The application of eikonal functions in the practice of optical design

I III I i i i l i i

n

o o

THE APPLICATION OF EIKONAL FUNCTIONS IN THE PRACTICE OF OPTICAL DESIGN

THE APPLICATION OF EIKONAL

FUNCTIONS IN THE PRACTICE

OF OPTICAL DESIGN

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOG-LERAAR IN DE AFDELING DER TECHNISCHE NATUUR-KUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG, 27 AUGUSTUS 1969

TE 16.00 UUR

DOOR

CAREL ARTHUR JAN SIMONS

geboren te 's-Gravenhage

frSDoelenstr. 101"^

Delft

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. H. DE LANG.

C O N T E N T S

SUMMARY 9 Chapter I. INTRODUCTION 11 Chapter II. THE EIKONAL FUNCTION OF A REFRACTIVE OR

REFLECTIVE SURFACE

2.1 Fermat's principle 13 2.2 Brief description of the functions of Hamilton and Bruns 14

2.2.1 Sign conventions 14 2.2.2 The point characteristic function V 14

2.2.3 The mixed characteristic function W 16 2.2.4 The eikonal S, or angle characteristic function T . . 17

2.3 The relation between V and S 19 2.4 The eikonal of an arbitrary refractive surface 20

2.5 Examples 21 2.5.1 The eikonal of a plane refractive surface 21

2.5.2 The eikonal of a plane parallel plate 22 2.5.3 The eikonal of a spherical refractive surface . . . . 23

2.5.4 The eikonal of a surface with rotational symmetry . 24 2.5.5 The eikonal of a surface with rotational symmetry

used at a paraxial magnification P' 24 2.6 Series expansion of the eikonal of a surface with rotational

symmetry 25 2.6.1 General remarks 25

2.6.2 The surface equation 26 2.6.3 The eikonal of a rotationally symmetrical surface of

t h e f o r m x = F { ( / - l - z 2 ) * } 28 2.6.4 Series expansion of the eikonal <? of a surface which

is expanded into a series of powers of y^-I-z^ 32

2.7 The length a 34 2.8 Example of the application of the formulae found 35

Chapter III. THE EIKONAL FUNCTION OF AN OPTICAL SYSTEM CONSISTING OF MORE THAN ONE REFLECTIVE OR REFRACTIVE SURFACE

3.1 Description of several methods to evaluate the eikonal of an

optical system of more than one surface 39

3.1.1 Classical method (Bruns) 39 3.1.2 Numerical variant of the classical method 40

3.1.3 Ray tracing method 41 3.1.4 Features of the three methods 42

3.2 An automatic computing routine based on eikonal functions

to improve optical systems 43 3.2.1 General remarks 43 3.2.2 The merit function (f> 44 3.2.3 The first and second derivatives of 0 45

3.2.4 Procedure to minimize <j} 46 Chapter IV. THE SELECTION OF REPRESENTATIVE RAYS

OF LIGHT

4.1 General remarks 50 4.2 The selecting process 51 4.3 A method to evaluate the range of variation 2a 55

4.4 An illustration of the effectiveness of the selection method 56 Chapter V. APPLICATIONS

5.1 General remarks 58 5.2 Correction of astigmatism in a / / 3 reproduction lens used at

a paraxial magnification — 1 58 5.3 An aplanatic single lens used at a paraxial magnification —2

5.4 Schmidt projection system 65

5.5 Conclusion 66 SAMENVATTING (Summary in Dutch) 67

S U M M A R Y

It is shown in this thesis that eikonal functions can be applied in the practice of optical designing. An analytical method is developed in order to improve the performance of already fair well corrected axially symmetrical optical systems with aspherical and/or spherical surfaces.

In Chapter I a brief description is given of the principles underlying common optical designing methods. At the same time current opinions are given regarding the possibilities of characteristic or eikonal functions, which have resulted in an underestimation of their applicability.

Chapter II begins with a brief discussion of various functions which may charac-terize an optical system. By applying Fermat's principle, formulae are derived which link a homogeneous equation representing a surface to its angular characteristic function, the eikonal. Special attention is paid to the derivation of formulae which link coeflicients in the series expansion of an axially symmetric surface to coefficients in the series expansion of its eikonal, and vice versa. The surface concerned is not only given in the well-known form x = X"=i^i(>'^+^^)' (' ~ 1,2,...) but also as X = X"=2'^i(-y^ + ^^)''^ (' ~ 2, 3,...) in order to facilitate the computation of cor-rections of for instance comatic errors.

In Chapter III several methods for the evaluation of the eikonal function of an optical system consisting of more than one surface are dealt with. As the eikonal of a system is the sum of the eikonals of its parts, the system eikonal can easily be differentiated with respect to the system parameters, e.g. curvatures and thicknesses. These derivatives, which can be written as compact formulae are, as we shall show, very well applicable in automatic correction procedures.

In Chapter IV a method is developed which enables us to select representative rays of light through the optical system, in order to increase the reliability of the compu-tation results. The fact that rays of light in isotropic media are orthogonal trajectories of wave fronts leads us to an approximation method of the Chebyshev type.

In Chapter V practical applications of the theory are given. The examples given have eikonal functions as a computational basis.

C H A P T E R I

I N T R O D U C T I O N

Geometrical optics is a science of image formation. A basis of this science is the ray conception of light, which states that a luminous point emits light which travels along lines, the so-called rays of light. The trajectory of a ray of light through a medium is not influenced by other rays. The ray model can be derived from Maxwell's electro-magnetic theory when the wave-length of the light tends towards zero [1, p. 109]. In practical optics, this assumption is in general permitted, as the dimensions of the elements of the optical instruments, and of the objects to be imaged, surpass the actual wave-length of the light used by many orders of magnitude.

A second basis is Fermat's principle [1, p. 128; 2, p. 11], which states that given two points A and B, the optical path-length of a ray of light travelling from A to B is an extremum when compared with any adjacent curve joining A and B. By applying Fermat's principle it can be shown that light through homogeneous media travels along straight lines. Snell's law, which is a direct consequence of these concepts, forms the computational basis of geometrical optics [3, p. 4; 4, p. 64].

Optical designing was originally dominated by analytical geometry. Most systems, designed by applying this mathematical discipline, whether they employ the focal properties of conic sections or not, have in common the existence of two conjugate points, the object and its image, both lying on the rotational axis of the system which generally contains at least one aspherical refractive or reflective surface [4, p. 129]. It took many centuries before it came to be understood that image formation, which is the process of copying a given brightness structure, can be achieved without re-quiring a perfect union of the rays of light, originating from one object point and passing through an optical instrument, into one image point. Since then, the develop-ment of the theory of image formation could take place. This led to the paraxial theory, which describes the relationships between the mutual positions of object, optical system and image together with the object magnification, and to the theory of aberrations, the latter giving an approximation of the light-distribution in the image [5, pp. 192, 249; 6, p. 2]. The most important result is that nowadays the majority of optical systems contain only spherical surfaces, which are easier to manufacture and to test than aspherical ones [7, p. 39; 8, pp. 295,298]. An aspherical surface is in general only applied when it is important to restrict the number of surfaces in a system with a required state of correction, as the aspherical surface possesses more degrees of freedom than a spherical surface with the same paraxial power [9, p. 506]. In order to design optical systems with preset characteristic features, automatic iterative computing routines are employed. In these routines, besides paraxial and aberration theory, ray tracing procedures [10] and spot-diagram analysis [11, p. 272] are also

, applied. These automatic design programmes in general provide an improved version of an already fairly well corrected system which was given as a starting point. It follows readily that, in order to be able to judge whether a system resulting from a certain computing cycle is better than the system computed in the previous cycle, and to be able to stimulate the occurrence of improvements, the designer needs at least one quality criterium which is a function of the design parameters (e.g. lens thick-nesses, curvatures), together with its partial derivatives. These derivatives are usually approximated by applying finite differences, the relationships between design para-meters and quality criterium not being known in a concise analytical form [12, 13]. To avoid this time consuming method of finding derivatives, rather intricate differen-tiation formulae are advocated, which seem only applicable when using a large com-puter [6, p. 192; 14]. It is amazing that the applicability of a theory, producing differential relations between an optical system and its imaging properties, that was developed independently from each other by W. R. Hamilton and H. Bruns in the 19th century [15, 16, 17] was not tested until now.

