An Elementary Treatise on Optics

Front matter

.

AN

ELEMENTARY TREATISE

Introduction

INTRODUCTORY OBSERVATIONS.

1. Concerning the nature of light, very little is known with any certainty; fortunately it is not at all necessary in mathematical enquiries about it, to establish any thing about its constitution. The science of Optics reposes on three Laws, as they are technically termed, which depend for their proof upon Observation and Induction.

2. In the first place, as it is observed that an object cannot be discerned if it be placed directly behind another not transparent, we conclude that the action of light takes place in straight lines. These straight lines are called rays, and are the sole object of discussion in the following Treatise.

3. When a small beam of light admitted through a hole in the shutter into a dark room falls upon a plane polished surface, such as that of a common mirror, it is observed to be suddenly bent back, or reflected, according to the technical phrase, and as it has otherwise the same appearance as before, we conclude that each ray of light is bent at the point where it meets the surface, or that more properly for each ray that existed in the beam, we have now two, an incident and a reflected ray meeting in the surface.

Observation leads us to conclude that these rays are invariably in the same plane, and that they as invariably make equal angles with the reflecting surface, or with a line perpendicular to it at the point of reflection: the angles which the incident and reflected rays respectively make with this perpendicular are called the angles of incidence and reflection.

4. If again we present to the beam of light above-mentioned a very thick plate of glass or a vessel of water, or any other transparent substance, we shall find that part of the light is reflected on reach the surface, but part enters the glass or water, not however without deviating from its former direction. It is in fact bent or refracted, so as to be more nearly perpendicular to the surface, so that the angle between an incident ray and a perpendicular to the surface, called as before, the angle of Incidence, is greater than that between the refracted ray and the perpendicular; this latter angle is technically termed the angle of Refraction.

Observations similar to those alluded to in the former case lead us to the conclusion that the angles of incidence and refraction are always in the same plane, and that though they do not bear an invariable ratio to one another, their sines do, provided the observations are confined to one medium, or transparent substance.

5. We have then these three laws upon which to found our theory.

The rays of light are straight lines.

The angles of incidence and reflexion are in the same plane and equal.

The angles of incidence and refraction are in the same plane, and their sines bear an invariable ratio to one another for the same medium.

Note. Sir I. Newton attempted to explain the Theory of Optics on the hypothesis that light is a material substance emitted from luminous bodies, and that the minute particles of this matter are attracted by any substance on which they fall so as to be diverted from their natural straight course. He succeeded in demonstrating the laws above-mentioned upon that hypothesis, but not so as to set the question at rest. Other philosophers, probably with more truth, have supposed light to consist in undulations, or pulses propagated in a very rare and elastic medium which is supposed to pervade all space, and perhaps to have an intimate connexion with the electro-magnetic fluid.

The action of light is by no means instantaneous. It has been discovered by means of observations on eclipses of Jupiter's satellites, that light takes eight minutes, thirteen seconds of time, to come from the Sun to the Earth.

It will perhaps be as well to detail an experiment by which the Laws of Optics may be well illustrated.

Let a square or rectangle (Fig. 1.) of wood, or any other convenient material, have its opposite sides bisected by lines , , and be correctly graduated along the top and bottom, so that the divisions, which must be equal on both lines, may be aliquot, parts tenths or hundredths, for instance, of or .

Let this rectangle be immersed vertically in water up to the line in a dark room, so that a small beam of Sun-light admitted through a shutter may just shine along its surface in a line .

There will then be observed a reflected beam along a line on the surface of the rectangle, and a refracted one down through the water, also lying just along the surface of the rectangle. Now if the distances be observed, it will be found that and are equal, and that and , which are respectively the tangents of the angles , to the radius or are so related, that if the sines of the same angles be calculated by the formula , these sines will be found to be in a certain ratio which in the case of pure water is about that of 4 to 3, or more correctly, 1,336 to 1. The experiment should be repeated several times when the Sun is at different heights, and the ratio of the sines of the angles , will be found invariably the same.

The fact of the incident, reflected, and refracted rays, , and , all lying precisely along the same plane surface, shows that those rays are all in the same plane, which is one circumstance mentioned in the Laws.

It may be necessary to observe that it is indifferent as to the directions of connected rays, which way the light is proceeding, that is, whether forwards or backwards, as any causes that act to produce a deflection from the straight course in the one case, would produce corresponding effects in the other.

Chapter 1

CHAP. I.

