THE THEORY AND DESIGN OF ZOOM LENSES

A. T. O'Leary

A Thesis submitted for the

Degree of Master of Science

at The Australian National University,

Canberra.

January 1972.

i

ACKNOWLEDGEMENTS

The author expresses his thanks to Professor H.A. Buchdahl

and Dr. P.J. Sands for suggesting the topic of this thesis and for

many useful and valuable suggestions, discussions and comments.

He also thanks his wife, Jacquellyn, for her patience,

understanding and help with proof reading, the Department of Education

and Science for the award of a research scholarship and Miss Norma

Chin for her excellent typing.

Except where due acknowledgement has been made in the

the work described in this thesis is the candidate's own.

A. T. O'LEARY

iii

ABSTRACT

This thesis contains a new approach to the paraxial and

third order design and theory of zoom lenses, especially those con­

sisting of thin elements. Equations describing the paraxial and

third order behaviour of zoom lenses are derived and are used to

design a series of three element zoom lenses.

In the paraxial theory the derivatives, with respect to the

axial separations, of the paraxial coefficients are required.

Optimization processes for the automatic design of lenses require the

availability of the derivatives, with respect to axial separations and

curvatures, of the third and fifth order aberration coefficients.

These derivatives can be calculated from formulae given herein and

these form the basis of a FORTRAN programme for calculating such

derivatives.

The derivatives, with respect to axial separations, of the

paraxial coefficients, occur in the formulae describing the paraxial

properties as functions of the zoom parameter. These formulae may be

solved to obtain the powers and initial (z = 0) separations of the

elements of the zoom lens. For two and three element zoom lenses the

equations are solved analytically.

Formulae are obtained for the third order aberration coef­

ficients as functions of the zoom parameter. Using the thin lens

approximation the shapes of the individual elements are introduced and

from the resulting expressions sets of simultaneous equations are

obtained in the particular case of the three (thin) element zoom lens.

The solutions of these sets of equations give zoom systems for which

iv

selected third order aberrations are substantially corrected over the

whole zoom range. Whereas optimization processes give one solution to

a particular problem, the present method gives all possible solutions

by a simple analytic or semi-analytic approach.

V

INTRODUCTION

This thesis is divided into two parts. Part I deals with the aberration coefficients and the derivatives of aberration coef­ficients for symmetric optical systems (§2). Part II, which deals

with the theory and design of zoom lenses, uses many of the results of

Part I. A zoom lens is an optical system able to produce, in a fixed plane, a corrected image with variable magnification. This effect is produced by the controlled movement of the constituent elements relative to each other (§14).

Essentially the material of Part I is taken from the work of Buchdahl+ [1], but some new material is presented. The notation (§§2 and 4) used throughout this thesis is basically that of L.

The equations of paraxial optics (§2) are linear and the paraxial coefficients occurring in these equations can be used to

build up expressions for the contributions of surfaces to the aberra­tion coefficients (§5). With the correct choice of ray co-ordinates

(§2) the 'a1 aberration coefficients (§5) describe the aberrations of an optical system. However when calculating the derivatives of aber­ration coefficients (and higher order aberration coefficients), the

'b* coefficients (§5) are also required. Although formulae can be

written down for these, it is simpler in practice to employ the so-

called 'principle of duality' (§6).

The theory presented in this work will be referred to frequently. The entire work will be referred to as L (it presents a Lagrangian optical theory) and any section or equation preceded by L refers to a section or equation in L.

vi

The paraxial, third and fifth order 'a' and 'b* coefficients of a system of surfaces can be obtained from an essentially simple computer programme employing the principle of duality. Such a

programme, COEFF, has been written (Appendix 2) and is used as a sub- programme in programme DERIVATIVES (§12 and Appendix 2).

In the design of optical systems a knowledge of the effects of variations of design parameters (chapter 2) is of great importance. In zoom lens theory and design this knowledge is especially useful

since the axial separations between lens elements are continuously variable. The derivatives of the final paraxial coefficients (§8), with respect to axial separations, can be used to describe the

variation of the first order (paraxial) properties of zoom lenses (§§17 and 18). Equations (8.11-13), though derived from the theory of L Part III, do not appear in L.

In Part II of this thesis an approach to the design of zoom lenses is presented. In this we allocate the initial axial

separations, powers and curvatures of thin elements in such a way that the paraxial and third order terms in the expansion of j|' are con­trolled. The results of this design procedure can be viewed only as a

first approximation to be used, say, as the starting point in an

optimization process for obtaining a final system, consisting of real

(thick) lenses, with balanced aberrations throughout the whole zoom range.

Optimization processes generally require a knowledge of at least the first derivatives, with respect to thicknesses, curvatures, etc., of the third and fifth order aberration coefficients. These derivatives can be obtained by the method of finite differences, but

vii

are given more accurately by the exact formulae presented in chapter 2.

The first derivatives of the third order aberration coefficients (§10)

and the fifth order aberration coefficients (§11) are given in a form

which may be used directly in computing schemes. The derivatives of

the third order coefficients, in a slightly different form, are given

in L. The derivatives of the fifth order aberration coefficients are

not given in L, but the method of their production is given.

In view of the importance of the first derivatives of aber­

ration coefficients, a computer programme, called DERIVATIVES, was

written and is summarized in §12. The basis of this programme is the

formulae contained in §§10 and 11.

The calculation of derivatives of fifth order aberration

coefficients with respect to the first extra-axial curvatures requires

certain formulae for the derivatives of the intrinsic contributions of

order five. These formulae are given at the end of §12 and are

obtained directly from the theory of L §156, specifically equation L

(156.3).

Part II of this thesis discusses the paraxial and third

order aberration theory of zoom lenses, especially those consisting of

thin elements (§30), from a new point of view. The notation used is

essentially that of L, although modifications have been made in

special circumstances.

As previously stated the elements of zoom lenses are movable.

The form of this movement can be used to classify zoom lenses, which

in the past have been classified as either 'mechanically' or

'linearly' ('optically') compensated systems. The meanings of these

terms are explained in §14. These classifications are purely

viii

artificial, linear compensation being merely a special case of

mechanical (or non-linear) compensation.

The theory set out in chapters 4 and 5 holds for linear com­

pensation, but it must be emphasized that this restriction is not necessary. However, if we are seeking analytic solutions of

equations (19.5), (20.4), (20.5), it appears we must restrict our­selves to linear compensation. It must be pointed out that non-linear compensation will allow better control over the optical properties of zoom lenses. This is because if the separation between a particular

pair of elements is written as a power series in the zoom parameter (§17) then the coefficients of this series may be regarded as degrees of freedom to be allocated by the designer. The usual degrees of

freedom are the powers and the initial separations of the elements (§15). Allowing non-linear compensation, even of only a quadratic nature, greatly increases the number of degrees of freedom available.

The paraxial theory and design of zoom lenses may be con­sidered independently of the aberrational theory and design if the coefficients in the series for the separations are known numbers. In

optical compensation (§14) all coefficients are zero except those of the linear terms in the zoom parameter. The coefficients of the

linear terms in the separations are chosen so that each separation varies at the same rate.

The final paraxial coefficients can be written as polynomials in the zoom parameter (§18). From these polynomials conditions can be obtained which, if satisfied, ensure that the system has the desired

properties. It is found that the height of the final image can be

made equal to the initial ideal image height at n points in the zoom

ix

range (§§17,18)« n is the number of elements in the zoom part (z) of the zoom lens (§15). The variation of the image height can be con­trolled by a judicious choice of these n points which are called the points of full compensation.

The equations obtained for the paraxial design of zoom

lenses (§20) can only be solved in certain cases. If we restrict our­

selves to the consideration of thin elements (§22) we find that the

design parameters (powers and initial separations) can be allocated

analytifally for the two and three element zoom lenses. The work of

Bergstein and Motz [8] indicates that it should be possible to solve the four element problem analytically.

For systems consisting of any number of thick or thin elements numerical solutions are possible (§25). The method of numerical solution utilizes the theory of §§9,18.

Once the powers and initial separations of the elements have been chosen using the theory of first order design, it remains only to

allocate the shapes (i.e. curvatures) of the elements. Extra axial powers (§40), if allowed, must also be allocated.

The aberrations of zoom lenses can be written in terms of what it is proposed to call fixed aperture aberration coefficients (§26). These are aberration coefficients calculated for an entrance

pupil with fixed diameter and position. The actual entrance pupil varies in both diameter and position as the elements move (§28). The

dependence of the third order fixed aperture aberration coefficients

on z can be obtained explicitly through the use of certain polynomials

related to the paraxial coefficient polynomials before each element

(§27). The actual aberration coefficients can then be written in

X

terms of the fixed aperture coefficients, the pupil magnification and

the entrance pupil position, all of which are functions of z, the zoom

parameter (§28). It is found that the actual aberration coefficients

of order three are ratios of polynomials in z.

These aberration coefficients depend on the curvatures of

the elements of the zoom lens. Once powers are allocated to the

elements, only one set of parameters, the shapes, has to be specified

to obtain the aberrations at a particular value of z. It is possible

to leave the shapes as variables (§30) and then the actual aberration

coefficients can be written in terms of the shapes of the elements and

z (§§28,31). Control of the aberrations is exercised through the

allocation of the shapes of the elements (§32).

Whichever method of control of aberrations is used, the

designer obtains certain equations, involving the shapes as variables,

which have to be solved. We find that in the case of the three

element zoom lens these equations can always be solved, either

analytifally (§36) or semi-analytically (§§37,38). In certain cases

the solutions are only approximate, but one must realize that thin

lens design is only intended as a starting point for further, more

realistic, work.

The calculations for §§36, 37 and 38 were carried out on a

programmable desk computer. However, if necessary, the work could be

done with a machine of lesser capabilities, but to perform all the

calculations by hand (using, say, a slide rule) would be extremely

time consuming, if at all possible.

'

xi

I note in §40 that all third order aberration coefficients can be controlled if first extra axial curvatures are allowed. The calculations in this case would be similar to those involved in §38.

xii

CONTENTS

Page

ACKNOWLEDGEMENTS i

STATEMENT OF ORIGINALITY ii

ABSTRACT iiiINTRODUCTION v

PART IABERRATION COEFFICIENTS AND THEIR DERIVATIVES

CHAPTER 1. ABERRATION THEORY OF OPTICAL SYSTEMS

1. Introduction. Aberration coefficients and their

derivatives 12. Notation. Paraxial optics 23. The aberration of a ray 64. Notation for aberration coefficients 7

5. The third order aberration coefficients 86. The principle of duality 9

CHAPTER 2. THEORY OF VARIATION OF DESIGN PARAMETERS

7. Equivalent variation of co-ordinates 118. The derivatives of the paraxial coefficients 129. The derivative of the augmented aberration 16

10. Derivatives of the third order aberration coefficients 1711. Derivatives of the fifth order aberration coefficients 18

12. The DERIVATIVES programme 20

xiii

Page

PART IITHE THEORY AND DESIGN OF ZOOM LENSES

CHAPTER 3. THE ZOOM LENS

13. Introduction. The theory and design of zoom lenses 2614. Properties of the zoom lens 27

15. Design considerations for zoom lenses 2916. Usual methods of design 3017. Proposed theory of zoom lenses 32

CHAPTER 4. FIRST ORDER THEORY OF ZOOM LENSESEMPLOYING LINEAR IMAGE COMPENSATION

18. The variation of the first order image height 3719. The magnification range of a zoom lens 3920. Control of the first order image height 4021. Choice of the points of full compensation 4122. The two element zoom lens 42

23. The three element zoom lens 4524. Two three element zoom lenses 48

25. The n element zoom lens 50

CHAPTER 5. THE ABERRATIONS OF ZOOM LENSES

26. Fixed aperture aberrations 5427. 7T, 7T, p, p as functions of the zoom parameter 5728. Effect of entrance pupil variation 5930. The third order aberration coefficients of a thin lens

in air 63

31. Fixed aperture aberrations as functions of shape 6532. Control of the third order aberrations 66

Page

CHAPTER 6. CONTROL OF ABERRATIONS IN A THREE ELEMENT ZOOM LENS

33. Use of fixed aperture aberration coefficients 69

34. Polynomials for a three element zoom lens 7035. Characteristics of the fixed aperture aberration

coefficients 7336. Control of spherical aberration 74

37. Control of spherical aberration and coma 78

38. Control of spherical aberration, coma and astigmatism 8339. Aberrations of systems 1 - 1 2 88

APPENDIX 140. Use of aspheric surfaces for controlling aberrations 95

APPENDIX 241. The programme DERIVATIVES 98

APPENDIX 342. The expansion of je' 141

APPENDIX 4

43. Table of Symbols 142

xiv

REFERENCES 143

PART IABERRATION COEFFICIENTS AND THEIR DERIVATIVES

CHAPTER 1

ABERRATION THEORY OF OPTICAL SYSTEMS

1. Introduction. Aberration coefficients and their derivatives

The theory of optical aberration coefficients and their

derivatives is contained in L. The purpose of chapters 1 and 2 of

this thesis is to give a summary of the basic results of L and to

present certain formulae not appearing in L.

In the notation of L (L § §1 and 7) the aberration of a ray

is

h; - m ' H, . ~k k

The notation is explained in §3 below. may be expanded in series

of ascending powers of suitable ray co-ordinates. These series will

be the sums of homogeneous polynomials of degree 3, 5, 7, . .., 2n+1.

The coefficients occurring in the polynomials of degree 2n+l are the thmonochromatic n order aberration coefficients. Formulae for the

primary and secondary aberration coefficients are given in L. The

formulae used in calculating the primary aberration coefficients are

reproduced below in §5.

The aberration coefficients of a system depend on the axial

separations of surfaces, the axial and extra-axial curvatures of

surfaces, the refractive indices of glasses, etc., of the system.

2

These quantities we call the parameters of the system. Any change in the constitution of the system we speak of as a 'variation of parameters'. When such variations are small we have as a first

approximation for the change of any aberration coefficients G'

bG' = Z 5t ,OT

the summation being extended over the varied parameters t .

In L equations for the first derivatives of the primary and some secondary aberration coefficients are given. The full set of

derivatives of the secondary coefficients appears in §11 below.

2. Notation. Paraxial optics

In Part II of this thesis many results obtained in L will be utilized. In order to familiarize the reader with the notation and basic results contained in L, I attempt, in this chapter and the next, to summarize Parts I and III of L. L deals, in the main, with mono­chromatic light propagating through symmetric optical systems. A symmetric optical system is a system which is axially symmetric and

which is invariant under reflection in any plane containing the axis

of symmetry.

The basic notation of L can be explained as follows. We

consider a single ray, with direction cosines (a,-ß,-7), intersecting a symmetric surface. A primed symbol (e.g. a') denotes the quantity

represented after refraction at the surface. We will use a set of co-ordinates (x,y,z) such that the origin coincides with the axial

point A (the pole) of the surface and the x-axis lies along the axis

3

of symmetry. The plane x = 0 is tangent to the surface at the pole A and is called the polar tangent plane. Any ray intersecting the surface may be specified by variables Y, Z, V, W where Y and Z are the

y- and z- co-ordinates of the point of intersection of the ray with the polar tangent plane. V and W are defined* by the relations

V := ß/a , W := 7/ß , (2.1)

iso that a = (1 + V2 + W2) 2.

Any symbol with a tilde underneath it will stand for both the y- and z-components of the quantity represented. For example

Y = (Y,Z) , V = (V,W) . (2.2)

The equation of a ray, before refraction, can then be written as

x = t and y = Y - Vt , (2.3)

where (x,y,z) are current co-ordinates on the ray and t is a variable parameter. If we use d' to denote the axial separation (or simply separation) between two consecutive surfaces then

Y = Y' - d'V'+V = V'+

Y, V are Y and V at the second surface of the pair.4 - +

The shape of a spherical surface (we shall generally be dealing with spherical surfaces) can be described by its radius of

curvature c := 1/r. The quantity _I, which is related to the angle

* The equations A := B and A =: B are defining relations for A and B respectively.

(2.4)

4

o f i n c i d e n c e o f a r a y , i s d e f i n e d by

I = c Y - V . ( 2 . 5 )

P a r a x i a l o p t i c s d e a l s w i t h r a y s which l i e i n an i n f i n i t e s i m a l

n e ig h b o u r h o o d o f t h e o p t i c a l a x i s . T h i s means t h a t t h e v a r i a b l e s

s p e c i f y i n g a p a r a x i a l r a y a r e o f i n f i n i t e s i m a l m a g n i tude . I f y , y and

i. a r e t h e p a r a x i a l e q u i v a l e n t s o f Y, V and I , i t i s shown i n L §§4 and

5 t h a t t h e b a s i c r e l a t i o n s o f p a r a x i a l o p t i c s a r e

A(Ni) = 0

cAy = A(jL + y) = 0

Combining ( 2 . 6 ) w i t h ( 2 . 4 ) g i v e s t h e r e c u r r e n c e r e l a t i o n s

[1 - (1 - k) c d ' ] y - kd ' v

(1 - k) cy + kv

( 2 . 6 )

y+ ( 2 . 7 )

v+

w here k, t h e r e f r a c t a n c e , i s d e f i n e d by k := N/N ' . The s u r f a c e s o f an

o p t i c a l sys tem can be l a b e l l e d 1, . . . , j , . . . , k. R e l a t i o n s ( 2 . 7 )

mus t t h e n h o ld f o r s u r f a c e s j and j+1 ( s i n c e y^ = y ^ ^ ) and so we see

t h a t y , ~j+1 a r e l i n e a r l y r e l a t e d t o y , v , f rom which i t f o l l o w s

t h a t y , Vj a r e l i n e a r l y r e l a t e d t o , v ^ . T h i s i s i n d i c a t e d by

w r i t i n g

\

ii>*l y pj+ y . v

qj ~l> • ( 2 . 8 )

l< L-i. II

VPJ y i + V . V .qj

7

The c o e f f i c i e n t s y .PJ

e t c . i n ( 2 . 8 ) a r e c a l l e d t h e ' c a n o n i c a l 1 p a r a x i a l

c o e f f i c i e n t s o f t h e sys te m . The te rm 1 c a n o n i c a l 1 r e f e r s t o t h e

i n i t i a l r a y c o - o r d i n a t e s used t o p r e s c r i b e r a y s e n t e r i n g t h e sy s te m

(L §§3,6). Canonical co-ordinates are a special case of the more general 'paracanonical' co-ordinates (L §12). For paracanonical co-ordinates the relations equivalent to (2.8) are

5

j

yaj * + ybj £

V . S + V, . t aj ~~ bj ~(2.9)

where s_, t are the paraxial equivalents of the paracanonical co-ordinates

S = cr Y + cr) •

T = T Y + t V

The choice of cr, cr, t , t determines the type of paracanonical co-ordinates. Generally subscripts p or q indicate canonical co-ordinates while subscripts a or b indicate paracanonical co-ordinates.

(2.10)

Although the paraxial variables are infinitesimal, the canonical (or paracanonical) paraxial coefficients are finite. For example, with canonical co-ordinates the initial paraxial coefficientsare, from (2.8), y = 1 , y =0, v =0, vPi qi Pi qi 1 .

Consider two arbitrary paraxial rays, one of which is meridional (distinguished by a bar). Let

A := N(y v - v y) . ( 2 . 11)

As the two rays propagate through the system A is invariant. This

invariance is demonstrated in L §5. A is called the paraxial invariant. This invariance is of great importance in the theory of

aberration coefficients.

6

3. The aberration of a ray

It is shown in L §5 that paraxial optics is stigmatic, that

is, all rays issuing from a particular object point intersect at onethimage point. If h^ is the initial object height then the j object

height is, according to L (5.101),

~ j = m j * i •(3.1)

where the paraxial magnification associated with the conjugate object

planes is

m. = N v /N. v J 1 01 j Oj (3.2)

The subscript 0 indicates that v ^ , v _. are angles associated with an

axial object point. If we consider non-paraxial rays, a ray from a

particular object point will not, in general, pass through the ideal

image point. Instead it will pass through some other point whose

co-ordinates H. ^ m. H , where H„ represents the initial co-ordinatesJ ~1 ^1

of a non-paraxial ray. The aberration of a ray, for a given object

height is

Hj - m H] (3.3)

It is shown in L §5 that

h = Nj voj *j ’ (3.4)

where is given by (3.1). Thus, from (3.3) we have, since h^ = ,

that

Sj h . + e . (3.5)

If (3.5) is multiplied by N vQ then we may write

7

A A + i j ’(3.6)

where A := N. v_ . H., e. = N. v . e.. e. is called the 'augmentedJ Oj j Oj

aberration' and N v ^ is called the 'augmenting factor'. Since A is

invariant it follows that

Ae. = AA. . (3.7)

The symbol A is defined in L §2. AX(= X' - X) denotes the change in a

quantity X, associated with a ray, due to refraction at a surface.

Summing over all surfaces we have

kII = E AA (3.8)j = l J

Eq. (3.8) is the basic equation of the theory of aberrations

of symmetric optical systems as presented in L, and is used to obtain

formulae for the aberration coefficients of such systems (see L

§§8,9,10,11 and L chs.II and VIII).

4. Notation for aberration coefficients

The following notation will always be employed.+

(i) The third order coefficients are denoted by a,b,c;A,B,C as

follows

g(1)Boo = a , g(1)810 = b , s(1) S11 II o

i(1)800 = ä , I (1)s10 = b , I (1)S11 = c ,

p(1)Goo = A , P0)

1 0 = B , pO)Gn = C ,

G(1)Goo = Ä , G0)G10 = B , G(1)G11 = C .+ See Appendix 3.

