Applied Optics and Optical Design, Part One

Part One copyright © 1957, 1985 by Dover Publications, Inc. All rights reserved.

This Dover edition, first published in 1992, is a reissue of the edition first published by Dover in 1957 (Part I) and 1960 (Part II). The Dover edition of Part I was an unabridged and corrected republication of the work originally published by the Oxford University Press, London, in 1929. Part II was originally published by Dover Publications, Inc., in 1960; this part was a posthumous publication edited and completed by Dr. Rudolf Kingslake (then Director of Optical Design, Eastman Kodak Company, Rochester, New York) with the assistance of Dr. Fred H. Perrin of the Eastman Kodak Company Research Laboratories.

Library of Congress Cataloging in Publication Data

Conrady, A. E. (Alexander Eugen)

Applied optics and optical design / A. E. Conrady; edited and completed by Rudolf Kingslake.

p. cm.

Includes bibliographical references and index.

9780486151229

1. Optics. 2. Optical instruments. I. Kingslake, Rudolf. II. Title.

QC371.C62 1991

535—dc20

91-33198

CIP

Manufactured in the United States by Courier Corporation

67007404

www.doverpublications.com

PREFACE

THIS book is the final result of about thirty-five years of continuous devotion to the study and practice of applied optics, beginning about 1893 with first attempts, purely as a scientific hobby, to fathom the mysteries of the design of telescope and microscope objectives. The hobby developed into a profession some five years later, and a large number of new types of telescopic, microscopic, and photographic lens systems were the result, followed during the great war by the design of most of the new forms of submarine periscopes and of some other Service instruments. It was probably the uniform success of this work which caused me to be invited to the principal teaching position in the new Department of Technical Optics founded in 1917 at the Imperial College of Science and Technology. The increased opportunities for research thus provided, and especially the close contact with students, following upon a long period of widely varied practical experience, have been invaluable in determining the form in which the subject of applied optics is presented in the present work.

Every effort has been made to limit the subjects dealt with and the methods employed to what the late Silvanus P. Thomson called ‘real Optics ’ and to exclude the purely mathematical acrobatics, which he called ‘examination optics ’.

Progress in applied optics has been retarded to a most deplorable extent by the widely accepted belief that the two principal methods of attacking individual problems, namely the elegant but approximate algebraical method and the more empirical but rigorously exact method of trigonometrical ray-tracing, must necessarily be antagonistic and mutually exclusive. It is hoped that this book will permanently establish the opposite doctrine that the two methods are ideally fitted to be applied in closest conjunction, the analytical method readily finding a rough solution (which, however, is hardly ever close enough to admit of actual execution), while the trigonometrical method quickly and systematically adds the necessary finishing touches.

In order to raise this co-operation of the two methods to the highest possible efficiency, the algebraical expressions have been put into such a form and the adopted variables have been so chosen as to render the progression from the rough analytical solution to the exact trigonometrical calculations as smooth and as simple as possible, the aim throughout being to reach the exact final result with the least total expenditure of time and trouble. In some cases this last demand has led to the final adoption of equations which obviously could be algebraically transformed into apparently simpler expressions, the reason being, as is explicitly shown in one or two instances, that the algebraically simplest formula is by no means always that which can be most quickly evaluated numerically.

Considerable attention is devoted throughout the book to a subject which is hardly mentioned elsewhere. As the final prescription for a new optical system can only be executed within certain limits of precision, and since, moreover, in many cases—probably the majority—it is impossible to correct all the aberrations, the important question always arises in practice : At what magnitude does any one aberration become a serious menace to the proper performance of a given optical system ? It is this question which is answered in this book in the case of all the ordinary aberrations, chiefly on the basis of the important quarter-wave limit laid down by the third Lord Rayleigh in 1878, but strangely neglected until comparatively recent years.

The present volume includes all the ordinary ray-tracing methods, the general theory of perfect optical systems, the complete theory of the primary aberrations, and as much of the higher aberrations as is required for the design of all types of telescopes, of low-power microscopes, and of the simplest photographic objectives. A second volume will give the necessary additions to the theory, largely on the principle of equal optical paths, to extend the scope of the complete work to the systematic study and design of practically all types of optical systems, with special attention to high-power microscope objectives and to anastigmatic photographic objectives.

A. E. C.

Table of Contents

Title Page

Copyright Page

PREFACE

INTRODUCTION

CHAPTER I - FUNDAMENTAL EQUATIONS

CHAPTER II - SPHERICAL ABERRATION

CHAPTER III - PHYSICAL ASPECT OF OPTICAL IMAGES

CHAPTER IV - CHROMATIC ABERRATION

CHAPTER V - DESIGN OF ACHROMATIC OBJECT-GLASSES

CHAPTER VI - EXTRA-AXIAL IMAGE-POINTS

CHAPTER VII - THE OPTICAL SINE THEOREM

CHAPTER VIII - TRIGONOMETRICAL TRACING OF OBLIQUE PENCILS

CHAPTER IX - GENERAL THEORY OF PERFECT OPTICAL SYSTEMS

CHAPTER X - ORDINARY EYEPIECES

APPENDIX

INDEX

INTRODUCTION

ALTHOUGH even the simplest properties of lenses are proved in this book, the proofs and their discussion are brief, being included chiefly for the purpose of establishing a uniform and consistent system of nomenclature and of sign conventions. For that reason some acquaintance with elementary general optics will be distinctly helpful to the beginner, and the optical section of a good modern text-book on Physics will supply the desirable information. As nomenclature and sign conventions are almost sure to clash with those adopted in this book, this preliminary study should be, as far as possible, limited to the descriptive parts, keeping an open mind with regard to signs and symbols.

