Geometrical Optics Refraction

Providing more than one quickest route

When light crosses the boundary between two media, it changes direction. This phenomenon is called refraction. In this chapter we study the rules and applications of refraction. The basic rule is the same as always: it is Fermat's principle of least time. We show that the principle leads to the experimentally established Snell's law of diffraction.

Lenses are the most common example of the application of the laws of refraction. In making a lens, the trick is to make the shape such that all routes from a source A to a destination B on the other side of the lens take the same time, despite the fact light traverses different thicknesses of glass on different routes.

We spend the remainder of this section dealing with the geometry of the paths, and derive some simple formulae for lenses.

Making visible things we cannot see

We study the effect of various combinations of lenses which make up optical systems. One fascinating example is the optical system of the human eye. We discuss some common eye defects and how these may be corrected using suitable lenses.

Finally, we describe optical systems which enable us to look at things which are either too small or too far away to be seen with the naked eye.

3.1 Refraction

3.1.1 The refractive index

As we all know, from experience of city traffic, if the speeds along different routes are not the same, the shortest route is not necessarily the quickest. We have seen already that in vacuum the speed of light is fixed at c = 2.99792458 x 108 ms-1, equivalent to travelling a distance of approximately 7.5 times around the earth in one second. Light can also travel through certain 'transparent' media such as air, water, glass or quartz, and there the speed is less than c. The factor by which it is smaller is specified by the refractive index n of the medium.

The definition of the refractive index of medium 2 with v

respect to medium 1 is ln2 = v , where v1 and v2 are the speeds of light in medium 1 and medium 2, respectively.

Glass

The most common man-made substance which is transparent to visible light is, of course, glass. It is interesting to note that the Egyptians were able to make glass as far back as 1500 BC. Glass is manufactured by fusing silica (sand) with alkali; the Egyptians used soda and natron (sodium carbonate). About 1000 years later we find early Greek references to glass by Aristophanes (~ 400 BC).

The following table shows the refractive indices with respect to vacuum of some materials (for light of wavelength 589 nm):

 Material n air (1 atm, 15°C) 1.00028 water 1.33 quartz, (fused) 1.46 glass (crown) 1.52 glass (flint) 1.58

In practice it is usual to omit the subscripts when writing the refractive index 1n2 if medium 1 is either vacuum or air, and simply to refer to the refractive index, n, of the denser medium.

The difference between the speed of light in vacuum (or air) and its speed in liquids or solids is quite significant. In crown glass, for example, light travels 1.52 times more slowly than in air. The practical consequence of this is the phenomenon of refraction — light changes direction as it crosses the boundary between one medium and another. It is not obvious why such a change of direction should occur, but perhaps not surprising.

Experience may tell us that if a car hits a muddy patch at an angle, it will tend to swerve. Similarly, waves in water tend to change direction as their speed changes in shallow water. Actually, we do not have to refer to any particular model for the mechanism which causes the change of angle; it is required by Fermat's principle as illustrated in the lifeguard problem.

3.1.2 The lifeguard problem

There is an elegant way in which we can derive an exact expression for the change in angle (Snell's law) directly from the fundamental principle of least time, which we illustrate using the example of 'the path of the lifeguard'.

To find the quickest path

Consider the problem faced by a lifeguard who sees a swimmer in difficulties. He wants to get to the swimmer in the shortest

Figure 3.1 The lifeguard problem.

time, though not necessarily by the shortest route. His running speed is greater than his swimming speed, so the shortest route does not necessarily take the shortest time. What route should he choose?

Probably his first instinct would be to go in a straight line, but this cannot be the quickest route as the distance in the water (where he moves more slowly) is too long. Neither is it best to run to the bank opposite the swimmer and swim straight across, so that he covers the maximum possible distance on land, because the overall distance is too long. Instinct tells us that a compromise route is the quickest, and the lifeguard should dive into the water at some point in between his starting position and the point opposite the swimmer.

It turns out that the lifeguard reaches the swimmer in the shortest time if he runs directly to the point P in Figure 3.1, and then swims directly from there to the swimmer. The sizes of the angles i and r fix the position of P and the time is shortest when sin i _ speed in water sin r speed on land

(See Appendix 3.1)

3.1.3 Snell's law

Light follows the quickest path between two points. As it changes speed on crossing the boundary from a less dense to a denser medium, it is bent towards the normal. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for light crossing from one given medium to another. This fact was discovered by Willibrord Snell in 1621. (At that time the speed of light was unknown and indeed generally considered to be infinite.) Fermat was able to relate Snell's constant to the ratio of the speeds of light on the assumption that these are finite and fixed for different media. We have now shown that:

The reverse journey

Assuming that the speed of light is the same in either direction, the quickest route from A to B will also be the quickest route back from B to A. If we reverse the direction of a light ray at any point, we might expect it to retrace its way back along the path by which it came. This is sometimes known as the principle of reversibility. Accordingly, for rays coming from a point object immersed in a denser medium such as water, the ratio sin i /sin r is the same as before. Now, of course, the roles of the angles i and r are reversed (r represents the angle of incidence and i the angle of refraction). We will see shortly, however, that the reversed ray may 'decide' not to retrace its path back to B, but be reflected back into the water.