8

(ii) The fifth order coefficients are denoted by s_ , s0, s„ . . . s,;\ £ 3 o

S.,S0,S . . . as follows I l 3 o

(2)g0Q = S1 ’

(2)810 = s2 . g(2)S11 = s3 , etc. ,

i(2)g00 = 7 1 ’ g(2)g10 = I2 . I (2)S11 = 73 ’ etc. ,

G(2)G oo ■ S1 ’G(2)G10 ■ s2 *

G(2)G11 " S3 ’ etc. ,

g(2)G oo = S1 ’ g (2)G10 ■ ¥2 > g(2)G11 " G3 • etc.

(iii) Intrinsic contributions will be represented by German

characters (here written X).

5. The third order aberration coefficients

The third order aberration coefficients are the sums of the

third order contributions. In terms of the intrinsic contributions of

order three these are (see L §10)

R0) = S0)&|UV ’.0) :pv '

Each contribution may be written as

(5.1)

g s a + gb

as usual. The intrinsic contributions may be written as (see L ch.IV)

a-a

= |N(1 - k) y i2 (i' - V ) ,a a a a Sb = qa - J n ,+-a 1 a

a=a = qa ,-a lb = qib ’

b-a = 2S a ’ = 2a + u) ,= b 1b=a = qb ,=a Sb = •

t For interpretation of N when using paracanonical co-ordinates, see L §12.

9

c = !(b - co) ,=a -a 1 cu = i hu ’=b =bc = qc ,=a -a

OilcrIIJ=l I OH (5.2)

where

w := cAO/N) , q : = i,/i . (5.3)b a

6. The principle of duality

Let u be any linear paraxial variable, and let u , be theparaxial coefficients associated with it. Then the 'dual' of u isadefined to be u, while the dual of u, shall be u . Buchdahl [2] b b adiscusses the duals of the various aberration coefficients. Specific­ally g^n . , is the dual of .J °n-V, n-pb &pVa

n-p (jl - V VConsider the coefficients of s_S| r] p and of V p-v n-ptT| t] P in the expansion of AA. The second combination of

variables arises from the first by the formal interchange of s with tand of S with T. The coefficients referred to are gv , and ~ ~ pVag^n , , i.e. g^n and its dual. We see that whatever function g^n &p-V,n-pb °pVa &pVais of the paraxial coefficients at the surface in question and those

/ \preceding it, g ' . must be the same function of the dual° °p-V,n-pbcoefficients. Thus we have the principle of duality:

'If any function of the paraxial and aberration coefficients vanishes, the dual function also vanishes.1

The principle of duality may be used to calculate 'b'

coefficients if a method for calculating 'a' coefficients is available.

If a formula is available for a particular 'a' coefficient, one

replaces each quantity in the formula by its dual. The formula then

10

g i v e s t h e d u a l o f t h e ' a ' c o e f f i c i e n t i n q u e s t i o n . T h i s w i l l be one

o f t h e ' b ' c o e f f i c i e n t s r e q u i r e d . I f a comput ing scheme i s a v a i l a b l e

f o r comput ing ' a ' c o e f f i c i e n t s , t o c a l c u l a t e 1b ' c o e f f i c i e n t s one

m e r e ly u s e s t h e d u a l s o f t h e o r i g i n a l d a t a . For example i f t h e

o r i g i n a l d a t a a r e t h e i n i t i a l p a r a x i a l c o e f f i c i e n t s , one r e p l a c e s yal

by y , v by v and v i c e v e r s a . The ' b ' c o e f f i c i e n t s a r e o b t a i n e d

i n what i s c a l l e d d u a l o r d e r . For example t h e u s u a l o r d e r f o r t h e ’ b '

s u r f a c e c o n t r i b u t i o n s o f o r d e r t h r e e i s (a, , a, , b , b , c , c ) . The=b =b =b =b =b =b

d u a l o r d e r i s ( c , , c, , b, , b, , a, , a, ) . Thus t h e fo rm u la which g i v e s =b =b =b =b =b =b

a i n t h e o r i g i n a l c a l c u l a t i o n g i v e s c when t h e d u a l c a l c u l a t i o n i s =a =b

p e r fo rm e d . The d u a l o r d e r f o r t h e f i f t h o r d e r s u r f a c e c o n t r i b u t i o n s

i s g i v e n i n L §11.

11

CHAPTER 2

THEORY OF VARIATION OF DESIGN PARAMETERS

7. E q u i v a l e n t v a r i a t i o n o f c o - o r d i n a t e s

The d e s i g n p a r a m e t e r s o f an o p t i c a l sys te m a r e t h e

s e p a r a t i o n s d' be tw een a d j a c e n t s u r f a c e s , t h e a x i a l c u r v a t u r e s c„ . o f J Oj

t h e s u r f a c e s and t h e r e f r a c t i v e i n d i c e s N. o f t h e media o f which t h eJ

s ys te m i s c o n s t r u c t e d . The e x t r a a x i a l c u r v a t u r e s ( s e e L §56) c -|j>

c 2 y e t c * ° f s u r f a c e s and t h e d i s p e r s i o n s o f g l a s s e s a r e o t h e r d e s i g n

p a r a m e t e r s .

Le t Tj be a d e s i g n p a r a m e t e r a t t h e j s u r f a c e o f a sys tem

o f s u r f a c e s . I f x , i s changed by an amount c)t . , y ' . , v ' » Y' and V(J J Oj Oj

w i l l change by d y ^ , ^ V0j* ^~j anc ^~j r e s Pect:i‘-v e -*-y» t h e i n i t i a l

c o - o r d i n a t e s b e i n g k e p t c o n s t a n t . We a sk what c h a n g e s , c)s, c)t, clS, c)T,

i n t h e i n i t i a l c o - o r d i n a t e s must be made t o p ro d u ce t h e same change i n

y ^ j , vq j > Y ' , Vj as t rie change i n p ro d u ce d . The r e q u i r e d changes

ÖS and c)T i n t h e p a r a c a n o n i c a l c o - o r d i n a t e s a r e c a l l e d t h e e q u i v a l e n t

v a r i a t i o n s o f t h e c o - o r d i n a t e s . These may be w r i t t e n as

öS = (T S + T T) Öt

ÖT = (r s + r t ) öt ~ V t ~ t

( 7 . 1 )

The F . . . r a r e f u n c t i o n s o f t h e p a r a c a n o n i c a l c o - o r d i n a t e s ( s e e L s t

§ 121) .

I t i s p o s s i b l e t o w r i t e , i n t e rm s o f t h e e q u i v a l e n t

v a r i a t i o n s , an e x p r e s s i o n f o r t h e change i n £ ' when t . i s c hanged . I f~ k j

we w r i t e

12

(S,T) (7.2)

then the derivative ( cie '/cE) ( cH/dx .) is given by equation L (121.6)k: ~ ~ JThis is very lengthy and so is not reproduced here.

8. The derivatives of the paraxial coefficients

In L §§125,135,146 it is shown that the algebraic equations

for the equivalent co-ordinate variations have the general form+

\dS ^'^bj ä'tbj ^

ÖTj abj ~j + cE St .J

ÖT ^ ' A aj ä'Iaj■ fcj 1 aaj - Öt j

. (8.1)

a, U are quantities such as, for example, ' Nv, V. When = d^ the

first term in each of (8.1) does not appear, i.e. we take <3/<3d . = 0

for these terms. From (8.1) we see that

dSc)t - abj(uaj £ + Ubj S> + 0(3)

ÖTÄ— = a .(u . J3 + u T) + 0(3)or . aj aj bj

(8.2)

Comparing (7.1) and (8.2) we have

“ a u , b a

a u a a

ab Ub

a uu a b

(8.3)

The anteprime in (8.1) indicates division by the paraxial invariantA = N(y v - v y ). a b a b

13

w h e r e t h e 0 , . . . , 0 a r e c o e f f i c i e n t s i n t h e p o w e r s e r i e s e x p a n s i o n s s t

o f t h e T . . . F o c c u r r i n g i n ( 7 . 1 ) ( s e e L 1 2 1 . 2 ) ) . s t

I n t h e p a r a x i a l r e g i o n

c)sdr 0 s + 0 t ,

s td tch s + 0 t ( 8 . 4 )

o r

d s

d scH

a b ( u a S + u b t ) ’

a fa u ,

d t

<3t_

a ( u S + u t ) , a a b( 8 . 5 )

a u a

T h u s we o b t a i n

&T.d s

y . ä— + y, .a i O T . b i

Ji > J

( a . y, . a j b i a , . y •) b j a i u j '

I n L §5 t h e n o t a t i o n

( f I g) = f g - f g , a b b a

w h e r e f a n d g a r e m o r e o r l e s s c o m p l e x s y m b o l s , i s i n t r o d u c e d . U s i n g

t h i s we h a v e

d y ,

S T = u j - 1 > J • ( 8 -6)

I n a s i m i l a r m a n n e r

dvC= ( a J b i ) u , i > J • ( 8 . 7 )

j

I n g e n e r a l t h e d e r i v a t i v e s o f may b e o b t a i n e d f r o m t h o s e f o r v ' by

w r i t i n g y . w h e r e v e r v ' a p p e a r s . I t i s t h e r e f o r e n e c e s s a r y t o

c o n s i d e r o n l y t h e d e r i v a t i v e s o f v ' .l

14

L §§125 ,1 35 ,146 g i v e s t h e form o f a , U f o r t = d, c^ , k.

The r e s u l t s a r e summarized i n T a b l e I . I n T a b l e I N ;= AN.

T a b l e I

X a U

d ' Nv V

c o s Ny Y

k ' N 'y I ' / k

We now p r o c e e d t o form h i g h e r d e r i v a t i v e s o f p a r a x i a l

c o e f f i c i e n t s . We have

dv.

and so

For i > j ,

j( a j I \ ) Uj ’ k > J

dx . dx .t J

( a . j v ) u . .

da

's v o. . V J u .O T i J 1 k j

du.and lj

( 8 . 8 )

so t h a t

d2 v;

d r . d r .i J

dv.l I -V / CX •

j y j( 8 . 9 )

However

d v 'bkd r .

u d v ' b i ak

u . d x . a i l( 8 . 10)

S u b s t i t u t i n g ( 8 . 1 0 ) i n t o ( 8 . 9 ) we o b t a i n

d2v,'kd x . dx .

i J( a . u . ) a v ; k u j

j ' V aT .f o r i > j . ( 8 . 11)

F u r t h e r m o r e , f o r Ü > i

15

ö^ bk u ö2v'bi _akÖt -Öt . u . Öt „Öt . ’

£ i ai £ i

and so, by a process similar to that above

a Vdr »ör . Öt .£ i j

(a,Iu.) &2vak Ujj1 i' ax.ax. u . ’£ l aifor Ü > i > j . (8.12)

If i > i , > .. s ~ s-1 —

d vkÖl . • • . ck . Öt .1 10 1,s 2 1

• > ^ > 1-j > then

-nS-1 ✓o Vaxk ax, 7 -1" •(8-13)1 2 is i l2 ai2

For t = d, a = sNv and U = V which means that if any sub­

script is repeated, i.e. if i = i thenXj X/i I

(a |u ) £ £

N(v|v) 0 ,

and so if any subscript is repeated the derivative is zero. A

similar situation occurs for t = c^.

The formulae above hold for mixed derivatives; care must

then be taken if any subscript is repeated. For example, in the first

order design of zoom lenses, it is necessary to calculate derivatives

such as

as+V,U)

(11)

(Hi)

öd. ... öd. öc1 in Ojs 1 J"\S+1 y o V ,

öd. ... öd. öc„.ödis

j < 1.

j > 1i2 °j 1t

as+Vad. ... 3d. ac0.ad. ^ ’ V t > J * S-t-i

1s-t 0J 1s-t-l ~'i

These are the three possibilities which occur when calculating the

derivative

16

Care must be taken to ensure that all possibilities have been accounted

for. In the present case the three derivatives (i)-(iii) cover all

possibilities. Using (8.13) these derivatives may be obtained.

9. The derivative of the augmented aberration

It is shown in L §§128,142,152 that in general,

de' de dE dAA= aj(Ujli) +a|57- + 57- (9.1)

Certain additions of subscripts etc., must be made on the right hand

side for different t .. These additions areJ

(i) t = d : _e takes a subscript (kj). This notation is defined

by G.. .. := G' - G.. The last term on the right(kj) k jmust be omitted since A is explicitly independent of

d..J

(ii) t = Cq : £ takes a subscript [kj]. This notation is defined

by Gr. . 1= G' - G '.[kj] k j

(iii) t = k : jz takes a subscript [kj] and an anteprime, and A an

anteprime.

Once the k derivatives have been calculated the derivatives

with respect to refractive indices are obtained as follows. We write

J = j+1. Then we have

17

Mi = Mi , dG* kj+idN T dk . dN _ dk . , , dN _ J j J j+1 J

k . J

Mld k . + k dG

j+i dkj+i

( 9 . 2 )

10. D e r i v a t i v e s o f t h e t h i r d o r d e r a b e r r a t i o n c o e f f i c i e n t s

From ( 9 . 1 ) we o b t a i n , by e q u a t i n g c o e f f i c i e n t s o f ,S| . . . T£,

t h e d e r i v a t i v e s o f t h e t h i r d o r d e r a b e r r a t i o n c o e f f i c i e n t s . U s u a l l y

o n l y t h e d e r i v a t i v e s o f ’ a ' c o e f f i c i e n t s a r e r e q u i r e d . These a r e

g i v e n by ( s e e L §§128 ,142 ,152)

dA‘

dT

da(30 - 0 ) A + 0 (A, + A + B ) +s t a t b a a o t

Ikd t

_ _ _ _ da0 A + 2 0 A + 0 (A + B ) +

s a s a t b a ot

dB'ak

dr

I kdr

db20 A + 2 0 B + 0 ( B + B + 2C ) +

s a s a t b a a ot

_ _ _ _ _ _ db0 (2A + B ) + (0 + 0 ) B + 0 ( B + 2C ) + ^ s a a s t a t b a ot

I kdt 0 - B- + ( e s + V c a + 9 t ( c b + c a ) + 5 7s a

I kdr

dc0 (B + C ) + 20 C + 0 C + -ss a a t a t b oi ( 10 . 1)

As i n §9 c e r t a i n m o d i f i c a t i o n s must be made f o r t h e

d i f f e r e n t t . These a r e e x a c t l y t h e same as i n §11. The fo rm u la e f o r

.0 )=adc.

, g ( 1 >=adk

a r e g i v e n i n L §§137 ,147.

18

11. Derivatives of the fifth order aberration coefficients

(a) The derivatives of the fifth order aberration coefficients

are not given in L. In the expression for these derivatives certain

coefficients, which occur in the expressions of F ... T , ares tpresent. These coefficients ag> ag ••• 7 , 7 are given by equation L

(122.5). The derivatives of the fifth order contributions to the

aberration coefficients also appear. These may be calculated by

taking the derivatives of equations L (11.3) and equations L (218.7).

(b) The derivatives of the fifth order !a' coefficients are then

given by (see (9.2) and L §§129,143,153)

1 akd'T (5es ' V S1a + et(Slb + Sla + S2a) + 3as Aa +

+ cl.(A + B ) +tv~a T V T ÖT

1 akes sia + 40S sia + (sib + S2a) 0t + as Aa +

+ (2a + a. ) A + a B + la^ u. ) 2-v t a. u t -s s t a t a or

öS2ak4es sia + 4es s2a + 0t(S2b + S2a + 2S3a + 2S4a} +

+ (2a + 3 ß ) A + ß A + (2a + a. + ß. ) B +s s a t a s t t a

_+ a (B + 2C ) + y t a a ot

19

c>S' _ _ _ _- ^ T ~ = 6 ( 4 S i + s 9 ) + (30 + G j S + 0 (S + 2 S + 2S. ) +oT s l a 2a s t 2a t 2b 3a 4a

+ ß A + (2a + 2ß + ß ) A + a B + ( a + 2a + ß ) B + s a s s t a s a s t t a

ds.+ 2a C +t a <3t

ä s ; ,3ak3 t ' 0 s s 2a + {30s + et) S3a + 0t(Sb + S3a + S5a) + 3 ?s Aa +

+ 7 . A + (a + y ) B + ( a + 2a J C + a C +t a s t a s t a t a dr

ösr3akck :(i2a + Sa5 + 2(0s + V Sa + VSb + Sa5 + \ +

+ (27 + 7 ) A + (a + 7 ) B + a C + 3a C + -r*s t a s t a s a t a or,

4akck 20 S_ + (30 + Q j S. + Q ( S . , + S. + 2S_ ) + 2ß A +s 2a s t 4a t 4b 4a 5a ' s a

+ (2ß s + ßt> Ba + V S + 2Ca) +4a

öS4ak

Öt0 (2S + S ) + 2 ( 0 + 0 ) S + 0 (S + 2S ) +s 2a 4a s t 4a t 4b 5a

+ S (2Aa + V + <S + 24> Ba + 24 Ca + 17

5akÖt 2 6 s (S3a + S4a> + 2 ( 0 s + V S5a + VSb + Sa + 4 S6a> +

+ 2S Aa + (S + 2S + V Ba + 1 S + (S + 2ßt + 2V S +

+ ß c +t a ck

20

ÖS' _ _ _ - - - -= 0„(2Sq + 2S,_ + Sc ) + (Q_ + 30J Sc_ + 0^(SCU + 4St J +s 3a 4a 5a" s t 5a t 5b 6a

+ y (2A + B ) + (ß + 7 + 2y ) B + ß C + (3ß + 2y ) C +s a a s s t a s a t t a

+ [5aör

öS6aköx 0 S + (0 + 30 ) S + 0 (S + S, ) + 7 B +s 5a s t 6a t 6b 6a s a

+ (7S + 27t) Ca + 7t Ca + ÖT6a

öS6akÖt Ös(S5a + S6a) + 4®t S6a + 9t S6b + ?s(Ba + Ca} + 3?t Ca +

Öt 0 1 .1)

Once again modifications, as in §9, must be made accordingto whether t is d, c^ or k.

12. The DERIVATIVES programme

A computer programme, written in FORTRAN, and calledDERIVATIVES, has been prepared to calculate the derivatives of the

third and fifth order monochromatic coefficients of orders one and two.The derivatives are calculated with respect to the parameters d , c^,k , N , c . and c at each surface of an optical system. The j J lJ 2jprogramme utilizes the results obtained in Part III of L and summarized

in §§7-11 above. A listing of programme DERIVATIVES is contained in

Appendix 2.

21

DERIVATIVES c o n s i s t s o f a MAIN programme and s e v e r a l sub-

programmes. These subprogrammes a r e l i s t e d below, a lo n g w i t h t h e i r

f u n c t i o n s .

( i ) COEFF. T h i s s u b r o u t i n e c a l c u l a t e s t h e ' a ' and 1b 1 a b e r r a t i o n

c o e f f i c i e n t s o f o r d e r s t h r e e and f i v e f o r a sys tem o f s p h e r i c a l

s u r f a c e s . COEFF i t s e l f c a l l s s e v e r a l subprogrammes.

(a) PARAX. T h i s s u b r o u t i n e u s e s t h e l i n e a r r e l a t i o n s o f

p a r a x i a l o p t i c s t o c a l c u l a t e t h e p a r a x i a l c o e f f i c i e n t s a t t h e

j*”*1 s u r f a c e g i v e n t h e p a r a x i a l c o e f f i c i e n t s a t t h e j - l ^ s u r f a c e .

t h(b) PRIM. The p r im a ry i n t r i n s i c c o n t r i b u t i o n s a t t h e j

s u r f a c e a r e c a l c u l a t e d i n PRIM u s i n g e q u a t i o n s ( 5 . 2 ) f o r t h e ' a '

c o e f f i c i e n t s . I n p u t t o PRIM i s t h e p a r a x i a l c o e f f i c i e n t s a t t h e

j t h s u r f a c e .

(c) SECIN. In t h i s s u b r o u t i n e t h e f i f t h o r d e r i n t r i n s i c

c o n t r i b u t i o n s t o t h e a b e r r a t i o n c o e f f i c i e n t s a r e c a l c u l a t e d .

The f o rm u la e used a r e t h o s e o f L §218. Only f o rm u la e f o r t h e

' a ' c o e f f i c i e n t s a r e used .

(d) EXTRA. Once t h e i n t r i n s i c c o n t r i b u t i o n s a r e c a l c u l a t e d , i t

i s n e c e s s a r y t o u se e q u a t i o n s L ( 1 1 .3 ) t o c a l c u l a t e t h e a c t u a l

c o n t r i b u t i o n s to t h e f i f t h o r d e r a b e r r a t i o n c o e f f i c i e n t s . The

e x t r a t e r m s needed a r e c a l c u l a t e d i n s u b r o u t i n e EXTRA.

Only t h e f o rm u la e f o r t h e ' a ' c o e f f i c i e n t s ( p a r a x i a l , t h i r d and f i f t h

o r d e r s ) a r e programmed. The ' b ' c o e f f i c i e n t s a r e c a l c u l a t e d by

e n t e r i n g d u a l d a t a i n t o t h e v a r i o u s s u b r o u t i n e s , a s e x p l a i n e d i n §6.

The r e s u l t s o f t h e c a l c u l a t i o n s o f COEFF a r e s t o r e d and r e t u r n e d t o

MAIN.