REFLEXION AT PLANE SURFACES.

6. Prop. To find the direction which a ray of light, emanating from a given point, takes after reflexion at a plane mirror in a given position.

Let , (Fig. 2.) represent a ray of light, proceeding from the point ; , the section of the reflecting surface by a plane perpendicular to it containing the line ; , the reflected ray making with an angle equal to the angle which makes with the same line: let be perpendicular to , and let meet it in .

Then since the angle is equal to , that is, to , the right-angled triangles , , having the side in common, are equal in all respects. Therefore is equal to .

Any other reflected ray will of course intersect in the same point ; so that if several incident rays proceed from , the reflected rays will all appear to proceed from , which as we have seen is at the same distance behind the mirror as is before it.

7. Suppose now that a ray (Fig. 3.) reflected into the direction by a plane mirror , meet in another mirror inclined to the former at an angle . It will of course undergo a second reflexion, and returning to meet the first mirror be reflected again, and so on; so that the course of the light will be the broken line .

Let perpendiculars be drawn to , , at the points , , , , … meeting each other successively in , , , , … Each of the angles at these points will of course be equal to the angle at .

Let the angle of incidence at | be called | 1,

|that at | …… | 2,

and so on.

Let also represent the angle at

The reader will find no difficulty in following these equations,

or

If now be any multiple of as we shall have somewhere that is, some reflected ray will be perpendicular to one of the mirrors, and these of course will end the series of reflexions.

If be not a multiple of some value of will make greater than and then will become negative. The geometrical fact indicated by this is that the broken line will at length be turned back upon itself, and the light after coming down the angle will go up again.

Let be the intersection of and

… and

… and

Then it will immediately be seen that the value of the angle at is or that at is the same; that at is double of these or so that if we represent the lines by and the angles between them by &c., we shall have

(provided be an odd number.)

Let be the intersection of and .

It will be easily seen, that

Let us resume now the first equation

Suppose then or

The first and nth angles will then be equal.

We must observe, that there is a limit to the angle of incidence after it becomes negative, namely, the double right angle; if it becomes exactly equal to this, the last ray will be parallel to one of the mirrors; if greater, it would meet it if produced backwards.

If

that is, if represent the number of degrees in must be a whole number.

Errata

Chapter 2

CHAP. II.

REFLEXION AT SPHERICAL SURFACES.

8. Prop. Rays meeting in a point being incident on a spherical reflecting surface; it is required to determine the directions of the reflected rays.

Let Fig. 4, represent the spherical surface, which we will suppose concave, or rather a section of it by a diametric plane containing an incident ray being the point from which that and the other rays are supposed to proceed.

Draw through the centre of the surface.

We must here extend the acceptation of the second law, which was understood only with respect to plane reflected surfaces, but which is just as true for curved ones, since we may consider the point of such a surface where reflexion takes place as belonging either to it, or to its tangent-plane, and the angles of incidence and reflexion will be those made by the incident and reflected rays with a normal to the surface at the point of reflexion.

In the present case, the surface being spherical, the radius is a normal, and if be drawn making with this an angle equal to , it will represent the reflected ray.

The line passing through the centre, is technically termed the axis of the reflecting surface, and the question is now to determine the point , the intersection of the reflected ray and the axis.

We may be supposed to have given,

the radius of the surface , (),

the distance , (),

the angle , (),

and we might calculate the angle , ().

We will call the distance , .

Now | ; that is, ;

|and | ; that is, ;

|.

Hence , or .

9. It appears then, that the place of the point or the distance depends on the value of , the angle , and is therefore not the same for different rays It diminishes as increases, that is, as diminishes, or approaches to coincidence with . It is important to know what its final value is, which is in fact to determine the point of intersection of the reflected rays when the incident rays are nearly coincident with the axis of the surface, forming consequently a very small pencil.

If we suppose , we shall have and 1/ or

10. If moreover we suppose infinite, which is supposing the rays parallel, we shall have

or

This is what is technically termed, the principal focal distance of the reflector, the place of being then which is called the principal focus, and if we call we shall have in general, that is, for rays nearly coincident with

11. These formulæ might easily have been obtained directly by supposing equivalent to , and we will make use of this method to deduce them in another form which is often more convenient.

Since , (Fig. 5.) bisects the angle , we have

.

And in the extreme case

that is, if we call , ; , ; , as before,

so that in fact , , and are in harmonic progression, and we have

, or ,

and if be supposed infinitely distant, or ,

then , or ,

which agrees with the preceding formula.