The basis of this theory is, that the application of Fermat's principle leads to the notion that, exceptional cases excluded, at the most only one ray of light can travel through any two given points which lie respectively in the object and image space of an optical system. Hamilton and Bruns have shown that the imaging properties of an optical system can be characterized by a set of functions of coordinates in both the object and image space, which uniquely determine a ray of light travelling through the system. The functional value equals the optical path-length measured along the ray between the two points, determined by the coordinates. However, until today the theory has not been considered a valuable contribution to practical designing, al-though some fervent admirers of Hamilton took great pains in showing otherwise (see for a discussion of the practical merits of the theory the disputes between M. Herz-berger and J. L. Synge [18, 19, 20, 21]). The main reason for the impopularity of Hamilton's theory of geometrical optics is that it is thought impossible easily to evaluate functions which characterize complex systems. In addition to this, the original theory can only be applied when analyzing, whereas the designer requires means which can assist him in the synthesis of an optical system.

In the following chapters we shall develop and test a practical method of attaining final correction in optical instruments, making use of the theory of Hamilton and Bruns and the suggestions of T. Smith [22]. The method enables us to determine uniquely a refractive or reflective surface, once the desired characteristic function is known, and vice versa. The method leads to a computing routine which, owing to its simplicity, is well suited for practical use in automatic optical designing [14].

CHAPTER II

THE EIKONAL FUNCTION OF A REFRACTIVE OR REFLECTIVE SURFACE

2.1 Fermat's principle

This principle, which was mentioned in Chapter I, requires stricter formulation, as it has to be fully appreciated before the discussion of the functions of Hamilton and Bruns can take place.

In geometrical optics, we characterize a medium by its refractive index n, where n is the ratio of the speed of light in vacuum c, and the speed of light in the medium V, so

" = I

(2.1)

Fermat's principle can be stated as follows: The path chosen by the light in travelling from one point to a neighbouring point is always such, that the time required is stationary with respect to small variations in the path connecting the two points. See fig. 1.

ds>

Fig. 1. Fermat's principle: ö f nds = 0 for the path chosen by light travelling from point A to a "* neighbouring point B.

The time T required for the light to travel from A to Bis ? d5 ? nds

A V \ c

As ^T = 0, we see that we may omit the constant factor 1/c. Thus we find

^ J nds = 0 (2.2)

A %•

\^ nds is called the optical path-length between A and B. Snell's law is one of the consequences of Fermat's principle [5, p. 51].

2.2 Brief description of the functions of Hamilton and Bruns 2.2.1 Sign conventions

Light incident upon an optical system travels from left to right, unless otherwise stated.

Distances measured from left to right are positive.

A spherical surface has a positive curvature when its centre of curvature lies to the right of its vertex point.

The refractive index « of a medium is said to have the negative value —n, when traversed by light, which has undergone an odd number of reflections.

The directional cosine of a ray of light, in the direction from left to right, is always positive.

Coordinates in the image space will be indexed. 2.2.2 The point characteristic function V

Consider two points A and B, respectively situated in the object and image space of an optical system 5. See fig. 2. In each of these two spaces we have a rectangular coordinate system. For the sake of convenience, the systems are assumed to be parallel

Fig. 2. A ray of light traversing the optical system S connects two points A and B. The optical path length between A and B, measured along the ray, is a function of the coordinates of A and B. The

function is called the point characteristic function.

with respect to each other when S is axially symmetrical. If this is not the case a different orientation of the two coordinate systems can be more recommendable.

The points are connected by a ray of light traversing S. As the path of the ray is governed by Fermat's principle, it is clear that in general there exist a number of discrete rays through A and B. In many cases only one ray is relevant. Thus there exists a unique relationship between the optical path length measured along the ray between A and B, and the coordinates of these points. The function which describes this relationship is Hamilton's point characteristic function V, often named the point characteristic [4, p. 94]. Thus: ,

B

yixA,yA,ZA,x'B,y'B,z'g) = ^ nds (2.3)

A

It is readily seen that function V can also be defined if we use only one rectangular coordinate system.

We can easily show that F characterizes the imaging properties of optical system S. Consider a small change dK in the value of V as caused by a change in the positions of A and B, which can be written as:

, , , dV, dV , dV ^ dV ^ , dV ^ , dV ^ , dV = — d x + — d y H dz 4- —-dx A dy -I dz

dx dy dz dx' dy' dz'

(2.4)

If the directional cosines of the ray of light in the object and image space are respec-tively called L, M, N and L', M', N', and the refractive indices respecrespec-tively n and n', we see that the change in optical path-length can also be written as:

dV = - nLdx - nMdz - nNdz + n'L'dx' + n' M'dy' + n' N'dz' (2.5) See fig. 3.

(x-tdx,y+dy,z+dz) (x,y,z

Fig. 3. Changes of the positions of two points, connected by a ray of light through optical system S give rise to a change d Fof the value of the point characteristic function V.

Comparing (2.4) with (2.5) we see that: dV = -nL dV = -l-n'L' dx dV dy dV dz dx' dV — nM and — = +n'M dy' dV -nN — = +n N dz' (2.6)

It can easily be seen that, if V is known, the relations (2.6) completely determine the imaging properties of 5 for a pencil of rays in the neighbourhood of the selected ray 15

of light between A and B. However, if A and B are exactly conjugate, the ray of light between A and B is no longer uniquely determined. In this case the function F cannot be applied. This seems rather paradoxical for a function which characterizes imaging properties, but it should be borne in mind that cases of exact imaging are pathological exceptions to the present formalism. We shall consider, for example, an ellipsoidal mirror. See fig. 4.

Fig. 4. When two points are exactly conjugate (the two foci of an ellipsoidal mirror) the point characteristic function becomes useless.

The focal point F, is exactly imaged on the focal point F2. The optical path-length between Fi and F2 is, as is well-known, equal for all rays of light passing through these points. Thus the optical path-length in this case is not an extremum, but a constant. We do not find merely one ray, but a continuous set of rays, all with different directional cosines, which all give the same value for V. Therefore, the relations (2.6) become useless because the directional cosines are indeterminate.