The mathematical knowledge which is assumed in the book does not, as a rule, go beyond ordinary geometry, algebra, and trigonometry, but analytical geometry and elements of the calculus are also employed in certain sections.

Serious students should bear in mind that the book is arranged in strict logical order and that it should therefore be worked through systematically from beginning to end. Moreover the present volume is almost entirely devoted to the development of general methods for the solution of optical problems, with very little specialization for isolated cases of restricted interest ; hence there are very few sections which could be safely omitted. Owing to the large amount of ground to be covered repetitions or needless elaborations have been avoided, and even facts or conclusions of the highest importance may be found stated only once and in the fewest words compatible with clearness and completeness.

There is a deplorable tendency among students to concentrate on the mathematical equations and their proofs. These, whilst necessary and highly useful, are merely the dry bones of applied optics. The soul of the subject will be found in the numerous pages of plain letterpress with hardly any mathematical intermezzos, where the more important equations are discussed in order to discover their true optical significance and the best methods of applying them in the solution of practical problems. Students who intend to take up optical design as a profession should also realize from the very beginning that skill and ingenuity in numerical calculations will play a prominent part in determining their value to an optical establishment ; they should therefore devote a considerable part of their available time to actual numerical work and should master as soon and as completely as possible the numerous hints and suggestions on this much neglected subject which will be found in almost every chapter of the book.

Whilst everything included in the text will find valuable applications in solving optical problems, it is not necessary, and would indeed be quite impossible for the vast majority of people, to remember the whole contents of the book. To assist the student in singling out the principal fundamental facts which should be firmly fixed upon the memory so as to be instantly available in thinking out or discussing optical phenomena or problems when books are not at hand, brief ‘special memoranda’ have been added at the end of each chapter. These should prove useful when their stated purpose is borne in mind ; but they should on no account be looked upon as representing everything that is of real value or importance in the particular chapter.

CHAPTER I

FUNDAMENTAL EQUATIONS

SPHERICAL surfaces (including the plane as part of a spherical surface of infinite radius) are the only ones which can be produced by the optical grinding and polishing process with a sufficient approach to the necessary accuracy. Very slight departures from the spherical form, amounting at most to a few wave-lengths in depth, can indeed be secured by the process of ‘figuring’ ; such small amounts can be taken care of in our computations by allowing for the slight departure from strictly spherical form by approximate corrections. We therefore limit ourselves in our general formulae to exact spherical surfaces.

A spherical surface is the simplest of all possible, curved surfaces, for it is perfectly defined by its radius and by the location of its centre. A radius drawn from the centre through any point of a spherical surface stands at right angles to the tangent-plane in that point and is therefore the normal of the surface. As by the laws of reflection and refraction both the reflected and the refracted ray lie in the plane defined by the incident ray and the normal at the point of incidence (‘incidence-normal’) we can at once conclude that the plane of reflection and refraction in the case of spherical surfaces always contains the centre of the sphere and can thus be determined with the greatest ease.

In all ordinary optical instruments we have another vast simplification by reason of the centring of all the surfaces. It is intended, and usually achieved with sufficient accuracy, that the centres of curvature of all the component spherical surfaces shall lie on one and the same straight line, the optical axis of the instrument. That evidently means that any ray which originally cut the optical axis and therefore entered the system in a plane containing the optical axis will permanently remain in the same plane, and can therefore be traced right through the whole system by plane geometry or trigonometry. As this saves us the decidedly considerable trouble of determining a new incidence plane separately for each successive surface it will be one of the chief aims of our theoretical discussions to develop computing methods which avoid as far as possible the complication of tracing ‘skew-rays’.

LAWS OF REFLECTION AND REFRACTION

By the law of reflection the reflected ray lies in the plane defined by the incident ray and the incidence-normal and forms the same angle with the latter as the incident ray ; but as the two rays lie on opposite sides of the normal, the angles have the opposite clock-sense and we express this by giving them the opposite sign : we therefore state the law of reflection as

(I)

By the law of refraction the refracted ray also lies in the plane defined by the incident ray and the normal of the refracting surface, but on the other side of both the surface and the normal : hence the angle of refraction has the same clock-sense or sign as the angle of incidence, and if N is the refractive index of the medium containing the incident ray, N′ that of the medium containing the refracted ray, the law of refraction states that

FIG. 1.

Small corresponding changes of I and I′ are frequently of interest, and are found with sufficient accuracy by differentiating (I)* with the result or

and this equation will be included under (I)*.

When angles become very small their sines become equal to the angles themselves expressed in radians, hence the law of refraction for ‘paraxial’ rays which enter a refracting surface at very small angles with the incidence-normal becomes, using small letters for ‘paraxial’ angles,

(I)*p

Comparing the two fundamental laws, we at once see that we may treat the law of reflection mathematically as a particular case of the law of refraction, resulting from putting N = – N′, for this gives

N′ sin I′= – N′ sin I, or sin I′= – sin I,

which with the necessarily acute angles can only be if I′ = – I. This is a very important deduction because it enables us to apply practically all our refraction-formulae to problems of reflection by simply putting N′= – N or N′/N = – 1.

THE FUNDAMENTAL FORMULAE AND SIGN-CONVENTIONS

It has been shown above that any problem of refraction at a spherical surface can be reduced to one of plane trigonometry by first finding the plane containing the ray to be traced and the centre of curvature.

In the diagram, Fig. 2, let the paper represent this incidence-plane, OP the ray to be traced through the refracting surface, and let the latter cut the incidence-plane in the circle AP with C as centre. Let ACB be a straight line passing through the centre C to which it is convenient to refer the ray in the particular problem : very often it will be the optical axis of a complete system of lenses, but we do not restrict ourselves at all to that assumption. A reference-axis ACB which does not coincide with the principal optical axis of a centred system will be referred to in future as an auxiliary optical axis.