( i i ) COLOUR. T h i s s u b r o u t i n e u s e s e q u a t i o n s L ( 9 3 .2 ) and L

(9 8 .2 - 4) t o c a l c u l a t e t h e c h r o m a t i c p a r a x i a l a b e r r a t i o n c o e f f i c i e n t s

22

o f c h r o m a t i c o r d e r one and two. T h e r e i s an o p t i o n t o e x c lu d e a l l

c a l c u l a t i o n s r e l a t e d to c h ro m a t i c c o e f f i c i e n t s and t h e i r d e r i v a t i v e s .

( i i i ) THETA. E q u a t io n s ( 8 .3 ) a r e e x e c u t e d and v a l u e s ° f 0 g . . . 0^

a r e r e t u r n e d t o MAIN.

( i v ) PDERIV. D a ta t o PDERIV a r e 0 . . . 0 which have beens t

c a l c u l a t e d i n THETA. These a r e used i n PDERIV t o c a l c u l a t e t h e

d e r i v a t i v e s o f t h e p r im a ry monochromat ic c o e f f i c i e n t s u s i n g e q u a t i o n s

( 1 0 . 1 ) . The d e r i v a t i v e s o f t h e i n t r i n s i c c o n t r i b u t i o n s da / d r , e t c . ,=a

a r e c a l c u l a t e d i n MAIN and added t o t h e o u t p u t o f PDERIV.

(v) ALFA. T h i s c a l c u l a t e s t h e c o e f f i c i e n t s a • • • y m en t ioned i ns t

§11 and needed f o r c a l c u l a t i n g t h e d e r i v a t i v e s o f t h e f i f t h o r d e r

a b e r r a t i o n c o e f f i c i e n t s . E q u a t io n s L ( 1 2 2 .5 ) a r e used . The d e r i v ­

a t i v e s o f t h e p r im a ry i n t r i n s i c c o e f f i c i e n t s needed a r e added i n MAIN.

( v i ) SDERIV. T h i s e x e c u t e s e q u a t i o n s ( 1 1 . 1 ) e x c e p t f o r t h e d e r i v ­

a t i v e s o f t h e c o n t r i b u t i o n s t o t h e f i f t h o r d e r a b e r r a t i o n c o e f f i c i e n t s .

T h e s e a r e added i n MAIN.

( v i i ) BITS. The d e r i v a t i v e s o f t h e f i f t h o r d e r i n t r i n s i c

c o n t r i b u t i o n s a r e c a l c u l a t e d i n MAIN. To t h e s e we must add o t h e r

t e rm s c a l c u l a t e d by t a k i n g t h e d e r i v a t i v e s o f e q u a t i o n s L ( 1 1 . 3 ) .

T h i s i s done i n BITS and t h e r e s u l t s a r e r e t u r n e d t o MAIN.

( v i i i ) CDERIV. T h i s c a l c u l a t e s t h e d e r i v a t i v e s o f t h e c h r o m a t i c

p a r a x i a l a b e r r a t i o n c o e f f i c i e n t s o f c h r o m a t i c o r d e r s one and two w i t h

r e s p e c t t o s e p a r a t i o n s , c u r v a t u r e s , r e f r a c t i v e i n d i c e s and d i s p e r s i o n s .

T h i s s u b r o u t i n e c a l l e s s e v e r a l o t h e r s u b r o u t i n e s .

( a ) THETA1. T h i s c a l c u l a t e s t h e c o e f f i c i e n t s 0 . . . 0 g i v e n by1S 1 _

e q u a t i o n s L ( 1 6 4 .5 ) c o n t a i n e d i n P a r t I I o f L. The 0 . . . 0 a r eos 0

23

t h e 9 . . . 0 c a l c u l a t e d f o r monochromat ic l i g h t by s u b r o u t i n es t

THETA.

(b) COLDER. E q u a t i o n s L ( 1 6 6 .4 ) a r e e x e c u t e d . These can be

used f o r d e r i v a t i v e s w i t h r e s p e c t o f d . , c . , k . i f t h e c o r r e c tJ J J

s u r f a c e c o n t r i b u t i o n d e r i v a t i v e s a r e added i n CDERIV.

The d e r i v a t i v e s w i t h r e s p e c t t o r e f r a c t i v e i n d i c e s and d i s p e r s i o n s a r e

c a l c u l a t e d i n CDERIV.

( i x ) PRINT. When a l l d e r i v a t i v e s ( f o r c h r o m a t i c and monochromatic

a b e r r a t i o n c o e f f i c i e n t s ) have been c a l c u l a t e d t h e y a r e t r a n s f e r r e d t o

PRINT which c o n t r o l s t h e p r i n t i n g o f r e s u l t s .

The c a l c u l a t i o n o f d e r i v a t i v e s w i t h r e s p e c t t o r e f r a c t i v e

i n d i c e s i s p e r fo rm e d i n MAIN u s i n g r e f r a c t a n c e d e r i v a t i v e s as

e x p l a i n e d i n §9. The c a l c u l a t i o n o f t h e d e r i v a t i v e s o f t h e mono­

c h r o m a t i c a b e r r a t i o n c o e f f i c i e n t s w i t h r e s p e c t t o t h e e x t r a - a x i a l

c u r v a t u r e s c ^ , c2 j *-s pe r fo rm e d l a r g e l y i n MAIN u s i n g e q u a t i o n s L

( 1 5 7 .2 ) and L ( 1 6 1 . 1 ) . The c o e f f i c i e n t s a g • • • 7t needed i n

ÖS' , / dc e t c . , a r e c a l c u l a t e d u s i n g L ( 1 5 6 . 1 ) . These a r e t h e nl a k i j

u s e d as d a t a f o r SDERIV. The d e r i v a t i v e s ds / c)c , , e t c . , a r el a 1

c a l c u l a t e d i n MAIN u s i n g t h e f o l l o w i n g e q u a t i o n s f o r t h e d e r i v a t i v e s

o f t h e f i f t h o r d e r i n t r i n s i c c o n t r i b u t i o n s .

i k^ + [cQ Ya (7 - 8k) - v ' J + ^ k ^ (3 k + 1)

= J q k 1 + k2 [Cq Ya O - k) q + ( (3 - 4k) c Q y & - v ' } q] +

+ k 3 [kq + | ( 1 + k) q]

24

ö |= 25 k 1 + k2 [cQ y a ( l - k) q + {(6 - 7k)

+ k3 [ ( 4k + 1 ) q + (2k + 1) q]

äs= 252 k 1 + q k2 3 c o y a (1 " 5 + '

+ k ^ t O + k) q2 4- (4k + 1 ) qq + k q2 ]

ö s= q2 ^ + 5 k2 [c0 ya ( l - k) q + | { ( 5 -

+ \ k3 [ (3k + 1) q2 + 2kqq + (1 + k) q2 ]

= 5 3 k 1 + q2 k2 [c0 ya (1 - k) q + i ( ( 5 -

+ J q k 3 [(1 + k) q2 + 2kqq + (3k + 1 ) q2 ]

- 2l*2 k , V 2c0 ya ° ' k) q + ( (5 ’

+ 2q k3 [kq + (2k + 1) q]

^ 7 ^ = 2q3 k i + q2 ^ [ 2 cq y ^ ( l - k) q + {(5

+ 2qq k 3 [ (2k + 1) q + kq]

ös= 2q3 k 1 + q2 k2 [3 c 0 y a O - k) q + [ ( 4

+ q k 3 [kq2 + (4k + 1 ) qq + (k + 1) q2 ]

ös= 2q4 k + q k [c y (1 - k) q + [ (6

de . 1 2 0 a

c A y - v ' } q ] + 0 a a

5k) c y - v ' } q] + 0 a a

6k) c y - v ' } q] + 0 a a

6k) c y - v ' } q] + 0 a a

6k) c y - v ' } q] + 0 a a

“ 6k) c y - v ' } q] + 0 a a

- 5k) c y - v ' } q] + 0 a a

7k) c y - v ' } q] + 0 a a

+ qq2 [ (2k + 1) q + (4k + 1) q]

25

ös^ = I 3 4 + q k 2 [cQ y^( 1 - k) q + £ { ( 3 - 4k) - v ' } q] +

+ q3 k 3 [ | ( k + 1 ) q + k q]

öc. iq 5 k1 + -Jqq4 k 2 [c 0 y a ^7 " 8 k " v a^ + ^ q2q3 k 3 ( 3k + D

where

k , = N c . i ' y5 0 a a

k 0 = N y4 i 2 a a

k_ = N k y4 i 2 3 a a

( 12 . 1)

and 5 - V ya ’ q = i , / ib a

E q u a t i o n s ( 1 2 . 1 ) a r e d e r i v e d from e q u a t i o n L ( 1 5 6 .3 ) b u t a r e n o t g i v e n

i n L. The d e r i v a t i v e s o f t h e c o n t r i b u t i o n s t o t h e f i f t h o r d e r

a b e r r a t i o n c o e f f i c i e n t s a r e o b t a i n e d u s i n g e q u a t i o n s ( 1 2 . 1 ) and t h e

d e r i v a t i v e s o f e q u a t i o n s L ( 1 1 . 3 ) .

I n s t r u c t i o n s f o r t h e use o f DERIVATIVES a r e c o n t a i n e d i n t h e

l i s t i n g i n Appendix 2.

26

PART II

THE THEORY AND DESIGN OF ZOOM LENSES

CHAPTER 3

THE ZOOM LENS

13. Introduction. The theory and design of zoom lenses

In chapters 3-6 below we study the paraxial and third order

theory of zoom lenses and apply this theory to the problem of the

three element zoom lens.

A zoom lens is an optical system able to produce a corrected

image, with a variable magnification, in a fixed plane. This is

achieved by moving the constituent elements relative to each other and

the image plane.

The paraxial theory is based on the axial separation

derivatives of the final paraxial coefficients v/, and y . We canak akuse this approach because the derivatives of all orders of these

quantities can be written down in a simple form. This is not so for

the derivatives of the primary aberration coefficients so we therefore

consider formulae for the 'addition' of optical systems and consider

zoom lenses as systems of movable thin elements.

This approach gives the primary aberration coefficients as

functions of a simple zoom parameter. These functions are built up

from certain simple polynomials in the zoom parameter. These

polynomials, which are related to the intermediate paraxial

27

coefficients can be calculated quite easily, especially for the three

element zoom lens.

From the formulae for the primary aberration coefficients we

can obtain simultaneous equations which, in the case of the three element zoom lens, may be solved without recourse to optimization.All possible zoom lenses, with the characteristics demanded by the

designer, can therefore be obtained.

14. Properties of the zoom lens

A zoom lens is an optical system which produces, in a fixed

plane, a corrected image whose magnification is continuously variable in a chosen range. Conventionally this variation of magnification is produced by the movement of the elements of the system relative to one another. In this work 'element' will mean a single component of a system, whereas 'lens' will, in general, refer to the whole system. Lohmann [3] has described a new class of zoom lens which utilizes

lateral movements of non-rotationally symmetric elements, but in this work zoom lens will be taken to refer to the conventional type of

system.

Zoom lenses have been placed in one of two categories depending on how the first order behaviour of the image is controlled.

The ideal image produced by a zoom lens is not stationary, that is,

either the ideal image plane moves, or the image height varies (in a

fixed plane) as the zooming process occurs. The control of the image is called image compensation. There are two types of image compensation, namely

28

( i ) m e c h a n ic a l image c o m p e n s a t io n

( i i ) l i n e a r image c o m pe nsa t ion .

I n m e c h a n ic a l c o m p e n s a t io n t h e movements o f t h e e l e m en t s a r e

c o n t r o l l e d by a sy s tem o f cams i n such a way t h a t t h e e l e m e n t s move

n o n - l i n e a r l y w i t h r e s p e c t to each o t h e r , t h a t i s , t h e p o s i t i o n s o f t h e

e l e m e n t s a r e n o t l i n e a r l y r e l a t e d t o each o t h e r . A more d e s c r i p t i v e

name f o r t h i s t y p e o f image c o m p e n s a t io n would be n o n - l i n e a r image

c o m p e n s a t io n .

I n l i n e a r c o m p e n s a t io n t h e movements o f t h e e l e m e n t s a r e

l i n e a r l y r e l a t e d t o each o t h e r . Th is means t h a t i f one p a r t i c u l a r

e l e m e n t i s i n a p o s i t i o n c h a r a c t e r i z e d by a p a r a m e t e r z^ , t h e n t h e

p o s i t i o n o f a second e lem en t would be c h a r a c t e r i z e d by

Z2 = a 2 + b2 z l ’ ( 1 4 - D

where a^ and b^ a r e c o n s t a n t s . A p a r t i c u l a r l y s im p le c a s e o f l i n e a r

c o m p e n s a t io n i s c a l l e d o p t i c a l c o m p e n s a t io n . I n t h i s a l t e r n a t e

e l e m e n t s a r e f i x e d r e l a t i v e t o one a n o t h e r and move be tw een t h e

r e m a in i n g e l e m e n t s which a r e f i x e d r e l a t i v e t o t h e image p l a n e . I n

t h i s c a s e ( 1 4 .1 ) becomes

z 2 = a ± z l . ( 1 4 .2 )

I t i s l i n e a r c o m p e n s a t io n , and s p e c i f i c a l l y o p t i c a l compen­

s a t i o n , t h a t t h i s work d e a l s w i t h . I t must be em phas ized , however ,

t h a t t h e methods a r e c o m p l e t e l y g e n e r a l and may be a p p l i e d t o non­

l i n e a r c o m p e nsa t ion .

A c o m p le te zoom l e n s may be c o n s i d e r e d t o c o n s i s t o f two

p a r t s :

29

(i) That part whose elements move. This part produces the zoom

effect and we shall denote it by (z) . The terms zoom system or zoom

lens will generally refer to (z). The aberrations of (Z) will vary

during the zooming process. One of the designer's aims will be to

control this variation.

(ii) A part whose elements are fixed relative to each other. This

we denote by (K). (K) is designed to have aberrations which balance

the average values of the aberrations of (Z). The properties of (K)

are chosen so that the whole lens will have a focal length which

varies between two specified values. There will be an aperture stop

somewhere between (z) and (K) (or in (k) ) . This ensures that the

brightness in the final image is constant as zooming occurs. We will

not deal further with (k).

15. Design considerations for zoom lenses

In designing a zoom lens one must take into account several

factors other than those considered in designing lenses of fixed focal

length.

In an ordinary design one aims to produce an optical system

with a specified focal length and image distance and with a particular

state of correction of aberrations at the numerical aperture and field

angle for which the system is to work. Cost and the size of the lens

must be taken into account.

For a zoom lens the magnification, and hence the focal

length, is variable. Although the aperture of the stop is constant,

the entrance pupil aperture and the field angle vary. Taking these

30

variations into account, the designer must produce a lens with a

satisfactory state of correction of aberrations for all positions of

the movable elements. The ratio of the maximum magnification to the

minimum magnification must have a particular chosen value. The

maximum (or minimum) focal length must have a particular chosen value.

The change in the length of (z) must be as small as possible since a

large change in the length of (z) will increase the length of the whole lens. Cost must be taken into account and this will determine

the number of elements to be used and whether aspheric surfaces may be

employed.

16. Usual methods of design

Several approaches to first order design are described in

the literature. The first zoom lenses used mechanical compensation.

Yamaji [4] describes first order design by discussing several classes

of zoom lenses. The more primitive types employ two movable elements;

one to vary magnification and one to compensate for the resulting move­

ment of the image plane. If the first movement is a simple displace­

ment then it is necessary for the second movement to be quite

complicated. This necessitates use of cams to control the movements

of the elements. Wear in these cams causes deterioration of the

performance of the lens.

A small number of elements allows control over relatively

few aberrations. Each element added to control aberrations must be

controlled by a cam thus adding to the complexity of the lens. Apart

from difficulties in producing cams with sufficient wear reducing

properties, the problem with previous methods for design of

31

mechanically compensated systems is that there is no one design method

applicable for all problems. Yamaji also considers linear and optical

compensation, but better discussions of these are contained elsewhere.

Bergstein [5] and Bergstein and Motz [6,7,8] consider the

problem of optical compensation in general with emphasis on certain

specific cases, namely the two, three and four element zoom lenses.

t hWriting F for the focal length of the V element and X ,V ° V’X' for the V object and image distances measured from the respective

focal points, Bergstein uses the Newtonian image equation

X'V - f2/xV V V = 1 ... n ,

to produce a terminating continued fraction which expresses the final

image distance in terms of the focal lengths and separations of the

system. He shows that if n is the number of elements, the deviation

of the image plane from the ideal image is zero for n positions of the

elements. Bergstein produces a set of 'varifocal equations' which,

when solved, give the values of the focal lengths and separations

necessary to produce a lens with the chosen properties.

Jamieson [9] considers optical compensation but points out

that the proposed method of design is applicable to linear compen­

sation. The arrangement of the elements at a particular position

during the zoom process may be specified by a zoom parameter z.

Jamieson considers a ratio of polynomials in z (the numerator

polynomial being of degree n) for the image position. If this ratio

of polynomials is decomposed into a terminating continued fraction,

then, taking into account subsidiary equations involving the ratio of

the maximum and minimum magnifications, the possible first order

solutions of the problem are obtained.

32

Other discussions of this subject are presented by Pegis and

Peck [10], Wooters and Silvertooth [11] and Back and Löwen [12].

Discussions of the problem of control of aberrations are

given by Yamaji [4], Bergstein and Motz [13] and Jamieson [9].

Jamieson uses the aberration formulae of Wynne [14] and by a simple

change of variables shows that each aberration may be expressed as a

quadratic form in the shape factors

CH + C2i ’ 1 ... n ,

where li

'2 i

first curvature of the i lens

second curvature of the i lens

The coefficients of and depend only on the first order design

and the zoom parameter z. By setting each third order aberration equal

to a desired value at chosen values of z, it is possible to control the

third order performance of the lens. The points at which the

aberrations have the chosen values are called the points of full

compensation for the aberration in question. Jamieson applies this

method to the design of three and four element systems. The systems

of non-linear simultaneous equations obtained are solved using a least

squares optimization method with a computer.

17. Proposed theory of zoom lenses

The design of a zoom system entails the assignment of values

to the parameters describing the system. For the first order design

these parameters are the powers (or focal lengths) of the elements,

the initial separations of the elements and the initial distance of

33

t h e i d e a l image p l a n e from t h e l a s t e l e m e n t . For t h e c o n t r o l o f t h e

a b e r r a t i o n s t h e a v a i l a b l e p a r a m e t e r s a r e t h e sha pes o f t h e e l e m e n t s ,

t h e r e f r a c t i v e i n d i c e s o f g l a s s e s and any e x t r a - a x i a l c u r v a t u r e s

a l l o w e d .

The r a t e s o f change o f t h e s p a c i n g s o f t h e e l e m e n t s ( t h e

c o e f f i c i e n t s b^ i n ( 1 4 . 1 ) ) can a l s o be v a r i a b l e s f o r u se i n d e s i g n .

I f t h i s i s so t h e f i r s t and h i g h e r o r d e r d e s i g n p r o c e d u r e s c a n n o t be

c a r r i e d o u t i n d e p e n d e n t l y . A l though t h i s poses no problem i n computer

d e s i g n , i t does p r e s e n t d i f f i c u l t i e s when a n a l y t i c s o l u t i o n s a r e

s o u g h t . Because o f t h i s r a t e s o f change o f s p a c i n g s w i l l no t be

c o n s i d e r e d as v a r i a b l e s .

The v a r i a b l e s m en t ioned must s a t i s f y c e r t a i n c o n s t r a i n i n g

r e l a t i o n s . For example , i t ha s been s t a t e d t h a t t h e r a t i o o f t h e

maximum and minimum m a g n i f i c a t i o n s must be a chosen number. I f we

d e f i n e t h e r e l a t i v e m a g n i f i c a t i o n o p e r a t i n g r a n g e as

mR := JEai5- , ( 1 7 . 1 )m . min

where m and m . a r e t h e maximum and minimum m a g n i f i c a t i o n s max mm

r e s p e c t i v e l y , t h e n t h i s i s one c o n s t r a i n t . The p o i n t s a t which t h e

d i s p l a c e m e n t o f t h e i d e a l image p l a n e from t h e cho s en image i s z e r o

a r e c a l l e d t h e p o i n t s o f f u l l image s h i f t c o m p e n s a t io n , o r s im p ly t h e

p o i n t s o f f u l l c o m p e n s a t io n . T h e re a r e n o f t h e s e p o i n t s f o r a sy s tem

c o n s i s t i n g o f n e l e m e n t s . The zoom r a n g e i s t h e r a n g e o f v a l u e s o f z

which i s encompassed by t h e zoom p r o c e s s . The r e q u i r e m e n t t h a t a l l n

p o i n t s o f f u l l c o m p e n s a t io n l i e i n t h e zoom r a n g e g i v e s r i s e t o n

c o n s t r a i n t s . F i n a l l y , i t i s u s u a l l y r e q u i r e d t h a t t h e l e n g t h o f t h e

l e n s be a minimum. T h i s r e q u i r e s t h a t each s e p a r a t i o n be a mimimum.

From t h i s we o b t a i n a f u r t h e r n-1 c o n s t r a i n t s . The minimum l e n g t h

sy s te m i s t h u s d e s c r i b e d by 2n e q u a t i o n s i n 2n v a r i a b l e s (n powers ,

n-1 s e p a r a t i o n s and t h e i n i t i a l image d i s t a n c e ) .