2.2.3 The mixed characteristic function W

As in section 2.2.2, we take two points A and B lying respectively in the object and image space of an optical system S. Here A is defined as the point of intersection of the ray of light and the perpendicular to that ray, originating from the origin O of the coordinate system in the object space. See fig. 5.

Fig. 5. The mixed characteristic function W(L, M, N, x'B,y'B' ^'B) = ƒ n^- ^> ^ . ^ ^'^^ 'he

A

directional cosines of the ray in the object space.

Point A is completely defined by the directional cosines L, M, N of the ray in the object space, together with the coordinates of point B in the image space. Analo-gously to 2.2.2, we here define the mixed characteristic function W [4, p. 100] as: 16

W{L,M,N,x'B,y'B,z'B)= J nds (2.7) As L^ + M^ + of course also A being given relations as in As in section define one ray an exact conju

A'^ = 1, we see that W has only five independent variables. We can define a function W^, using only one rectangular coordinate system, or

by point coordinates, and B by directional cosines. A similar set of section 2.2.2 can be derived characterizing the imaging properties of S. 2.2.2, we find here, too, coordinates for which W does not uniquely . Thus it is in the case in which there exists, for a continuum of rays, gation between a point and a direction. See fig. 6.

Ai A2

Fig. 6. The mixed characteristic function does not uniquely define a ray in the case of an exact conjugation between a point and a direction. F is the focal point of a paraboloidal mirror. All rays of light passing through the focal point F of the paraboloidal mirror are reflected in the same direction. As in 2.2.2 we do not find the essential extremum, but a constant value for W, the optical path-length between Ai...A„ and F.

2.2.4 The eikonal S, or angle characteristic function T

The eikonal of Bruns [17, p. 353] is essentially the same function as Hamilton's T [4, p. 104]. Here, too, point A lies in the object space, and B in the image space of an optical system 5. A and B are both defined as the feet of two perpendiculars to the ray originating from the origins of the coordinate systems used. See fig. 7.

!^

B

'0'

Fig. 7. The angle characteristic function ê(L, M, N, L', M\ N') == ƒ nds. L, M, N and L', M', N' are directional cosines of the ray in the object and image space.

The ray of light, together with points A and B, is defined by the directional cosines of the ray in the object and image space. The value of S' is again the optical path-length measured along the ray between A and B. Thus we can define:

B

S{L,M,N,L,M',N')= \nds (2.8) Analogously to sections 2.2.2 and 2.2.3 the function loses its usefulness in the case of

an exact conjugation between two fixed directions L, M, N and L', M', A^'. See fig. 8.

0 / 0

r A-^

Fig. 8. A pencil of parallel meridional rays is refracted by cone C. The refracted pencil too consists of parallel rays. Hence the angle characteristic function can not be applied.

Consider for example a cone C. A pencil of parallel rays, all lying in a plane which contains the rotational axis of C, is refracted as a pencil of parallel rays. We imme-diately see that in this case the functional value of S, together with the directional cosines before and after refraction does not uniquely determine one ray of light.

Of all characteristic functions, we consider S most suitable for practical optical work. The first reason for this choice is, that because

L^ + M^-l-iV^ = L'^ + M'^-l-iV'^ = 1 (2.9)

Fig. 9. The eikonal of optical system S does not change when the positions of object and image plane are altered.

W(L,M,N,x'B,y'B,z'B) = | nds (2.7)

As L^-l-M^ + yV^ = 1, we see that W has only five independent variables. We can of course also define a function W, using only one rectangular coordinate system, or A being given by point coordinates, and B by directional cosines. A similar set of relations as in section 2.2.2 can be derived characterizing the imaging properties of S. As in section 2.2.2, we find here, too, coordinates for which W does not uniquely define one ray. Thus it is in the case in which there exists, for a continuum of rays, an exact conjugation between a point and a direction. See fig. 6.

Ai A2

Fig. 6. The mixed characteristic function does not uniquely define a ray in the case of an exact conjugation between a point and a direction. F is the focal point of a paraboloidal mirror. All rays of light passing through the focal point F of the paraboloidal mirror are reflected in the same direction. As in 2.2.2 we do not find the essential extremum, but a constant value for W, the optical path-length between Ai...A„ and F.

2.2.4 The eikonal S, or angle characteristic function T

The eikonal of Bruns [17, p. 353] is essentially the same function as Hamilton's T [4, p. 104]. Here, too, point A lies in the object space, and B in the image space of an optical system S. A and B are both defined as the feet of two perpendiculars to the ray originating from the origins of the coordinate systems used. See fig. 7.

B

* -0

Fig. 7. The angle characteristic function S{L, M, N, L', M', N') = J nds. L, M, N and L', M', N' are directional cosines of the ray in the object and image space.

The ray of light, together with points A and B, is defined by the directional cosines of the ray in the object and image space. The value of S is again the optical path-length measured along the ray between A and B. Thus we can define:

/ ( L , M, N, L', M', N') = ƒ nds (2.8) Analogously to sections 2.2.2 and 2.2.3 the function loses its usefulness in the case of an exact conjugation between two fixed directions L, M, N and L', M', N'. See fig. 8.

Fig. 8. A pencil of parallel meridional rays is refracted by cone C. The refracted pencil too consists of parallel rays. Hence the angle characteristic function can not be applied.

Consider for example a cone C. A pencil of parallel rays, all lying in a plane which contains the rotational axis of C, is refracted as a pencil of parallel rays. We imme-diately see that in this case the functional value of S", together with the directional cosines before and after refraction does not uniquely determine one ray of light.

Of all characteristic functions, we consider S' most suitable for practical optical work. The first reason for this choice is, that because

L^ + M^ + N^ = L^ + M'^ + N'^ = 1 (2.9)

Fig. 9. The eikonal of optical system S does not change when the positions of object and image plane are altered.

ê has only four independent variables, whereas W and V respectively have five and six independent variables. The second reason is a purely optical one. See fig. 9.

Consider the practical case of an optical system S, with an axis a of rotational symmetry. We choose as origins of the two parallel coordinate systems in the object and image space respectively the vertex points of the first and the last surface of the system. We immediately see that the value of S is not affected when the positions of the object plane O and image plane ƒ are changed from position 1 to 2. Thus S does not change when the same optical system is used at different magnifications. This advantage is not to be expected from V and W.

For these two reasons we shall prefer S to V and W in our calculations. However, we still require a method of transforming S into F and vice versa, because S does not inform us of the situation at the place in which we are most interested, the image plane.

2.3 The relation between F and S

Consider a rectangular coordinate system where the x-axis coincides with the axis of rotational symmetry of a refractive surface 5. S separates two media with refractive indices n and n'. The two planes O and ƒ are both perpendicular to the axis. See fig. 10.

Fig. 10. The relation between the point characteristic function V(C, D) and Sjis When choosing the vertex point of S as the origin of the coordinate system, the optical length measured along the ray of light between A and B is equal to S. The optical length between C and Z) is F (C, D). If the coordinates of C and D are respec-tively given as x, y, z and x', y', z', and the directional cosines of the ray through C A B D before and after refraction are L, M, N and L', M', N' respectively, we see immediately that

F(x, y, z, x', y', z') = S{L, M, 11, M') - n{Lx + My + Nz) + n'{Lx' + M'y' + N'z') (2.10) 19

2.4 The eikonal of an arbitrary refractive surface

We shall follow the method given by T. Smith [22] with some modifications, in order to derive the eikonal in a convenient way. Consider two homogeneous isotropic media with refractive indices n and n', separated by a surface ƒ (x, y, z, a) = 0.

