Optics Made Plane

Rays and waves at plane interfaces epitomise the science of Optics


That light -- or vision -- moves in straight lines is axiomatic. In ancient times, it was studied as a part of geometry, its rays represented by lines. Regular reflection and its laws were known, supporting the science of catoptrics. Refraction was known and explained, but not its accurate laws, so the science of dioptrics was limited. The apparent breaking of a stick partly in water, burning glasses and mirrors, and the colours produced by refraction in prisms were recognised. There was no mental conflict in thinking of light as the rapid movement of subtle particles, for which the property of a trajectory was natural, as well as the bouncing off of mirrors at equal angles. These views continued with little modification until after the time of Newton, who explained refraction by forces acting normal to the interface on the particles of light. The accurate law of refraction was known by this time, that the sines of the angles made by the rays with the normal to the refracting surface were in a constant ratio depending on the natures of the two media. The late discovery of this law by Snell is remarkable; Claudius Ptolemy made tables of refraction, and that the angles were in a constant ratio for small angles were known, but not the complete law for any angle. Part of the problem may be that the law is completely arbitrary, with no satisfactory explanation, and extremely elegant. Descartes plagiarised Snell to give a partial explanation of the rainbow, but could not explain the colours except as a modification of the light due to edge effects.

But this was only one of the many problems with light rays before Newton's work. Colour was one of them, and the double refraction of Iceland spar (calcite), but most curious were the coloured fringes seen at the edges of shadows under certain conditions, first remarked by Grimaldi. These fringes, and colours, were explained with a wave of the hand and a remark on the nature of edges as modifying the properties of light. Newton showed that the colours were inherent, not a product of modifications of qualities, which made the explanation of Snell's Law more believable. He did not successfully explain the fringes or double refraction. Meanwhile, Huyghens gave a partial explanation of double refraction on the novel hypothesis that light was a vibratory disturbance like sound, as well as a very clear model of the mechanism of refraction that led to Snell's Law. On Newton's hypothesis, light particles travelled faster in a denser medium (being accelerated normal to the interface when entering) but on Huyghen's the waves travelled more slowly. The question could have been decided by a measurement of the speeds of light in air and in water, but although this was clearly recognised, it was far beyond the technique of the time. Incidentally, the index of refraction of glass is taken as 3/2, and that of water as 4/3, in the estimates made in this paper. The index of ice can be taken the same as water, or for more accuracy, 21/16 or 1.31.

So rays were safe during the 18th century, the noncritical student pointing to the great difference in behaviour between sound, which was a wave, and light, which was particles. The lack of a correct theory did not stand in the way of creation of serviceable optical instruments, from eyeglasses and the camera obscura to theodolites and compound microscopes. No sooner had the 19th century begun than Thomas Young measured the wavelengths of visible light by means of interference, and suggested the true explanation of the many observations of diffraction that had been made by then. As usual, Young's brilliant intuition was disregarded in England, so it was left to the equally brilliant young Frenchman, Augustin Fresnel, to put the wave theory of light on an unshakeable foundation by explaining all known diffraction phenomena, and predicting even more that were duly found. Malus enlightened the world about polarization, so the picture of light as a transverse wave was fully recognised.

A striking consequence of polarization is the double refraction in crystals, quite noticeable in calcite, and first described by Bartholinus in 1669. A cleavage rhomb of calcite shows a double image, one as expected, the other displaced a little. When the crystal is rotated, the displaced image rotates about the ordinary one. As shown in the figure, the two images are polarized at right angles to each other. Huyghens gave a plausible explanation on the wave theory in 1690, but the correct explanation had to wait for the discovery of polarization. The index of refraction of calcite for 589nm (Na yellow) light is 1.658 for light polarized normal to the axis (ordinary), 1.486 for light polarized parallel to the axis (extraordinary).

What did this do to rays? Mathematical analysis showed that the traditional rays were normals to the wavefronts, the surfaces of constant phase. If light is emitted from a point, the wavefronts are expanding spheres, and the rays are radii. Now, rays are much easier to handle than wavefronts. For one thing, they are local and retain their identity, while wavefronts are always spreading. As long as we are not near an edge, or places where the wavefront is kinked or cusped, rays will give a pretty good account of what the waves are doing. In the remaining cases, wave theory is necessary for a good description. Therefore, rays give us a simple account of light where they are applicable, but wave theory is the final arbiter. The use of wave theory is essential in the understanding and design of really good optical instruments.