34

The shape o f a t h i n l e n s i s

S ( 1 7 .2 )

S p e c i f i c a t i o n o f power and shape g i v e s t h e c u r v a t u r e s o f t h e t h i n l e n s .

These a r e

c s+i) <p = ( s - mC1 2 ( N - 1) ’ C2 2 (N-1) *

where <J> i s t h e power o f t h e l e n s and

<t> = (N-1) ( C] - c 2 ) ,

f o r a t h i n l e n s .

( 1 7 .3 )

( 1 7 .4 )

The r e f r a c t i v e i n d i c e s o f t h e e l e m e n t s o f a zoom l e n s can

a lw ays be c h o s e n so t h a t t h e P e t z v a l c u r v a t u r e i s z e ro o r some sm a l l

r e s i d u a l . T h i s l e a v e s o n l y t h e sha pes to be a l l o c a t e d ( i f we a r e

c o n s i d e r i n g e l e m e n t s c o n s i s t i n g o f s p h e r i c a l s u r f a c e s ) . T h e r e a r e n

sh a p es and so n c o n d i t i o n s may be s a t i s f i e d . For i n s t a n c e a

p a r t i c u l a r a b e r r a t i o n migh t be s e t t o a cho s en v a l u e a t n p o i n t s o r

t h e a b e r r a t i o n and i t s d e r i v a t i v e may be s e t e q u a l t o c h o s e n v a l u e s a t

n / 2 p o i n t s . O bv io u s ly t h e r e a r e many d i f f e r e n t c o n d i t i o n s t h e

d e s i g n e r m igh t w ish h i s sy s tem to s a t i s f y .

The t h e o r y o f zoom l e n s e s w i l l be d e s c r i b e d as f o l l o w s . The

c r i t e r i o n o f f i r s t o r d e r s t a b i l i t y w i l l be t h e v a r i a t i o n o f t h e i n t e r ­

s e c t i o n h e i g h t o f an ' a ' r a y w i t h t h e i n i t i a l i d e a l image p l a n e . Th i s

i s s i m p l e r t h a n c o n s i d e r i n g t h e v a r i a t i o n o f t h e p o s i t i o n o f t h e i d e a l

image p l a n e .

35

As we shall always be dealing with lens elements rather thant hsurfaces, a subscript i will denote the i element rather than the

thi surface as in L. A subscript n will denote the last element. A

primed symbol will denote the quantity represented after the element

in question. As in L intersection heights will be denoted by y and

ray angles by v. The n-1 initial separations between the elements

will be denoted by d_ , d , ..., d and the initial image distanceI Z n-1

will be denoted by d . The ideal image plane will mean the initialnideal image plane and the intersection height of a ray in this plane

will be simply y ' . In this notation the intersection height of an 'a'

ray with the ideal image plane is initially

y ' = y - d v' a an n an (17.5)

We shall use a zoom parameter z to specify the configuration

of the zoom lens at any particular time. If Z is the displacement of

a chosen lens from its initial position and if Z is the largestmaxallowed value of Z then

z := Z/Zmax (17.6)

All distances will be scaled in this manner, i.e. by division by ZmaxThese distances are said to be normalized and the resulting system is

called the normal system. All the theory contained herein relates to

the normal system.

g(z) will denote the quantity g when the system is in the

configuration specified by z. In general, (17.5) may thus be written

as

y'(z)a y (z)an d (z) v' (z) . n an (17.7)

36

For a system employing linear compensation we write

d.(z) = d. + a.z , i = 1, ..., n . (17.8)l l i

Since y (z) and v" (z) depend linearly on each of the d.(z) an an lt h(i = 1, n-1), both of these are (n-1) degree polynomials in z.t hBecause of this we find that y'(z) is an n degree polynomial in z.

cl

It thus has n zeros and by suitable choice of the parameters of the zoom lens we may constrain all of these zeros to lie in the zoom

range. This is the basis of the first order design.

37

CHAPTER 4

THE FIRST ORDER THEORY OF ZOOM LENSES

EMPLOYING LINEAR IMAGE COMPENSATION

18. The variation of first order image height

The intersection height of an 'a' ray with the ideal image

plane is given by (17.7) as

y^(z) = yan(z) - dn(z) 4 n (z) ' (18.1)

yan(z), v^ (z) are (n_l) degree polynomials in z and so

y (z)an

v' (z)an

n-1 / ^ y (z)z — j=0 J* V dzJ /z=0

n-1 / c^v' (z)E3=0 j! <^z' z=0

(18.2)

From (18.1) it follows that

y ' (z)an-1E —j=0 J

y (z)anSzJ z=0 - dn(z)

c) v' (z) anözj z=0 -

(18.3)

Write

/ djy (z) / anj: V äzj /z=0

d^v' (z)

V ^ z=0

X j)

.,(j)>

(18.4)

and in this notation (18.3) becomes

y'(a) an-1Ej=o

y(j) - d (z) v ,(j) an n an (18.5)

38

Once the quantities y ^ \ v ' ^ (j = 0, n-1) are knownan an VJy'(z) is determined. These quantities can be obtained as follows. We aconsider only v' (z). Since there are n-1 separations we have

öv' (z) n-1 öd,(z) öv' (z)an _ y l_______anöz öz öd.(z)i=1 i

(18.6)

where, from (17.8)öd.(z)l

öz a i ‘ Therefore

öv' (z) anöz

n-1zi=0

(18.7)

since d^ = d^(0) and = v^(0). Assuming linear compensation,

taking the derivative of (18.6) gives

or

c)2v ' (z) anöz2

n-2z£=1

n-1zi=i+1

öd.(z) öd (z) ö2v' (z)l_______£__________ an_____öz öz öd.(z)öd (z)i £

n-2z£=1

n-1zi=£+l

02v'anai ai öd.ödi £

This process may be continued to give

5k n (z)5zJ

n-jz n-j+1 n-1Z ... Z a. a.i2=i]+l ij=ij--|+1 L] ±2

ah'v _________an______öd öd^ ... öd^

a . X i .J

(18.8)

If (18.8) is written out in full it is seen that it is merely the

weighted sum of all separation derivatives of differential order j

These separation derivatives are readily calculated using (9.13),

while making sure that only air space separations are considered.

Given a paraxial ray trace through the zoom lens in its initial

39

configuration it is possible to write y'(z) as a polynomial with knowncl

coefficients.

19. The magnification range of a zoom lens

According to L(5.103) the magnification of a system of n

elements is

N_ v i_!_HiN ' v' n pn

(19.1)

and so m corresponds to the minimum value of v' (z). Therefore max pn

(v' )K pn max( v ' ) .V pn'min

(19.2)

The number r is defined by

r :v 'O)pnv' (0) pn

(19.3)

If the maximum value of v' (z) occurs at z = 1 thenpn

v" (1)pn (v' ) v pn max = R .r — v ' (0) pn (Vpn min

If the maximum value of v' (z) pn occurs at. z = 0 then

v' (1) pn (v ' ) .N pn m m 1r - v' (0) pn (v' ) v pn^max R *

(19.4a)

(19.4b)

To obtain a lens with a particular value of R we may have either r = R1or r = - .

40

From ( 1 8 . 4 ) , r i s g i v e n by

v ' (0) + v ' (1) + an an . v ' ( n -1 ) an

v ' ( 0 )an( 1 9 .5 )

20. C o n t r o l o f f i r s t o r d e r image h e i g h t

We a r e g o ing t o p l a c e c o n s t r a i n t s on t h e v a r i a b l e s b , d i nl i

such a way t h a t a l l n z e r o s o f y ' ( z ) a r e r e a l and l i e i n t h e i n t e r v a la

0 < z < 1 . To do t h i s we w r i t e

y ' ( z ) = 5 ( z - z ) ( z - z ) . . . (z - z ) . ( 2 0 .1 )a 1 2 n

5 i s a s c a l e f a c t o r chosen so t h a t t h e l e f t hand s i d e o f ( 2 0 .1 ) e q u a l s

i t s r i g h t hand s i d e . The numbers z . ( i = 1, . . . , n) a r e chosen by t h e

d e s i g n e r i n t h e r a n g e 0 < z^ <; 1. ( 2 0 .1 ) may be r e w r i t t e n as

n ,y ' ( z ) = s E ( - 1 ) K r . (n) z . ( 2 0 . 2 )

a k=0 n ‘ k

The f u n c t i o n y , (n) o f t h e z e r o s z i s d e f i n e d as t h e sum o f a l l c r o s s k l

o f k r o o t s , e . g . f o r n = 4

S-'s -!> v _' II

Z1 + Z2 + Z3 + 2 4

II—sC\1 Z1Z2 + Z1Z3 + Z1Z4 + Z2Z3 + Z2Z4 ■

73 (4) = Z1Z2 Z3 + Z1Z2Z4 + Z1Z3Z4 + Z2 Z3 Z4

74 (4) - Z1Z2Z3 Z4 • ( 2 0 .3 )

7 ^ (n ) i s d e f i n e d t o be u n i t y . ( 2 0 . 2 ) i s now t h e d e s i r e d

form o f y ' ( z ) w h i l s t ( 1 8 .5 ) i s t h e a c t u a l form. E q u a t in g ( 2 0 .2 ) t o a

( 1 8 . 5 ) we o b t a i n

41

y(0) - d v'(0) an n an 5 7 (n) n (20.4)

(j) d „'<■)> y - d v - a van n an n an (-1) 5 7n_j(n)

for j = 1, 2, n-1, while for j = n

6 = (-1)n+1 a v'(n-1}n an

(20.5)

(20. 6)

At z = 0 we have that y'(0) = 0 and so (n) = 0. (20.4)a nthen becomes

dn. , (0) 0 .

(20.5) and (20.4) constitute n constraints which must be

satisfied by the variables <K and d^. Equation (19.5) is one further

constraint. When the powers and separations are chosen in such a way

that (19.5), (20.4) and (20.5) are satisfied, the lens will have a

relative operating range of R and the variation of the image height

will have zeros at the chosen values of z.

If the conditions

d.(0) = (d.)m .n , i = 1, n-1 , (20.7)

are also imposed on the variables then a minimum length system will be

obtained.

21. Choice of the points of full compensation

The variation of the image height as given by (20.1) assumes

that the radius of the entrance pupil is constant. In general this is

42

not the case, and it will be shown later than the radius of the

entrance pupil is given by a polynomial of degree n. For the present,

however, we shall assume that the radius of the entrance pupil can be

approximated by a linear function of z. It is shown in §27 that the

pupil magnification range is the same as the image magnification

range and so if the radius of the entrance pupil is h^ when z = 0,

then the radius of the entrance pupil may be approximated by

h (z) = hQ [l + (1 - 1) z] . (21.1)

Using this approximation, the actual variation of the image height is

y'(z)ar5 (z-z )(z-z ) ... (z-z ) ____ I_____ £__________ n_

[r + (1 -r) z] (21.2)

If z ... z are chosen, initially, to be equally spaced,1 n(21.2) may be graphed (ignoring the scale factor — ) and from this

hograph it is easily seen how the compensation points should be shifted

to obtain balanced variation of the image height. The ideal result is

to have the same value for the image height variation at each extremum

of y'(z)•a

22. The two element zoom lens

Cruickshank [15] shows that for thin lenses (i.e. lenses of

zero thickness)

l

v . ai

y

is the power of the

ai+1 . thl

b . y . + v l ai

y . - d. v ai l

lens and d. isl

ai

• (2 2.1)ai

the separation between the

43

i and i+1 l e n s e s . These t r a n s f e r e q u a t i o n s may be used to o b t a i n

t h e c o e f f i c i e n t s i n t h e p o ly n o m i a l s f o r y^2 (z) and v ' 2 ( z ) . I f we

c o n s i d e r a l e n s w i t h o b j e c t a t i n f i n i t y , we may s e t y = 1 and v = 0.

R epea ted u se o f ( 2 2 . 1 ) g i v e s

(0)y a2 ■ 1 - d l

(1) = - a , ♦ ,y a2

, ( 0 )Va2 " *1 + *2

> ( l ) = - a , ♦,va2( 22 . 2 )

S u b s t i t u t i n g from ( 2 2 . 2 ) i n t o ( 2 0 . 4 ) , ( 2 0 .5 ) and (1 9 .5 )

g i v e s t h e f o l l o w i n g r e s u l t s . From ( 2 0 .4 )

(1 - d 1 b 1) - d2 ( ct)-| + 02 ' d l ^ = 0 • ( 2 2 .3 )

From ( 2 0 .5 )

a i 1 + d2 a i 1 ^2 " d2 ^ l + d2 " d l 1 ^2 = ” 5 7-j » ( 2 2 *^)

where 5 = a^ a^ 0^ 4>2 and 7-j = 7-j(2 )* From ( 1 9 .5 )

- 0 $2 = ( r - 1 ) (0 + <J>2 “ d i ^ * ( 2 2 .5 )

For a minimum l e n g t h sy s tem d^ = 0. I n p r a c t i c e we r e q u i r e d^ > 0,

f o r example, d^ = 0 . 1 .

I t i s p o s s i b l e t o s o l v e ( 2 2 . 3 - 5 ) d i r e c t l y f o r 0 , <t>2 and d2

The p rob lem becomes s i m p l e r i f we make t h e s u b s t i t u t i o n s

d , = s, + ~ + ~ 1 1 $1 02

s2 + 0

44

E q u a t i o n s ( 2 1 .2 ) become

. ( 0 )

y.( i )

- v s i }

- a ,

, , ( 0 )a2

- (1 )

- S] * , 4>2

- 3l 6, . ( 2 2 . 6 )

and s u b s t i t u t i o n i n t o ( 2 0 . 4 ) , ( 2 0 .5 ) and ( 1 9 .5 ) g i v e s

S1 S2 ■ V * 2

a ! ( s 2 ' V *1 * 2

s 1(1 - r ) + a 1

The minimum l e n g t h c o n d i t i o n g i v e s t h e f u r t h e r e q u a t i o n

1 , _Ls i + ~ + .1 6, * 2

where (d„) . i s t h e c h o s en minimum v a l 1 min

The s o l u t i o n o f e q u a t i o n s ( 2 1 . 7 - 1 0 ) i s

0 , ( 2 2 . 7 )

- 5 7 1 » ( 2 2 .8 )

0 . ( 2 2 .9 )

t h e r e q u a t i o n

li> •1 min( 2 2 .1 0 )

o f d^ ( f o r example d 1 = 0 .1 )

a / (r - 1)

a 1/ ( r - 1) - a 2 y^

[a 1 - a2 (r - 1) y^ + (r - I ) 2 ]

(r - 1) [ ( r - 1) d 1 - a 1 ]

(r - l ) [ a 1 - a2 (r - 1) y^ + (r - l ) 2 ]

[ ( r - 1) d 1 - a 1 ] [a1 - a£ (r - 1) y ^ ](22.11)

The v a l u e s o f t h e s e p a r a t i o n s d 1 and d 0 a r e d^ + (d - | )m^n an<^

1d 0 + s„ + — . I f we s u b s t i t u t e f o r s „ , 4). and t h e n t h e s o l u t i o n 2 2 $ 2 2 1 Z

45

to the minimum length two element zoom lens is

dl " <d1>min

cL = - a 7 +2 1(r - 1)(d1 - a2 di y^) + d.]

1 (r - 1) [(r - 1) d1 - a1 ]

(r - 1)02 [(r - 1) d1 - a1] [a1 - a2(r - 1) y^] ( 22 . 12)

where 0 = a.j - a2(r - 1) 7 + (r - l)2.

For a particular value of (d_) . , this is the only1 minsolution to the problem of the two element zoom lens.

23. The three element zoom lens

For the three element zoom lens, solution of equations

(19.5), (20.4) and (20.5) directly becomes rather involved. If we

make the changes of variables

..+-L3 d>3

, 1 , 1S2 + *7 ~2 3

, 1 , _L (23.1)

the problem becomes easier to solve. Using (22.1) it is found that

ya3 = - V *2 + S1 *1 *2 * * 3 + S1 S2 *1 *2

^ = a1 S2 *1 *2 + a2 S1 *1 *2 + al % *2/*3

46

(2)y a3 = a l a 2 *1 *2

, ( 0 )a3 = - *, 4>3 /0»2 + s 2 h *2 *3

. , ( 1 )a3 = ( a l S2 + a 2 S1) *1 ♦2 *3

^ ( 2 )a3 = a 1 a 2 *1 * 2 * 3 ’

a 3 0 1 *2 0^ . S u b s t i t u t i o n i n t o ( 2 0 . 4 ) ,

( 2 3 .2 )

( 1 9 . 5 ) g i v e s

S1 S35f + if ' S1 S2 s 3 = °

$ 0 (j)

a i + a 3 “ I T ' ( a l S2 S3 + a 2 s i s 3 + a 3 S1 S2 ) *1 *2 *3

- 5 7 /

( a i a 2 s 3 + a l a 3 S2 + a 2 a 3 V *1 *2 *3 = S h

1 - r(1 - r ) s -| + a ] a 2 *

( 2 3 .3 )

I n e q u a t i o n s ( 2 3 .3 ) 7 = 7 ^ ( 3 ) and 7^ = 7^ ( 3 ) . For t h e minimum l e n g t h

sy s tem we r e q u i r e

d 1 ■ (<V n ,in

d2 ( d 2 ) min *

For t h e moment we do no t c o n s i d e r t h e minimum l e n g t h sys tem .

I t w i l l be s e en t h a t i f we s u b s t i t u t e f o r 5 i n e q u a t i o n s ( 2 3 . 3 ) , 0

does n o t a p p e a r . The v a l u e o f ^ i s d e t e r m i n e d s o l e l y by t h e c h o ic e

o f ( 2 3 .3 ) i s a s e t o f f o u r e q u a t i o n s i n t h e f i v e v a r i a b l e s 0

47

<t>3 > s i> s 2 » s 3 * F ° r t i e moment t a k e t o have a f i x e d v a l u e .

E q u a t i o n s ( 2 3 . 3 ) can now be s o l v e d f o r <t> , $3 , s ^ , i n t e rm s o f S3 .

The s o l u t i o n o f ( 2 3 .3 ) f o r g e n e r a l a ^ , a^ and a^ can be w r i t t e n down,

bu t i t i s r a t h e r d i f f i c u l t t o o b t a i n . S u b s t i t u t i n g s p e c i f i c v a l u e s

f o r a ^ , a2 and a^ makes t h e p rob lem v e r y much s i m p l e r . For o p t i c a l

c o m p e n s a t i o n a^ = - 1 , a^ = 1 and a^ = -1 . The s o l u t i o n o f e q u a t i o n s

( 2 3 . 3 ) i s

s 2 = (S3 - 7 , ) +S3 ( S 3 ' h + 1}

(1 - r )

s2 - s 3 + 7 , .

s3 ’ S3 7 1 + ?2 +S3 ' 7 1 + 1

1 - r ( 2 3 .4 )

( 2 3 .5 )

(1 - r )(1 - r ) s s 2 - s2 + S1 - 1 ( 2 3 .6 )

s 3 ( s l s 2 *2 '( 2 3 . 7 )

b i s d e t e r m i n e d as f o l l o w s . The l a s t o f e q u a t i o n s ( 2 3 . 1 ) i s

a , 1 , J_d i = S 1 + + * 2 •

From t h i s , c h o o s in g a p a r t i c u l a r v a l u e f o r d^ , we have

(<*i - s <t>2 - r ( 2 3 .8 )

The minimum v a l u e o f d^ i s 1 .0 so a r e a l i s t i c c h o i c e would be

d^ = 1 .1 . The s e p a r a t i o n d2 i s o b t a i n e d from t h e second o f e q u a t i o n s

( 2 3 . 1 ) . I t i s

a - , 1 . 1d2 = S2 + T2 + ^

( 2 3 .9 )

48

We may obtain the system with d„ = (cL) . as follows.2 2 minCalculate for several values of s . By graphing versus s it ispossible to find where d = (d_) . (see figure 1).2 2 min

There will be several solutions for each choice of s ,

corresponding to the choice of sign of amd <t>. The different

solutions will be named according to the signs of cj and 0 . The

solution which takes the + sign for both an< $3? for example, will

be called the (+, +) system. Each solution must be investigated when searching for minimum length solutions.

24. Two three element zoom lenses

To obtain a particular value of R we may have either r = Ror r = ~. We consider both of these solutions. Jamieson [9] has Kconsidered solutions of the three element zoom lens when R = 3. We reproduce his results using the theory of §23.

(i) First consider the r = 3 solution. Jamieson has chosen the points of full compensation to be at z = 0, 0.42, 1. This gives

= 1.42 and 7 = 0.42.

It is found that values of s less than 1.5 give rise to

imaginary powers. Figure la shows a graph of d^ versus s for the

four possible solutions. From this it is seen that only the (+, +)

and (-, +) solutions are physically possible. Only the (-, +)

solution gives a minimum length solution. This occurs when s = 2.46.

If we consider d^ =0.1 rather than d^ = 0.0, s = 2.485. Choosing

1.1 we find that the only minimum length solution is

49

F ig u re 1. v e r s u s s^ f o r (a) r = 3 .0

(b) r = 0 .333.

50

<D0 d d d1 2 3 1 2 30.286426 -0.884442 0.700642 1.10 0.10 3.913

This is the same as the system obtained by Jamieson.

(ii) The other way of obtaining R = 3 is to put r = Jamieson haschosen the points of full compensation to be at z = 0, 0.58, 1 and so 71 = 1.58, 72 = 0. 58.