We suppose ƒ to be homogeneous, as any surface can be described by a homo-geneous function. Therefore, if the variable a has the dimension of a length, we have:

f(x,y,z,a) = f(kx,ky,kz,ka) = 0 (2.11) See fig. 11.

©

Fig. 11. The evaluation of the eikonal of an arbitrary refractive surface/.

A ray of light / strikes f in P (x, y, z), where it is refracted. The directional cosines of the ray, before and after refraction are L, M, N and L', M', N' respectively. The incident and the refracted ray are perpendicular with respect to two planes Q and Q', both of which contain O, the origin of the rectangular coordinate system used. As the two media are supposed isotropic, the rays of light will be the orthogonal trajectories of the wavefronts. If A and B are the points of intersection of / and the planes Q and Q', then the optical path-length between A and B is the eikonal S of the surface.

<ƒ = n(Lx + My + Nz)-n'{Lx + M'y + N'z) (2.12) By applying Fermat's principle we find that

S^dx^iydy^S^dz = 0 (2.13) when

/ , d x + / , d y + / , d z = 0 (2.14) As ƒ is homogeneous in its variables, we also find

x/. + y/. + z/. + a/, = 0 (2.15) \ /

\ / \ /

With the aid of (2.12), 2.13) and (2.14) we find

nL-n'L _ nM-n'M' _nN-n'N' _ xinL-n'L) + y(nM-n'M') + z(nN-n'N')

fx ~ fy ~ fz ~ >^fx+yfy+Z.fz

(2.16) The numerator of the last fraction of (2.16) is equal to the right hand side of (2.12). The denominator is equal to — a/"„ according to (2.15). The functions/j„/j,,/j, ƒ,, too, are homogeneous in the variables x, y, z and a. This, together with (2.15) .enables us to eliminate x, y and z. The result will be a homogeneous equation

<P(fx,frLJa) = 0 (2.17)

Defining

L = n'L — nL

M = n'M'-nM • (2.18) N = n'N'-nN

we can, using (2.16), write (2.17) in the form

< / . K M , N , - j = 0 (2.19)

Thus we have found an expression for the eikonal S of one refractive surface, in an implicit form. Equation (2.19) is of course also applicable when ƒ is a reflective sur-face, by substituting the value ~n for n'.

We shall now give some examples of the computation of S by means of equation (2.19).

2.5 Examples

2.5.1 The eikonal of a plane refractive surface The plane is given by the homogeneous equation

f(x,y,z,a) = Ix + my + nz + a = 0 (2.20) ƒ, = /; ƒ, = m; ƒ, = n; / , = 1 (2.21) Thus JjL =, It = II = k (2.22) / m n 1 21

(2.22) can be written as: Ifx _ mfy _ nf,

l^ m^ n^ = / a (2.23) and, as the directional cosines /, m, n of the normal to the plane fulfill the condition

l^ + m^ + n^-l = 0 (2.24) we see that (2.23) together with (2.24) leads to:

Ifx+mL+nf-f, = 0 (2.25) Because of the identity between (2.17) and (2.19) we see that (2.25) can be written in the form Thus IL + mM + nN = 0 a S' = a(lL + mM + nN) (2.26) (2.27) 2.5.2 The eikonal of a plane parallel plate

In this case we have a direct check on formula (2.27). Consider a plane parallel plate, of thickness T and refractive index n', which is surrounded by a medium with re-fractive index n. See fig. 12.

The eikonal of the first surface is, according to (2.27):

<?i = (a-T){l(n'L-nL) + m(n'M'-tiM) + n(fi'N'-nN)} (2.28)

Fig. 12. The evaluation of the eikonal of a plane parallel plate.

For the second surface we find:

S2 = a{l(nL-n'L) + m(nM-n'M') + n(fiN-n'N')} (2.29) g = ^^+£2 = r{lifiL-n'L) + m(nM-n'M') + n(nN-n'N')}

= x{ii{l,m,ny{L,M,N)-n'{l,m,n)-{ll,M',N')]

= x{n cos / — n' cos ('} (2.30) This is the well-known formula for the increase in the optical path-length of a ray of

ligth traversing a plane paraUel plate.

2.5.3 The eikonal of a spherical refractive surface The surface equation is:

f{x,y,z,a)=- x^ + y^ + z^-a"- = 0 (2.31) Following t h e procedure given in section 2.4 we find:

Jx Jy Jz J a

X y z a (2.32)

By using (2.31) we see that (2.32) transforms into

i.fxfH.fyf+ifzf ^(faf (2.33) Thus, according to the identity between (2.17) and (2.19), (2.33) becomes:

L^ + M^ + N^=~ (2.34) a

Which leads to

^ = ±a{L^ + M^ + N^}* (2.35) The + sign indicates that there are two solutions. See fig. 13.

According to the sign conventions of section 2.2.1, together with eqs. (2.10) and (2.35), we find (see fig. 13) that the optical path-length between A and C via B is:

S'^^ = r{± {1} + M^ + N^f- L) (2.36)

Fig. 13. The eikonal SAC of a spherical refractive surface. Mis the centre of curvature of the surface.

In the limiting case of a refracted ray coinciding with the rotational axis, the lengths AB and BC equal zero. Hence (f^c equals zero. Thus we see that in the case of Lbeing positive, we have to take the positive root, and when Lis negative, the negative root. 2.5.4 The eikonal of a surface with rotational symmetry

If we define w by

w^ = y^-l-z^ (2.37) then the surface equation, in the case of the x-axis being the rotational axis, becomes

/(x, w, fl) = 0 (2.38) Defining W by

W"^ = M^ + N^ (2.39) we find, following equations (2.17) and (2.19):

</> ifx,LJa) = 4> U, ^A = ^ (2-40)

2.5.5 The eikonal of a surface with rotational symmetry used at a paraxial magnification fi'

The origins of the coordinate systems of object and image space are transferred from the vertex point of the surface to the axis points of respectively the object and image plane. Let A be the paraxial power of the surface. Then, using well-known paraxial theory [5, p. 200] together with (2.10), we find the expression

^I>' = ^surface + ^ ( l ' ~j + ' ^ ( l " i ? ' ) ( 2 . 4 1 )

When P' = + 1 , (2.41) simpHfies to S^^ = (^'surface» which is of course to be expected.

2.6 Series expansion of the eikonal of an arbitrary surface with rotational symmetry 2.6.1 General remarks

When applying arbitrary surfaces with rotational symmetry, the equations of the surface are generally given in the form of a series expansion. In most design methods it is profitable to consider the deviation of the actual surface from a spherical surface as given by the so-called asphericity or deformation terms [1, p. 138; 5, p. 274; 10]. Thus the eikonal, too, can usually only be evaluated by means of a series expansion. Therefore we considered it to be of practical interest to find formulae relating the series expansion of the surface equation to the series expansion of the eikonal func-tion and vice versa. The surface equafunc-tion has to be written in a homogeneous form, in order to be able to follow the procedure given in sections 2.4 and 2.5.4. We shall call the axis of rotational symmetry the x-axis. The authors who have hitherto contri-buted to the subject of characteristic functions and image formation in general automatically accepted the idea, that a surface of rotational symmetry is to be ex-panded in a series of powers of y^+z^. Thus only even powers of the distance from the X-axis were considered.

