What Optics is, why to study it, and some fundamentals
The word comes from the Greek for eye, οψ , but the study of the eye itself is physiology and neurology, the study of vision a branch of psychophysics, and the study of illumination and colour an applied field of its own. Optics is descended from the ancient Greek science of mathematical perspective, in which the behavior of the rays of vision was discussed by deductive geometry. When it was realised that light was a physical, objective thing, Optics became the study of that thing, between the lamp and the eye, but not including the source or the receiver except as necessary to the argument. Light is an electromagnetic wave phenomenon with a firm basis in electromagnetic theory. Optics encompasses not only light in the strict sense, as that radiation that stimulates vision, but also those neighbouring parts of the electromagnetic spectrum in which essentially the same methods of study and description are effective, ranging from microwaves on one hand to X-rays on the other. It is divided into two main parts, geometrical optics, where wave propagation can be described by rays and including optical instruments, and physical optics, which is the study of light by means of wave theory, often without explicit reference to its electromagnetic basis.
A necessary reason to study anything is that it gives pleasure and entertainment. Without this, the study of anything is not worth the effort, however profitable the knowledge might be. I presume that you will proceed into Optics only on this condition. Now let us list some practical reasons.
I. Optics will explain how the optical instruments encountered in daily life function and can best be used. A general recently told an aide reporting a situation 'not to explain how a watch worked, just to tell him the time.' Such military twits stand slack-jawed with amazement when they observe cock-ups that could have been avoided with a little knowledge of how people and things work. For example, cameras and binoculars can be selected and used much more effectively with a knowledge of how they function.
II. The study of the eye, vision, and illumination all depend on some knowledge of Optics.
III. Optics is an excellent example of the nature of wave motion.
IV. Optics is widely used in practical measurement of all kinds: surveying and mapping, factory and machine shop, medicine and optometry, geology and mineralogy, and, of course, in science.
V. The historical development of Optics is a long and fascinating story.
VI. A knowledge of optical phenomena will allow you to appreciate nature and your own senses more deeply.
VII. Laboratory experiments in Optics are some of the most beautiful and fascinating of all such work. Many experiments can be done with simple equipment at home by the amateur.
You probably can be convinced easily that light travels in straight lines. Contemplate the shadows cast by the sun and the moon, or a lone streetlight. Make shadow-puppets on the wall using a bare bulb, then see what happens when you put a piece of cardboard with a small hole in it in front of the bulb. Observe a searchlight, or the crepuscular rays radiating from the clouds near sunset, or the cone of light from a cinema projector, or from your own hand torch. Get a laser pointer and shine it about.
Take two index cards. Fold one in half and cut a small notch so that you have a hole about 2mm square when the card is flat. Now let the sun shine on the card with the hole, and use the second card to look at the light coming through the hole, with the cards an inch or so apart. Now slowly separate the cards, observing the shape of the patch of light. What does it look like when the cards are about a yard apart? Try the same thing with some small pinholes. Try to make the pinholes different sizes, but not over a millimetre in diameter. What is the difference in the patches of light from pinholes of different sizes? Now try to explain your observations, drawing diagrams if necessary.
By this time you will have become sensitive to the relative sizes of the source of light and the object casting a shadow. The pinholes make patches of light of the same size, the brightness increasing with the size of the pinhole. The square hole makes a round patch of light that is slightly fuzzy at the edges. You may have concluded that what you are seeing is not the shadow of the hole, but an image of the sun. This will have been strengthened if a wisp of cloud was passing in front of the sun, and you saw it moving the opposite way across your circular patch of light. Dividing the diameter of the disc of light by the distance from the pinhole, you will find the apparent size of the sun in angular measure, about 1/115 radian. This relation, that radius times radians gives arc length is very useful, and you are probably already aware of it. 360° is 2π radians, of course, which can be used to convert from one to the other.
Such observations will make the concept of the ray of light quite concrete. The direction of rays of light is altered by a mirror, or by entering or leaving a transparent substance like glass or water. These two cases are called reflection and refraction, and are observed to occur with predictability. Find a small mirror that you can stand upright, and stand it on some surface you can stick common pins into. A bulletin board is ideal. Take two pins, and use them to define a line striking the mirror at an angle. Now look in the mirror and find the place where the reflections of the two pins line up, and stick in two more pins to mark this direction. Draw a line representing the position of the mirror. If you draw the lines defined by the pins, they will meet at the mirror, and make equal angles with it. In Optics, one generally measures angles with respect to the normal to the surface. The Law of Reflection is that the incident and reflected rays make equal angles with the normal, and the reflected ray lies in the plane containing the incident ray and the normal.