One of the many transitional areas between rays and waves concerns the concept of optical path, F ,which is the summation of path lengths weighted by the index of refraction of the medium, F = ò ndl. Since the index of refraction n is the ratio of the speed of light in vacuum to the speed of light in the medium, the optical path is the distance in vacuum that the light would travel in the same time. The really important thing is that it is proportional to the number of wavelengths in the interval. Light that travels to a point on two different paths with equal optical paths will be in phase there, and interfere constructively. Wavefronts are a constant optical path from their sources.

Suppose light is to travel from a source P to a point Q that is on the far side of a plane boundary with a different medium of index n. To have any intensity, at least a small cone of light must be emitted, and the different rays will pass close to Q. Since there must be a wavefront at Q, the optical paths along any such rays must be equal. We conclude that the optical path must be the same (to a reasonable approximation) on all neighbouring paths on the actual path from P to Q. This is, in fact, Fermat's Principle. The requirement of sameness means in analysis that the optical path length is an extremum (maximum or minimum), or stationary with respect to other neighbouring paths connecting P and Q.

The diagram on the right shows the rays involved in the plane of refraction. The red rays are a small variation from the blue rays, and are drawn with a greatly exaggerated difference for clarity. For a small change in direction, the point of intersection with the interface moves a distance dx from V to V'. The thick red line is the increase in distance in the first medium, and the thick blue line the decrease in distance in the second medium. The purple lines show the wavefronts above and below the interface. An expression for the change in optical path in this small variation is easily written down. Equating it to zero gives the requirement of stationary path, and Snell's law is the immediate result. Many textbooks clumsily illustrate this by writing an expression for the total optical path length PVQ in terms of the abscissa x of point V, then differentiating the expression with respect to x and setting the result to zero. Here, I have not used Calculus (explicitly), only trigonometry, and the algebra is much easier.

We can further analyse this by trying to discover the shape of the wavefront in the lower medium. In the upper medium, near P, it is, of course, a sphere. The bundle of rays coming from P is said to be homocentric, and could be brought to a point image by a lens. After refraction, the rays in the plane of refraction meet at P' if projected backwards. The distance from V to P' is r' (in our approximation V and V' are the same; remember the great exaggeration). The magnitude of r' can be worked out, and the result is as shown. Now consider a small rotation of the plane of refraction about the vertical axis through P. The rays in this plane meet those in the previous plane at P" when projected back. The distance along the projected ray from V to P" is r", which is always equal to nr. P' is actually beyond P" when n is greater than unity, but this does not make any difference. The rays of the bundle, after refraction, no longer meet at one point when projected backwards; they are not homocentric, and could not be brought to a point by a lens. When point Q is almost directly below point P, the angle of incidence is small, and the cosine factor in the expression for r' is very close to unity. Then P' and P" are the same distance away, so they coincide. This makes the bundle homocentric, and able to be focussed to a point.

A small enough area of any surface can be described by its curvatures in sections at right angles, one a minimum, and the other a maximum, as the plane of section is rotated about the normal. These are called the principal curvatures. In refraction by a plane surface, the two sections are called tangential and sagittal ('arrow-like'), and the corresponding centres of curvature are P" and P'. The bundle of rays is called astigmatic ('not point-like') because it cannot be brought to a point (stigmatic) image. We have shown that oblique refraction in a plane surface creates astigmatism. As an example, let us imagine we are looking upward from beneath the waves. The whole bright hemisphere is packed into a cone of 48.6° radius from the zenith, with a silvery sheen beyond. We seen objects 10° from the zenith at 7.5° , and P" is only 1% farther away than P'. Coming down to 45° , P" is 1.44 times farther away, so the astigmatism is becoming appreciable. At 60° , we are looking up at about 40.5° , and P" is 2.3 times farther away. The astigmatism is large and very noticeable. Finally, at the silvery edge P" has receded to infinity. Even if we had powerful glasses to correct our underwater farsightedness, we would still be troubled by powerful astigmatism except for a small cone over our heads.

