IV Aspheric Surfaces

E. WOLF,PROGRESS IN OPTICS XXV

0 ELSEVIER SCIENCE PUBLISHERS

B.V., 1988

IV ASPHERIC SURFACES BY G~SNTERSCHULZ Zentralinstitutfur Optik und Spekirofkapie Akademie der Wissenschaftender DDR 1199 Berlin-Adlershoj; GDR

CONTENTS

1 . INTRODUCTION

....................

PAGE

351

$ 2. TYPES OF ASPHERICS AND THEIR MATHEMATICAL REPRESENTATION . . . . . . . . . . . . . . . . . . . 352

. . . . . . . . . . . . . . . . 357 5 4 . FABRICATION AND TESTING METHODS . . . . . . . 387 8 5. FIELDS OF APPLICATION . . . . . . . . . . . . . . . 391 $ 6. LIMITS OF THE IMAGING PERFORMANCE OF 403 ASPHERICS . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 410 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . 410 3. DESIGN OF ASPHERICS

6 1. Introduction Aspheric surfaces, or “aspherics”, are optical surfaces that are neither spherical nor plane and are used in imaging and nonimaging systems. A sphere has only one shape parameter, the radius of curvature, R (the limiting case is the plane with R + co). An aspheric, however, in principle may have an infinite (in practice a large) number of shape parameters. Mathematically an aspheric is generated, for example, by the rotation of an axisymmetrical plane curve about its axis. Spherical surfaces appear here as a special case of aspherics in which the rotating curve is a circular arc. Thus, because aspherics offer a much larger variety of imaging possibilities, they were considered by early scientists such as Kepler in 1611, Descartes in 1638, and Huygens in 1678 (RIEKHER[ 19641). For example, using the law of refraction, Descartes calculated aspheric lens surfaces imaging an axial point stigmatically, that is, sharply. As is well known, a real imaging of this kind is not possible with spherical surfaces. Descartes also designed a machine for grinding aspheric lenses. The first aspherics used in practice appear to date from the middle of the 18th century. Their shape was generated by retouching or figuring a polished surface, thus improving the imaging performance when compared with that of spheres (KILTZ[ 19421). This technique was not only used for telescope objectives but also for telescope mirrors, for example, the parabolic mirrors made by Short at about 1750 (RIEKHER[ 1957, p. 831). These surfaces, however, departed from spheres only by very small amounts and could be generally regarded as quadric surfaces. More complicated shapes of various types (with accuracies required for imaging systems) could neither be calculated nor made at that time. This has essentially changed only in the last few decades, in particular by the progress in computer development and by new manufacturing techniques and testing methods for aspherics. Thus the treatment of more complicated aspherics is now becoming more and more possible and effective. On the other hand, the demand for such surfaces is steadily increasing, and new applications are resulting, since many optical problems are easier to solve by using aspherics and others can be solved by using aspherics only. For example, one aspheric can often replace several 35 1

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[IV, 9 2

spherical surfaces, and certain aberration types can be removed only if aspherics are used. The optics of spherical surfaces has undergone a lengthy development. By comparison the optics of aspherics is only at its beginning, and although many problems are still unsolved, many results have already been obtained. This article will give a survey of the progress made in this field, ranging from fundamental questions and theoretical results to practical applications. The relevant literature is already so extensive that often only examples can be quoted. On the other hand, unfortunately some results that would be of interest have not been published, for example, because of their economic implications. This article on aspheric surfaces discusses their theoretical treatment and design methods, optically effective properties, and possibilities of application, with emphasis on methods and properties differing from those of spherical surfaces. Section 2 surveys the types of aspherics and their mathematical representations. Section 3 deals with methods for designing aspheric surfaces, beginning with optical purposes or goals they can achieve. Section 4 summarizes manufacturing and testing methods for aspherics. The development of effective manufacturing methods is very important, but the problems involved are less optical in nature. Current review articles and compilations of testing methods can be found in the references. Section 5 discusses the applications of aspherics in a number of fields. In view of progress so far, questions'arise regarding fundamental possibilities and general limits of the performance of aspherics. These questions are discussed in Q 6.

Q 2. Types of Aspherics and Their Mathematical Representation 2.1. SURFACES OF REVOLUTION ABOUT THE OPTICAL AXIS

Since surfaces of revolution about the optical axis are used in most cases, this article describes primarily this type of aspherics. The surfaces are used as reflecting or as refracting surfaces. Simple types are quadric surfaces that are generated by the rotation of a conic section about one of its axes. Figure I shows examples of such surfaces and their imaging properties in reflection and refraction (HERZBERGER [ 1958, p. 441). Usually the surfaces are represented in Cartesian coordinates, for example, in the form

h2 = 2R.2 - (1

+ b)z2,

(2.1)

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TYPES OF ASPHERICS AND MATHEMATICAL REPRESENTATION

REFLECTING ...

REFRACTING ....