Values of s greater than -0.5 give rise to imaginary solutions. The four possible solutions are graphed versus s in figure lb. The only realizable solutions are the (+, +) and (+, -) solutions. Once again it is the alternating sign solution which has minimum length. This occurs at s = -1.23. If, instead of d^ = 0 we

choose a small positive value we find, for s = -1.21 that:

-0.344683 0.692099 -0.887040 1.10 0.083 -2.337344

The negative image distance indicates that the image is virtual. These results agree with those of Jamieson.

25. The n element zoom lens

The n element zoom lens is described by 2n equations in 2n

variables, and so, even for relatively small n, it soon becomes

difficult to solve the equations analytically. It is, however, possible to solve the equations numerically. For this purpose a

computer programme, utilizing damped least squares optimization, was

written. The programme minimizes a function of the form

51

2n . v£ (7i - 7 °° )i=1

(25.1)

The 7 are functions of the parameters of the zoom lens. The y .(°°) are

the required final values of these functions. Each of (20.4), (20.5)

and (19.5), as well as (20.7) is used to obtain the function y ^ and

the corresponding y (°°)

In the optimization process the derivatives of F with

respect to separations and curvatures are required. The y ^ are

functions of the paraxial coefficients and their derivatives with

respect to z. It is thus necessary to calculate the derivatives^7i

and ^— . Whether the y . depend on paraxial coefficients or the öc0

derivatives of paraxial coefficients it is necessary to evaluate

expressions such as

Ö7.

0s v 0s,öd öd^ • • • öd_ and öc öd. ... öd .0 i l,s 1

The former can be calculated directly using (8.13); with the latter

more care must be taken. As pointed out in §8 there are three

distinct cases to be considered,

akas+V(i) öd. ... öd. öc .Ls L1 °J

j < 11 .

We consider

öv'öd. öc

Li Oj 3d7~rT5d-'VVv£) "JLS 1

The result is

52

ÖS+V , öVöd. . . . ö d . öc .

xs L1 ° J

' N . ( y . / v . ) a jö d ± ...ad v

s

, ( 2 5 .2 )

f o r j > .

( ü )

>,s+l , s v a k

. . öd . öc öd.l 2 ° J l l

J > i

We c o n s i d e r

äd. . . . äd. 3c . 3d.i s i 2 Oj i ,

3d. . . . 3d. 3c Vai,1 i 0 Oj 1 1 1s 2

and t h e f i n a l r e s u l t i s

3S+V .öd. . . . öd. öc .öd. i i 0 Oj i ns 2

ö V . a i' N. ( v . / y . )

V 1, ^ 1 • - ddl , dc0j yajs 2 ( 2 5 .3 )

f o r j > i . The d e r i v a t i v e on t h e r i g h t o f ( 2 5 .3 ) can be c a l c u l a t e d

u s i n g ( 2 5 . 2 ) .

>,s+1 ö v'

( i i i )Sdi • • • Sdi r Sco j Sdi

s s - t s - t - 1. öd - V t > j > 1s - t - 1

Here we c o n s i d e r t h e e x p r e s s i o n

3S‘V3di . . . 3d. 3c 3d.

s s - t s - t - 1. . . öd

3s-t ‘ V .X ' N . ( v . / v . ) T—ll l, l 2 3d.

3di • • • d d l ^ O js s - t

X

1. . . ö d . v a .

s - t - 1 12 L2

The f i n a l r e s u l t o f t h i s i s

53

ÖS+V,öd ... öd öc .ödi i „ Oj i . .s s-t s-t-1

... öd N. (v. /v. ) X L1 L1 x2

SV.X 1

öd. ... öd. öc öd. ... öd. v .i i Oj i i ais 1 s-t-1 l l

(25.4)

for i > J >-t ~ J s-t-1

By repeated application of (25.4) we obtain an expression on the

right which resembles the left of (25.3). The left of (25.4) can then

be evaluated.

54

CHAPTER 5

THE ABERRATIONS OF ZOOM LENSES

26. F ixe d a p e r t u r e a b e r r a t i o n s

A b e r r a t i o n s o f zoom l e n s e s may be c o n s i d e r e d f i r s t l y f rom

t h e p o i n t o f view o f what i t i s p ro p o se d t o c a l l f i x e d a p e r t u r e

a b e r r a t i o n s . I n zoom l e n s e s t h e s t o p i s g e n e r a l l y p l a c e d somewhere

a f t e r t h e l a s t e lem en t o f (z) . As t h e e l e m e n t s o f (Z) a r e moved t h e

p u p i l m a g n i f i c a t i o n , and he nc e t h e d i a m e t e r o f t h e e n t r a n c e p u p i l ,

v a r i e s . At t h e same t im e t h e p o s i t i o n o f t h e e n t r a n c e p u p i l , r e l a t i v e

t o t h e f i r s t e l e m e n t , c h a nges . F ixed a p e r t u r e a b e r r a t i o n s i g n o r e t h e

c ha nges i n t h e p o s i t i o n and d i a m e t e r o f t h e e n t r a n c e p u p i l .

C o n s i d e r two o p t i c a l sys tem s p l a c e d w i t h r e s p e c t t o one

a n o t h e r i n such a way t h a t t h e y p o s s e s s a common a x i s o f symmetry.

Le t t h e f i n a l p a r a c a n o n i c a l c o e f f i c i e n t s o f t h e two sy s te m s be

y ' , y \ > • • • , A' , . . . ; and y ' , y ' , . . . , A' , . . . ; r e s p e c t i v e l y , pi ql pi p2 q2 p2

I n t h i s c h a p t e r s u b s c r i p t s ' p 1 and * q 1 r e f e r t o t h e p a r a ­

c a n o n i c a l c o - o r d i n a t e s o f component o p t i c a l sys tem s on t h e i r own.

S u b s c r i p t s ' a ' and 1b ' r e f e r t o p a r a c a n o n i c a l c o e f f i c i e n t s o f

c o m p o s i t e s y s te m s . L e t t h e s e p a r a t i o n be tw een sys te m s 1 and 2 be d ^ .

Le t t h e c a n o n i c a l c o e f f i c i e n t s o f sys te m 2, on i t s own, be <J2 ’

T2 ’ T2* A c c o rd in g t o L ( 4 8 .4 ) t h e c o e f f i c i e n t o f s p h e r i c a l

a b e r r a t i o n f o r t h e c o m p o s i t e sy s tem i s

A 'a Ap1 + W2 Ap2 + ^2 P2 (Aq2 + Ap2 + Bp2) +

3 / r ,+ pl (Aq2 + Bq2 + BP2 + CP2 } + ^2 p2 (Bq2 + + Cq2} +

+ P2 Cq2( 2 6 .1 )

55

The c o e f f i c i e n t s 7 and a r e

a2 y pl + (<t2 ' d l (J2 ) Vpl

t 2 y p2 + ( t 2 ‘ d l t2) v pl

• ( 2 6 . 2 )

I f we choose 7T = 1 and p^ = 0 t h e n by a n a lo g y w i t h ( 2 6 .1 )

we may w r i t e , f o r an n e le m en t sy s tem

A' = E 4 tt? A' . + TT. p . ( A ' . + A' . + B ' . )an 1 p i 1 1 q i p i p i

+ 7T: p^(A' . + B' . + B \ + C ' . ) +77. P3 ( B' . + C' . + C ' . ) + 1 1 q i q i p i p i 1 1 q i p i q i

+ p? C' . 1 q i ( 2 6 .3 )

where

(J. y ' . , + ( cr. - d o’.) v '1 a i -1 1 1-1 1 ai -1

Ti y a i - 1 + (Ti " d i -1 Ti^ Va i - 1

( 2 6 .4 )

d^ i s t h e s e p a r a t i o n be tw een e le m e n t s i -1 and i . o \ , o \ ,

a r e t h e p a r a c a n o n i c a l c o - o r d i n a t e s used t o o b t a i n t h e p a r a c a n o n i c a l

c o e f f i c i e n t s o f e lem en t i on i t s own. The e l e m e n t s o f t h e co m p o s i t e

s y s te m may be e i t h e r i n d i v i d u a l l e n s e s o r may be g roups o f l e n s e s . We

w i l l t a k e t h e e l e m e n t s t o be i n d i v i d u a l t h i n l e n s e s . I f we w r i t e

^ i = a i y b i - 1 + ((Ji ' d i - l (Ji ) Vb i - 1

Pi Ti yn i - 1 + (T i d i -1 Ti^ Vb i -1

, ( 2 6 .5 )

t h e t h i r d o r d e r a b e r r a t i o n c o e f f i c i e n t s o f t h e t o t a l sys tem may be

w r i t t e n as

56

A'an i= l

3 - * / 3 - 7 ,E -faj\ 7r. A ' . + TT? TT. p . ( A ' . + B ' . ) + 7 7 p. A ' . + l l p i l l l q i p i l l p i

+ TT: p . p. ( A ' . + B ' . ) + TT.TT,' p ? ( B ' . + C' . ) + l l l q i p i l l l q i p iq i pi '

+ 7r . P ^ P . ( B' . + C ' . ) + TT. p3 C ' . + P 3 p C ' l l i q i p i l l q i l l q i

B'i= lE < 2T: TT: A ' . + 7r. 7tE p . ( 2 A ' . + B ' . + tt? p? B ' , I l l p i l 1 1 q i p i l 1 qq i p i q i

+

+ TT: 7r . p . ( 2 A ' . + B ' . ) + TT. 7T. p . P . ( 2 A ' + B ' + B' + l l l p i p i l l l l q i q i p i

+ 2C' . ) + TT: p? B ' . + TT. p. p ? ( B ' . + 2 C ' . ) + p i l l p i l i l q i p i

+ T T . P . ( B \ + 2 C ' . ) + 2 p 2 p2 c ; l l l q i q i l l q i

C'i=1E . K ’ l Ap i + 7 ’I Pl (Aq i + V + ’I P! Aqq i pV q i

+

+ 1 * TT. p . Bp . + TT. TT. P . P l ( B ' . + B ' . ) + P ? p. Bq . +

+ 7 Tfp f Cp . + TT. p t P ? ( C p . + Cq l ) + p f p f Cq . I ,

c 'an E, K E V + E Pi Aq i + \ *1 Pi (Ap i + Bp i ) +i= l

73 7x+ TT. TT. P ^ ( B ' . + C ' . ) + TT. p . p ? ( B ' . + C ' . ) + 7 T P C ' + i i 1 p i p i i i i q i q i l p i

+ P . P . C ' . l l q i( 2 6 . 6 )

57

27. TT, TT, p, p as functions of the zoom parameter

The functions 77\, 77\, p^, P^ depend on the paracanonical

co-ordinates of the element i, and those of all elements preceding it.

They also depend upon the paraxial paracanonical coefficients after

element i. According to L (32.5), using the convention established

in §26, we have

y'. = Tr. y'. + p. y'. .ai l pi l qi \

v ' . = TT. v ' . + p. v ' . ai l pi l qi

y ' . = tt. y ' . + p. y ' . bi l pi l qi

v ' . = tt. y ' . + p. v ' . / bi l pi l qi

(27.1)

Substituting these relations in (26.4), (26.5) gives

(or. y ' . _ + [cr. - d cr. ] v ' . ) tt. , +l pi-1 l l-l l pi-1 l-l

+ (cr. y ' . , + [o'. - d. o'.] v ' . ) p.l qi-1 l l-l l qi-1 l-l

( t . y ' . , + [t . - d t . ] v ' . ) TT. . +l pi-1 l l-l l pi-1 l-l

+ (t . y'. + [t . - d t . ] v ' . ) p.l qi-1 l l-l l qi-1 l-l

(<Ti ypi-l + [ai ' di-l ^i1 vpi-l) Tl- +

+ (cr. y' . , + [o'. — d o' . ] v ' . ) p.l qi-1 l l-l l qi-1 l-l

(t v7 + (t - d T ] ) 7r ■}*V i ypi Ti l-l iJ pi-r ' i-1

+ ( t . y ' . , + [t . - d. T . ] v ' . ) p.i qi-1 l l-l l qi-1 l-l (27.2)

In a zoom lens, each separation d^ is a function of the zoom

parameter z. If we substitute d^(z) = d^ + a^ z whenever a

separation occurs, we find

58

\ = (bal 2 + Cal} V i + (bbi Z + cbi) Pi-1

p. = (g . z + h .) TT. + (g Z + h ) p. l ai ai l-l bi bi l-

7r. = (b . z + c .) TT. - + (b z + c ) p. i ai ai l-l bi bi l-l

Pi = (Sai 2 + hai} V i + (8bi 2 + hbi> pi-1 (27.3)

We wish to find 77\, 77\, p p ^ as power series in z. Theretti _are i-1 separations preceding the i element and so each of 7T^, 1[J\,

P i ’ P i a P°lynomial of degree i-1 at most. Write

i-1Z P j=0 ij

i-1 Z P j=o ij

i-1Z r z j=0

i-1Z r z j=0 1J

(27.4)

Substitution of these series into (27.3) gives, for example,

i-1Z p z j=0 1J

i i_2 i i_1= (bai 2 + Cai} ^ ^ + (bbi - + cb.) _Zq rM > J

Equating coefficients of z gives

P. . = p. . . c . + r . c + p. . . b . + r b . (27.5)ij l-l,J ai l-l,J bi i-l,J-1 ai l-l,J-l bi

This expression holds true for j =0, 1, ..., i-1 if we put

p = r = p = r = 0 . (27.6)pi-1,-1 i-1,-1 Pi-1,-1 i-1, -1 V '

In a similar manner we find

r = p h + r h + p g + r gij Pi-1,j ai i-1,j bi Pi-1,j-l ai i-1,j-l Sbi(27.7)

59

Once a g a i n t h i s h o l d s f o r j = 0 , 1, i - 1 i f ( 2 7 .6 ) i s t a k e n i n t o

a c c o u n t . The e x p r e s s i o n s f o r t h e b a r r e d c o e f f i c i e n t s a r e

= p c + r . , . c . + p , b . + r „ b .i j

1h

< i V« a i l - 1 , j b i * i - 1 , j -1 a i i - 1 j j -1 b i

i j

ii

H* 1 (_*. h . + r . h + p

a i i - 1 , J b i i - 1 , J -1 8a i i - 1 , j - 1 8b i

( 2 7 .8 )w i t h t h e u n d e r s t a n d i n g t h a t

p = r = p = r-1 i - 1 , - 1 * i - 1 , i -1 i - 1 , i - 1 0 . ( 2 7 . 9 )

From ( 2 7 . 5 ) , ( 2 7 .7 ) and ( 2 7 .8 ) i t i s p o s s i b l e , g i v e n t h e

p a r a c a n o n i c a l c o - o r d i n a t e s w i t h which t h e a b e r r a t i o n c o e f f i c i e n t s o f

each e le m en t have been c a l c u l a t e d , t o o b t a i n 7T., 7T., P . , P. asl i i if u n c t i o n s o f z.

28. E f f e c t o f e n t r a n c e p u p i l v a r i a t i o n

Once t h e f u l l s e t o f p o l y n o m i a l s 77 , 77\ , p^ , p_ ( i = 1, . . . , n)

a r e known, t h e f i x e d a p e r t u r e a b e r r a t i o n c o e f f i c i e n t s may be c a l c u l a t e d

t h r o u g h t h e u s e o f ( 2 6 . 3 ) and ( 2 6 .6 ) (we a r e as suming t h e a b e r r a t i o n

c o e f f i c i e n t s o f t h e i n d i v i d u a l e l e m e n t s a r e known). I f c a n o n i c a l

c o - o r d i n a t e s have been used i n c a l c u l a t i n g t h e a b e r r a t i o n c o e f f i c i e n t s

o f each e l e m e n t , t h e e n t r a n c e p u p i l f o r t h e whole sy s tem w i l l have u n i t

r a d i u s and w i l l be i n c o n t a c t w i th t h e f i r s t e le m e n t .

A c c o rd in g t o L ( 3 3 . 1 ) , t h e a c t u a l p o s i t i o n o f t h e e n t r a n c e

p u p i l w i l l be

P ( 2 8 .1 )

60

where d is the distance of the stop behind the i refracting surface.

In zoom lenses the stop is generally placed after the last movable

element and so

P v' d) 5 an(28.2)

where we have suppressed the dependence upon z of y , ..., v' , dbn an(= d(0) - z). We do this whenever no confusion is likely to be caused.

L (33.2) gives the radius of the entrance pupil. If e is the radius

of the stop, the radius of the entrance pupil will be

h (28.3)

Since y, , etc., are functions of z, it follows that p and h are bnfunctions of z.

The paraxial invariant must be constant during the zoom

process. For an object at infinity v = 0 and so the paraxial

invariant is

A = h v . (28.4)b 1

To keep A constant, v must be a function of z. For the system withblunit aperture and stop in contact with the first surface, we have,

taking co-ordinates in the object plane and the first tangential

plane

S

T(28.5)

for an object at infinity. For the actual system, which has aperture

61

h and whose stop is at a distance p from the first element,

*S h S\

(28.6)

Using L (32.1) to make the transformation from co-ordinates

(28.5) to co-ordinates (28.6) we obtain

*A'an h4 A'

*A'an Ah2(A' + p A' )an an

*B'an A2(B' + 4p A' + 2p2 A' )an an an

*C' = A2(C' + 2p A' + p2 A' )an an an an- Tv3 - - - o

*C' “ (C'n + P[BL + CL1 + 3P2 Kn + P K J <28'7>an n an an an an an

The are the actual aberrations of the zoom lens for an object atp.V Jinfinity.

29. A theorem concerning pupil magnification

We prove the following theorem.

'If, for a zoom lens, there is full image compensation at

both ends of the zoom range (i.e. at z = 0 and z = 1), then

the pupil magnification range is the same as the image

magnification range.'

The pupil magnification of a zoom lens is obtained from

(28.3). Written in full it is

62

h ( z )e yan(z)an

1 _________d ( z ) v ' ( z )

________ 1____________ __y (z ) - (d - a z) v ' (z)

an n an

Because we have f u l l c o m p e n s a t io n a t bo th ends o f t h e zoom

f o l l o w s t h a t a t z = 0

and a t z = 1

y ( °)anv ' ( 0 )an

y O )anv ' (1)an

d + a n n

From e q u a t i o n ( 2 8 .4 )

h (0) = __________1_________e y (0) - d v ' (0)an an

and

h( 1) ______________ 1_____________e y (1) - (d + a ) v ' (1) *an n an

T h i s means t h a t

h( 1)h ( 0 )

v anv 'an

( 0 )

(Tj1r

I f m(z) i s t h e image m a g n i f i c a t i o n , we a l s o have

d e f i n i t i o n o f r

m( 1) m(0)

v anv an

( 0 )

oTir

( 2 9 .1 )

r a n g e , i t

( 2 9 .2 )

( 2 9 .3 )

( 2 9 .4 )

, f rom t h e

( 2 9 .5 )

From these results we see that the theorem stated above is proved.

30. The third order aberration coefficients of a thin lens in air

When canonical co-ordinates are used, the third order

aberration coefficients of a thin lens in air are particularly simple.

Use of canonical co-ordinates does not imply any loss of generality.

As in [15] we use the shape factor S to specify the curvatures of

lenses. S is defined by

S (30.1)

The contribution to the coefficient of primary spherical

aberration from a single surface is, according to (5.2)

=a ■§N( 1 - k) y i2 (i' - v ) a a a a (30.2)

For canonical co-ordinates, applying this formula to a thin lens in

air gives

i 3 "C1

- 1N2

-Jn (n D (c2 ci] N +i) ’(30.3)

where N is the refractive index of the lens and c and c^ are the

curvatures of the first and second surfaces respectively. Using

(30.3) and substituting from (17.3), the coefficient of primary

spherical aberration is

64

A' = P = Pl+ ap2

. N + 2 2N(N - I)2 b

4(N + 1) 4N3 - 4N2 - N + 2 "N(N - 1) N(N - I)2

(30.4)

In a similar manner, using (5.2) and (30.3) the other

aberration coefficients are found to be

A 'P

r (N + 1) 2N + 1 4 N(N - 1) N (30.5)

B'Pf (N + 1) 2N + 1

2 ‘ N(N - 1) N (30.6)

B' = 0P

t (N + 1) 2 N (30.7)

C' = 0 . (30.8)P

Using the identities L (20.41-47) and the equation L (26.4), the ’q'

coefficients are found to be

A' b2 (N + sq 4 N(N - l)

A 'q 2N

B' = B' = C' = C'q q q q

(30.9)

(30.10)

0 (30.11)

65

31. F ixed a p e r t u r e a b e r r a t i o n c o e f f i c i e n t s as f u n c t i o n s o f shape

I f , i n c a l c u l a t i n g t h e a b e r r a t i o n c o e f f i c i e n t s a p p e a r i n g i n

( 2 6 . 3 ) , ( 2 6 . 6 ) , we u se c a n o n i c a l c o - o r d i n a t e s t h e n we may w r i t e t h e

t h i r d o r d e r f i x e d a p e r t u r e a b e r r a t i o n c o e f f i c i e n t s d i r e c t l y as

f u n c t i o n s o f t h e shapes o f t h e i n d i v i d u a l e l e m e n t s . S u b s t i t u t i n g from

( 3 0 . 4 - 1 1 ) i n ( 2 6 . 3 ) , ( 2 6 .6 ) we o b t a i n

A'an Z | tT4 ~ K ^ 2 + TT3 “ ■ K0 (IT “ + p) S +4 2 2

+ 7T2 j(7T2 ^ K3 + TTP0 K4 + p2 K )

A'ann r 3 - 03 *2Z -j TTJ TT S2 + 7T2 K2 (777T0 + | 7Tp + ~ TTp) S +

0 o — 02 — 0 r,— 0 — 0+ 7r 2 (IT2 7T — K3 + TTJTp J K5 + 7 ^ J K6 + ^ P 2 Ky)

_ _ ,2 _ _ _B ' = Z ^ 7 r 2 7T2 -7_ K S 2 + 7 n r — Ko (T77r0 + 7Tp + TTp) S + an i _ i I 4 1 4 2

_ * 2+ 7T0 (TTTT2 — K_ + 7T2 P0 K + 777TP0 K + 27Tpp K_ + 7Tp2 ) 4 3 7 6 /

c' = Z -I '/(^tt2 ~r k s2 + tt k (tnr20 + t p + thtp) s +an I o I o 20‘

+ “ (7T2 7T2 + T77T2 P0K7 + 772 7Tp0 + TT^p2 Kg +

+ 777TPP0 + 7T2 p2 “ K )

66

c;„ ■ E, i t v 2 + Vs + “ +t s +1=1~ 4>2 — m . 1

+ ir j (7 n r k 3 + 2 Kg + 37J7TP f K6 + TTPP Kg +

+ 7TP2 [K4 - Kg]) ( 3 1 . 1 )

■where

N + 2‘1 N(N - I ) 2

4(N + 1) N(N - 1)

3N + 2 / NN - 1

3N + 2

4N + 3 2N + 1

N + 1 ( 3 1 . 2 )

I n ( 3 1 . 1 ) a s u b s c r i p t i h a s been d ropped from e v e ry symbol,

We s e e t h a t each f i x e d a p e r t u r e a b e r r a t i o n c o e f f i c i e n t i s a

2 1 0q u a d r a t i c form i n each shape f a c t o r . The c o e f f i c i e n t s o f S^, S^, S.

a r e a l l p o l y n o m i a l s o f d e g r e e 4 ( i - 1) a t mos t . I f t h e r e f r a c t i v e

i n d i c e s have been c h o s en , we can , u s i n g ( 3 1 .1 ) and ( 2 8 . 7 ) , w r i t e down

t h e a c t u a l a b e r r a t i o n c o e f f i c i e n t s a s f u n c t i o n s o f t h e sha pes and z .