If, as before, we put for , we shall have

, and , that is, ,

from whence it appears, that

.

12. Upon the whole we may collect that if a small luminous body be placed before a spherical concave mirror, at some distance from it, the distance of the focus will always be something more than half the radius of the surface, which is its accurate value for the light of the Sun, the rays of which are considered as parallel; that if the luminous point be moved towards the mirror, the focus will come forwards to meet it, and at the centre , they will coincide, (Fig. 6.) as the formula will easily show, and as one might naturally expect, for the rays in that case being all normal to the surface, would be reflected back upon themselves. When the light is brought between the centre and the surface, or between and , (Fig. 7.) and in a manner change places, as one might expect from observing that in the formula

,

and are quite similarly involved, and therefore may be commuted without altering the equation. When the light is at the middle of the line , namely, in the principal focus , (Fig. 8.) the formula shows that we must have , that is, infinite, which answers to what we found before, namely, that when the incident rays are parallel, or is at an infinite distance, the reflected rays meet in .

When is brought between and , (Fig. 9.), or is made less than , the formula , or , shows that , and consequently , must be negative, that is, goes to the other side of the reflector, and the reflected rays instead of converging, diverge.

When comes to , meets it there.

As a converse to the last case but one, we may take that of rays converging to a point behind the reflector, and reflected to a focus in front, (Fig. 10.) To accommodate the formula to this case, we must make negative, and we have then

, or ,

which shows to be essentially positive.

The student will find no difficulty in examining particular cases; the one most likely to occur, is that in which the radiant point is at the opposite point of the sphere from the centre of the mirror.

Making , we find here .

It will occur to every one, that of the two foci and , that which lies between and moves much more slowly than the other, when their places are changed; in fact, we have seen that by merely bringing up from to , was sent from to an infinite distance, and that when moved on from towards , came back from an infinite distance on the reverse side of the reflector to meet at .

13. We have hitherto considered only one species of spherical reflector, the concave; let us now take the convex, (Fig. 11.) where as before, is the centre, the radiant point, , an incident and a reflected ray, making equal angles with the radius or normal. Let cut in .

Then we have, keeping the same notation as before,

that is, as before, ,

that is, ;

,

and finally, ,

which is the same result as before, except the sign of the 2d term; it will however immediately occur that they may be reconciled completely by supposing the radius to be positive in the one case, and negative in the other, which is exactly true in algebraical language as applied to Geometry. This is however more clearly distinguished in the case of the other formula, which here becomes for, as before,

for, as before, that is,

whence and dividing by

or as above.

Here then and become both negative, which the Figure plainly shows.

Making infinite, or we find , or as before, so that the principal focus is here at the bisection of the radius of the reflector, and of course behind it. In fact, as long as is positive, or is in front of the mirror, so long must necessarily be behind it, and the reflected rays diverge.

If, however, the incident rays converge to a point behind the mirror, that is, be negative, the formula will become

or

and may be positive, provided that is, so that if the focus of the incident rays lie between the principal focus and the back of the mirror, the reflected rays will converge. Of course, if the incident rays converge to the principal focus, the reflected rays are parallel.

14. It will easily be seen that the effect of a concave mirror is to give convergence to rays, that is, to increase convergence when it exists in the incident rays, to give convergence to parallel rays, and even to divergent within a certain limit, and beyond that to lessen their divergence.

Convex mirrors on the contrary give divergence to parallel rays, increase previous divergence, and make even convergent rays diverge, or at least diminish their convergence.

Errata

Aberration in reflexion at Spherical Surfaces

CHAP. III.

ABERRATION IN REFLEXION AT SPHERICAL SURFACES.

15. We found in the beginning of the last Chapter, that a cone of rays incident on a spherical surface were not so reflected as to meet in a point, but that the point which we called the focus of the reflected rays was, in fact, the mathematical limit of the intersections with the axis of rays chosen more and more nearly coincident with it. Let us now examine how much the intersection of the axis with a ray at a small but sensible distance from it, differs from this.

Taking the centre of the surface as the point to measure from, (Fig. 12.)

and being the actual intersection of the reflected ray and axis,

let .

Then since is the limiting value of when , we must have

being made equal to in every differential coefficient, or if we consider as a function of the versed sine of , which we will call

The brackets indicating that is made in each coefficient.

and so on, whence

This in geometrical terms amounts to

The Aberration is represented by this series without its first term; and when the angle , and a fortiori its versed sine, are but small, the second term of the series will give a near approximate value.