An astigmatic bundle of rays can be made stigmatic again by using a nonspherical lens to compensate for the differences in curvature. Eyeglasses are sometimes made nonspherical to cause astigmatism in an entering bundle of rays, that is straightened out by astigmatism introduced by refraction in the eyes. If the cornea does not have equal principal curvatures, it will create an astigmatic bundle from a stigmatic one. Sometimes it is the lens of the eye that is at fault (as in Thomas Young's eyes). The eyeglass lens for an eye is given an additional power, in diopters, in a plane whose normal is in a specified direction, which is called cylinder. The image of a line produced by an astigmatic bundle changes in form with the orientation of the line. In some pair of orientations at right angles, the lines can be sharply focussed (but not as pointwise images), but in general only a blur circle of some minimum size can be achieved.

Suppose we have two surfaces, a slab of material, with air on both sides. The conditions on exit are exactly the same as those on entrance but reversed. Therefore, the astigmatic bundle is rendered stigmatic again on exit. We observe no astigmatism when looking obliquely through a thick pane of glass. However, the image is moved somwhat. Looking perpendicularly, if P is a distance a above the top surface, its image is a distance na + t from the bottom surface, where t is the thickness of the slab. In the final medium, the image is (na + t)/n = a + t/n from the bottom of the slab, instead of the actual distance of P, which is a + t. The image is closer by a distance of (1-1/n)t. For glass, this is about t/3; for water, it is t/4. This also holds for objects seen directly from above at a given depth in glass or water.

Newton used a prism to separate light into its colours. He did not have to make a prism, because he could go to his local jeweller's and buy one. The colours made by a prism had been known since antiquity and were an object of curiosity and delight. When you see a halo encircling the moon or the sun, 22° from it, you observe light refracted by the tiny hexagonal prisms of ice crystals high in the troposphere. The halos are not strongly coloured because the ice prisms are so small (raindrops are large, and make the colourful rainbow). A prism has plane faces intersecting at a certain angle A. The light enters one face, and exits the other in the normal method of use. Prisms used as reflectors are a special case, and may have more complicated forms. The usual prism used to disperse light has A = 60° , and is an equilateral prism. Since the faces of entrance and exit are not parallel, we certainly expect a prism to introduce astigmatism, which will be larger at larger angles of incidence on the faces. The two faces will compensate to some degree, but the problem is not a simple one, although a straightforward calculation will give the result. This calculation is most easily done in the spirit of our treatment above, where a narrow bundle and the optical path are considered..

When the bundle encounters the entrance and exit surfaces symmetrically, we may expect that the astigmatism introduced by the first refraction is cancelled by the second refraction, and this is indeed the case. In all other cases, the astigmatism created by one surface will dominate. If the light does not pass symmetrically through the prism, the angle of entrance will be different from the angle of exit. Now, light rays are reversible, so there will be two angles of entrance that give the same deflection, or deviation, of the ray. At some angle of entrance the angle of exit will be the same, so there will be only one angle of deviation, and since it is an extremum (any slight change in the angle of entrance either way will give the same angle of deviation), the deviation will be a maximum or a minimum. It cannot be a maximum, as any approximate calculation will show, so it must be a minimum. This means that as the angle of entrance varies, there is a sharp edge to the movement of the ray that exits as it turns around, and the light is concentrated somewhat in this direction. The rainbow and the halos are also minimum-deviation phenomena. Because of the edge, ray theory is not applicable near it, and wave theory must be used to understand the details. The diagram at the right shows how the angle of minimum deviation D can be calculated. If the angle of minimum deviation is measured, the index of refraction can be found. If you use the index for ice, 1.31, the minimum deviation in a 60° prism comes out as 22° , the diameter of the halo. In a 90° ice prism, the minimum deviation is 46° , the radius of the large (and rare) halo.

Several marvels are manifest when one looks in a mirror. The first is, how can any material surface reflect light so regularly, so that it seems to come from behind the mirror, as if there were no surface there at all? If we start with a raw metal surface, light is scattered diffusely as expected. Then we begin to polish, and the surface remains matte, though smoother and smoother. If our abrasive is fine enough, a remarkable thing eventually starts to happen. The surface begins to show lustre, and finally becomes a mirror. What has happened is that the irregularities have become smaller than a wavelength, and then the roughness of the surface is averaged away over the wavefront. The fact that the colour of copper or gold remains shows that the light still does penetrate the metal enough to receive the colour (by absorption). The original coarse metal was probably shiny to radar, which has a much larger wavelength. This could never happen unless light were a wave. The same thing happens when polishing a piece of glass, such as a lens. It appears frosted until laboriously smoothed, and then transparency breaks out miraculously. Window glass is both affordable and transparent because great care has been taken to manufacture its surface sufficiently smooth without polishing. Cheap mirrors are made by coating one surface with metal that adheres to the already smooth glass, which protects the shiny metal surface from corrosion. The surface of still water is naturally smooth, and so is transparent.