In our case, in order to ensure that the equation is in a homogeneous form, we would obtain a series of powers of (y^ -|-z^)/a^, where a has the dimension of a length.

This expansion was suggested by T. Smith [22]. Later on we shall work out this suggestion, and derive formulae which form a link between the surface and its eikonal, as to our knowledge these formulae were not derived before. We shall first give a more general approach for rendering the surface equation in the form of a series ex-pansion of powers of [(y^ + z^)/ö!^]*. There are two reasons which justify this type of expansion. Firstly the approximation by means of a truncated power series can be more accurate when not only even powers of the distance from the rotational axis are considered, but odd power terms too. Secondly there is an optical reason. It is well-known that the total aberration of a ray of light, as we call the difference between its point of incidence in the paraxial image plane, and the paraxial image point, is a function of the height of incidence of the ray in the system and of its angle with the rotational axis [1, p. 211; 5, p. 249]. Taking H for incidence height, A for angle, and C for aberration coefficient, the difference d is:

d= t Y C^j.u-ti''""-'-^" (2.42)

j = i i = i

It is easily seen that H isa linear function of (y +z ) \ It can be shown that an aber-ration term in d, which contains H'' can in principle be corrected when the refractive surface contains a term with H'*^. As p is not always an odd number it is evidently of practical importance to investigate the case, in which the homogeneous surface equation is expanded into a series of powers of (y^4-z^)*/a.

2.6.2 The surface equation Defining w = (y^ + z^)^ _ ^ a (2.43) (2.44)

we can write the surface equation in the homogeneous form:

" 1 " 2 2 " 1 •?

X = a<5ao + Y « + y r + y r + ... (2.45)

, •. ^ 1 ^ 2 2

(2.46) The coefficients in (p{t) have been written in such a manner that differentiation of (p{t) with respect to t provides a simple expression. We shall see later that (p'(t) is used when deriving the formulae which relate the surface to its eikonal.

The influence of ÖQ, which only gives the translation of the origin of our coordinate system, was already considered in section 2.3. Thus, for the sake of convenience we may in future assume a^ to be equal to zero. The coefficient a^ in (2.45) indicates that, when it is non-zero, the surface equation contains a conic term. Thus at its vertex point, the direction of the normal to the surface is undefined. See fig. 14.

Fig. 14. In the vertex point of an axially symmetrical surface containing a conical term, the direction of the surface normal is undefined.

It is clear that at ? = 0 we cannot expect a unique relation between ray directions before and after refraction. If, moreover, a2 • • • a„ are zero there will be no unique relation between ray directions and optical path-length, as we are in the situation mentioned in section 2.2.4. Therefore the expansion of S is in principle not possible in the case of the number a^ being non-zero. In practice the conical vertex point is avoided by applying a surface of revolution without a central circular section of radius Q. See fig. 15.

Fig. 15. The conical vertex point is avoided in practice by leaving out a circular section with radius Q. In this case instead of t we take

(2.47) t' X = = w

"1

+

-¥•

' + .

but we see that for / ' equal to zero still the normal to the surface is not uniquely defined. In principle this problem can be solved by using a sufficient approximation of the original surface equation (2.45) beginning with second degree terms. This is possible as the central part of the surface with radius Q is not used in practice. The solution is of the following kind. See fig. 16.

Fig. 16. The avoiding of a linear term in the surface equation.

We require that the equation of the dotted curve is such that there is no discon-tinuity in dx/dw. It is always possible to apply such a dotted curve, its form being immaterial, as the central part of the surface is not to be used.

Thus the surface equation to be applied assumes the form:

x = aht' + ^t'+^t'+..\ (2.49)

2.6.3 The eikonal of a rotationally symmetrical surface of the form X = F{(y^ + zy}

When following the procedures of sections 2.4 and 2.5.4 we find, using

X = a j ^ r + • ƒ / • ' - h ...J = aKO (2.49) (v^ + z^)* w t = ^^ ' = — (2.43) (2.44) a a f{x,w,a) = a(pii)-x = 0 (2.50) ƒ. = - 1 ƒ . = <p'(t) (2.51) ƒ, = (p(t)-t(p'{t)

Now we have to apply equation (2.40) in one way or another, in order to obtain a series expansion of the eikonal <f. Therefore we make use of the following theorem [23, p. 129; 24, p. 14]: Given

il/{z) = 4)(z)-v

(^(M) = v; <l)'iu) # 0,

an analytical function ƒ can, in a certain domain of values of z, be expanded in the form:

f{z) = f{u) + Z W f L J ^ . - l — [/'(u){^(«)n + «„ (2.52)

m=i m\ du

In our case, for instance, following (2.17), (2.19), (2.40), we wish to expand/J/j^ into a series of powers of fJf^.

Thus, according to (2.52), we have: J a ƒ ^/a

Tx'Kfx

, = 0 ^\fj n!dr"-'Ldf V/J V/J/J J' = o

= -(p'it)= -{a2t + a3t^ + a^t^ + ...}

f)

=^

JxJl = 0

We see that the general term can be written as

fJfxJ J

which can easily be transformed into ^{tcp'it)-<p{t)}-\-^^

dt [-(Pit) < = 0

•{-<p'{t)}" (2.56a)

n! df" ' L (,<P(OJ J' = o which gives, by applying (2.54):

G„ = 1 d" t(p"it)

n! dr" ' L \a2 + a3t + a^t"... Defining

g{t) = a2 + a3t + a^t^ + ... it is easy to see that:

t(p"{t) = tg{t) + t^g'it) Thus the general term becomes

^" = n7d7^L"(ö^^7(^J, = o-^^'^^'^^

In order to transform G„ into a more convenient form, we apply Leibniz' rule of repeated differentiation:

(u-vy-' = t;-M""' + ['J'y^'-u""-'> + ('2y^>-u ,(2).„('"-2) _{4-... (2.61)}

Then the coeflrtcient of G„ becomes nlldf-'Kgit)/ df-^Xgit)) i = o

+

£WV2,(„-„^('?Ï5V

L(o""d'=o

As the transformation of (2.38) into (2.40) is reversible we shall, in order to obtain symmetrical transformation formulae, write the series expansion of the eikonal <f in a similar form to the surface equation.

We have found that fjf^ is a power series of fjf^ and thus, according to (2.40) S/{aL) is a power series of W/L. Defining

- ?

^(^) = h^^ + ^ ^ 3 ^ ^ ^ 4

+

...

(2.63)

(2.64)

and when the surface equation is given in the homogeneous form:

« 2 2 ^ 3 3 X = a{^t^+-^t^ + ... where t =

(/ + 0

Thus, according to (2.52), we have:

''-"'^ , r f / ~ Y . - •

fx \fxMo ^\fj n! dt"-'

d ffa\ ( t V

We see that the general term can be written as G„ = 1

d"-n! dt"

-^{tcp'it)-cp{t)}-\-L^^

.dt [-(Pit)

which can easily be transformed into G„ = -1 d"

n! dt" t(p"it) (p'it) •Wit)}"

(2.53)

(2.54)

(2.55)

[-(p'it)}" (2.56a)

(2.56b)

which gives, by applying (2.54):

G =

1 d ^

n! dt"-' t<p"it) _{fli + a^t + aj ...} •Wit)}" Defining

git) = cf2 + a3' + «4<^ + ... it is easy to see that:

t(p"it) = tgit) + t^g'it) Thus the general term becomes

G „ = i ^ r ^ + ^ -Wit)}"

n\dt"-'lgit)"-'^ 0(0" J-o ^^^^^

In order to transform G„ into a more convenient form, we apply Leibniz' rule of repeated differentiation:

(u-t;)^'"'= t;-t/""> + ('J'y'>-u<'"-" + ( 2 V"-"""'" + .