32. C o n t r o l o f t h e t h i r d o r d e r a b e r r a t i o n s

I n a zoom sys tem c o n s i s t i n g o f t h i n e l e m e n t s w i t h s p h e r i c a l

s u r f a c e s , t h e v a r i a b l e s a v a i l a b l e f o r c o n t r o l o f t h e t h i r d (and h i g h e r )

o r d e r a b e r r a t i o n s a r e t h e sha pes and r e f r a c t i v e i n d i c e s o f t h e t h i n

67

elements. It is always possible to choose the values of the refractive

indices in such a way that the Petzval curvature (i.e. *C' - ) isan aneither zero or some chosen small residual. This done, the only

variables available to control the remaining primary aberrations are

the shapes. Increasing the number of elements in a zoom lens will,

aside from allowing an increased magnification range, allow better

control of the aberrations.

It is necessary for each aberration of the zoom lens

((Z) + (k)) to be substantially corrected throughout the zoom range.

This may be achieved by requiring that the value of each aberration be

the same at several chosen points in the zoom range. System (K) is

then designed to balance this level of the aberration. (K) must be

sufficiently complex that the value of each of the primary aberrations

can be chosen independently.

The points for which a particular aberration has the same

value are called the points of full compensation for the aberration in

question. The number of these points, for all aberrations, depends on

the number of elements in the zoom lens (in (z)). If the levels of

aberrations are to be chosen by the designer then the total number of

points of full compensation for all aberrations will be n. Thus if

each of T aberrations is to be controlled at P (K = 1, ..., T) pointsKin the zoom range, we succeed if (see [9])

TZ P„ = n • (32.1)K=1 K

If the levels of each aberration are allowed to be variable, we

succeed if

68

TZ ( P - 1 ) = n (32.2)

It is possible to control the aberrations of (z) in other

ways. We might, for example, require that the first derivative, with

respect to z, of each aberration be zero at certain points in the zoom

range. Choice of these points would be dictated by experience.

Another possibility is that the first and higher derivatives (if

possible) of aberrations be made zero at one point in the zoom range.

Obviously many different conditions can be imposed. The best method

of control will only be determined through experience or a deeper

theoretical understanding of the aberrations of zoom lenses.

69

CHAPTER 6

CONTROL OF ABERRATIONS IN A THREE ELEMENT ZOOM LENS

33. Use of fixed aperture aberration coefficients

Equations (28.7), which give the actual aberration coef­ficients in terms of the fixed aperture aberration coefficients, may

be solved to give the fixed aperture coefficients. The results are

A'a A *a;h a (33.1)

A'a*A' _aAh2 *A' (33.2)

B'a*B'_aA2 M e *X' + 24 *A,Ah2 a a (33.3)

C'a*C' __aA2

2p - p2— § *A' + ^4 *A' Ah a h a (33.4)

C'ah2 — p —— _ — rkn'3 aA

3 p 2 — P J.. o (*B' + *C') + ~ T 4 *A' - - 4 *A" A2 a a Ah a h^ a (33.5)

The fixed aperture aberrations are known functions of the shapes and z. They may thus be used to obtain equations for the shapes. For example, for a three element zoom lens, there are three shape factors to be allocated. Choosing a value for *A' and threeclvalues of z, we have three equations involving the three shape factors.

By allowing *A', *A', etc., to be variable it is possible to control

more than one aberration.

70

34. Polynomials for a three element zoom lens

We obtain the functions 7T , T7\, p^, p^, (i = 1,2,3) assuming

canonical co-ordinates and optical compensation. Once these

polynomials are available it is possible to obtain the fixed aperture

aberration coefficients as functions of shape and z.

The initial separations between the elements will be denoted

by d and d^. For optical compensation the rates of change of these

separations are a. -1 and a. = 1. For canonical co-ordinates er = 1,1 ' 2

cr = 0, T = 0 and t = 1. Using these values we find, comparing (27.2)

and (27.3), that

Cai y'ypi-l - di-1 v'pi-1

Cbi = Yqi-1 - di-l v *qi-1

b . ai = "ai-1 Vpi-1

Vbi = ”ai-l Vqi-1

h . ai = Vpi-1

hbi = Vqi-1

8ai = gbi = 0 and i = 2,3

For canonical co-ordinates the final paraxial coefficients

are, using (22.1)

1

1 0 (34.2)

71

If we agree that y _ etc. are the paraxial coefficients aObefore the first element, then for canonical co-ordinates ir , 7T j ,

are given by (26.4) and (26.5). We find TT.J = 1, 77\| = 0, p = 0,

p^ =1, and so we must have

P10 = 1 ’ P10oiiouoII , r1Q = 1 . (34.3)

We may now use (27.5), (27.7) and (27.8) to (

With the aid of (33.1-3) it is found that

calculate t etc.

P20 0 - d1 ♦,) ;20■ -4

P21 = ♦l P21 = 1IIoCMJ-l *1 720 = 1

r21 r21 = 0 ’ (34.4)

P30 — (1 - d1 ^>(1 - d2 * 2 )

P31 = <t> (1 " *2 " * 2 ^ ”

p32 = — cj> 01 2

P30 = -[dpi - d2 ( D 2 ) + d2)

P31 = *2(d1 - d2)

p32 = -♦2

r30 = (1 " d - j b - j ) 0 2 +

r31 =* 1 * 2

r32 0

r30 = 1 - di 4r31 =

* 2

r32 = 0 (34.5)

72

From t h e s e c o e f f i c i e n t s t h e r e q u i r e d p o l y n o m i a l s a r e

~ (1 - d ,

* 2= - d + z

1

p 2= ♦i *

1 CL = i ,

01 1) ( 1 - a 2 * 2 2 2 '

- 0 2 (1 - d 1 - <J>2 ] z “ <l>1 <t>2 z2 ,

tt3 - - ' V 1 - d 2 2 + d2 + ^2^1

P3 ■ O - dj ♦ , ) 0 2 + ^ 1 + 0 0 2 z

II

1 00

1 a O - d] »p + 0 2 2 ( 3 4 .6 )

Us ing t h e s e r e s u l t s i n e q u a t i o n s ( 2 7 . 1 ) we can w r i t e down

t h e f i n a l c a n o n i c a l p a r a x i a l c o e f f i c i e n t s o f t h e o p t i c a l l y compensa ted

t h r e e e l e m e n t zoom l e n s . They a r e

y a3

ii

Va3

II

*3+

P3

yb3

ii

u?

1

Vb3

ItTII

* 3+

p3 ( 3 4 .7 )

We s e e t h a t u s i n g c a n o n i c a l c o - o r d i n a t e s t h e p o l y n o m i a l s a s s o c i a t e d

w i t h t h e t h i r d e le m en t a r e j u s t t h e p a r a x i a l c o e f f i c i e n t s ( a s

f u n c t i o n s o f z) a t t h e f i r s t s u r f a c e o f t h e t h i r d e l e m e n t . The method

a p p l i e d above can be u sed t o f i n d t h e f i n a l p a r a x i a l c o e f f i c i e n t s f o r

any zoom l e n s . The method i s n o t r e s t r i c t e d t o t h e s p e c i a l c a s e o f

t h i n e l e m e n t s and can be used i n s t e a d o f t h e method d i s c u s s e d i n §18.

Given a f i r s t o r d e r d e s i g n , f o r example t h e one o b t a i n e d i n

§24 ( i ) , t h e p o l y n o m i a l s ( 3 4 .6 ) can be c a l c u l a t e d . These p o l y n o m i a l s ,

when s u b s t i t u t e d i n t o e q u a t i o n s ( 3 1 .1 ) g i v e t h e f i x e d a p e r t u r e

73

aberration coefficients as functions of the shapes and z. For even as

few as three elements the substitution is quite lengthy, numerous

multiplications of pairs of quadratic and quartic expressions being

necessary. To facilitate this lengthy, but essentially simple,

process a FORTRAN programme can be prepared. Such a programme carries

out the procedure set out in this section and then performs the

multiplications necessary to produce the coefficient polynomials

appearing in (31.1). Input to the programme consists of the results

of a first order design, namely the powers and separations, as well as

the diameter and position of the stop. The output is the coefficient

polynomials mentioned above, the pupil radius polynomial and the pupil

position polynomial. Once these polynomials are available it is

possible to proceed with the control of the third order aberrations.

35. Characteristics of the fixed aperture aberration coefficients

Certain important features of the fixed aperture aberrations

can be deduced from the constancy of 7T , 7T , p^, p . The actual

values of these when canonical co-ordinates are used give rise to

other features.

Because 7T , 7 , etc., are constant it immediately follows

that the coefficients of and are constant. This is important

because in any simultaneous equations obtained by sampling the fixed

aperture aberration coefficients, the coefficients of S , are the

same in each equation.

For canonical co-ordinates we have

7T = 1 , 7T1 = 0 , p1 = 0 , P1 1

74

A look at the equation for A' shows that the coefficient of S2 isan 1zero since 7T = 0. Similarly we find that the coefficients of S2 and

S„ are zero in the expressions for B' , C' and C' .1 an an an

These features simplify the solution of simultaneous

equations obtained from equations (31.1).

36. Control of spherical aberrations

As pointed out in §32, there are many possible ways to con­

trol aberrations. Here we consider what is perhaps the simplest

method. We do not control the amplitude of the variation of spherical

aberration, but merely require that the value of spherical aberration

be the same at certain chosen points in the zoom range. If we specify

a value for the coefficient of spherical aberration then we may

exercise control at three points in the zoom range. If we relax con­

trol on the level of this coefficient we may, theoretically, control

it at four points.

As a test system, we take the one studied by Jamieson and

reproduced in §24 (i). The first order design parameters of this

system are

*1 *2 *3 dl d20.286426 -0.884442 0.700642 1.100 0.102

The zoom lens will be designed to work at a numerical

aperture of f/8. The full field will be 5° at the maximum focal

length.

75

The focal length of a lens is, according to L (5.104)

f 1N' v' (36.1)n pn

where is calculated using canonical co-ordinates. Using (33.7) and

the data for the system under consideration, it is found that

v' = 0.18330 + 0.18923 z + 0.17748 z2 .P3

The maximum focal length of the lens is thus f = 5.45554 units. A

full field of 5° corresponds to v = 0.04363. With these values forblthe focal length and field angle it is found that a numerical aperture

of f/8 corresponds to an initial entrance pupil radius of 0.3195 units.

The stop is placed at an initial distance of 1.1 units [16], this

being the minimum realistic value. It is found, using (29.1), that

for z = 0 the pupil magnification is 1.93843 and so it follows that

the radius of the stop must be 0.16482 units. The paraxial invariant

is A = 0.0140.

We will duplicate the results obtained by Jamieson^ for his

system 6d (in [9]). Jamieson has chosen a value of 0.001 for his

coefficient of spherical aberration. Because of a difference in

formulae we aim for = 0.0005. Following Jamieson we choose the

points of full compensation for spherical aberration to be z = 0, 0.3,

1.0. With these points it is found that the simultaneous equations

for S.j, and S^, obtained from (33.1), are

0.012353 S2 - 0.025912 S.j - 0.15009 S| + 0.012668 S2 +

+ 0.047929 S| + 0.027218 S3 = 0.105225 ,

+ Systems in [9] will be referred to by J followed by the number given in [9], e.g. System 6d in [9] is J6d.

76

0.012353 - 0.025912 S - 0.240798 S| + 0.059652 S2 +

+ 0.135159 S2 + 0.060647 S3 = 0.219556 ,

0.012353 S2 - 0.025912 S - 0.607084 S| + 0.313535 S2 +

+ 1.192477 S2 + 0.050612 S3 = 2.893702 . (36.2a,b,c)

These equations may be solved analytically. The result of elimination of and S2 from (36.2c) using (36.2a,b) is a quartic equation in S3. The roots of this quartic are possible values of S3« The roots are

1.831789 -1.692984

1.919399 -1.645075 . (36.3)

Similarly elimination of and S3 gives a quartic in S . The roots of this, in corresponding order to (36.3) are

1.904647 -1.489476-0.709821 1.155313 . (36.4)

Substitution of a value of S3> and the corresponding value of S2> into (36.2a) gives a quadratic equation for S . Both roots are possible values of S . Because of this there are eight possible solutions to

equations (36.2a,b,c). The solutions occur in pairs (conjugate pairs) which differ only in the value of S . Both solutions in such a pair

have exactly the same primary spherical aberration at every point in

the zoom range. This occurs because the two shapes give rise to the same contribution to the fixed aperture spherical aberration. The

full set of solutions to equations (36.2a,b,c) is contained in

table II. The curvatures of the various solutions are given in table III. Solution number 4 in table III is the solution obtained by

77

Table II

S3 S2 S!

1.831789 1.904647 6.9383461.831789 1.904647 -4.8407181.919399 -1.489476 5.4735111.919399 -1.489476 -3.375883

-1.692984 -0.709821 4.061690-1.692984 -0.709821 -1.964062-1.645075 1.155313 5.264882-1.645075 1.155313 -3.167254

Table III

Lens 1 Lens 2 Lens 3

ci C2 ci C2 ci c2

System 1 1.584802 1.185524 -2.378871 -0.740894 1.382897 0.406201System 2 -0.766256 -1.166634 -2.378871 -0.740894 1.382897 0.406201System 3 1.292364 0.893086 0.400875 2.038851 1.425680 0.448985System 4 -0.474318 -0.873596 0.400875 2.038851 1.425680 0.448985System 5 1.010510 0.611232 -0.237652 1.400323 -0.338419 -1.315111System 6 -0.192464 -0.591741 -0.237652 1.400323 -0.338419 -1.315111System 7 1.250714 0.851436 -1.765176 -0.127199 -0.315020 -1.291715System 8 -0.432667 -0.831945 -1.765176 -0.127199 -0.315020 -1.291715

Jamieson. The advantage of the present method is that all possible

solutions to the particular problem are obtained.

If we set *A' = 0.0 at z = 0, 0.3 and 1.0, it is found thata3there are no real solutions.

78

Relaxing control on the level of *A' we may demand thataispherical aberration be equal at say z = 0, 0.3, 0.6, 1.0. The resulting simultaneous equations are

0.012353 - 0.025912 S

+ 0.027218 S3 -

0.012353 - 0.025912 S

+ 0.060647 S3 -

0.012353 - 0.025912 S

4- 0.100378 S3 -

0.012353 S* - 0.025912 S

+ 0.050612 S3 -

- 0.150090 + 0.012668 S2 + 0.047929 S| +

95.965814 *A' = 0.057243a3

- 0.240798 S| + 0.059652 S2 + 0.135159 +

374.542056 *A' = 0.032431a3- 0.367480 S| - 0.139017 S2 + 0.359830 +

1394.700100 *A\ = -0.134601 a3- 0.607084 S| + 0.313535 S2 + 1.192477 +

7773.230910 *A' = -0.991301 .a3

If *A'3 is eliminated from these equations we are left with three equations similar to (36.2a,b,c). These may be solved as before and we find that in this case there are no real solutions.

37. Control of spherical aberration and coma

If we allow the values of spherical aberration and coma, at the points of full compensation, as variables, then five conditions

may be satisfied. We may, for example, demand that spherical aberra­

tion and coma have three and two points of full compensation

respectively. Jamieson has chosen this approach and has used z = 0,

0.3, 1.0 and z = 0, 1 as the points of full compensation for spherical

aberration and coma respectively. The five equations controlling spherical aberration and coma at these points are

79

0.012353 S

0.012353 S

0.012353 S

-0.045235

-0.045235

* - 0.025912 S - 0.150090 + 0.012668 S2 + 0.047929 S| +

+ 0.027218 - 95.965814 *A' = 0.0572433 a32 - 0.025912 S - 0.240798 S| + 0.059652 S2 + 0.131590 S| +

+ 0.060647 S0 - 374.542056 *A' = 0.0324313 a32 - 0.025912 S - 0.607084 + 0.313535 S2 + 1.192477 S| +

+ 0.050612 S0 - 7773.230910 *A' = -0.9913013 a3

S + 0.241040 - 0.300620 S2 - 0.086920 S| - 0.188720 S3 -

- 669.729589 *A\ + 459.205441 *A'_ = -0.146348a3 a3

S] + 0.062494 S| - 0.595951 S2 - 0.966947 S| - 0.785230 S3 -

- 6297.566305 * A \ + 6616.527511 *A' = 1.050266 .a3 a3(37.la,b,c,d,e)

These equations are obtained by substituting z = 0, 0.3, 1.0 in (33.1)

and z = 0, 1 in (33.2).

These equations cannot be solved analytically, but we can,

as a first step eliminate *A'_ and *A' . The result is threea3 a3equations in S^, S2 and S3* The three equations are

0.012353 - 0.025912 S - 0.118842 S| - 0.003517 S2 + 0.017879 S| +

+ 0.015702 S3 = 0.065790

0.012353 S2 - 0.025912 S - 0.144378 S| + 0.008907 S2 + 0.033622 S| +

+ 0.026926 S3 = 0.070350

0.017877 S2 - 0.016266 S - 0.335070 S2 + 0.136318 S2 + 0.059043 S3 +

+ 0.090472 S3 = 0.215252 . (37.2a,b,c)

We use the fact that the coefficients of and are the

same in (37.2a,b) to develop a graphical method to find solutions.

80

Elimination of S. and between (37.2a,b) provides one quadratic

equation in and S^, namely

S| - 0.486529 S2 - (0.616502 S| + 0.439536 S3 - 0.178571) = 0

(37.3)52 will be real if

S2 + 0.712951 S3 - 0.193663 > 0 . (37.4)

Thus S^ is real when

S3 > 0.209862 or S3 ^ -0.922813 .

We seek solutions of equations (37.2a,b,c) in the following manner.

Choose three values of S3 in one of the regions above (i.e.

53 > 0.209862 or S3 £ -0.922813).

Now carry out the following steps for each value of S3*

(1) Solve (37.3). There are two solutions; call them S2+ and S^-.

For example, choosing S3 = 0.3, 1.0, 1.5 we obtain the results

exhibited in table IV.

Table IV

S3 s2+ V0.3 0.5039416 -0.0174131.0 1.211069 -0.7245401.5 1.631442 -1.144914

(2) Solution S2+ and solution S^- must be investigated. By way of

example we consider the S2+ solution. Values of S3 and the

corresponding values of S2+ are substituted into equation (37.2a).

Since this is quadratic in S there are again two solutions, S^+ and

81

S1

R e s u l t s o f s o l u t i o n o f ( 3 7 .2 a ) a r e c o n t a i n e d i n t a b l e V.

(3) We now u s e e q u a t i o n ( 3 7 . 2 c ) . From t h i s we o b t a i n t h e f u n c t i o n

Y = 0 .017877 S2 - 0 .017266 S - 0 .335070 S | + 0 .136318 S2 +

+ 0 .059043 + 0 .090472 S3 . ( 3 7 .5 )

Values o f and t h e c o r r e s p o n d i n g v a l u e s o f S2 and a r e s u b s t i t u t e d

i n t o t h i s and v a l u e s o f Y o b t a i n e d . The S^, S^+, S^+ s o l u t i o n g i v e s

v a l u e s o f Y which a r e shown i n t h e f i f t h column o f t a b l e V.