Note. The above is perhaps the neatest way of obtaining the series for the aberration: it is sometimes done by a method simpler in its principle

16. There is another series a little different from this, but which amounts nearly to the same thing.

If we take the letter to represent the aberration,

Now ; so that being half the radius, is the algebraical value of half the excess of the secant over the radius, and if be drawn touching the surface at , we shall have the following series in geometrical terms,

The first term of which differs but little from that found before, as or differs but little from when is so nearly equal to

When the incident rays are parallel, the aberration is or 1/2 accurately.

If , for for so that the point coincides with

REFLEXION AT CURVED SURFACES NOT SPHERICAL.

CHAP. IV.

REFLEXION AT CURVED SURFACES NOT SPHERICAL.

17. t will be recollected that the common parabola has the peculiar property that two lines drawn from any point on the curve, one to the focus, the other parallel to the axis, make equal angles with the tangent or normal; whence it follows that rays proceeding from the focus of a paraboloid will be reflected accurately parallel to its axis, and vice versâ rays coming parallel to the axis will be made to converge accurately to the focus or from it according as it is the concave or convex surface that reflects.

18. In like manner rays proceeding from one focus of an ellipsoid will be reflected accurately to the other focus, or if the outward be the reflecting surface, rays converging towards one focus, will diverge after reflection as if they proceeded from the other.

19. The analogous property of the hyperbola leads one to the conclusion that the surface generated by the revolution of that figure about its major axis, is such that rays meeting in one focus, will after reflexion diverge from or converge to the other.

20. These are the only surfaces that reflect light accurately in the manner we have described, which the simplest of all curved surfaces, the sphere, does not do; as to other surfaces, if we had occasion to treat of their reflexion, we should proceed in a manner similar to that used for the sphere, but in most instances if it were required to investigate the simple case of a small pencil of rays incident perpendicularly, it would be easiest to substitute for the surface its osculating sphere, that is, provided there were one, which is always the case at the vertex of a surface of revolution.

All this will, however, be much better understood after going through the next Chapter, which places the subject in a more general point of view.

Chapter 5

CHAP. V.

OF CAUSTICS PRODUCED BY REFLEXION.

21. We have hitherto considered only the places where reflected rays intersect the axis of a spherical mirror; we will now examine their intersections with each other, and treat the question in a more general manner, that is, we will suppose a cone of rays to be reflected at any surface, any how disposed with respect to the radiant point. We will, however, simplify the question a little by taking only a plane section of the surface through the radiant point.

In Fig. 15, represents a point from which proceed rays , , … which are reflected at the curve … into the directions , ….

The question being merely one of Plane Geometry, it will immediately occur to the reader that these reflected rays must be very analogous to normals drawn at different points of a curve, and that as these by their mutual intersections produce a broken line which becomes a regular curve, when their number is increased without limit, so the reflected rays should give rise to a similar broken line and curve, which is, in fact, the case, the curve being what is technically termed a caustic.

22. Prop. Given a point from which a thin pencil of rays proceeding, fall on a spherical reflector, to determine their intersections after the reflexion.

Let , (Fig. 16.) be two incident rays, , the reflected rays meeting in , , the normals at , meeting in , which if we suppose the distance to be, according to the phrase, infinitely small, will be the centre of the osculating circle.

Let

|

|

|

, or

|.

Then since a small variation in the place of causes an infinitely less variation in that of , we may establish the following equations by differentiating those above,

.

Moreover, since , , , make respectively equal angles with the curve as in an ellipse, of which and would be the foci,

or ,

so that our equations now stand

,

,

and if we multiply the first by , and the second by , and subtract, we shall have

and , or ,

and if we put for the expressions will be rather simpler,

, or .

23. Here we may observe, that if be infinite, that is, the incident rays parallel, we have simply , which referred to the geometrical figure, shows that is in that case one-fourth of the chord of the osculating circle so that if be a tangent, and perpendicular to it, calling , , we have in general bc|

, and

.

24. Prop. Given the radiant point, and the reflecting surface, to describe the caustic.

In order to determine the nature of the caustic curve, that is, the section of the superficial caustic, we may consider it as a spiral having for its pole.

Let (Fig. 18.) be drawn perpendicular on which is of course a tangent to the caustic at . Call , ; and ,

|,

|,

,

.