Then the coefficient of G, becomes

1 r d""' / 1 \"-' d"-^ f 1 y-' 1

n ! L d f " - ' V W dt"-'\git) J, = o

+

+ —

^>V2.(n-i)^r^-:«V

nlL dt"-Agit)) dr-^V9(07J'=o

L(0""'i=o

As the transformation of (2.38) into (2.40) is reversible we shall, in order to obtain symmetrical transformation formulae, write the series expansion of the eikonal ê in a similar form to the surface equation.

We have found that f Jf^ is a power series of f Jf^ and thus, according to (2.40) SJiaL) is a power series of W jL. Defining

L

^ ( ^ ) = ^ ^ 2 + ^ ^ 3 ^ ^ 4 ^ 4

+

2 ' 3 4

and when the surface equation is given in the homogeneous form:

(2.63) (2.64) X = a < ; y ( ' + y / ^ + ... where t =

(/ + z¥

we find as the surface eikonal:

\

S = aLil/i3^) = aL\^^^ + ^^^ + ...\ (2.65)

b„ in (2.65) being the coefficient of / " " ' in the series of ascending powers of

-L^(a2 + a3t+...y-" (2.66) 02 for instance is the coefficient o f / ° in ia2+a3t+ ...)-'/I etc.

As was stated before, the transformation is made symmetrically reversible, thus when (2.65) is given, (2.49) is found by interchanging a„ and b„ in the transformation procedure (2.66). We shall give the transformation formulae for the first six coeffi-cients in table 1. TABLE 1 «2 b. = -"

ai

, _ —5ai + 5a2a3a4 — asa2 «2

_ \Aa^ — 2Vala^a2 + 'iala2+6ala3a^ — a^al

06 - -"i

«2

b,= — 42a3 + 8403^0402 ~ ^^ai<JsCii — ISa^a^a^ + 7a;ia^ai + la^a^al — a-^ai

„ 11 a , 1 "2 = TT 02 0 , 31

It is seen from the tabulated results that ^2 must not be equal to zero. This is in accor-dance with the mathematical theorem applied, which required that <^'(0) 7^ 0. It is easily verified that

(l>iz) = a2t + a3t^ + aj^ + ... (2.67) Thus the series expansion of <f, as given in this section, is only possible when the

radius of curvature in the vertex point of the refractive surface has a finite value. 2.6.4 Series expansion of the eikonal & of a surface which is expanded into a series of

powers of y^+z^

As observed in section 2.6.1, surfaces of rotational symmetry were hitherto only expanded into series of ascending powers of y ' + z'. T. Smith [22] gave the procedure for finding S for this case. As the actual transformation formulae were hitherto not worked out in detail, we shall derive them below. Here the derivation of the trans-formation formulae of the coefficients in the series expansions is somewhat simpler than in section 2.6.3. We shall, for the sake of convenience, as much as possible follow the procedure of section 2.6.3.

Defining

w^ = y' + z^ (2.68) w'

V = ^ (2.69) a^

the homogeneous surface equation is of a similar type to (2.45) and (2.46)

X = ^cpiv) (2.70)

The factor | is introduced to avoid numerical factors below. We find:

fix,w,a) = a(piv)-2x = 0 (2.71)

fx= - 2

ƒ„ = 2vW'iv)

fa = (Piv)-2v(p'iv)

(2.72)

As in section 2.6.3 we apply the theorem given there in order to expand fjf^ into a

power series, but as the surface is given here as a power series of y ' + z ' it will appear useful to expand into a series of powers of

ifjfx)^-fa

= V(p'iv)-i(piv) (2.73)

fx = v(p'\v) (2.74)

Looking at (2.74) it becomes clear why expansion into powers of if J f^Y is preferable. For we find, in a manner analogous to that in section 2.6.3, the coefficient C2„ of

ifjfx?"-t^2n — 1 d" --{v(p'iv)-\(piv)}

dv iv(p n i d i ; " - '

The factor v" automatically disappears, and we find:

n!d."-'Ll 2 "^ jV(<;)j .

-11

(2.76)

Formula (2.76) already contains a numerator in a convenient form which saves us the trouble of finding a relation between numerator and denominator, as (2.57) and (2.59). The application of Leibniz' rule gives us, as the final result:

^In 1 1 Defining ^ = 2 n ( 2 n - l ) ( n - 1 ) ! W' _dv"-' Wiv) 1 (2.77) (2.78)

we find by following the same procedure as was given in (2.65) and (2.66):

H^) = di3r + ^ ^ 2 _^ ^ _ ^ 3 _ _(2.79) ^ 1 Cj 2 C3 3 x = Ac,v + ^v^ + ^v^

^^LHT) = ^LU^ + '-^^' +

The transformation formulae are computed in an analogous way. Here d„ in (2.81) is the coefficient of y"-' in the series of ascending powers of

2 ; i ^ { c i + C 2 ! ; + C3.'+ . . . } ' - ' " (2.82)

We shall give the transformation formulae for the first six coefficients in a simpler form than (2.82) suggests, after having dealt with the length a in the surface equation.

2.7 The length a

The method of finding the eikonal of a refractive surface makes use of the possibility of writing the surface equation in a homogeneous form (2.11), (2.17), (2.19). The con-stant a in the equation, which has the dimension of a length was hitherto not given a value except in sections 2.5.1, 2.5.2, 2.5.3. There the procedure of choosing a value did not give rise to any difficulties, as the adequate choice was strongly suggested by the surface equation itself. In section 2.6 we do not, at first glance, have any preference. We shall here deal with two lengths in geometrical optics, favourable with respect to their possibilities of application. The two choices for this length are the radius of the circular entrance pupil of the system and, for instance, its focal length. As our system consists of only one surface it is clear that we shall in the first case take the maximum possible height of incidence, and in the second case, in order to avoid ambiguity, the radius of curvature at the vertex point of the surface.

We see from (2.44), (2.45), (2.69), (2.70) that, in order to achieve a quick convergence TABLE 2

[ ( /

+

z')M

of the series, it is important that t and v attain only small values. Therefore it is in general unadvisable to use the maximum height of incidence as at the rim / and v will have the value 1. If we select the radius of curvature, we are in practice always certain of finding absolute values of / and v which are less than 1, except in the rare case of a hemispherical surface. Moreover, the paraxial properties of an optical surface are determined by the radius of curvature.

The substitution of the radius of curvature /"vertex ' " the surface equation does not usually give rise to any difficulties. Only when r attains a very large positive or neg-ative value shall we make use of another value for a.

An advantage of the application of /"venex ''^s in the well-known fact that the coefficients Oj in (2.45) and Ci in (2.80) both become equal to 1. As a result, the trans-formation formulae can be simplified. They are given in tables 2 and 3.