3.1.4 Apparent depth

To an angler looking down at a fish beneath the surface of water, the fish appears to be closer to the surface than it actually is.

Rays from the fish are bent away from the normal as they leave the water surface. Figure 3.2 illustrates how this changes the perceived position of the fish.

The rays appear to diverge from the point O' rather than from O. The object appears to be situated at an 'apparent depth AO' which is related to the real depth AO:

When the angles i and r are small,

AO sin i n real depth apparent depth

This relationship holds for small angles only when the angler is more or less directly above the fish. For larger angles, the angler sees the fish at an even smaller apparent depth.

3.1.5 The dilemma faced by light trying to leave glass

Rays coming from the denser medium are refracted away from the normal. As we consider rays striking the surface at greater and greater angles (Figure 3.3), we will come to the point where the angle of incidence r = dc (the critical angle). At that angle sin i/sin dc is such that sin i = 1. The refracted ray, at least in principle, skims along the surface at an angle of 90° to the normal. In practice, as we will see shortly, the intensity of the refracted ray decreases, becoming very small at that angle.

The dilemma is this: since sin i cannot be > 1, what is to become of a ray reaching the surface at angles r > dc?

A ray from the denser medium striking the surface at an angle greater than the critical angle dc cannot get out, and will be totally internally reflected.

If the ray passes from the glass into air, the value of dc can be calculated from:

sin90o sin dc

 Critical angles Substance n 0c water glass 1.33 1.52 48.8° 41.1°
Figure 3.3 Total internal reflection.

Reversal through 180° — rays return inverted.

Periscope — rays emerge to give an upright image.

Figure 3.4 Totally reflecting prisms.

3.1.6 Practical applications of total internal reflection

Since the critical angle for glass is just under 45°, right angled prisms with the remaining angles each 45° are ideal for turning a light beam through 90° or 180°, as we see in Figure 3.4. Since no loss of intensity occurs in the process, totally reflecting prisms are used rather than mirrors in most optical instruments.

Light pipes and optical fibres

In a light pipe, the rays of light keep hitting the surface at angles greater than dc and cannot get out. They will travel around corners which are not too sharp and arrive at the other end of the pipe without loss, as illustrated in Figure 3.5.

Figure 3.5 Light pipe.

Since the rays from different parts of the original source arrive at the other end completely scrambled, such a light pipe can be used for transmission of illumination only, and for creating images.

An optical fibre is thinner than a human hair, but is very strong. Optical fibre cables can transmit more data, and are less susceptible to interference than traditional metal cables. To transmit images, a bundle of very fine glass or plastic fibres, each of the order of a few microns in diameter, can be used. Each fibre transmits light from a very small region of the object, to build up a well-resolved image. Fibre-optic bundles are used widely in telecommunications.

In medicine, flexible endoscopes are used to look at internal organs, in a reasonably non-invasive manner. The endoscope functions as a viewing device inserted through a very small incision, while the surgical instrument is inserted through another small incision, and manipulated within the tissue. Particularly common operations include gall bladder removal, knee surgery and sinus surgery. In laser surgery, fibre optics can be used both to guide the laser beam and to illuminate the area.

3.1.7 Freedom of choice when a ray meets a boundary

The principle of reversibility applies only when both the starting point and the destination of the light ray are known, in situations such as the lifeguard problem (Figure 3.1). Often, when light comes to the boundary between two media it exercises freedom of choice' as to whether it is going to be reflected or refracted. This applies to rays going from a less dense medium such as air to a denser medium such as glass, or vice versa, as illustrated in Figure 3.6.

Figure 3.6 shows that when the light goes from glass to air, the intensity of the transmitted beam becomes fainter as the angle of incidence increases, becoming zero at the critical angle. At that point, freedom of choice is gone and the ray can only be

reflected and partially refracted.

reflected. It is difficult to understand what determines the fate of a particular ray — whether it is reflected or transmitted.

The 'shop window effect'

An example of partial reflection occurs when we look at a shop window. We see our own image in light reflected by the glass, and the goods on display, in light transmitted by the glass. In the first case light from outside the shop 'bounces back', and in the second light from the inside comes through. A similar 'freedom of choice' exists for light rays coming from the other direction. (In the case of the shop window this freedom is exercised twice, first at one glass surface and then at the other.)