FIG. 2.

We then define the position of the ray by the distance AB = L from the pole or vertex of the surface to the point where the ray (produced if necessary) intersects the adopted axis ACB, and by the angle of obliquity or convergence U under which the ray meets that axis. It will be seen at once that both these quantities leave an ambiguity. We can find a point to the left of the pole A which is also at the given distance L from it, and we may in either of these intersection-points apply the angle U either above the axis AB as shown in Fig. 2 or below that axis. To remove these ambiguities we stipulate that the intersection-length AB shall be given the positive sign when it falls to the right of the pole A, and the negative sign when it falls to the left of the pole. In the case of the angle of obliquity U we stipulate that it shall always be an acute angle, and that it shall have the positive sign when a clockwise turn will bring a ruler from the direction of the adopted axis into that of the ray : and the negative sign if the turn is counter-clockwise. The only additional datum which enters into the calculation is the radius of curvature r of the refracting surface. Obviously this also requires a sign-convention to distinguish concave from convex surfaces. In agreement with the convention for the intersection-length L we stipulate that r shall have the positive sign when the centre C lies to the right of the pole A, and that r shall be negative when C lies to the left of A. Fig. 2 shows all the quantities in their positive position. The formulae to be deduced are such as will invariably bring out the new data for the refracted ray in accordance with the above sign-conventions.

The sign-conventions as stated are optically the most convenient, and therefore almost universally adopted. It should be noted at once that whilst the convention with regard to L and r agrees with that of co-ordinate geometry, the convention as to the sign of angles is the reverse of that usual in co-ordinate geometry. This occasionally will call for adjustments of the sign in equations derived in the first instance by the methods of analytical geometry.

The nomenclature employed in all our computing formulae has been carefully selected so as to be easily memorized, and also to be within the capacity of the ordinary typewriting machine. It is based on the following simple principles :

Only English letters are employed, capitals for the data of rays at finite angles with, or at finite distances from, the optical axis, small letters for rays so close to the optical axis or to a principal ray as to allow of the use of simpler formulae.

Vowels are invariably used for angles ; Consonants for lengths. Y is treated as a consonant.

Quantities which are changed in value by the refraction at a surface are distinguished by the use of ‘plain’ and ‘dashed’ letters respectively.

Suffixes such as I1, U4, &c, are only used when really necessary. Plain letters are usually retained for the surface actually under consideration, and the suffix ( – 1) then applies to the preceding, the suffix 1 to the following surface. In general formulae for a whole series of surfaces the latter are numbered successively 1, 2, 3, &c.

In lens-systems the surfaces will always be numbered in the order from left to right. The refractive index ¹ of the medium to the left of any surface will be denoted by plain N, that of the medium to the right by N′, and plain L and U will be used for the ray in the left medium, L′ and U′ for the ray in the medium to the right.

It should be clearly realized and borne in mind that the use of ‘plain’ or ‘dashed’ letters is determined by the location of the actually existing ray, and not by that of the intersecting point B or B′. Thus in Fig. 2 the ray OP only exists in the medium to the left of the refracting surface, and its co-ordinates therefore are plain L and plain U, although the actual intersecting point B, found by producing the ray beyond the surface which really intercepts it and alters its subsequent course, lies to the right of the surface.

In Fig. 2 (a) (which again shows all the quantities in their positive sense) we have the arriving ray OP, when produced, meeting the axis at B at the distance

FIG. 2(a) .

AB = L from the vertex A, and making with the axis the angle U. Drawing the radius CP through the point of incidence we have the angle of incidence OPR, for which we adopt the symbol I, also appearing at CPB in the triangle CPB. In this triangle we know the side CP = r and the side CB = AB – AC = L – r, also the angle CBP = U ; dropping a perpendicular CE from C upon BP, we have

and introducing the value of CE by the second equation into the first we obtain equation

(I)

by which we calculate the angle of incidence. A transposition of equation (I)* next gives

(2)

by the law of refraction, and we thus determine the direction of the refracted ray PB′ in Fig. 2(a)

We now have to determine L′ and U′ for the refracted ray. Referring to the diagram we see that the angle PCA at the centre of curvature is external angle to the triangle PCB and therefore = (U + I), and is also external angle to the triangle PCB′, corresponding to the refracted ray and therefore = (U′ + I′). Consequently we have the important relation (which should be remembered as it is very frequently employed subsequently) :

U + 1 = U′+ I′

and by transposition of this we obtain the value of U′, namely

(3)

In the triangle PCB′ we now know the side PC = r and the two angles I′ and U′ : we therefore can determine the side CB′ = L′ – r. Employing the same reasoning already applied to the triangle PCB, we easily obtain

(4)

(5)

which completes the work for the surface under consideration.²

In accordance with a generally accepted custom the formulae have been deduced for a ray travelling from left to right. The formulae are not, however, limited in validity to this usual direction, for our definition of the meaning and of the signs of the L, U, and r is quite free from any reference to the direction in which the light is travelling. If we required to trace the same ray in the reverse direction we should have as given quantities its intersection-length AB′ = L′ and its inclination U′ to the optical axis in the medium of index N′ to the right of the surface, and we could compute its course by a simple transposition of the above formulae taken in inverse order. Equation (4) transposed would give

(1)

By transposing equation (2) we should then have

(2)

Then by transposing (3)

(3)

and finally by transposing (1)

(4)

giving

(5)

It will be noticed that these equations are mathematically identical with those for left-to-right calculations, from which they are obtainable by simply exchanging ‘plain’ and ‘dashed’ symbols.