When you look at one of the pins in the mirror, you see something that looks just like the pin and is as far behind the mirror as the pin is in front. This is called an image of the pin. Draw a diagram that shows any two rays from the pin reflected at the mirror according to the Law of Reflection. Then extend the rays backward until they meet. This is the position of the image. You should be able to prove geometrically that the image is as far behind the mirror as the object, in this case the real pin, is in front of it, and on the same normal to the mirror. It is called a virtual image because the rays never actually pass through the image, only seem to do so where they actually exist. Do not despise this simple experiment; it will help you to understand clearly.
Fill the kitchen sink, stick a straight rod in it slantways, and then stoop down and look at it. Of course, it will appear to be bent abruptly at the water line but straight above and below. This ancient observation shows that light rays are bent when passing obliquely into a transparent medium. The effect is not as strong as that of reflection, and is very difficult to measure accurately unless special apparatus is made for the purpose. It was even more difficult to understand the regularity of the phenomenon, and to come up with a simple law describing it. Willebrord Snell's assertion that the ratio of the sines of the angles of incidence and refraction at a plane interface was constant for any pair of transparent media did not appear until 1621, certainly a remarkable delay. The law of refraction was certainly not obvious, and its discovery was a major advance. This law is all that is needed for the design of serviceable optical instruments. Once you know the law, it is easy to verify it experimentally, but I do not know any easy experiments that suggest it. The law also includes the fact that if the direction of the light is reversed, the ratio is inverted. The smaller angle with the normal is always in one of the two media, which is called the denser of the two, from an old and incorrect assumption that a denser material was always the more refracting.
The ratio of the sines in Snell's Law is a quantity known as the index of refraction. Suppose we know the indexes for two different substances relative to air. Consider the two substances almost in contact, but with a thin layer of air between them. Snell's Law says for the two substances individually that sin i / sin r' = n' and sin i / sin r" = n", where the unprimed i refers to the angle of incidence in air. Dividing these two expressions, we find sin r' / sin r" = n" / n'. The relative index of refraction for the two substances is simply the ratios of their indexes with respect to air, so all we need to know is an index for each substance. The law can now be expressed as n' sin i' = n" sin r" (there is no significance to the use of i or r for an angle when the primes give the necessary information of which medium the ray is in). Since n sin i is equal in the two media, we can extend this to the statement that in a layered medium where n varies (such as the atmosphere), the product of the index of refraction and the inclination of the ray is a constant. It is now usual to quote indexes of refraction with respect to vacuum and not air. The index of refraction of air at 0°C and one atmosphere pressure is 1.00029, so this does not make a lot of difference to approximate measurements.
There is a home experiment for measuring the index of refraction of water that is instructive, and can be fun. First, find a bottle of whisky that is a rectangular prism, with four flat sides, purchase it, and empty it. Fill it with water, and make a mark with a china marking crayon about halfway up on one of the vertical corners. Now sight along the line formed by this mark and the lower back edge of the bottom along the side of the bottle and note how far back the back edge appears to be through the water. In fact, measure how far back it is using a millimeter scale along the base. Finally, measure the height of the mark, and the width of the base. With these three distances, you can calculate the angles of incidence and refraction, and so the ratio of their sines, which should be the index of refraction of water.
The index of refraction of water is close to 4/3, and that of glass close to 3/2. These fractions are easy to remember. Ice has a slightly smaller index than water, around 1.31. Observe clear ice floating in water, and compare with pieces of glass in the water. Ripples on the surface of water refract light, as does the heated air above a fire when sunlight passes through it. Drops of water, and spherical glass flasks, also refract light. One notices that any interface both reflects and refracts light, and the relative amounts vary with the angle of incidence. The effect is stronger the greater the difference in indexes of refraction.
Flashes of colour may sometimes be seen in making these observations. Diamond has a high index of refraction, and gems show sparks of colour. This colour arises only on refraction, not on reflection, and its source was long a mystery, since light was regarded as a single uniform thing. Colour was attributed to an interplay of light and darkness, and prisms to see the colours were a common toy. Newton showed that colour was an inherent property of light, and that seemingly white light contained an array of colours from indigo to red. By analogy with sound, this was imagined to be like the sound of an orchestra, composed of different pitches produced by the instruments individually, the pitch representing the colour. Each colour had a different index of refraction, largest for indigo and least for red. The effect is small; for glass, the index varies only by 0.02 - 0.01 over this range, which only spreads the refracted ray out over about 1°. In a prism, this is doubled, so a spectrum of sunlight thrown on a wall 20 ft from the prism is about 8" long. If you have a 60° prism handy, you can reproduce Newton's spectrum by using the sunlight that comes through a small hole in the blind. The change of index with colour is called dispersion, since it causes the different colours to be dispersed.