3.1.8 The mystery

Partial reflection was a deep mystery to Newton. He realised that it is not determined for example by some local microscopic surface characteristic. The reason, he wrote, was 'because I can polish glass' — the fine scratches due to polishing do not affect the light. Whether or not bumps and hollows on the microscopic or even the molecular scale can be affected by polishing is questionable. Experimentally, we find that the path of a light ray at the boundary between the two media is unpredictable, regardless of the scale at which we perform our experiments. We now know that Newton was exploring a most basic phenomenon of nature.

'The photon decides'

The phenomenon is particularly interesting from the point of view of light photons. As photons reach the surface one by one, some of them are reflected, and some continue on through the surface. Which happens is not predetermined but is purely statistical; in fact, the photon decides. We have here an example of the rules of quantum mechanics. The photon is governed by basic laws of probability. What is more, the actual probabilities may depend on seemingly absurd things such as what may or may not happen in the future, for example, whether or not it might meet further reflecting surfaces later!

Such mysterious behaviour of photons forms the starting point of quantum electrodynamics (QED), co-founded by Richard Feynman (1918-1988).

3.1.9 A practical puzzle — two-way mirrors

In spy movies and identification parades we often see two-way mirrors which behave as transparent glass windows from one side and mirrors from the other side. This seems to be impossible from physical principles, so how do they work?

Many such mirrors are partly silvered so as to reflect not all rays, but a greater fraction than reflected by normal glass. They are based on the principle of greater brightness on one side, and little light on the other.

For light hitting the glass from either side, a certain fraction of light rays are reflected, and the remainder transmitted.

Because of the difference in brightness, the vast majority of rays emerging from the glass on the bright side are reflected rays which originated on that bright side. The opposite is true on the dark side, where most of the light coming through is transmitted light. The effect may be enhanced by coating the dark side of the glass with a very ^^ thin layer of silver, turning it z^M into a 'half-silvered mirror' { | which causes about 50% of the l_J J light to be transmitted and the I M other 50% to be reflected. [Mj The human eye can adapt to an enormous range of intensities. The eyes of the members of the identification parade are adapted to bright lights and the Two-way mirror in an identification very small number of rays com-parade. ing from the other side of the two-way mirror is not noticeable.

3.2 Lenses

3.2.1 The function of a lens

The basic function of a lens is to gather all the rays falling on its surface and to re-direct them by refraction in a prescribed fashion. Figures 3.7 and 3.8 illustrate how converging and diverging lenses behave.

Parallel rays converge to a point called the focus:

Rays from a point source O (object) converge to a point I (image) beyond the focus:

image image

A diverging lens causes parallel rays to diverge so that they appear to come from a point in front of the lens.

3.2.2 Fermafsprinciple applied to lenses

The principle of least time is most easily seen to apply directly to Figure 3.7. The job of the lens is to gather all light rays which strike it on the side facing the beam, and bring them to a focus on the other side. The lens does this by providing a number of paths of equal length in time. Light travels more slowly in glass than in air. The path along the straight line from O to I is actually made longer in time by slowing down the light as it passes through the thickest part of the lens. Rays going from O to I by one of the more roundabout routes have to traverse a smaller thickness of glass. Light rays are presented with these other routes which take the same time as the central route.

The problem is to find the shape of the lens to ensure that the smaller width at any point off-axis exactly compensates for the extra length of the journey!

Expensive lenses are complex in shape and may have many components, but it turns out that a single lens with spherical surfaces works quite well, particularly for rays close to the optical axis. The focal point of a converging lens is defined as the point at which incoming rays parallel to the axis are brought together at the other side of the lens. Conversely, a source of light at the focus will give rise to a beam of light parallel to the axis at the other side.

3.3 Objects and images: converging lenses

3.3.1 Ray tracing through a thin lens

A lens which is thicker in the middle than at the ends always acts as a converging lens, even when the curvature is convex on one side and concave on the other. It can also be shown that turning the lens around will not change its focusing properties. This means the focal length of a lens is the same on both sides, regardless of its shape, as illustrated in Figure 3.9. Accordingly, lenses are designated by a single value of f which applies to either side of the lens.

 parallel rays entering from left parallel rays entering from right A ^- V-4—

Figure 3.9 Focal points are at the same distance from both sides of a thin lens.

Figure 3.9 Focal points are at the same distance from both sides of a thin lens.

3.3.2 Principal rays (thin lenses)

We can also draw rays which pass through certain selected points for lenses, as we did for mirrors. In Figure 3.10 we have just three such rays and not four, because there is no such thing as the centre of curvature for a lens as a whole; each surface has its own centre of curvature and neither has direct significance in ray tracing for the complete lens.

Ray (1) enters parallel to the optic axis, and passes through the focus on the other side.

Ray (2) strikes the centre of the lens, and passes through without deviation.

Ray (3) passes through the focus, on its way to the lens, and exits parallel to the axis on the other.