T a b l e V

S3 S2+ V VY (S.|+)

0 .3 0 .503942 3.964441 -1 .866813 0 .228576

1.0 1.211069 5 .310544 -3 .212916 0.235633

1.5 1.631442 6.276910 -4 .179282 0 .195094

(4) A g raph may now be drawn showing Y v e r s u s S^. A l t e r n a t i v e l y a

p a r a b o l a may be f i t t e d t h r o u g h t h e p o i n t s shown i n t a b l e V. The

q u a d r a t i c a p p r o x i m a t i o n t o Y i s

Y = - 0 .2 5 7 0 0 4 + 0 .344186 S3 + 0 .148450 . ( 3 7 .6 )

For ( 3 7 .2 c ) t o be s a t i s f i e d r e q u i r e s Y = 0 .2 1 5 2 5 2 . P u t t i n g

Y = 0.215252 i n ( 3 7 .6 ) g i v e s a q u a d r a t i c e q u a t i o n , t h e r o o t s o f which

g i v e v a l u e s o f S3 f o r which Y = 0 .215252 ( i f t h e q u a d r a t i c a p p ro x im a ­

t i o n i s good) . One o f t h e s e r o o t s may be i n a r e g i o n o f S3 which i s

f o r b i d d e n . I f t h i s i s so i t does n o t g i v e r i s e t o a s o l u t i o n o f t h e

o r i g i n a l e q u a t i o n s . For each r o o t i n t h e chosen r e g i o n o f S3 we may

c a l c u l a t e S2+ and t h e n S^+. S u b s t i t u t i o n o f t h e s e v a l u e s i n t o t h e

o r i g i n a l e q u a t i o n s (37.1 a , b , c , d , e ) g i v e s t h e c o r r e s p o n d i n g l e v e l s o f

82

spherical aberration and coma.

(5) A check should be carried out to see if the quadratic approxima­

tion holds. Substitution of an allowed root of (37.6), and the

corresponding values of and S^+, into (37.5) will show whether the

approximation gives a value of Y close to the required value. A

comparison of the values of the levels of spherical aberration

calculated from (37.1b,c) and a comparison of the values of the levels

of coma calculated from (37.1d.e) will show how good the approximation

is. If the approximation is not good, three points chosen in the

neighbourhood of the approximate solution should be used to repeat

steps (l)-(5).

In the > 0.209862 region there are four possible search

combinations which we label

S3 S2+ VS3 Y VS3 Y VS3 V Y

There are another four combinations in the < -0.922813 region. To

find all real solutions each of these combinations must be considered.

In the present case only two real solutions were found. These

solutions are shown in Table VI. The curvatures given by these

solutions are shown in Table VII.

83

Table VI

S0 S *A 0 *A3 2 1 a3 a3

0.222943 0.339383 3.746070 ioXLO -1.06 X 10-41.302119 1.467281 5.890979 3 . 2 2 x 1 0 ‘4 -4.10 X 10’4

Table VII

Lens 1 Lens 2 Lens 3

ci c2 CMoo

C1 c2

System 9 0.947499 0.548221 -1.096938 0.541037 0.597721 -0.379473System 10 1.375707 0.976429 -2.020673 -0.382696 1.124233 0.147538

The solutions shown are, of course, approximate. However it

would be possible to refine these solutions even further by repeating

steps (1)-(5) for points closer and closer to the actual solution.

System 9 agrees (to within 5fj0) with the solution given by

Jamieson (system J7).

38. Control of spherical aberration, coma and astigmatism

If we allow the levels of spherical aberration, coma and

astigmatism, at their points of full compensation, to be variables,

six conditions may be satisfied. That is, we may control each aberra­

tion at two points (we choose the end points) in the zoom range. The

six equations are obtained by putting z = 0, 1 in (33.1,2,3). Thus

84

0.012353 - 0.025912 S - 0.150090 + 0.012668 S2 + 0.047929 +

+ 0.027218 - 95.965814 *A' = 0.057243a3

0.012353 - 0.025912 - 0.607084 S| + 0.313535 S2 + 1.192477 S| +

+ 0.050612 S0 - 7773.230910 *A' = -0.9913013 a3

-0.045235 S1 + 0.241040 S| - 0.300620 S2 -.0.086920 S| - 0.188720 S3 -

- 699.729589 *A' + 459.205441 *A' = -0.146348a3 a3

-0.045235 S + 0.062494 S| - 0.595951 S2 - 0.966974 S| - 0.785230 S3 -

- 6297.566305 *A' + 6616.527511 *A' = 1.050266a3 a3

-0.774210 S| + 1.865800 S£ + 0.315260 S| + 1.189900 S3 -

- 5102.040785 *B' + 13393.087436 *A' -a3 a3

- 4394.682338 *A' = 0.009590a3

-0.012870 S2 + 0.238779 S2 + 1.568234 + 2.321167 S3 -

- 5102.040785 *5' + 21441.802544 *A' -a3 a3

- 11263.892694 * A % = -2.212227 (38.1 a,b,c,d,e,f)a3

Once again the equations controlling the aberrations cannot

be solved analytically. However, because of certain features of the

equations, a graphical method, similar to that employed in §37, may be

used. As in §37 the coefficients of and Sj in (38.1a,b) and the

coefficients of are the same in (38.1c,d). does not appear

explicitly in (38.1e,f), but we note that the coefficient of *B^3 is

the same in these equations. We may therefore eliminate and

between (38.1a,b), S, between (38.1c,d) and VfB' between (38.1e,f).1 a3This leaves three equations in S ., S0, *A' and *A' . These are2 3 a3 a3

85

0.456994 + 0.300867 S£ + 1.144548 S| + 0.0233940 S3

- 7677.265096 *A' = -1.048544a3

-0.178546 S| - 0.295331 S2 - 0.880054 - 0.596510 S3

- 5597.836716 *A' + 6157.322060 *A'a3 a3 1.196614

0.761340 - 1.627021 S2 + 1.252974 + 1.131267 S3 +

+ 8048.715108 *A' - 6869.210356 *A'a3 a3 -2.221817 .

(38.2a,b,c)

From (38.2b,c) we can eliminate *A' and this givesa3

0.206996 - 0.841592 S2 - 0.005082 S| + 0.112226 S3 +

+ 813.818272 *A' a3 -0.205631 (38.3)

,VA' may now be eliminated between (38.2a) and (38.3). The a3result of this is a single equation quadratic in S2 and S3* This is

- 5.106824 S2 + (0.733160 S3 + 0.723450 + 1.997956) = 0.(38.4)

Equation (38.4) has real roots if

Sf + 0.986755 S0 - 6.167761 < 0 .3 3 —

The last condition holds if S3 lies in the region

-3.025408 < S3 < 2.038654. We confine our search for solutions to

(38.la,b,c,d,e,f) to this region alone.

The procedure for finding solutions is as follows. First

choose three values of S3 in the allowed region and repeat each of the

following steps for these values.

(1) Substitute S3 into (38.4) and solve for Call the two

solutions S2+ and .

86

(2) Substitute and (either S^+ or S^-) into (38.3) and thenceobtain *A' .a3

(3) Now use the results obtained so far to solve (38.2c). This gives

(4) S is now calculated by substituting known values of S , S , *A' ,I 3 2 a3*A^2 into equation (38.1c).

(5) From (38.1a) we obtain the expression

Y = 0.012353 - 0.025912 S - 0.150090 S| + 0.012668 S2 +

+ 0.047929 + 0.027218 S_ - 95.965814 *A' . (38.5)3 3 a3

Using the values of the variables obtained in steps (1)-(4) a value ofY can be obtained.

(6) A graph of Y versus can be plotted for the chosen values of S . Alternatively we may fit a quadratic through the three points. From (38.1a) we see that we require Y = 0.057243. Putting this into (38.5) gives a quadratic equation the roots of which are the values of for which Y = 0.057243. Using these values of S , the values of the other

variables can be obtained by repeating steps (1)-(4).

As in §37 we must investigate several possible solutions. In

this case there are only two possible combinations, namely

s2+

S3

However these must be checked throughout the whole of the allowed

region. One combination may give rise to several solutions.

87

Taking = 0,1,2 we find that only the S^- combination gives real solutions. Table VIII contains values of the various variables and Y for three values of S .

Table VIII

S3 V *A'a3 *A'_a3 si Y

0.0 0.426923 1.42 X 10-4 -0.85 X IO“4 4.125632 0.0677771.0 0.802600 2.81 X 10’4 -2.31 X 10"4 1.685884 -0.0469102.0 2.176032 5.42 X IO-4 -7.25 X 10‘4 14.692406 1.796901

(7) The solution obtained by fitting a quadratic to the values for

and solving for Y = 0.057243 must be checked by substitution in (38.1a,b,c,d,e,f). If the approximation is not sufficiently accurate then the steps outlined above must be repeated for three points in the neighbourhood of the solution. In the example above (table VIII) we see that there must be one root near zero and another between = 1 and = 2. The quadratic fitted through = 0,1,2 is not satisfac­

tory but does give an indication of where the solutions lie.

After a few repetitions of steps (1)-(6), for each solution,

accurate results are obtained.

It is found that in all of the allowed region of there

are only two real solutions. These are set out in table IX and

table X.

Table IX

S S n S *A' *A' *B'3 2 1 a3 a3 a3

0.049787 0.435838 4.003185 1.42 X 10'4 -0.92 X IO"4 -2.2 X 10“41.792207 1.620422 5.872532 5.28 X 10"4 -4.98 X IO-4 -9.3 X 10“4

88

Table X

Lens 1 Lens 2 Lens 3

ci C2 ci C2 ci C2

System 11 0.998830 0.599552 -1.175934 0.462041 0.512660 -0.464034System 12 1.327705 0.928427 -2.130221 -0.492244 1.358230 0.381535

Although only two points of full compensation were specified

for spherical aberration, it is found that there are in fact three.

The extra one occurs at z = 0.5. It will be noticed that system 11 is similar to system 9 and system 12 is similar to system 10. Neither system 11 or 12 is the same as system J8. An analysis of system J8

using a computer programme for aberrations (see §39) indicates that J8 is in fact not a solution of the problem.

39. Aberrations of systems 1 - 1 2

The primary aberrations of systems 1 - 1 2 can be obtained by substituting values of S , and into equations (31.1). Through the use of equations (28.7) graphs of the primary aberrations can be drawn. In each of the three cases considered (i.e. in §§36,37,38)

there is more than one solution with the required properties. It is

possible to choose between alternative solution on the basis of the

behaviour of the primary aberrations, but it is more satisfactory to

take into account the secondary aberrations.

To do this a computer programme, which was written to calculate aberration coefficients (up to the ninth order), was

modified for use with zoom lenses. At a chosen number of points,

89

equally spaced in the zoom range, the programme calculates the

diameter and position of the entrance pupil and uses these to calculate

the actual aberration coefficients of orders one to nine. Using this

programme graphs of certain of the fifth order aberration coefficients

were obtained for systems 1 - 12.

We now consider each of the sets of solutions 1 - 8, 9 and

10 and 11 and 12, separately.

(i) Figure 2 shows the primary spherical aberration of the

conjugate pairs of systems 1 and 2, and 5 and 6. These are,

respectively, the 'worst' and 'best' of the eight solutions (considering

primary sperhical aberration as the only criterion of quality). On

the figures the primary aberrations are denoted by cr , cr , etc., and

the secondary aberrations by etc.

Graphs of primary spherical aberration for all eight systems

show that system 4, which is the solution given by Jamieson, is far

from being best.

Let us assume that for the zoom lens being designed only

spherical aberration is important (we have corrected only primary

spherical aberration). We can then choose between the different

systems on the basis of third and fifth order spherical aberration.

The fifth order spherical aberration for each of systems 1, 2, 5 and 6

is shown in figure 3. Average (or d.c.) levels of aberrations are not

important because we can design (K) to compensate for these. To

obtain a balance between third and fifth order spherical aberration

requires that the overall ( (z) + (K)) third order coefficient be

negative when the overall fifth order coefficient is positive and vice

versa, i.e. the third and fifth order coefficients of spherical

90

0.04

0.02 "

Figure 2. Primary spherical aberration for (I) systems 1 and 2, and(II) systems 5 and 6.

0.04

O. 02

0.02

Figure 3. Secondary spherical aberration for systems 1, 2, 5 and 6.

91

a b e r r a t i o n s h o u ld be 180° o u t o f p h a s e . I n none o f t h e sys tem s

o b t a i n e d i s t h i s t r u e o r even a p p r o x i m a t e l y t r u e and so we must f i n d

a sy s tem i n which n e i t h e r t h e t h i r d no r t h e f i f t h o r d e r c o e f f i c i e n t s

v a r y w i l d l y . System 6 i s one such sy s tem and sys te m 8 i s a n o t h e r ( n o t

show n) .

( i i ) F i g u r e 4 shows s p h e r i c a l a b e r r a t i o n and coma f o r sys tem s 9 and

10. A g l a n c e a t t h i s shows t h a t , f rom t h e v i e w p o i n t o f p r im a ry a b e r r a ­

t i o n s , t h e sy s tem o b t a i n e d by J a m ie so n , sys tem 9, i s t h e b e t t e r . The

f i f t h o r d e r s p h e r i c a l a b e r r a t i o n and coma o f t h e s e sys tem s a r e shown i n

f i g u r e 5. T h e s e c o n f i r m t h e view t h a t sy s tem 9 i s t h e b e t t e r o f t h e

two.

( i i i ) F i g u r e s 6a and b show s p h e r i c a l a b e r r a t i o n , coma and a s t i g m a t i s m

f o r sys te m s 11 and 12 r e s p e c t i v e l y . Because o f t h e s i m i l a r i t y be tween

sy s te m s 9 and 11 and 10 and 12 we e x p e c t t h a t sys tem 11 w i l l be t h e

b e t t e r o f t h e p a i r . C o n s i d e r a t i o n o f b o th t h e t h i r d and f i f t h o r d e r

a b e r r a t i o n s c o n f i r m s t h i s . The f i f t h o r d e r a b e r r a t i o n s a r e shown i n

f i g u r e s 7a and b.

92

0.02

0.04 -

F i g u r e 4. P r im a ry s p h e r i c a l a b e r r a t i o n (cr^) and coma (o^) f o r

sys tem 9 ( f u l l ) and sy s te m 10 ( b r o k e n ) .

0.02 V

F i g u r e 5. Seconda ry s p h e r i c a l a b e r r a t i o n (p^) and coma ( jjl )

sy s te m 9 ( f u l l ) and s y s te m 10 ( b r o k e n ) .

f o r

93

0.04

0.75

0.02 -

0.04

F ig u re 6. Prim ary s p h e r i c a l a b e r r a t i o n (cr^), coma (o^) and

a s t ig m a t ism (cr^) f o r (a) system 11, and (b) system 12.

94

xio3/A

F ig u re 7. Secondary s p h e r i c a l a b e r r a t i o n (m ) , coma ( ij ) and

a s t ig m a t is m ( | jl ^ q ” M»-j -j ) f or ( a ) system 11, and (b) system 12.

95

APPENDIX 1

40. Use o f a s p h e r i c s u r f a c e s f o r c o n t r o l l i n g a b e r r a t i o n s

The c o n t r i b u t i o n s t o t h e t h i r d o r d e r a b e r r a t i o n c o e f f i c i e n t s

f rom an a s p h e r i c s u r f a c e a r e , a c c o r d i n g t o L ( 6 7 .1 )

a o ,= a + a b °b + 2q2a=a =a =a =a

i a° - ~

= §a + qa c = =aO ~pc + q^a =a

b=aO _ «V

= b + 2qa - a c = =ao— ^3c + q a =a ( 4 0 .1 )

I n ( 4 0 .1 ) a = c,(AN) y4 and q = y. / y . The t e rm s such as ° a a r e t h e 1 a b a =a

c o n t r i b u t i o n s c a l c u l a t e d f o r a s p h e r i c a l s u r f a c e . These c o n t r i b u t i o n s

a r e g i v e n by e q u a t i o n s ( 5 . 2 ) and ( 5 . 3 ) . c^ i s t h e f i r s t e x t r a a x i a l

c u r v a t u r e o f t h e s u r f a c e i n q u e s t i o n . The meaning o f t h i s i s e x p l a i n e d

i n L §56. I t s u f f i c e s t o s a y h e r e t h a t c^ i s a measure o f t h e d e p a r ­

t u r e f rom s p h e r i c i t y o f a s u r f a c e .

For a s i n g l e t h i n l e n s i n a i r , u s i n g c a n o n i c a l c o - o r d i n a t e s ,

we have ( u s i n g p and q t o i n d i c a t e c a n o n i c a l c o - o r d i n a t e s ) y = 1 andP

y = 0 a t bo th s u r f a c e s . T h i s means t h a t q = 0 a t bo th s u r f a c e s and soq

t h e c o n t r i b u t i o n s t o a l l t h i r d o r d e r a b e r r a t i o n c o e f f i c i e n t s , e x c e p t

t h a t o f s p h e r i c a l a b e r r a t i o n , a r e t h e same as f o r s p h e r i c a l s u r f a c e s .

Let c ^ and c ^ be t h e f i r s t e x t r a a x i a l c u r v a t u r e s o f t h e f i r s t and

second s u r f a c e s r e s p e c t i v e l y . The c o e f f i c i e n t o f s p h e r i c a l a b e r r a t i o n

f o r t h e l e n s i s

A 'P

a + a pi p2

V - (N - l ) ( c - c )p 1 1 1 2

( 4 0 . 2 )

96

$ = (N - 1) (c - c-|2 t ie f^rst extra axial power of the thin

lens.

Equations (26.3, (26.6) give the fixed aperture aberrations

for a system of thin lenses and we find, substituting (40.2) for A'Pi

wherever this occurs, that the fixed aperture aberration coefficients

are

A'an

A'an

B'an

C'an

C'an

°A'annZ 7r40i=l

°A'ann 3-Z TT 7T6.

i=l

°B/an - Z 27T27T2$i=1

C' - Z TTTT2^an i=1

C - Z 777T3b1an . . 11 = 1(40.3)

The terms such as °A' are the fixed aperture aberrationancoefficients calculated for spherical surfaces, i.e. using equations

(31.1) . If equations (42.3) are used in (33.1), instead of equations

(31.1) , we see that three extra variables, occurring linearly, have

been introduced. Considering the levels of the third order aberrations

to be variables, this gives us ten degrees of freedom, i.e. three

shapes, three first extra axial powers and four aberrational levels.

Using these we can control the four third order aberrations (the five

Seidal aberrations except cr which is determined by our choice of

refractive indices) at ten points. For example we might require that

the number of points of full compensation for each aberration be

97

spherical aberration: 3 ,coma: 3 ,astigmatism: 2 ,distortion: 2

Substituting the values of z at the chosen points of full compensation, we obtain ten equations in ten unknowns. Because seven of these occur

linearly, and because of the characteristics remarked upon in §35, we

may use a process similar to that described in §38 to find solutions. Simplification occur because the coefficient of <t> behaves similarly

to the coefficient of S .

98

APPENDIX 2

41. The programme DERIVATIVES

A programme, called DERIVATIVES, was described in §12. This

programme, which calculates the derivatives, with respect to design

parameters, of chromatic paraxial and monochromatic third and fifth order aberration coefficients.

The theory used in the programme is contained in Part I of this thesis and is discussed in §12. Instructions for the use of the

programme are contained in the comments at the start of the programme itself. Comments throughout the programme explain the purpose of the different sections and subroutines. A listing of DERIVATIVES follows.