This is very important, for many calculations of complicated lens-systems can be considerably shortened and simplified by using left-to-right and right-to-left calculations alternately.

As has already been pointed out, the fundamental computing formulae are applicable to every case of refraction or reflection at spherical surfaces as soon as the incidence-plane has been determined. In the general case of a so-called skew-ray the determination of the incidence plane calls for a calculation of some difficulty at each successive surface. To avoid this, one of our principal aims will be to avoid the tracing of skew-rays as far as possible, and to derive the required results from rays proceeding in the plane of the optical axis of a centred lens-system. For such rays the transfer of the data from surface to surface is of the simplest kind.

FIG. 3.

Assume that AkPk represents the kth surface of a centred system of lenses and that a ray OPk in the plane of the optical axis of the system has been traced through the k surfaces. We shall know its intersection-length e9780486151229_i0020.jpg and its angle of convergence e9780486151229_i0021.jpg . Let Ak+1 Pk+1 be the next, (k+1)st, surface at an axial separation e9780486151229_i0022.jpg from the kth surface. In accordance with our last stipulation as to signs and symbols e9780486151229_i0023.jpg will necessarily lie to the right of surface AkPk and is therefore a positive quantity. We shall usually employ the separation in this sense of a ‘dashed’ quantity. But we must note that for a right-to-left ray-tracing the same separation will more naturally be looked upon as the distance at which the kth surface lies to the left of the (k+1)st, and that it should then be called dk+1 (without a dash) and be treated as intrinsically negative. In the case of general theoretical formulae this alternative aspect of the separation is sometimes preferable, and the double name, implying a reversal of sign, of any given separation or thickness should therefore be clearly realized and well remembered. It means that in any formula we may replace e9780486151229_i0024.jpg by – dk+1 and vice versa.

The refractive index of the medium between the two successive surfaces also has two names : it is e9780486151229_i0025.jpg when thought of in connexion with the kth surface, but Nk+1 when associated with the (k+1)st surface. But as refractive indices are absolute positive numbers we have in this case e9780486151229_i0026.jpg

We can now read the transfer-formulae directly from the diagram. The convergence-angle e9780486151229_i0027.jpg of the ray from or to the kth surface is obviously identical with that of the ray to or from the (k+1)st surface, therefore e9780486151229_i0028.jpg . But the intersection-lengths measured from the two surfaces are different by the separation. We thus arrive at the transfer formulae

(5)*

or for right-to-left calculations the transposed formulae

(5)* right-to-left

It should again be noted, as a rule which is in fact universally applicable, that in the equations for right-to-left ray-tracing ‘plain’ and ‘dash’ are exchanged and also k and (k+1), the latter on account of the reversed sequence in which the surfaces are met by the light.

Finally we must discuss the most specialized of all applications of our standard [6] formulae, namely that to an object-point situated on the optical axis of a centred system. If B is such an object-point and P the point of incidence of a ray, we shall find the intersecting point B′ of the refracted ray by the formulae in the ordinary way. But in this case we can draw a most important conclusion. By reason of the perfect symmetry of a spherical surface with reference to its centre it is at once obvious that if we calculated a ray from B through P’, at the same distance Y below the optical axis as P is above it, we should get a diagram for the new ray exactly congruent to that for the first ray, and therefore, that we should find precisely the same intersecting point B′. This argument we can at once extend, for obviously the same reasoning applies to any section of our spherical surface laid through the optical axis AC,.at any angle whatever with the plane of the diagram. We thus conclude that the image-point B’ found in the first calculation must apply to a whole cone of rays which has its apex at B before refraction and at B′ after refraction, and which passes through the refracting surface at the distance Y from the optical axis. The one calculation thus answers for a complete circular zone of the surface : and this evidently holds for any number of successive surfaces provided their centres of curvature lie on the original axis AC.

FIG. 4.

FIG. 5.

Important Note.

It should be observed that the change in the direction of a ray which is produced by refraction always depends on the relative refractive index N′/N. The indices determined by the spectrometer and communicated by the glassmakers are the relative indices of glass to ordinary air, and therefore allow automatically for the refractive index of air (about 1·00029) in glass-air refraction.

This fact justifies the universal optical practice of treating air as having a refractive index equal to ‘one’ exactly, at least in so far as the air may be treated as an unvarying medium. Really its index is approximately proportional to its density, and also varies with the humidity of the atmosphere. These disturbing causes may render the refractive indices of glass relative to air inaccurate by several units in the fifth decimal place, and are alone sufficient to demonstrate the futility of excessive accuracy in optical calculations.

NUMERICAL CALCULATIONS

A. MATHEMATICAL TABLES AND THEIR USE

Optical calculations can be carried out by a good calculating machine, and this method may sometimes save time and always reduces the mental strain of the work owing to the infallible accuracy with which the machine will produce correct results, provided the data put into it are correct.

For the present we will assume, however, that the calculations are to be carried out on paper with the aid of suitable mathematical tables. A good table of logarithms is then the primary requisite. For the majority of optical calculations the accuracy afforded by logarithms to five places of decimals is both desirable and sufficient. Among the innumerable tables of this type first place for optical calculations must probably be given to one at present almost unobtainable in England, viz. Bremiker’s table of logarithms and trigonometrical functions to five decimals. The strong points of the table are that it employs decimal division of the degree instead of the usual sexagesimal minutes and seconds. This greatly simplifies interpolation and reduces the risk of error in additions and subtractions of angles : moreover the table gives the trigonometrical functions at closer intervals than those of sexagesimal tables : the log sines of the first five degrees are given at intervals of 0·001° and the logs of all the functions of larger angles at intervals of 0·01°. The best five-figure table with sexagesimal division of the degree is Albrecht’s ; its strong points are the excellent and complete arrangements for interpolation and the inclusion of secants and cosecants in the trigonometrical part. The numerous small five-figure tables with big intervals and average differences for interpolation should be avoided on account of the great loss of accuracy and the waste of time in the inconvenient interpolations. Worst of all are those among this type of table which give sines, cosines, &c., in separate tables, for in optical (as in many physical and astronomical) calculations we frequently require several functions of the same angle almost simultaneously, and it is a gross waste of time to have to hunt these up on different pages instead of finding them all in one horizontal line.