Then if for be put its value , or , and for the proper function of given by the equation to the original curve, and be then eliminated, we shall have an equation in and , which will be that of the caustic.

25. In order to obtain an equation in rectangular co-ordinates, we may proceed as follows:

Reasoning as before, since the caustic is formed by the continual intersections of the reflected rays, two of these are necessary to determine one point of the caustic, and the point where one of them meets the caustic, is that which it has in common with the next; so that if we refer the two reflected rays to the same abscissa, their ordinates, differing in general, coincide at this point, and as far as that point is concerned, a change in the point of the reflecting curve, or in its co-ordinates, takes place without any alteration in the co-ordinates of the reflected ray (Fig. 19.)

We have therefore only to find the equation to the reflected ray belonging to an assumed point of the curve; to differentiate this, considering the co-ordinates of the curve as the only variables, and eliminate these co-ordinates between this equation, its primitive, and that of the curve.

An Example will make this more intelligible.

To diminish the length of the process, we will confine ourselves to the simple case of parallel rays, and take one of them for our principal co-ordinate axis. Let (Fig. 20.) be this axis, , co-ordinates of the curve, a normal, , an incident and a reflected ray. The question is first to determine the equation of the line . Since this line passes through the point whose co-ordinates are , , the equation must be

,

being the tangent of the angle .

Now,

.

And since is a normal,

.

The equation is therefore

(1);

and we have to put for its value in terms of the co-ordinates given by the equation to the curve, and eliminate and between this, the equation (1), and its derivative.

26. The process is sometimes facilitated by taking for the variable a function of the angle , as its tangent which is equal to . The quantity we have called α is the tangent of twice this angle, and if we put for this angle, the equation to the reflected ray is

.

Example. Suppose the curve to be a common parabola, its equation is ,

.

Then if we put for , bc|

|;

| (1).

Then differentiating with respect to ,

(2);

and substituting this value of in (1), we have

, or ;

the equation to a point, namely, the focus, where .

27. The method pursued in the following example is perhaps less elegant than that just given, but it has often the advantage of being simpler and less prolix.

Required the form of the caustic, when the reflecting curve is a common parabola, and the incident rays are perpendicular to the axis.

Let (Fig. 21.) be a point of the curve , an incident and a reflected ray. Then, taking for granted that when is a point of the caustic, is one-fourth of the chord of the circle of curvature at , we have the following easy method of determining the co-ordinates of . bc|1

Let | | ,

| | ,

| | ,

|,

|,

|;

| ;

| ;

bc|

,

.

So far all is general; in the particular example proposed,

, , ;

bc|

,

|

.

From this it appears that , or the curve crosses the axis, where , which answers to the point in the parabola for which , bc|

math|3√3a2//{9a−X/2X2/−X2/}

.

When this is infinite, so that the caustic like the reflecting curve is perpendicular to the axis at its origin: when

, ; .

The angle at which the caustic afterwards cuts the axis is therefore that having for its natural tangent , which shows it to be one of .

The curve extends without limit in the same directions with its generating parabola.

28. Required the form of the caustic when the reflecting curve is an ellipse, and the radiating point its centre. Fig. 22, &c.

The polar equation to an ellipse about its centre being

,

we have

,

and ;

.

bc|

Hence, when , | ,

|and when , | .

The former of these values is always essentially positive, since is supposed to represent the semi-axis major, and therefore must be greater than ; but may be equal to, or greater than , so that when , may be infinite or negative.

When , the form of the caustic is such as that shewn in Fig. 22.

When , gives , and the curve is that of Fig. 23.

When , we have infinite branches asymptotic to the axis minor, as in Fig. 24.

When , there are asymptotes inclined to that line (Fig. 25.).

29. There are some simple cases in which it is easy to determine the nature of the caustic by geometrical investigation.

Prob. To find the form of the caustic, when parallel rays are reflected by a spherical mirror. (Fig. 26.)

Taking as usual a section of the mirror, let be one of the reflected rays, touching the caustic in . Then, since we know that in this case is one quarter of the chord, if be bisected in , and be joined, will be a right angle, and if a circle be described through the points , will be the diameter of it. Let be the centre, join . Then since , if a circle be described with centre and radius , cutting the axis in , the principal focus, the arc which measures the angle to the radius , must be equal to the arc , which measures twice the angle to the radius , which is half of the other radius .