TABLE 3 d, = 1 di = -Ci dj = - C 3 + 3C2' ^4 = — C4 + 8f2C3 —12C2^ d^ = — f5+5C3'—55C2^C3+10C2C4 + 55C* de = -C6+12c2C5 + 12f3C4-78c2C3'-78c2'c4 + 364c2^C3-273c2' c, = 1 C2 = —d2 etc.

When b2„+1 = 02„+1 = 0 where n is a natural number, we observe that Z)2„ = d„ and ^2n = c„, as was to be expected.

2.8 Example of the application of the formulae found

We shall consider a spherical surface, for which we already derived expression (2.35). See fig. 17.

The radius of curvature according to our origin conventions is + a . We shall ex-pand (2.35) into a series of powers of ^•.

+ o(L' + M' + iV')* = all 1 + ^ t ^ I = oL(l+.r)* =

= oL(l + V 2 ^ - ' / 8 . ^ ' + ' / : 6 - ^ ' - ' / l 2 8 ^ ' + ' / 2 5 6 . ^ ' - ' V . 0 2 4 ^ * + - ) (2.83)

© / ©

Fig. 17. The eikonal of a spherical surface with radius of curvature +a. Translation of the origin of the coordinate system from the centre of curvature to the vertex point.

When applying the formulae of section 2.7 we should bear in mind, that we must either translate the origin of the coordinate system from the centre of curvature to the vertex point of the surface, or add a term to the series expansion of S. It is immediately seen from (2.36) that this constant is equal to at.

The surface equation is: x' + y' + z' = a'

X = - o ( 1 - ^-^] = - 0 ( 1 - i ; ) *

= -ciii-'l2V-'lsv'-'i\e'^'-'U2,v^-'i256v'-''ho2y-.:) (2-84) Following the method of section 2.6, and adding the constant — 2 to account for the

displacement of the origin of the coordinate system, the surface n is also:

X = ^(piv) = ^[ -2 + CiV + ^v^ +^v^ + (2.85)

and by using table 3 we find: Cl = 1

Cl = / 2

d, = 1 di = - V 2

Ci C4 Cs Cé — = = = / 8 ' / 1 6 3 5 , / l 2 8 6 3 ; / 2 5 6 ^3 = 78 " 5 = / l 2 8 d^ = / 2 5 6 Thus: o . / , . ^ . - ' / 2 _ ^ 2 ^ ^ _ ^ 3 ^ = ^ L l 2 + ^ + ^ S = 0L(1 + V 2 ^ - V 8 ^ ' + V , 6 . ^ ' + - ) which is in fact identical to (2.83).

The fact, curious at first sight, that c„ and d„ alteratively differ in sign is caused by the circumstance that the surface equation and the eikonal formula are of the same type: a(l±iS)*.

2.9 The eikonal of a surface with an infinitely large rvenex

As we have seen, the formulae of section 2.6, which link the surface to its eikonal, cannot be applied in this case. However, we can always employ the method given in sections 2.4 and 2.5.4. We shall show this for the only practical case existing, viz. the so-called Schmidt plate of the first kind [1, p. 247]. The homogeneous equation is:

X = — a -4 (2.86) fix,w,a) = - - p w ^ - o ' x = 0 (2.87) fx = -a' fy, = a^w^ fa = - 3 o ' x

After some simple calculations we find, following (2.40):

w Y

a-^-a^-L

OAL

This result can be extended to the case of

(2.88)

(2.89)

X = a-a„ w (2.90)

Then we find:

n - 1 . / W 1 / ( 1 - 1 )

« " • ' - • h r T - (2-91) n " \a„L

In the case of n = 2 we can easily check the identity between (2.91) and

where

38

= a L < ' ^ ^ H (2.65)

W

CHAPTER HI

T H E E I K O N A L F U N C T I O N O F A N O P T I C A L S Y S T E M C O N S I S T I N G O F M O R E T H A N O N E R E F L E C T I V E

O R R E F R A C T I V E S U R F A C E

3.1 Description of several methods to evaluate the eikonal of an optical system of more than one surface

3.1.1 Classical method iBruns)

In section 2.2.4 we have shown that the eikonal function S for one surface can be found, when the equation of the surface is given together with the refractive indices of the two adjacent media. For one surface we already found

S = SiL,M,i:,M') , (3.1) where L and M are two directional cosines of the incident ray of light and L' and M'

two directional cosines of the refracted ray.

If we have a system consisting of more than one surface, we can write the eikonal Sj of the J-th surface as:

Sj = SjliLj, Mj, Lj, M'j) = SjiL„ Mj, Lj^„ Mj^,) (3.2) as the directional cosines in the image space of the y-th surface are the object space

variables of the ij+ l)-th surface. The eikonal function of the complete optical system of/ surfaces is in general equal to the sum of the eikonal functions of they surfaces, to which has to be added the sum of/— 1 contributions due to the distances between the vertex points. As shown in section 2.3 this latter contribution can be written as:

I « ; L ; / / (3.3)

i

where /,' is the distance between the vertex points of the /-th and the (/+ l)-th surface. Therefore, if we, for instance, take a thick lens consisting of two surfaces, we find for the eikonal S°, when the origins of the perpendiculars to the incident and the refracted ray are respectively the vertex point of the first and the last surface:

ê' = S,iL„M„L2,M2) + ^,(L2,M2) + ^,(L2,M2,L3,M3) (3.4) See fig. 18.

The eikonal S of an optical system was earlier defined as a function with only four 39

1-2 , M

Fig. 18. The eikonal of a thick lens. The existence of intermediate variables.

independent variables, two directional cosines of a ray in the object space, and two in the image space. Thus we are faced with the task of eliminating the intermediate variables. In the case of the thick lens, we have to eliminate L2 and Mj. This can be done by requiring:

dS

cLi dM, = 0 (3.5)

This relation is a direct result of Fermat's principle, which states that the optical path-length measured along a ray of light in the optical system, is stationary with respect to infinitesimal displacements [I, p. 128; 17, p. 367], which in this case are dis-placements of points an infinite distance away from the system. It is clear that this procedure can be extended to the case of n surfaces. We then find the eikonal function

<^"(Li,M,,L„+i,M„+i)

by eliminating the intermediate variables, requiring that dS dS dS

dL^ dM-, 8L„ dM„ = 0

(3.6)

(3.7)

In principle the problem is solved, as we always find just sufficient equations to elim-inate the intermediate variables. This does not, however, mean that in practice this is always easily done.

3.1.2 Numerical variant of the classical method

Methods exist of developing a given function of, say, n independent variables into a series expansion of / independent variables, while requiring that the first derivatives of the function with respect to the remaining n-i variables equal zero [25]. The exact values of these latter variables for which the function has to be stationary need not be 40

known. Therefore, when applying this method to evaluate the eikonal function, we could select the paraxial values of the intermediate variables, as these approximative values can be expressed as functions of the variables in the object and image space, without the necessity of ray tracing. As an example we take the case of a thin lens with two surfaces and achieve the elimination of the intermediate variables by means of the paraxial law:

n' n , .. _,

- = - + A (3.8)

where n and n' are the refractive indices on either side of the lens, s and s' are the distances between the axial point of the lens and respectively the object and the image and A is the paraxial refractive power of the lens. In our example we find for the two surfaces:

^ = ^ + ^.