The table used by optical students at the Imperial College and employed in calculating all the examples given in this treatise is Bremiker’s six-figure table of logs and trigonometrical functions (English publisher : D. Nutt, London). The drawback of a bulk of about 600 pages is amply compensated by the closer intervals : the logs of numbers are given directly for five significant figures, and the logs of trigonometrical functions at intervals of 10 seconds of arc for the whole quadrant, with a most useful additional table of log sines and tans for every second up to 5°. It is employed in our optical practice so as to obtain rather more than the usual five-figure accuracy by roughly interpolating numbers beginning with 4 or a lower figure for a sixth significant figure and by interpolating angles to the nearest second. After making these interpolations the sixth decimal place of the logs may be either rounded off to the nearest five-figure number (that practice is followed in the examples given here), or it may be retained with a slight further gain in precision.

The three tables specifically mentioned are printed from stereotype-plates, and may with considerable confidence be regarded as absolutely free from misprints or errors.

In calculations by approximate analytical formulae, reciprocals, roots, and powers of natural numbers are of frequent occurrence ; for that reason every optical computer should have close at hand Barlow’s Tables of squares, cubes, square roots, cube roots and reciprocals of all integer numbers up to 10,000 (London : E. & F. N. Spon, Ltd.), which is also a stereotype-edition and free from errors.

A few golden rules in the use of tables may be impressed on computers. Although good tables are almost sure to be free from error, some of the figures are apt to be indistinct or damaged. Errors arising from that source are avoided by acquiring the habit of glancing at the two neighbouring values and noting that the tabular number to be actually employed is the mean of its two neighbours within at most a few units of the last decimal place. As an example of the wisdom of thus checking a doubtful figure it may be related that a student in an examination required the reciprocal of 1·683, the true value of which is 0·5941771. In the specimen of Barlow’s table employed the 9 had a damaged tail, and the student used 0·5042, and so spoilt an otherwise faultless calculation. As all the reciprocals in that neighbourhood begin with 59 he should have noticed this ; moreover he should (in the absence of examination-fever) have realized that 0·50, &c., could not be the reciprocal of a number so far short of 2·0 as the one looked up ! The most common errors in using tables arise in the interpolations. The amount resulting from the interpolation is apt to be added when it ought to be subtracted, or vice versa. The simplest and usually sufficient safeguard against this kind of error is to note that the interpolated value must necessarily fall between, and cannot lie outside, the tabular numbers between which interpolation was carried out. Thus, supposing log cos 48° 1′ 34″ to be required, the table gives for 48° 1′ 30″, the log cos = 9·825300, and for 48° 1’ 40 9·825277. Therefore we have a difference for 10 = 23 Units of the 6th decimal, and for 4″ 4 × 2·3 or 9 Units. This must be subtracted and gives 9·825291 as the correct value. If—from the habit acquired through working so largely with sines—the difference were added it would give 9·825309, and would be rejected mechanically if the common-sense rule given above is borne in mind. A determined effort should be made to learn to do the entire interpolation mentally, and not to make a written sum of it ; if this easy trick cannot be acquired the working out should be done on a separate piece of paper, not on the margin or in vacant spots of the actual calculation. Gross errors quite commonly arise from the turning up of the correct minutes and seconds, but a wrong degree, or the degree at the top of the page when that at the bottom should be taken. Another error particularly common in optical calculations on account of the vast predominance of sines in them is that the occasional tan is apt to be fetched out of the accustomed sine-column. All these warnings may appear trite and superfluous, but long experience has shown that most of the mistakes in calculations arise from such elementary blunders.

A minor worry in numerical calculations presents itself in rounding off superfluous decimals if the numbers to be cut off are 5, either alone or followed by zeros. If the 5 is merely dropped without raising the last retained figure, the numbers actually employed will on the grand average be too small. If on the other hand the last figure is always raised when the dropped figure is 5, then on the average our numbers will be too large. A widely adopted rule which avoids this onesidedness and has everything in its favour is that in such cases we always round off to the nearest even number. Thus, supposing we wanted log tan 41° 41′ 10″ to four decimals, the 6-figure table giving 9·949650, we should employ 9·9496 because 6 is even, and not 9·9497, for 7 is odd. But if log sin 41° 44’ 40" were required to four decimals, the 6-figure value being 9·823350, we should take 9·8234 because 4 is even. It would be equally justifiable to round off always to the odd figure, but the even one has the advantage that it leads to no further worry if one-half of the number turned up is really required, as is frequently the case. In some mathematical tables terminal fives are marked so as to indicate whether further decimals would lead to a value either larger or smaller than an exact 5 followed by endless zeros. In such cases the rounding off would of course be carried out correctly in accordance with the indication of a ‘small’ or a ‘large’ five.