99

CD LU Z L U lO

Za

UJ• »QC

33 ZD O h-

</>•—<

OLU

OC*—>UJ LOh— LU 2 L O O 0 2UJ JC •—X L U < 0 o LULU Ol 2 LUQ O h h ►— c C ►— J X O < xQC J O • h—ttl >- L/> O • H- o >O J<z j*-L U c r l LU aCLU O j H co

Oa£LU <h-C Q < Q O O -a io o s i Urn* *—X LL U -X 2 2 < a : < 2 U J q ,—. < h-h- LU3T o < CO O o a o N O ii —► o H-lulu a: *—ID J * — 2 •— cO *— 2 O H O oOO ^ f» ►* - X | o h -C *< J < h-UU J —% H ,w p4 ►H > JLU h~ 1 LU a o — LUCCOLU J o QQ w M w pH sO h- h-

1 ac O lu*—Qo >- o a: c c o »—4 O r \ j< O * h- 00OOoOZD LO h— lO OC 0 0 < < h -< L U •«Ql CNJ H X O O h- ►2H-LUH- LOUJCLO' O l u x a . lo • — ►Cl. ► rH—1 ► X< o < —• X O —t UJ O '—LU o lu LUO X o — <c — —— > h—

h - Z > h" X < LOXoO*— LU r» H Q •* CO-O LO—t CD COQOCQC CO » t o h- LULU UU o Wf—4*"» LO ► al Cl h- ►OCUJKO LULU 2 m 0C O O Z X * o O qO « o - O Lual h- “ 5*—0 . 0 0 > 0 2 0 O 2 O h- O Q r4 ►—4 CO —* r«J OLU h— H* CMX lO < *— 7D*— *— D N i / ) X — IDO * * * X c o * * * O w ►o ► CO —4h- UJOC J H-Q ( - LUh- lU V h - «OaC m ^r—4 CQ —* CO — ► > ►CD

o: lu< < O Q < JOLUQLU<oOOCL m OL pH — lO «o — h- *— ►► LU*— >0C UJO h -—* 0 2 * — aloe o O *CL < ► ►CM h- L——4

J X a C X ►—Q. 2 ►— *— •—0<LUO UU r—4 ► (M — O —• h- CD—4< i- | < a: •—j L— DC UJCLh-OCOC o — o — LOO pH #• ► ►CD•—•— i l UJO<CD 2 • LULUQ.UJUJCL< H < H O ► ► — o > X ►X 3 1 < O h -h- o LUOcOOoOoOX w *—•—*— — O Q- —< CD h-O< LOaC 2 0 . • X>— <UJ — Nl* x< — o —< a: — h- CD—4QClOLUI— QCvJO LO ZO h-cO h-a: a: CD < »—4 UJ 2 h- ►CDCUJOCX 2 O a l 1-4 • Z O O 0 0 < < 0 0 •* ► *— r— w < O O ► CD ►a > z > u j < 2 LU «/>< < a : o: cl < — Cl Ql < L 0 J ►LO > h-Ch

*—H- < < /> > UJ O a : < — oolu o r - 4 •> ► f— •> — J — ► h- CO CDO K - < a oOX^—O X 2 LU O lu 2 LU J p-4 LO — —* CM 0<M — h- ► »*—< t > Z 2 h- O iT iH QJ Q k— <t w — O O *-0<M —I LU—ivO ► vO LU 001—OOCO O LU o >.QC— • • V— m r—4 rH — -«r— ^ >* h- CD< •—G O •—OCX •*— UJ>-U_ -4-LUh-*—Q 3: > — sO —* •* -4-C J O O CO •si CO ►x a i o u j h-UJX J s O X Q C -Q u a h Z *• - < a: ►CD—• ► O —C *— h— ► •'f­O lu 1 LO < X < X - i h - < O Z < O Z < — CO f— CO *-H -4 (MO — — h— in lu CDa: a i a ih - a :< 1 LUUJ< LU < o .— —(CO ► •— cO—■ —<— —(O lO Z ► KJ h- ►X | o a r o o c a o o o j o l o ilu O oo r—1 d. * *— a. cD — — vO » 2 a: o > NJ cOsOO I - W Z uj o x - O '—- LO»— * D G C * < w ►*— *-0 CD » l o CO ►—4 ►UJ CL h— ► ►CD

o O Z < coaCQCo CD <rj»— LO i— ( M ^ < — -H X — D r l X —4 r—2 0 ► h- vf O ►o o a io uj cl o a i -c 0 > C D LO O LU D C r-> 0 — — O O LO ► 0 - 4 — X --------- ► XN h— LT\LU *— *—h— CD CD LUO • QC > X U J*—x LOweD ►—CMCM >H-rM CD CDh-LUh-lO O X L 0 3 3 _ jH < h— Oh— h— □Q — *— ►—i Cl — ►>—* LU CO O 2 vO —< —' cO •» •* ► ►< <aC *— O '— • X LU • o * — •• j <r o o o < — *-* ►CD —H-CD ►C.DL0 2 ► ► ► Oh-COCO o ^

lo j a i a i — LU O — < L U > 0 LU Cl *— J X LU O h - O — LOlO < 0 I -J > Cl I < *—U jZ Ia c x < r I ILUX I I Q < lO I

arzoo I

h Z I O c Q < < - 4 “ 5 < X — < H- « 2 — >-cc > ax > x 0 > - —« >LL QCOO 0 2 2 LULU CD cD OCDXLU< * < —. > - 0 2 2 0 2 U hX C D O O - m 7 X w I— O I I Q D m > * - LU>- O '—

—il u «— c o o < — r i ( \ j a i — cl ► » o o o _ j ► X 'M cd h - o•— ►—»*■ Cl *— O CL*— ►O ►—» Q, •— —I r—I *—*> h — CM X CD ►—» — CO ► — LO ► ► ►— LO - ( \ J Z — h - h - ► ► O »CO

— 2 ► —• — —4 0 U - 0 0 —4 0 —• * - 0 . 0 0 C\J o ►CM CM OO r Q O► ► *o x -,n Q H H ►cd-< - ►*:aia:LOcQ »v i n CLaoh- ►

— —4— —• ► o —* —4—» ^ l l - h h •• u j u j a : 0 < cdk- m » cD cocmO r J O — —»•— < ' - < ’0 0 2 * , < H r - l | - Q ^ : L U I 1 0 H ► ► < O 0 »»(J)

U J* —IH- lO « u l > O U _ Z h - - J —•—**-h < I O * — 2 C O X <—0 < X w X v . X c J ► ► ►—•—• X ►<? ► U .J L O □ Z L U < - I L Z * - ' Z J * - < ' - C O - i < D c O D < - L l O D « O I - ► ► < “0 > - i r v l I D < h - - 4OOZL0l-aiL0t-UMMUN<D3!O<Nwir\00(NL0KU.U„L0OO^J«''OSI-KN uOcDOOlD

i_) t—< t X 2 C L U ü UJ D U - 3 : u . > Q . O i 7 ) a . Q . L O c O \ \ U L Q . ^ N D v O N Q - O N 'v V v < C L ^ 'v N 2 0 V 2 0 LU C CL LU UJ Q- II O O 'H H -00 ►—IN O ^ U J '- ' IN J V S S t n M

L U O D Q C I » G Z C C U J oO J O O C L X /I 4 , Z Z Z Z Z Z Z Z 2 Z ' ^ f N I Z Z Z Z i i l O • ' H m I Q Q ^ ^ N O O I O m Uj Q —• o * - o O — ll o r — J > * O O O a O O G ü a C c u c c O O U - a : H O c ü 3 : h c Q c ü I l c Q i l i c C a 2 X a: h— h—l o a:h —> - L U o loo o r o o h * Q O o l a : M •— •— ►— *— »—*—i>—i*—>*— S \ h i ■«►—I h \ v .'n .'x w < Q - < lU U J U J O U -X * -L U U J O cl Q O Q C U < lJ U L O W ^ o O U )o O o O o o a )c O Z Z o O C O o O Z z a v Z Z Z Z Z Z Z Z Z Z a : r f i u i s j D « t - u . o : i o > < i ^ « n j o z z z z z z z z z z o o z 2 z a ü c i : H O o a o o o o a Q OO o o a :< —• C o L u c D u - M a : o < a : o z * - u j i u u j u j L u u J L u u j u j u J u j X x u j i i J u j X X i j u o o x x x x x x x x x xO —.L U a lC L U O L U L U L U Z h - O lU X h - X X X X X X X X X X X X X X X X X X ec X X X X X X X X X Xa : x c D < Z Q : a : ( i i n o u j u . z - * - * u j « u . D Z ' - ' - ' - M « M M M H M O O M M M O o x u j o o G o o o ü o o Qah<rQ.H<aiflhOUlOUJ(MrO(LinM<MQQQGQDQQQOOUDDQULJ ►LJÜOUUÜl )OUU(J

****

U U O Ü O U Ü Ü Ü U O O O U U U O Ü U U

100

GOaor\jCl.

< ► -J OO X <\l ► CL

35 ►► <CO

gOnO gO>tO *-< _ j

► «“(X 3:< < ► ►

>U -\C O ^►to*-» ac

CO *>CL ► >-> - < h-

3 H-t< tO C O ► oCy . *.rO . <►< CL 35 «J

COCO •> ► O<—«to <T XX ►cO 3 OC►<C CL ► O

< f \J ► O U-— t O < m 3x *-co 3 •> oca:►<(\J ►UL LULU

O-HCL >4-35 OO•«go ► 3 ► cc cc

-) •.< *LU OOXCOCSJ CO 3 ►20. 35 ► _J>

^ O - -CO <OCXOO< <\J3 0 <►toco 3: ► 0 2

2 ► ►O X<CL ^ 3 • Z • 3 •

O O C ►CO ►lOr-4 3 3

OtOCL 3 ►N \ S<

2 0►-*0C

O — i------- OC tOH*ÜJ —------«-* — — <03000.0 --------------—---------------I m OjC Z O J X O IIS :oc2 « o ^ ^ c o o j ^ z z i r r r r H ^ ^ a 'c o h - « o m v t T o c M ^ x x a c 2 2 x a : x x u j i j u o j< * —• ► • « ► ► ► ►UJUJUJUJUJLU ► » • ► » • > ► ► • > • ► ►UJLULULUUUUJ UJ LULU H« l—>|—

"vtOtO 1 0 3 <U-U- X Vi— ujuj ac—i< O Q C O U J U OO Cl cot—CO

2 - n 0 T 5 0 h l - f - h t - h - T 5 T m T T ) T ? l h l - k l “ t - h l - h h II II l| OOXiZ *— .w .—------• || || || || || I) w w — || || || || || || || || -------—

U -2F uj ► cocococococo— — — — cocoaocQcocoooaoaocOcoco— — — — — — — — o *-«cmU . D I H X X X X J E X ^ t N ic O ^ U ^ ^ X S I J L S I X X S E lS Z r E S iX j r ' - H r M C O s r m v O r '- a o O 'r - t r « , ^ UJOOH- II O O O O O O ► ► ► ► ► ► Q C O O O O O O O O O O O O ► ► ► ► ► ► ► ► ► ► ► • O tO - J > - tO tO iO tO tO L O - ) - ) - 3 - 3 - 3 ”-3 < roO tO tO (y )tO iO <y)tO tO tO tO tO ~ 3—J T O - J T O —J T T O O

Z Z Z a Z Z H f O U ►!— «~<ac Q. cl cl cl cl c l ^ * - — o o o to o o o o to to c o o o o o o o o o o o * -« —O O O ► O O II II C O O —t < II l| II II || II CO CO COCO 00 CO 2 II II II II II II II II II II II II CO CO CO CO CO CO 03 CO CO CO CD COx x x c o x x a i - j x a H £ < c o o o i L O S x r x x x o i M ^ ^ J X 2 ü O Q £ : ^ H r x z x x : x x x x x x 2 : r s : s c o s i : z D _ i D - t M r m r r a D D D D D o x x i x x x x r x r x x D ^ D ^ D O D D D D O a o a . O o *-« O < to id o cc lu lu uj uj uj id to </) to to to to UU LU uj UJ ID tu LU ID UJ UJ UJ LU ID to 00 to to 00 OO 00 to 00 to to to t > 0 0 O l > X ^ 0 0 . 3 0 CL I— I— I— H - l - l — CL Ol CL Q. CL OL tO I— I— h - h - l— h - f - h - K 0 0 tO lO lO tO 0 0 tO tO tO oO tO tO

OO o O

«— I

r - t

r -A

101

COom

i/>u- •>UL CO <UU Ul OCD 0 cOO • Z

a: Ul ZZX X X0 3 3 UJ

X X ao1 - 0 X X O< z x t—» 0 XIX X a XXX UJ cO m

UJO z X <co< X O O<X Q 1— •—* z X X

UJ «» — < m m>-0 X xm • x z XXcO < XX —» X 0< —1 < z 0 — ) *• z2 lX O X „ H id X X«X O CO*-* id id X X XXX -J UJ(J «— nj O Xa- 00 < CD i—4 X <\J O X X

0 0 UJ O ZX 0 O m XLUxx XX co XX X UJ XX co LU 0 0♦— — • X XO CO X O z

0 < XO X 0 z < XU_LU— 1 X 0 z X zO xX —2: z X X m X

UJUJ 0 0 0 X X X X Xl / ) I I UJ ► ►—t X X X 0 0UJ ?a. Oid X X X X X z X> 00 Cw — — z c X X XX X X Xx ~>< U_X HH XX X X x x O X XXX XO •> •“ — X ►—t «TH A 0 •• *• eg eg X X — X rnrn< 0 1 0 O XX o x 0 0 O -J—J x rH rH X XX 0> UJ 1 /3 CO —» ■*—# ►-«CO H H X --------- z H H X ► ► 0 H H*—< *»x UJ CfO x c X rH rH X C CO X rH rH X -JO rH rHX ~)x -IO is : X z II II X I I X II II a ■--------- 0 II IIXidO CZaQ 0 0 XX X HH HH 0 0 0 0 HH HH <00 X MM0 C1 o x O COCO OX X H •* 0 COCO X •» X ara: x H H

•»uu ►—x a XX 0 0 0 rH HH COCO X HH X 0 0 < n r.coox XUJ 1 1 X HH HH V 1 1 X HH HH X mm X HHMUJO uju. •» rH r** 00 X •* H X XX UJ H 0 xx 0 H Hh u Z I lU HH HH z z X -> < XX O "> rH 0 1 1 x< O X xO H H xc X HH W 0 *■ <__) W H — , x H Hr_J ►x moo. rH rH Jt 0 < CD z rH rH < CD rH 0 HH HH <CDDOh 4 - + O-J 0 O O 0 4 - + > O O a X H H OOOOO < u z z z a CL 0 ZZ —X LO 0 0 X X Z Z —-J LL LLX UJ u. x Z Z •—<-J Z INI >- X z z m < r nr 0 < idid>i-<cam Ouj ■— — Im O xinx 0 0 -HI co ——in 0 CO H 1 0 X —«—inx 0 0 -* —Oh UIOO CCOG'U-U.O'C 0 0 0 0 ecoerz 0 0 0 0 X c < CD O' X 0 0 0

co lo u i u m 1ms: 0 0 0 0 X x x m o 0 0 0 0 eg X n i n e OsT*>tZ l-M 0 X OO O'UJUJ O'*— r\jm<\icn X DDO'O rgmcNja • X OOO'XZn.t-4 -vfO u l m v o COCO►XX ►X H H•» H < com ►x H HH H 0 X 0 mm ►< ► ► ►

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a: • cxc .-'oooxc jcxc '-iu jxxxxo '-toooxc-sxxxxxo z>xOoox<oxxxOOIUJOI o O — w X 2E X X a: XX ►—I— ►—»— HH^whJOOhhhht-ZJU'J-'tw.whXOKI-H-mi- 3[ ii x o n j r g c cax cdx x c d i n • z 3 :inm<-cox o f tn x x x x ^ —»_j-jinincco —<x X *~i uj -■* x x z O CD o o a: a 30 x o □ o' a: x x o O n O o O X O n x X x x o ul < < n a O O x o n a: x x h i Q i : u 3 : h - < Q G a a i u . 1 0 . 0 3 3 j 3 o Z o ü o o o o 3 ; u . D ] C 3 : 3 E 2 u « i i O c i ü u . u . 2 u . n 3 : 3 3:

UJOz

zoO

O rH (\l(M in in

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O'inO'

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<f 4 - —< r\lin m o r~>

O' o —<in <#■ —1O'0 0 0 0 0 0 o o

102

oou.ULUJOO

XCL<Xt—ia:aLLO

tO LU > *—<X<>t-4QCUJa

00ULu.UJoo

00 CO

« —» QC — X ”0z:—o <o >

*o ••—•CMoo |Z - *►—< ~3 OC — X< Z h •— ' X

* X—* ad ”5 < — 5 : * :

— . U J — - Xz • z 0 L *

*—4 X Q . •• - — * #» m

L U — To O - > X - « - ► 0 0

w L U w X L UCO > c o " O X >

- o > Ä » —4 X U . W »»■4o •* S h •» 0 < r — > xo • • < 1 1-4 ~ 3 C 0 <

H X > - o c o X — . M X >

r O — - o e c w L U * <M o — f— Mf \ J < Q o < > 0 C < > - . * V O ► a cf O * —• L L > CO L U X — * - 0 > L UO X X ** i - o •* 1— -5 — — ( M < x o

# h - Z A * “ 9 < — < ■ — - > < — * c o h —1—4 O ” > > o O _ J “ 3 « r H - 0 * o n j

o — * »—« w h - O X — — X f \ l C O . — < * XCD ► X X c O c c 1 < — IO C C 0 X L J

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z : ^ + < 0 0 0 0 - 3 0 Z o Z O L 5 V < — Z O r - K M r n '4 - t n 'O r - o o O ' 'rsj 0 — - 5 0 0 0 II t| II II II l| o O Z - H > O 0 0 0 CD CD 0 CD CD 0 CD CD CDII *—4 •* ►«■4 f—4 —’ O LU LU 11 II II II II II 11 II II

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141

APPENDIX 3

42. The e x p a n s i o n o f %'

The augmented a b e r r a t i o n , e.', o f a r a y can be expanded as a

power s e r i e s i n t h e r a y c o - o r d i n a t e s Y, V.

I f we w r i t e

£ := Y2 + z2 , T) := YV + ZW , (; := V2 + w2 ,

t h e n i n t e rm s o f c a n o n i c a l c o - o r d i n a t e s AA. may be w r i t t e n

n py y y , (n) “ W . , n - p p - v V^ ^ ^ ^u.V ~1 + ^u.V ~1 ^1 ^1 ^1n=1 m=o V=0 ^ 1 ^

r ( n ) n -p p-V „V

The a r e c a l l e d t h e c o n t r i b u t i o n s t o t h e a b e r r a t i o n

from a p a r t i c u l a r s u r f a c e . From e q u a t i o n ( 3 . 8 ) t h e augmented a b e r r a ­

t i o n i s t h u s

K

00 n p , \ , .E E E (G(“i' Y + G(">' V ) I

n=1 |i=0 V=0 ^ Vk 1 ^ Vk 1

n - p p-V „V

where

. ( n )PVJ

V1^ SpVi * i=1 ^

f ( n )GpVj

j -1 _g,z -(n)

i=1 pVi

The a r e c a l l e d t h e a b e r r a t i o n c o e f f i c i e n t s o f t h epV * pV

sys tem .

S i m i l a r e q u a t i o n s may be w r i t t e n f o r p a r a c a n o n i c a l

c o - o r d i n a t e s .

142

APPENDIX 4

43. T a b l e o f symbols

The f o l l o w i n g t a b l e l i s t s t h o s e symbols commonly a p p e a r i n g

i n t h i s t h e s i s and which a r e n o t c o n t a i n e d i n L. The number i n

b r a c k e t s , a f t e r t h e meaning o f t h e symbol , i s t h e s e c t i o n i n which t h e

symbol i s f i r s t u sed .

Symbol Meaning

a

b

c

d

5

e

<D

g

V n)h ( z )

®

r a t e o f change o f a s e p a r a t i o n . (14)

c o e f f i c i e n t o c c u r r i n g i n f o rm u la e f o r 7T, 7T. (27)

c o e f f i c i e n t o c c u r r i n g i n f o rm u la e f o r 7T, 7T. (27)

s e p a r a t i o n be tw een two s u r f a c e s . (2)

s c a l e f a c t o r i n image h e i g h t e q u a t i o n . (20)

r a d i u s o f a p e r t u r e s t o p . (28)

power o f a t h i n l e n s . (17)

f i r s t e x t r a a x i a l power o f a t h i n l e n s . (40)

c o e f f i c i e n t o c c u r r i n g i n f o rm u la e f o r p, p. (27)

f u n c t i o n s o f r o o t s o f a p o l y n o m i a l . (20)

r a d i u s o f e n t r a n c e p u p i l . (21)

r a d i u s o f e n t r a n c e p u p i l a t z = 0. (21)

K f u n c t i o n s o f r e f r a c t i v e i n d e x o c c u r r i n g i nO

f o rm u la e f o r t h e f i x e d a p e r t u r e a b e r r a t i o n

c o e f f i c i e n t s . (31)

t h e a u x i l i a r y sys te m i n a zoom l e n s . (14)

A t h e p a r a x i a l i n v a r i a n t . (8)

143

Symbol Meaning

P(z)

P> P TT, 7T

r, r

P, P

r/( j)an,'<J>an

max©

the distance of the stop from the first surface.

(28)

coefficients in the expansions of 7T, 7T. (27)

polynomials occurring in formulae for the fixed

aperture aberration coefficients. (26)

the magnification range of a zoom lens. (19)

V' (1)/v' (0). (19)pn pncoefficients in the expansions of p, p. (27)

polynomials occurring in the formulae for the

fixed aperture aberration coefficients. (26)

the shape of a thin lens. (17)

the derivative of order j of v' • (18)anthe derivative of order j of y' . (18)anthe image height in the ideal image plane. (17)

the zoom parameter. (14)

distance of an element from its initial position.

(17)

maximum value of Z. (17)

the zoom part of a zoom lens. (14)

144

REFERENCES

[1] Buchdahl, H.A., Optical Aberration Coefficients (Dover

Publications, 1968).

[2] Buchdahl, H.A., J.O.S.A., 55 (1965) 641.

[3] Lohmann, A.W., Applied Optics (U.S.A.), _9 (1970) 1669.

[4] Yamaji, K. Progress in Optics, .6 (1967) 105.

[5] Bergstein, L., J.O.S.A., 48 (1958) 154.

[6] Bergstein, L., and Motz, L., J.O.S.A., _52 (1962) 353.

[7] Bergstein, L., and Motz, L., J.O.S.A., 52 (1962) 363.

[8] Bergstein, L., and Motz, L., J.O.S.A., 52 (1962) 376.

[9] Jamieson, T.H., Optica Acta, JJ7 (1970) 565.

[10] Pegis, R., and Peck, W . , J.O.S.A., 52 (1962) 905.

[11] Wooters, G., and Silvertooth, E.W., J.O.S.A., J55 (1965) 347.

[12] Back, F.G., and Löwen, H., J.O.S.A., ^8 (1958) 149.

[13] Bergstein, L., and Motz, L., J.O.S.A., 47 (1957) 579.

[14] Wynne, C.G., Optica Acta, ,8 (1961) 255.

[15] Cruickshank, F.D., Tracts in Geometrical Optics and Optical

Design No. 3, University of Tasmania, 1961.

[16] Jamieson, T.H., private communication, September 1971.