Ray optics treats light as rays that travel in straight lines, obeying the laws of reflection and refraction. It is the oldest and most intuitive model of light, adequate for studying mirrors, lenses, prisms, and optical instruments where the size of the apertures is much larger than the wavelength of light. This chapter applies ray optics to spherical mirrors, lenses, and prisms.
We begin with reflection from plane and spherical mirrors, the mirror formula, and magnification. We then study refraction through Snell's law, total internal reflection, and refraction through spherical surfaces and lenses. The lens maker's formula and the thin lens formula describe image formation, and the power of a lens measures its converging or diverging strength.
The chapter ends with optical instruments: the human eye, microscopes, and telescopes, which use combinations of lenses to magnify and resolve images. We also study dispersion of light and the working of prisms and spectrometers. Ray optics connects the physics of light with the design of instruments used everywhere from laboratories to astronomy.
The law of reflection states that the angle of incidence equals the angle of reflection, and both rays and the normal lie in the same plane. A spherical mirror is a portion of a sphere; concave mirrors converge parallel rays and convex mirrors diverge them. The centre of curvature C, principal focus F, and pole P define the mirror, with focal length f = R/2, where R is the radius of curvature.
The mirror formula relates the object distance u, image distance v, and focal length f:
1/v + 1/u = 1/f
The magnification produced by the mirror is:
m = h_i / h_o = -v / u
The sign convention follows the Cartesian convention: distances measured in the direction of incident light are positive, distances opposite are negative. The magnification is negative for real, inverted images and positive for virtual, erect images.
Refraction is the bending of light when it passes from one medium to another. Snell's law states that the ratio of the sines of the angles of incidence and refraction equals the ratio of the refractive indices:
n1 sin(i) = n2 sin(r)
The refractive index of a medium is n = c/v, the ratio of the speed of light in vacuum to that in the medium. The absolute refractive index of air is nearly 1, and that of glass is about 1.5. When light passes from a rarer to a denser medium, it bends toward the normal; from denser to rarer, it bends away from the normal.
The relative refractive index of medium 2 with respect to medium 1 is n21 = n2/n1 = v1/v2. Refraction at plane surfaces produces apparent shifts in depth, and in lateral displacement in parallel-sided slabs.
When light travels from a denser to a rarer medium, the angle of refraction increases with the angle of incidence. At a certain angle of incidence, called the critical angle C, the refracted ray grazes the surface at 90 degrees:
sin(C) = n_rarer / n_denser
For angles of incidence greater than the critical angle, no refracted ray exists; all light is reflected back into the denser medium. This is total internal reflection.
The critical angle for glass to air is about 42 degrees and for diamond about 24 degrees. Total internal reflection is used in optical fibres, where light is trapped inside a thin glass core by repeated total internal reflections, enabling lossless transmission of signals. It also explains the brilliance of diamonds and the mirage in deserts.
Refraction at a single spherical surface separating two media is described by:
n2/v - n1/u = (n2 - n1)/R
A thin lens is made of two spherical surfaces. The lens maker's formula gives the focal length in terms of the radii R1 and R2 and the refractive index:
1/f = (n21 - 1)(1/R1 - 1/R2)
where n21 is the refractive index of the lens material relative to the surrounding medium. The thin lens formula relates object and image distances:
1/v - 1/u = 1/f
The magnification is m = v/u. The power of a lens is:
P = 1/f (in metres), measured in dioptres (D)
A converging (convex) lens has positive power, and a diverging (concave) lens has negative power. For two thin lenses in contact, the combined power is P = P1 + P2.
When white light passes through a prism, it is dispersed into its constituent colours because the refractive index of the glass depends on wavelength. Violet light is refracted most and red least. The angle of deviation D for a prism of angle A is given by:
D = i + e - A
For minimum deviation Dm, the ray passes symmetrically through the prism, and the refractive index is:
n = sin((A + Dm)/2) / sin(A/2)
Dispersion is the separation of white light into its spectrum, seen in rainbows and prisms. The deviation produced by a prism is greater for shorter wavelengths. Prisms are used in spectrometers and binoculars, where total internal reflection prisms replace mirrors to give compact, high-quality images.
The human eye acts as a camera, forming a real image on the retina, with the ciliary muscles adjusting the focal length of the eye lens for viewing at different distances. The angular magnification of a simple magnifying glass with focal length f is:
M = 1 + D/f (image at near point), or M = D/f (image at infinity)
where D = 25 cm is the least distance of distinct vision.
A compound microscope forms a highly magnified image using an objective of short focal length and an eyepiece. Its total magnification is the product of the magnifications of the objective and eyepiece:
M = (L/f_o)(D/f_e)
A refracting telescope uses a large-aperture objective of long focal length and a short-focal-length eyepiece. Its magnifying power is:
M = f_o / f_e
Telescopes are used to observe distant objects with large angular magnification and better resolution.
| Quantity | Formula | Remarks |
|---|---|---|
| Focal length of mirror | f = R/2 | Concave f negative, convex f positive |
| Mirror formula | 1/v + 1/u = 1/f | Cartesian sign convention |
| Magnification (mirror) | m = -v/u | Negative for real images |
| Snell's law | n1 sin i = n2 sin r | Ratio of refractive indices |
| Critical angle | sin C = n2/n1 | Denser to rarer |
| Lens maker's formula | 1/f = (n21-1)(1/R1 - 1/R2) | Thin lens in air |
| Thin lens formula | 1/v - 1/u = 1/f | Convex f positive |
| Power of lens | P = 1/f(m) | Dioptre (D) |
| Prism deviation | n = sin((A+Dm)/2)/sin(A/2) | Minimum deviation |
| Simple magnifier | M = D/f | Angular magnification |
| Instrument | Formula | Use |
|---|---|---|
| Compound microscope | M = (L/fo)(D/fe) | Small objects, high magnification |
| Refracting telescope | M = fo/fe | Distant objects |
Ray optics explained the behaviour of light through mirrors, lenses, and prisms. Reflection from spherical mirrors obeys 1/v + 1/u = 1/f, and refraction follows Snell's law n1 sin i = n2 sin r. Total internal reflection, with critical angle sin C = n2/n1, underlies optical fibres and the brilliance of diamonds. Thin lenses are described by the lens maker's formula and the thin lens formula, with power measured in dioptres. Prisms disperse light and are described by the minimum deviation formula, while microscopes and telescopes combine lenses to magnify images. These principles are applied in cameras, the human eye, and the optical instruments of science and technology.