With regard to the characteristic of logarithms, which determines the position of the decimal point in the corresponding number, it will be found convenient and to lead to fewer mistakes to adopt the universal practice of the tables of logs of trigonometrical functions ; that is to increase the negative characteristic of all proper fractions by 10 so as to operate always with purely positive logs. Thus, for the log of 0·0273, which is really 2·43616, we write 8·43616. Strictly this requires the addition of – 10 to make it correct, but in practice it is not necessary to worry about this because it is unthinkable that the result of a calculation, at any rate of an optical system, should be in doubt to the extent of 10¹⁰ or 10 – 10 times the real value, and that is the only doubt that can arise. Thus, in the example, if the log of result 8·43616 had been found with utter disregard of positive or negative multiples of 10 in the characteristic we should know that the numerical result must be either

0·0273 (which is reasonable),

or

273,000,000

or

0·00000000000273,

or another 10, 20, 30, &c., zeros either before or after the decimal point, and all these numbers would be plainly absurd, being for practical purposes equivalent to either infinite or zero values. Doubts of that order could only arise in the occasional calculation of numbers beyond human conception, such as the number of vibrations which light performs in coming from a distant star to the earth ; or the ratio of the mass of a milligramme to that of the earth.

The rules to be observed in working with these logs of proper fractions with positive characteristic are :

When determining the log of a proper decimal fraction the characteristic is = 9 – number of zeros after the decimal point.

When determining the number, known to be a proper fraction, to a given log, the number of zeros after the decimal point is = 9 – the positive characteristic of the log. Note that this rule is merely an algebraical transposition of the preceding one, so that really only one needs remembering.

In adding up a number of logs of this kind we proceed by the usual rule, but put down only the unit-place of the characteristic : e9780486151229_i0033.jpg , the 2 of 28 being thrown away, being in fact cancelled by two of the omitted ( – 10)s.

In subtracting a log of a fraction borrow, if necessary, a ten in the minuendus ; it is really the omitted ( – 10) of the subtrahendus :

Integer powers of proper fractions are calculated by the usual rule ; supposing (0·0273)³ to be required, we put down log (0·0273)³ = 3 × log. 0·0273 = 3 × (8·43616( – 10) ) = (2)5·30848( – 30) ;

∴ (0·0273)³ = 0·000020346.

To calculate complicated powers of proper fractions, the quickest and safest logarithmic method consists in using the true negative log of the fraction. Assuming (0·0273)¹.⁷¹⁹ to be required, we use log 0·0273 = 8·43616 – 10 = – 1·56384, then log (0·0273)¹.⁷¹⁹ = – 1·56384 × 1·719 = 2.68825 = 7·31175( – 10) ;

∴ (0·0273)¹.⁷¹⁹ = 0·00204998.

When integer roots of proper fractions are to be evaluated by logs, the omitted – 10 of the logs of proper fractions must be considered ; in order to obtain the log of the root again with – 10 as the omitted part, the – 10 of the given log must be increased to – 10 . k if k is the index of the root, and the initial positive characteristic must, in compensation, be increased by 10(k – 1) before dividing by k. Supposing e9780486151229_i0035.jpg to be required, the procedure is

In working out formulae of a number of terms connected by multiplication and division a practised computer always makes a straightforward addition-sum of the logarithmic work by using the logs of the reciprocals of denominator-terms, or their ‘cologs’. As a rule these cologs = log 1—the tabular log are not obtainable directly. By noting that log 1 is zero and may be written 9·9999(10), (i. e. all nines, but a ten, treated as a unit-number, in the last place), the subtraction of the tabular log from zero may be done without any borrowing and the resulting colog may be written down from left to right practically as quickly and with as little risk of error as the log itself. Thus supposing cos 31° 5’ 17" to occur in the denominator of a formula and no secant-table to be available. Turn up the log cos, which is 9·93266, imagine written above it 9·9999(10) and subtract from left to right : 9 from 9 = zero, 9 from 9 = zero, 3 from 9 = 6, 2 from 9 = 7, 6 from 9 = 3, and in the last place 6 from 10 = 4. These straightforward subtractions can be done as quickly as the results can be written down, and 0·06734 is on the computing paper almost as quickly as if the tabular log itself had been copied. Should the log to be subtracted from nothing end with one or several zeros, then the terminal (10) of the imagined sequence of nines must of course be applied to the last significant figure of the log and the colog is finished with as many zeros as are at the end of the log.

The accuracy of practically all approximate optical formulae—such as the TL ones in the theoretical part—does not exceed that obtainable from a 10-inch slide-rule. Much time and paper may be saved without real loss of precision by becoming reasonably expert in the use of this most convenient tool, which will be frequently referred to subsequently. The now universal arrangement with uppermost or a scale from 1 to 100, second or b scale of the same extent on the upper edge of the slide, and c and d scales from 1 to 10 at the lower meeting edge of slide and rule, will always be assumed. With reasonable care in setting and estimating fractions of the actual divisions the probable error of a simple calculation by slide-rule is about e9780486151229_i0037.jpg of 1 per cent.

B. GENERAL HINTS ON NUMERICAL CALCULATIONS

A neat and orderly arrangement of each calculation is of primary importance ; it saves time and precludes mistakes even at the time when the work is done, but the chief advantage will be reaped subsequently when an old calculation is referred to as a guide in a new or modified design. Whilst it is not possible for most people to write with the regularity and precision of copperplate, no trouble should be spared in acquiring the habit of writing each figure so that it cannot be mistaken for another. 6 and 9 should have long tails so as to distinguish them clearly from 0 ; 2, 3 and 5 should have their characteristic differences well displayed, &c. Mistakes are made over and over again by a computer misreading his own 2 as a 3, even within a few minutes of writing it down, or a 7 with deficient upper works is read as 1, and so forth. Paper with impressed squares (about e9780486151229_i0038.jpg inch) will assist greatly in keeping numbers to be added or subtracted correctly alined and so to avoid confusion of the proper decimal places—another fruitful source of errors. It is a decided advantage to become used to writing rather small figures as it makes it possible to complete most calculations on a single sheet. Columns of e9780486151229_i0039.jpg -inch width are then sufficient for 5-figure work. A definite scheme should be adopted for each type of calculation so that each quantity is always found in a definite part of the column. Practically all optical formulae embody some algebraical sign-convention ; therefore a + b does not necessarily mean that the numbers expressing a and b are to be added together. To avoid mistakes from this source, which are very prevalent, it is best to get rid entirely of the idea that a + b means the sum and a – b the difference and to substitute the infallible rule that a quantity which appears in an algebraical formula with the positive sign is to be used with its own sign, whilst one which appears in the formula with the negative sign is to be introduced with reversed sign. Budding computers cannot be too pedantic in watching the signs of all the quantities introduced into a calculation.