The second marvel is in the very appearance of the mirror. The lustre and shininess is in our apperception, and is very different from the impression given by merely a smooth surface. It is as if the lustre were a special colour, a quality of the perceived light. Our visual sense gives us lustre if the surface looks like one that reflects regularly.

Reflection and refraction at a smooth plane surface depends not only on the angle of incidence, but also on the polarization. This is a good way to distinguish whether a phenomenon is due to refraction or to diffraction. Diffraction does not depend on the polarization of the light. The amounts of light reflected and refracted are given quantitatively by Fresnel's Equations (see a text on Physical Optics for these). One deduction that can be made from them is that the fraction of the incident energy that is reflected at normal incidence (head-on) is given by [(n-1)/(n+1)]2, where n is the relative index of refraction. As in any use of Fresnel's Equations, this is accurate only for a very clean, uncontaminated surface. For practical surfaces, it is a lower limit; usually more is reflected or scattered. For glass, this is 4%, and for water, 2%. If light is incident from the denser medium on the smooth, clean interface of a rarer medium at such an angle that there cannot be a refracted ray (its sine becoming greater than one when Snell's Law is used), all the energy is reflected as from a perfect mirror. This total internal reflection, which we mentioned above when looking from under water, has many curious properties. Light normal to a short face of a 90° glass prism is totally reflected by the hypotenuse. The corner of a cube reflects light back the way it came.

If light is incident onto a clean plane surface at an angle with the normal of about 56°, very little of the part polarized vertically (in the plane of the ray and the normal) is reflected, most of it passing through. The reflected light is highly polarized perpendicular to the plane of incidence. At this angle of incidence, called the polarizing angle, the reflected and refracted rays are at right angles. Therefore, the tangent of the polarizing angle equals the index of refraction. Before the easy availability of cheap polarizing filters, this phenomenon was used to create and analyze polarized light. Sunglasses with lenses of polarizing filters oriented to absorb light horizontally polarized considerably reduce the glare from water and other surfaces. What is called the direction of polarization is somewhat arbitrary unless the electromagnetic nature of the light is taken into consideration. The convention above makes the direction of polarization that of the electric vector. Long ago, the convention was the opposite one.

A rough plane surface acts like a new source of light, radiating the energy it receives, less losses, in all directions. This is scattering of the light, usually with loss of phase information: the light becomes scrambled, but its frequency content is not changed. In an ideal case, the light is not preferentially radiated in any direction, but the amount radiated in unit solid angle in any direction is strictly proportional to the projected area of the source, or cos q. Such a surface is called Lambertian, and is often assumed in illumination calculations. An actual surface may be anything between a specular reflector and a Lambertian scatterer.

Plane surfaces close together furnish direct evidence of the wave nature of light. Waves reflected from the upper and lower surfaces will be weak, but about equal in amplitude, so their interference is easy to observe. The plane surfaces can be formed by a thin layer of oxidised oil floating on water, or can be the space between two flat pieces of glass pressed together. If one of the pieces of glass is a lens or other spherical surface, the distance between the upper and lower surfaces will vary regularly with circular symmetry, producing what are known as Newton's Rings. If you put a small, rather weak lens on a flat piece of glass and look at the point of contact in reflected white light, you will see a black dot surrounded by a few coloured rings. In monochromatic light, many more dark and light rings can be seen. By measuring the rings, the wavelength of the light can be determined if the radius of curvature of the lens is known. The fact that the central spot is black shows that there is a phase difference of half a wavelength between rare-to-dense and dense-to-rare reflection.

Finally, we now know that exchanges of energy with light occur in discrete amounts, or quanta, that are proportional to the frequency of the light. These are the photons, which give light a kind of graininess that is very evident in its interactions with atoms and molecules. A definite trajectory cannot be ascribed to photons, so they do not make the ray theory again a fundamental one, nor a particle theory valid. In fact, waves are an aspect of photons, and rays an aspect of waves, so the relation between photons and rays is remote. The quantum theory of light is one of the most accurate theories in all of Physics, and light has many fascinating quantum aspects. As long as we are concerned with light by itself, however, it is not necessary to go beyond the classical wave theory.


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Composed by J. B. Calvert
Created 9 April 2000
Last revised 18 April 2000