This gives, according to the paraxial laws,

n,M2 = !LlMlél±!hM^ (3.11)

This procedure remains simple if all thicknesses in the optical system are zero, but in practice it very soon becomes tedious when lenses with non-zero thicknesses are in-troduced.

3.1.3 Ray tracing method

We shall finally deal with a ray tracing method which will prove to have much in fa-vour. The eikonal of an optical system of more than one surface can be found by tracing a number of representative rays of light through the system, and computing 41

the optical path-length of each ray between the perpendiculars originating, for in-stance, from the vertex points of the first and the last surface of the system. In an op-tical system with axial symmetry the eikonal function can be written as a function of three variables. Following Bruns we can choose «, v, w [17, p. 409] where

u = il} + M^)l2 V = iLL + MM') w = (L'^ + M'^)/2

(3.12)

This substitution facilitates computation of the series expansion representing the eikonal function. In most cases it is already sufficient to trace a very limited number (five to ten) of meridional rays [26].

3.1.4 Features of the three methods

The first two procedures, the classical method and its numerical variant both suffer from the circumstance, that with an increasing number of surfaces in the optical system, the process of finding an expression for the eikonal function rapidly becomes more and more difficult. Mathematically, the problem is as difficult for ten surfaces as it is for two, but the practical computing procedure becomes so clumsy that even with the aid of a high speed computer the calculation is quite time-consuming.

The third method appeared to us to be the best. The tracing of a number of rays of light is a simple and accurate procedure. The evaluation of <f from the information gained by ray tracing is simple too. With this method the number of variables is in-dependent of the number of surfaces in the optical system. Another fundamental asset of this procedure lies in the fact that ray tracing automatically reveals vignetting effects (see fig. 19). It is immediately noticeable if one of the rays selected does not traverse the entire optical system and which part of the surface is used, whereas with the first two methods we are compelled to consider all surfaces of the optical system as being of equal importance.

Fig. 19. Vignetting effects. 42

Now we come to the question as to whether the eikonal of a complete system is a good aid when designing the system. In the case of analyzing the system performance the system eikonal is useful, as we saw in section 2.2. For that purpose we can use the relations (3.13). aS dL dS dM dS dN = nx = ny = nz di dlL dS dM' dS dN' — n'x -n'y — n'z

which are found by differentiating equation (2.10) with respect to the directional co-sines of the ray in the object and image space. These relations link a change in the eikonal to a transverse aberration of a ray of light [6, p. 2].

The above-mentioned methods were conceived in the time before the development of electronic computers. In that period every method which in theory provided useful information concerning optical systems without the necessity of performing numerical calculations before a general discussion could take place, was considered a valuable contribution [2, p. I l l ; 4, p. 95; 25]. It is therefore in our opinion not surprising that in the literature we did not find any practical results obtained by following these methods. In an attempt to apply them we found, when following the first two methods, that in order to attain an acceptable degree of accuracy we had to cope with so many coefficients in the series expansions of the partial eikonal contributions, that it led us to the conclusion that in practice it is unadvisable to use these methods for the evalu-ation of the eikonal function of an optical system. The third method is applicable but in the designing stage we can more profitably, in analogy with the surface contribu-tions in the theory of aberracontribu-tions, apply the eikonal contribucontribu-tions of the separate elements of the optical system. The formulae derived in Chapter II, which enable us to compute these contributions, will be used to furnish a designing tool.

3.2 An automatic computing routine based on eikonal functions to improve optical systems

3.2.1 General remarks

In Chapter 1 we mentioned two essential parts of automatic designing routines. The first part, the quality criterium, is the functional value of a so-called merit function c^, which is a measure of the state of correction of an optical system. By computing the functional value of </> after each iteration, which results in the system parameters

being changed, we are able to judge whether the system is improved or not. The merit function is defined in such a manner that in order to transform an initial optical system into a system with a desired performance, we have to minimize </>. In the minimization procedure we make use of the first and second derivatives of 0 with respect to the system parameters. Thanks to the application of eikonal functions these derivatives can easily be computed.

3.2.2 The merit function ^

We evaluate tj) in the following way. Suitable rays of light, the /-th of which is given in the object space by six coordinates x,-, y,, z,-, L,-, Mj, A^,- (see section 2.2) are traced through the initial optical system. The /-th ray is given in the image space by the com-puted coordinates x / , y,', z{, L,', M{, N / . However, the optical system with the de-sired state of correction transforms the /-th ray into an image space ray characterized by coordinates x / ' , y,", z,", L,", M,", A^,". The optical path-length K,' between X;, y;, Z; attd x / , y,', z/ is computed. The value of the point characteristic K," between Xj, y,, Z; and x,", y,", z," is found by applying equation (2.6)

vi' = p^'+„'{L;(x;-x;')+M;(y;-y;')+/v;(z;-z;')} (3.i4) As the independent variables, the directional cosines of the /-th ray were found by ray

tracing, we are able, by using formulae derived in Chapter II, to compute the eikonal contributions of the elements of the optical system, and hence their sum, which equals the value of the system eikonal of the /-th ray. If, for instance, the y-th surface in the optical system is spherical and they+ 1-th surface aspherical, and if they are separated by a distance tf, we find by using equations (2.36), (2.64), (2.65), (3.3) and taking Ljj > 0 (see section 2.5.3) as the sum of their contributions

rjiiLlj + Mlj + ISlj)^ - L;,j\ + n'/jL,j + r^ +, L,.,. +, <A,,; +. (3.15) Thus we are able to compute the system eikonal Si of the /-th ray. The system eikonal

of the /-th ray in the system with the desired state of correction is approximately, using equation (2.10):

S'-: = V!' + n{L;Xi + M-j; + N.-z,-) - n'(L'/x;' + M'.'y',' + N'^z'i) (3.16) If there are m rays we define the merit function (/> as follows:

m

4>=Y^iS',-S'.f (3.17)

t = i

In the iterative correction procedure each new computing cycle leads to a new set of 44

"

S;' 's. It is immediately evident that the smallest value of (j) that can be attained is zero, which implies the identity between S/ and Si". The successive values of (j) after each iteration give at a glance an account of the effectiveness of the correction process. In the case of a series of decreasing values of (j) we can use the optical system found after several iterative steps as a new initial system. This leads to new values for the set S/', which better approximate the desired performance of the optical system.

3.2.3 The first and second derivatives of </>

We presume to be dealing with an optical system, traversed shall give the derivatives of ^ with respect to changes of the^ the y-th axial thickness, and the ^-th surface (aspherical). Change in radius of curvature of y-th surface.

4 ^ = I 2iSl-SmLf.j + Mlj + Nlj)^-L,j}

or J i=i

erf 1=1

(take positive root when L^j > 0, and negative root Change in the y-th axial thickness.

^ = f;2(^:-#;')(";A'„)

:.r,?,^<*-'>"

Change in 6„ (M = 3, 4, 5, 6, 7) of k-th surface (aspherical)

1 - = Z ^«-^/')r.L,,^,^,

COak i = l "

dbi Ita^^'^''"-'^'-*^

Change in ryenex of ^-th surface (aspherical).

It -i2(#; o(i..^.+ai.-/f')