In logarithmic calculations a difficulty arises because logarithms can only be found for positive numbers ; in order to avoid a separate investigation in each calculation as to the sign of the result, a mark should be put to each logarithm which belongs to a negative number and a small n is used for that purpose in the specimen calculations here given. When any one logarithmic formula (that is one containing only factors and divisors) has had all the logarithms extracted from the table and the n added to such as represent negative numbers, the sign of the result is at once found by counting the number of n’s occurring in the sum. If their number is even the numerical result must be positive, whilst if there is an odd number of n’s in the sum then the numerical result must be negative : this should be invariably indicated by putting the terminal n to the logarithm of the result, which will automatically lead to a minus-sign being put to the corresponding number found in the table of logs.

C. ACTUAL OPTICAL CALCULATIONS

The ray-tracing formulae given in the theoretical part are extensively used in practically every optical designing problem : they should therefore be thoroughly mastered, and they should be employed in working out numerical examples until the process becomes almost automatic. A few progressive examples will be given in extenso for use as patterns in further work, for which plentiful material will be supplied subsequently. The examples actually given have been selected so as to be suitable for the testing of theoretical conclusions in later chapters.

Example I

A simple biconvex lens of crown glass of the specification

receives light from a real object-point B1 at 24″ from the vertex of the shallower surface. For the usual left-to-right direction of ray-tracing we therefore have L1 = – 24. The refractive index of the glass is to be taken as = 1·5180. Rays are to be traced which leave B1 at angles of 1, 2, and 3 degrees with the optical axis. If the directly computed rays are to pass through the lens above the optical axis (another widely adopted custom) we must put U1 = – 1°, – 2°, – 3° respectively, because a ruler would have to be turned counter-clockwise to carry it from the direction of the optical axis into that of the ray. When several rays are to be traced through the same system it is best to do the work in parallel columns. The complete working out is then as given on page 17.

The two surfaces are here arranged side-by-side to fill the page. In actual practice it would usually be preferred to place the second surface calculation below that of the first ; the transferring of the last figures in numerical columns 1 to 3 to the heads of columns 4 to 6 would thus be saved.

DETAILED EXPLANATION OF THE CALCULATION

The calculation by standard schedule begins with equation (I) sin I = sin U (L – r)/r. For this we require L – r, and this is worked out in the first 3 horizontal lines, putting down L with its true sign as – 24 and r (given as 10) with reversed sign as – r = – 10, giving L – r = – 34.

As, at the first surface, all three rays come from the same point B, this calculation of (L – r) is only written down once ; at the second surface the rays have different L-values, and (L – r) has to be calculated separately for each one.

The logarithmic work then begins. Log sin U to the given angles is turned up and as the angles, and therefore also their sines, are negative, the logs found have the n added. Log (L – r) is similarly turned up and as (L – r) is also negative its log is also followed by an n. Addition of the two logs then gives log (L – r) sin U, and as the logs added carry 2—an even number—of n’s the product will be positive and its log does not receive an n. Division by r means subtraction of its log, and this completes the calculation of equation (1). We then compute equation (2) sin I′ = sin I. N/N′. Log sin I is already on the paper. We therefore form log N/N’ = log N – log N′, and as our rays pass from air (N taken as 1 exactly) into glass of index N′ = 1·5180, the log of which is 0·18127, we have

log N/N′= 0.00000 – 0·18127 = – 0·18127 ³ :

and as this has, by the formula, to be algebraically added to log sin I, we put it down, as found, namely – 0·18127 and find log sin I′.

We then proceed to evaluating equation (3) U′ = U + I – I′. Leaving 4 blank lines for the subsequent completion of the logarithmic work, we build up the angle-register, beginning by putting down the given U of each ray. We then turn up the 3 values of log sin I in the 5th line of the logarithmic work and enter the angles in a second line of the angle-register. In the present case sin I, and therefore I itself, is positive throughout. If log sin I had an n at the end we should have to put a minus-sign to the angles of incidence, as happens at the second surface. By equation (3) U and I have next to be algebraically added, but as the U are negative whilst the I are positive, the arithmetical operation amounts to subtracting U from I. We, however, must call the result by its algebraical name (U+1). To complete the working out of (3) we next turn up the angles to the log sin I′ found, and as the equation calls for – I′ we put down the angles found from the table with the reverse of their own sign, in this case with a minus-sign. We thus obtain U′, the angle which the refracted ray forms with the optical axis.

