large aperture (to collect as much light as possible from a faint, distant source), and an eyepiece of short focal length. Since the object is effectively at infinity, the objective forms a real, inverted, diminished image at its focus; this image is then viewed through the eyepiece, used as a simple magnifier. In normal adjustment, the final image forms at infinity, the objective's focal point coincides with the eyepiece's focal point, and the two lenses are separated by the sum of their focal lengths — the length of the telescope tube.

The magnifying power of an astronomical telescope in normal adjustment is the ratio of the objective's focal length to the eyepiece's focal length; a large f₀ and small f_e give high magnification but a longer, heavier tube. A large-aperture objective also improves resolving power and light-gathering ability, which is why observatory telescopes use very large objectives.

Grinding a single large lens free of defects is difficult and expensive, and a large lens suffers from chromatic aberration (different colours focusing at slightly different points, since refractive index depends on wavelength) and sags under its own weight. A reflecting telescope avoids these problems by replacing the objective lens with a large concave (parabolic) mirror, which focuses parallel rays by reflection rather than refraction, introducing no chromatic aberration. In the common Cassegrain design, light collected by the large primary mirror is reflected towards a small secondary convex mirror, which redirects it back through a hole in the primary mirror to an eyepiece, folding a long focal length into a compact tube. Because large mirrors are easier to support and manufacture without distortion than large lenses, essentially all major research telescopes today are reflecting telescopes.

Worked example: An astronomical telescope has an objective of focal length 75 cm and eyepiece of focal length 5 cm. Find its magnifying power and tube length in normal adjustment.
M = f₀/f_e = 75/5 = 15.
Tube length L = f₀ + f_e = 75 + 5 = 80 cm.