3.3.3 The lens equation

The object distance and image distances are related to the focal length of the lens:

— + — = — (The lens equation) u v f size of image = v size of object u

These equations are derived in Appendix 3.2.

Magnification is obviously related to the above ratio and can be defined in such a manner as to indicate the orientation of the image. In our convention we define magnification as m = -v/u. Notice that when the ratio v/u is positive, the image is upright, and when it is negative, the image is inverted.

3.3.4 Symmetry

The lens equation exhibits symmetry between the object distance and the image distance. The relation would be just as valid if object and image were interchanged. This is particularly obvious if we look at the Newtonian form (see Appendix 3.2), which shows that moving the object further away brings the image closer to the lens, and vice versa. In the extreme case of the object at infinity, the image is at the focus; placing the object at the focus will give an image at infinity, i.e. a parallel outgoing beam. We cannot place the object further away than at infinity, therefore the image of a real object is never closer to the lens than the focal distance.

3.3.5 Breaking the symmetry

We can now raise an interesting question relating to symmetry. The image is never closer than the focal point of the lens. We have just stated that image and object may be interchanged, and the relation remains valid. But there is nothing to stop us from placing the object inside the focus! Does this not break the symmetry? Let us see what happens when we do this:

The rays, having passed through the lens, appear to diverge from an image on the same side of the lens as the object. Instead of the object, we see a virtual image which is upright and magnified, as illustrated in Figure 3.11. The lens can be used in this way as a magnifying glass.

The magnifying glass will work provided we hold it such that the object is at or inside the focus.

We can derive relations for the magnifier directly from the lens equation:

111 uf

Figure 3.11 Object placed nearer to the convex lens than the focus — a simple magnifier.

The image distance, v, is negative if u < f and the image is virtual.

Magnification m = - — (m is positive since v is negative) u fi The image is upright

The closer the object is to the focus, the smaller is the denominator u - f and the larger the magnification.

How about object-image symmetry? Can we somehow 'force' an image to appear inside the focus? To do this we need to have a converging beam in the first place and light always diverges from all points on a normal physical object. We need a second lens to create a converging beam which is heading towards an image when intercepted by the lens in question. This image (which the rays never reach) will act as a virtual object and lead to a final image inside the focus of the second lens.

3.3.6 An intuitive approach — the task of a lens

We have derived the properties of the image and its distance from the lens from the lens equation and also by tracing principal rays. The results obtained can be visualised and understood in another way by considering the task which the lens is 'trying to perform'. The task of a converging lens is to collect the rays of a beam which is diverging from a point and to bring them together at a focus. If that is not possible, it will at least make the beam less diverging. The beam from a source at infinity is parallel as it reaches the lens, so that its divergence is zero and it is therefore most easily focused. As the object comes nearer to the lens, the beam as seen by the lens becomes more and more divergent and the task becomes more difficult. Finally, when the object is inside the focus, the best the lens can do is to make a beam which is less diverging, which will then appear to come from an object which is further away, on the same side as the actual object. It will also appear larger in size because, as seen in Figure 3.11, the ray through the centre of the lens has to be projected back and upwards before it meets the other rays.

Questions

1. A lens is masked so that only half of it is in use, as seen in Figure 3.12:

Which is correct?

1.1. We get an image of the right side of the object only.

1.2. We get an image of the left half of the object only.

1.3. We still get the entire image but it is of lower intensity.

2. A lens is masked so that only the central portion is in use, as seen in Figure 3.13:

Which is correct?

2.1. We get an image of the central part of the object only.

2.2. We get only a tiny image of the whole object.

3.3. We get the entire image as before, but of lower intensity and greater clarity.

Both questions above can be answered when we recall that from any given point on the object rays go to the corresponding point on the image via every point on the lens. Statement No. 3 is therefore correct in both cases. We still get the entire image, but fewer rays.

In question No. 2 the rays we get are more selective in that they better fulfil the paraxial approximation. The image is therefore clearer, if less intense.

On a bright day the aperture on a camera lens is reduced as we have sufficient light to use only the central spot on the lens. The same is true for the human eye where the iris accommodates, according to lighting conditions.

3.4 Objects and images: diverging lenses

The principal rays in Figure 3.14 have been drawn according to rules similar to the previous ones:

Ray (1) enters parallel to the optic axis, diverted so that backward projection passes through the focus. Ray (2) strikes the centre of the lens and passes through without deviation.

Ray (3) sets out from the object in the direction of the focal point on the right, and exits the lens parallel to the axis.

The rays in the diagram all come from the same point at the top of the object; similarly, rays which diverge from every other

on both sides, or on one side only provided the concave side has greater curvature.

Figure 3.14 Principal rays passing through a convex lens.

point are made more divergent by the lens. The image is always virtual, on the same side as the object, and always smaller than the object and nearer to the lens.

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