We now return to the logarithmic work and compute equation (4) L′ – r = sin I’. r/ sin U′. Log sin I’ is already down, hence we copy log r from the fourth line of the logarithmic work, but give it the + sign as it is now a factor. Addition gives log r sin I′. To complete (4) we have to divide by sin U’ or to subtract log sin U′. The latter is therefore turned up for each column and log (L′ – r) found by subtraction. The ray-tracing is then completed by (5) as shown under the angle-register, and as the rays are to be taken through a second surface we must prepare for this by (5)*, by deducting (algebraically) the given thickness d′ = 0⋅600 from the L’ found. We thus arrive at the intersection-lengths of the three refracted rays referred to the vertex of the second surface, i. e. the correct L-values to be used in tracing these rays through that surface. In accordance with the second of (5)* the U′ of the first surface are simply transferred as the U of the second surface, and similarly the log sin U′ in the last-but-one line of the logarithmic work of the first surface is copied to serve as log sin U for the second surface. The calculation through this surface is then an exact repetition of that described for the first, but instructive variations in signs should be noted. The final results are seen in the last line of the angle-register and the last line of the whole calculation :

The first ray cuts the optical axis at a distance L′ = 8 e9780486151229_img_1632.gif 7088 to the right of the vertex of the second surface and forms a clockwise angle U’ = 2° 44′ 20″ with it.

The writing of angles with hyphens separating degrees, minutes, and seconds will be found less likely to lead to mistakes (such as misreading 2° as 20 or 4′ as 41). It also acts as a constant reminder of the fact that in borrowing or carrying degrees, minutes and seconds the usual arithmetical rule does not hold, inasmuch as a degree is equal to sixty minutes and a minute equal to sixty seconds.

GENERAL VALIDITY OF THE STANDARD EQUATIONS

The standard equations have been directly proved in [4] and [5] for the case in which all the quantities are positive. We must now prove that they will give the correct result in every conceivable case if the sign-conventions are observed. For this purpose we must examine the geometrical relations between the quantities in the various cases and must show that our formulae, under the sign-conventions, lead to the same result.

Referring back to Fig. 2 a it is evident that all the quantities referring to the incident ray will be positive and that no essential changes can occur as long as B lies to the right of the centre of curvature C. But if we imagine point B moving from an original great distance towards the refracting surface, there will occur a very special case when B coincides with C ; the entering ray will then coincide with the radius PC which is the incidence-normal and will go straight on because the angle of incidence is zero. No calculation of any kind is therefore required and no question can arise as all quantities remain unchanged. If point B approaches the surface still more, it will fall to the left of C and we are confronted by a new case, Fig. 6.

FIG. 6.

In considering it geometrically we treat all the quantities as positive or absolute ; to distinguish these absolute values from the quantities in our trigonometrical computing formulae embodying our sign-conventions, we will enclose the symbols in a bracket, such as (L), in working out the geometrical result. Fig. 6 then shows that in the triangle PCB we have side PC = (r), side BC = (r) – (L) (because L is now smaller than r), and by the arguments used in deducing the computing formulae we find

For the important angle at C between the axis AC and the incidence normal PC we find, seeing that in the present case U is external angle to the triangle PCB,

angle at C = (U) – (I).

According to our computing formulae we have

both of which seem to contradict the geometrical result. But if we examine first the trigonometrical formulae for sin I, we can write it in absolute values of the given quantities, which in this case are all positive,

and this shows that our computing formulae will give the same numerical value of sin I as that above deduced geometrically, but will attach a negative sign to sin I and therefore also to the angle I itself : or mathematically put, we shall find I = – (I). If we then put this computed value into the computing formulae for the angle at C = U + I, we obtain in absolute value

angle at C = (U) – (I)

in precise agreement with the geometrical result above. Hence the entire result obtained from the computing formulae is in accordance with the geometrical deductions, simply because the angle of incidence is brought out by the formulae as a negative one.

These relations obviously hold for any position of B between the refracting surface and the centre of curvature.

A third class of cases arises when point B falls to the left of the refracting surface (Fig. 7).

FIG. 7

Geometrically we shall then find

The computing formulae will again be in the invariable form

but as B lies to the left of the surface, L will be negative under the sign-conventions, or L = – (L), and as U is a counter-clockwise angle it is also negative or U = – (U). Therefore the working out of the computing formulae for sin I will give

in precise agreement with the geometrical result. We shall therefore obtain a positive angle of incidence or I = (I), and if we now work out the angle at the centre of curvature = U + I we shall find, as U = – (U) and I = (I),

angle at C = – (U) + (I) = (I) – (U),

also in agreement with the geometrical result. This case obviously covers every position of B between the refracting surface and negative infinity, and therefore completes the study for a surface of positive radius pierced at a point above the axis by a ray from any point of that axis.

Geometrically we have exhausted the possible relations between the quantities, for the other nine cases which may occur lead to the same diagrams merely turned into various positions. Thus the turning of the three diagrams in their own plane through 180° (looking at them upside down) gives three cases for a concave surface passed by rays below the axis (Fig. 8).

FIG. 8.

By our sign-conventions these cases have the same sign of the angles but the opposite sign of L and r as the corresponding first cases. Turning the original three diagrams over like the page of a book and looking at them through the paper gives three more cases for a concave surface with the ray passing above the axis ; and these three diagrams viewed upside down (still through the paper) yield the last three possibilities of a convex surface passed by a ray below the axis. In the third set of three cases all signs are reversed as compared with the original cases. In the fourth set the signs of angles are reversed, but those of L and r are the same as in the original cases. Some of these additional cases may profitably be discussed by students in the manner employed above for cases 2 and 3 in order to verify still further the claim that the fundamental formulae may be applied with absolute confidence to every case that can possibly arise.

We have applied the proof to the point B ; obviously it applies equally to point B′. And as the connexion between the two points is derived from the simple and unequivocal fact that the angle at C is the same for the ray after refraction as it is before refraction : (U + I) = (U′ + I′), we may regard the proof as complete. It is important to be perfectly convinced of the universal validity of the fundamental formulae, as it enables us to accept as equally valid all deductions from them which depend only on the ordinary rules of algebra and trigonometry.

When the relations of the quantities before and after refraction are considered the total number of distinctive cases is raised from 12 to 20, for in the original