The part of the nature of light that explains dispersion was not known for a hundred years after Newton, until Thomas Young's realisation that light was a wave motion. This fact was even harder to come by than the law of refraction, since the evidence for it was even more deeply hidden. To reproduce Young's observation, take a piece of fine uninsulated wire about 1 mm in diameter, or a common pin, and hold it in front of the pupil of your eye while looking at a bright coloured point source of light (a small hole covered by coloured plastic film in front of a bright light, for example). This should be done in the dark, since what you are looking for will be dim. You will see equally spaced dark and light lines, or fringes. Young had only a candle, but nevertheless succeeded in projecting the fringes on a screen and measuring their separation. Young had noticed that when he looked at a wire with a bright light beyond it, the top and bottom edges of the wire appeared luminous, like line sources of light. Since these two sources were separate, the distances from each of them to some point on the screen were slightly different. Now, if light was a vibration, and the two sources were excited in step (as was possible when illuminated by a distant point source), then the light from the two sources falling on a point of the screen would be more or less out of step, since they had to travel different distances to get there. As one went up and down in a direction perpendicular to the wire, the waves would get out of step regularly, sometimes adding and sometimes subtracting, producing the observed fringes.
There is much more in this simple observation than is obvious on first sight. One had to carefully note the analogy with water waves, and assume that there was some varying magnitude to a light vibration, and that the magnitudes of several vibrations superimposed at the same point added algebraically (with regard to positive and negative sign) to produce a resultant, and that the intensity was the square of this resultant. This wave process is called interference, although the essential thing is that the several waves do not affect one another at all, but passively add up to a resultant that is observed. Young is the discoverer of interference in light, a very great discovery. Interference is used to make the most precise measurements of length, and the metre is now defined in terms of the wavelength of a certain light of constant frequency emitted by the cadmium atom.
Young thought light waves were longitudinal, like sound waves, because his experiments showed none of the directional dependence that might be expected for transverse waves, like those on water. The discovery of polarization soon after revealed that light had to be considered as a transverse wave, however. Many of the typical wave phenomena are the same for longitudinal and transverse waves, however.
A wave is a curious thing. Suppose an underwater volcano blows up somewhere in the ocean, creating a powerful wave that spreads out in all directions. A boat along its path in the deep ocean might not notice the gradual rise and fall in the sea as it passes. Far away on a sandy island shore the wave climbs up the shelving bottom, and its energy is concentrated in less and less water, which moves more and more violently, and a huge foaming wave destroys the cluster of huts on the island. The volcano has destroyed a village a thousand miles away a half-day later, with no hint of its rage in between. No water has moved from volcano to island, just a local movement of the water that is passed on leaving things unchanged after its passage. Energy is contained in the motion, so effects can be produced later at a distance.
There are many kinds of waves, that behave very differently. The one property that unites them is that nothing is carried with them but motion. At any point, there is a magnitude -- let us call it displacement, whether it is an actual displacement in space or not -- that varies with time, called the wave function. The wave function may move forward unchanged in shape, only delayed in time, or may change shape as it moves. Water waves change in shape as they move, unless they have certain special shapes. These special shapes are sinusoidal wave functions, in which the displacement varies as the sine or cosine of an angle that increases linearly with the time at a given point. This angle is called the phase. The phase also increases linearly with the distance of travel. Since a sinusoidal function repeats every 2π radians, or 360°, the wave function repeats in time every T seconds, called its period, and repeats in space every λ metres, called its wavelength. For a wave travelling to the right (increasing x), the phase is 2π (t/T - x/λ) at any point x and at any time t. The phase increases with time, and decreases as one moves right. A point of constant phase must move at a speed x/t = λ/T = v, called the phase velocity. Instead of the period, it is usually more convenient to use the frequency f = 1/T. We have the very useful relation that λf = v, showing how phase velocities can be measured, or how frequency and wavelength can be related. The unit of frequency is the hertz, Hz, which is one per second. All this that we have had so far is a lot of terminology, but it amply repays study, since waves are everywhere in modern life.
Young found that the waves of light to which the eye was sensitive had wavelengths between 400nm for the indigo, to 700nm for the red, roughly 2000 waves to the millimetre: very tiny. Combined with the speed of light, 300 000 km/s: very fast, this gives a frequency of about 6 x 1014 Hz: very large. That these three magnitudes are so far removed from everyday experience is a good reason why light does not look like waves to the casual observer. Now we can say that the index of refraction is a function of the wavlength, or frequency, which is the same thing, a precise physical quantity, instead of the fuzzy subjective quality of colour. How he measured the wavelength is shown in the diagram on the right. The approximate formula is satisfactory because x is very small compared to f (if a lens is not used, this is just the distance to the screen). The leverage factor is a/f, which should be made as large as is convenient. If a is 1 mm, then the fringe spacing will be 1 mm if f is 2 metres.
