Theory

Geometrical and wave optics

How light is reflected and refracted according to Snell's laws, how mirrors and thin lenses form images, and how interference, diffraction and polarization reveal the wave nature of light.

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Light, rays and index of refraction
Reflection and refraction

Light, rays and index of refraction

1. Light ray and geometrical optics

A light ray is an oriented line, perpendicular at every point to the wavefronts, that is, to the surfaces of constant phase, which indicates the direction of propagation; in a homogeneous medium the rays are straight. Geometrical optics describes light only by means of rays, and it is an approximation valid when apertures and obstacles have dimensions much larger than the wavelength. When the dimensions are comparable with $λ$ , as for slits a few tenths of a millimeter wide, interference and diffraction appear, and these are the subject of wave optics.

2. Index of refraction

In a homogeneous transparent medium light propagates with a speed $v$ less than $c$ . The index of refraction of the medium is the pure number

n = \frac{c}{v} \geq 1.

Vacuum has $n = 1$ , air $n \approx 1.0003$ , which in problems is approximated by $1.00$ , water $n = 1.33$ , ordinary glass $n \approx 1.5$ , diamond $n = 2.42$ . Of two media, the one with the larger index is called optically denser. In water, for example, $v = \frac{c}{1.33} \approx 2.26 \cdot 1 0^{8} m/s$ .

3. Frequency and wavelength in a medium

When light passes from one medium to another the frequency $ν$ stays the same, because it is imposed by the source and the fields on the two sides of the interface oscillate at the same rate. In a medium of index $n$ the wavelength is therefore

λ_{n} = \frac{v}{ν} = \frac{c}{n ν} = \frac{λ}{n},

where $λ$ is the wavelength in vacuum. Light of $600 nm$ in vacuum has in water $λ_{n} = \frac{600}{1.33} nm \approx 451 nm$ .

Reflection and refraction

When a ray strikes the interface between two transparent media, part of the light returns into the first medium and part enters the second, changing direction. Angles are measured from the normal to the surface at the point of incidence, and the plane of incidence is the plane that contains the incident ray and the normal.

1. Law of reflection

It is an experimental law. The reflected ray lies in the plane of incidence, on the opposite side of the normal from the incident ray, and the angle of reflection is equal to the angle of incidence:

θ_{1}^{'} = θ_{1} .

The law holds for every reflecting surface, flat or curved, and it is the basis of the theory of mirrors.

2. Snell's law

It is an experimental law. A ray that passes from a medium of index $n_{1}$ to a medium of index $n_{2}$ is refracted in the plane of incidence, on the opposite side of the normal, and the angle of incidence $θ_{1}$ and the angle of refraction $θ_{2}$ satisfy

n_{1} sin θ_{1} = n_{2} sin θ_{2} .

Together with the law of reflection it makes up Snell’s laws, and it allows one to compute the direction of the refracted ray. A ray that passes from air to glass with $n_{2} = 1.50$ and $θ_{1} = 4 5^{\circ}$ has $sin θ_{2} = \frac{sin 4 5 ^{\circ}}{1.50} \approx 0.471$ , so $θ_{2} \approx 28. 1^{\circ}$ .

Incident, reflected and refracted rays: the angles are measured from the dashed normal; with

n_{2} > n_{1}

the refracted ray bends toward the normal

Direction of the deviation

From Snell’s law, $sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1}$ , and the sine is increasing between $0$ and $\frac{π}{2}$ .

If $n_{2} > n_{1}$ , then $θ_{2} < θ_{1}$ : the ray that enters an optically denser medium bends toward the normal.
If $n_{2} < n_{1}$ , then $θ_{2} > θ_{1}$ : the ray bends away from the normal.
If $θ_{1} = 0$ , then $θ_{2} = 0$ : the ray perpendicular to the surface is not deviated.

Through layers with parallel faces the quantity $n sin θ$ is conserved at every surface, so the angle in the last medium depends only on the indices of the first and the last.

Dispersion of light

The index of refraction depends on the wavelength; in glass and in water it decreases as $λ$ increases, so violet light is deviated more than red light. Because of this dispersion a prism separates white light into colors. In problems in which the index is given as a single number, dispersion is neglected.

3. Slab with parallel faces

Consider a transparent slab of thickness $t$ and index $n$ , with two flat parallel faces, immersed in air, and a ray that strikes the first face at angle $θ_{1}$ . Then the ray emerges parallel to the incident ray, displaced laterally by the distance

d = \frac{t sin ( θ _{1} - θ _{2} )}{cos θ _{2}}, sin θ_{2} = \frac{sin θ _{1}}{n} .

For this reason a window pane does not alter the direction from which objects are seen. With $t = 1.0 cm$ , $n = 1.50$ and $θ_{1} = 4 5^{\circ}$ we have $θ_{2} \approx 28. 1^{\circ}$ and $d \approx 0.33 cm$ .

Proof

The two faces have parallel normals, so the inner ray strikes the second face at the same angle $θ_{2}$ at which it was refracted at the first. Snell’s law at the two faces gives $sin θ_{1} = n sin θ_{2} = sin θ_{3}$ , hence $θ_{3} = θ_{1}$ and the emerging ray is parallel to the incident one.

Let $A$ and $B$ be the points of entry and exit. The segment $A B$ forms the angle $θ_{2}$ with the normal and crosses the thickness $t$ , so $A B = \frac{t}{cos θ _{2}}$ . The displacement $d$ is the distance of $B$ from the extension of the incident ray, which passes through $A$ and forms the angle $θ_{1} - θ_{2}$ with $A B$ ; hence $d = A B sin (θ_{1} - θ_{2})$ .

The theory continues with 7 more sections (total internal reflection, plane and spherical mirrors, thin lenses…), plus all worked examples, the solutions and the PDF.

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