The phase velocity depends on the period with water waves, sometimes quite significantly. When a local storm roils the sea, the different phase velocites soon sort out waves of different periods. Those of long period move the most rapidly and come rolling in to shore first, to be followed by waves or shorter and shorter period. This is not observed with light. When one star eclipses another, the red light does not vanish first, followed by the blue, but it goes dark in all colours at once. There is a little dispersion in things like water or glass, but not in vacuum. When the phase velocity is independent of wavelength, the wave function travels without change of form, and this is the case with light.
White light has a very complicated wave function. At any given time, it can be expressed as a sum of sinusoidal waves. This is a very well-known mathematical procedure called Fourier analysis, that makes it possible to express the frequency spectrum of any given wave. For light, this spectrum remains the same as the light travels, so it is all right to say that the light contains these different frequencies that are separated by dispersion in a prism, although it is probably more precise to say that the prism makes a Fourier analysis of what is presented to it, and that is what we see.
Refraction in the wave picture is shown at the right. The two plane wavefronts are perpendicular to the rays, and are the same wavefront a time t apart. The intersection of the wavefront with the interface has moved a distance s to the right in this interval. The trace velocity s/t is the same in both media, so we find at once that the ratio of the sines is a constant, equal to the inverse ratio of the velocities. The index of refraction of a medium, as we have defined it above, is equal to c/v, where c is the speed of light in vacuum, and v the phase velocity in the medium. Now we have a physical meaning for the index of refraction.
The question remains as to exactly what kind of wave light is. It was first thought that any wave had to have a medium in which to travel (not an unreasonable idea at all), so light was a transverse vibration in some medium that permeated all space and matter, the luminiferous ether. The problem was that such a medium defied all attempts to observe it directly, and had properties like no other known form of matter. Starting from a different direction, Maxwell obtained a complete and comprehensive explanation for the phenomena of electric and magnetic forces by 1865, which were also thought of as mechanical stresses in some pervading medium. To everyone's surprise, this description predicted the existence of wave motions with a speed equal to that of light. Careful experiments showed that the agreement was exact, and in 1887 Hertz produced and detected such waves that acted just like light, but of much larger wavelength. There was no doubt that light was electromagnetic waves.
The theory of relativity showed that any attempt to detect an electromagnetic ether was futile, and when this theory was amply corroborated, one had to accept that electromagnetic waves, like no other wave motion that had been found, required no medium for their existence except space itself. The theory of relativity is based on the observation that the speed of light is the same in any frames of reference that move with constant speeds with respect to each other.
We have now made the first steps in the study of Optics, introducing the fundamental models that explain the behaviour of light, and the definitions and concepts used everywhere in further enquiry. We have not introduced all the rich banquet of light, things such as optical instruments, polarization, diffraction, and quantum behaviour, but have provided the basic tableware.
Surfing the shelves of a university library will reveal many books on Optics and related subjects. Here are a few of the works I have found specially worth using, and which are in my own library. All contain many references.
The most easily available modern Optics text is the one by Hecht, or Hecht and Zajac (Optics). It is well-illustrated and includes most of what a Physics graduate has been expected to know.
The older text of Jenkins and White (Fundamentals of Optics) was long a standard, and came out in several editions. The final edition was edited to include much modern material, but I think the earlier 4th edition is classic, and very well-written. Every advanced physics text writer of the 40's and 50's wrote an Optics text, and they are a mixed bag.
The even older text of R. W. Wood (Physical Optics) does not have many of the later 20th century developments, but is an absolutely fascinating and masterly review of the subject. It was reprinted by Dover.
Born and Wolf (Principles of Optics) is a wonderful, comprehensive advanced treatise, with excellent mathematical analysis.
The British texts of Longhurst (Geometrical and Physical Optics), Ditchburn (Light), and Lipson and Lipson (Optical Physics) are well-written and informative.
For electromagnetic theory, there is no better book than J. D. Jackson (Classical Electrodynamics). The 2nd edition uses Gaussian units, but the 3rd has descended to Giorgi (MKS). Many engineering texts treat electromagnetic waves, with the occasional howler.
B. K. Johnson's Optics and Optical Instruments is a small book filled with practical information. It was reprinted by Dover, and is well worth consulting.
The modern developments of coherent optics, lasers, computer lens design, and nonlinear (solid state) optics have their specialized literature, all of which is rather advanced.
Composed by J. B. Calvert
Created 11 April 2000
Last revised 15 April 2000