....... ,...ELLIPSOID

....:z

................. ....PARABOLOID

Fig. 1. Some rotationally symmetrical quadric surfaces and their imaging properties. A reflecting surface (left) images an axial point at another axial point. A refracting surface (right) images the infinite axial point at another one if the ratio of the refractive indices has been appropriately chosen. The axial points represented are foci of the conic sections, and the images are perfectly sharp (stigmatic). A stigmatically imaging single lens in air results if the refracting medium is bounded on the other side by a surface passed by the rays normally, that is, by a spherical surface or a plane, respectively.

that is, 2 =

1

+ ,/1

h2/R - (1 + b)h2/R2 ’

h2 <

~

R’ l t b

,/m;

(see, for example, STAVROUDIS [ 19721). Here h = x, y and z are the Cartesian coordinates of a surface point (origin at the vertex), and the z axis is the optical axis. 1/R is the vertex curvature of the surface (R 5 0), and b is a measure of the departure from sphericity (b # 0 for aspherics). Different values of b indicate different surface forms: hyperboloid (b < - l), paraboloid (b = - l), and ellipsoid (b > - 1). Quadric surfaces according to fig. 1 are examples of Cartesian surfaces (named after Descartes, see 5 l), and they have the property of forming a sharp image of an axial point. Refracting Cartesian surfaces of the fourth order are described by HERZBERGER [ 1958, p. 1861: [(h2 + z’) (n’2- n’) - 2z(n’L’ = 4(L

+ L ‘ ) [(h’

t

2’)

+ nL)]’ (n’L’+ n”L) + 2zLL’(n - n ’ ) ] .

(2.3)

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ASPHERIC SURFACES

Here n and n’ are the refractive indices in the object and image space, respectively, L is the optical path length from the axial object point to the vertex, and L’ is the corresponding length from the vertex to the image point. The two points are imaged at one another stigmatically. If in eq. ( 2 . 2 ) ~ is ~substituted for h2, one obtains the meridional curve of the aspheric defined by the intersection of the surface with the meridional plane x = 0. This curve is, with respect to y, differentiable an arbitrary number of times. More general aspherics with this property are often represented in the form (HERZBERGER and HOADLEY[ 1946]), see fig. 2a, z = a2h2

+ a4h4 + a6h6 + . . .

(2.4)

Expanding eq. (2.2) in a power series in h and comparing this series with (2.4) gives (HOPKINS[ 1950, p. 1511, BORN and WOLF[ 1964, p. 1381) 1 a2 = - , 2R

l+b

a4=-.

8R3

Instead of eq. (2.4), other forms of representation have also been used including a series representation in polar coordinates (RUSINOV[ 1973, p. 9]), a generalization of eq. (2.1) with higher powers of z, and hybrid forms of (2.2) and (2.4); and series with Chebychev polynomials (BRAAT [ 1983a]), Zernike’s polynomials (KROSSand SCHUHMANN [ 1985]), and with transcendental functions

(01

Ibl

Icl

Fig. 2. Rotationally symmetrical aspherics and mathematical properties of their meridian section representations z = z ( y ) ( y = & h). (a) z ( y ) is differentiable an arbitrary number oftimes (e.g., a polynomial with even powers ofy). (b) z ( y ) is everywhere differentiable at least once (twice, for example, for cubic spline functions), and in discrete points (nodes, represented in the figure) certain higher derivatives jump. (c) z ( y ) is, for y # 0, differentiable at least once, and at the vertex ( y = 0) the first derivative jumps; the surface is optically used except in the immediate vicinity of the axis or only at larger distances from the axis. The parameters of the aspheric in case (a) are of global character and in case (b) of local character; in case (c) either kind is possible.

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355

(RODGERS [ 19841). In principle, all these representations are equivalent, but for practical computation they are not. For example, it is advantageous ifa series can be used for which only a few series terms are necessary. Sometimes parameter representations

h

=

h(a) ,

z

= z(a)

for aspherics, which have been known for alongtime (SCHWARZSCHILD [ 1905, no. 2]), are also useful. In each case the appropriate form of representation depends on the problem to be solved. For this reason sometimes other representations are used that have a limited differentiability(affecting the wave aberration correspondingly). Examples are [ 1978]), which have power series with odd or nonintegral exponents (IZUMIYA a restricted differentiability at the vertex of the aspheric. For certain purposes representations with an even more limited differentiability are used. This is the case, for example, if aspheric shapes are obtained from differential equations that are solved numerically in small steps, point by point (see 3.6). In these and other cases (see 5 3.7) the meridional curve of an aspheric can be defined by a number of points (nodes) and curve sectionsjoining these points (fig. 2b). This can be done, for example, by cubic spline functions (RIGLERand VOGL [ 19711). Here z is, for each curve section, a polynomial of the third degree in y . Generally, the polynomial coefficients have different values in different curve sections. However, they have to fulfill the condition that the total meridional curve and its first and second derivatives are continuous everywhere. The third derivative usually jumps at the nodes. The positions of the nodes and the first derivative at the end node determine the shape of the meridional curve. If continuity of the second derivative (necessary for cubic spline functions) is not required, each curve section shown in fig. 2b can be represented by a part of a conic section, one axis of which coincides with the optical axis (SCHULZ [ 19851). This simplifies the ray tracing (see 5 3.4). If the continuity of the first derivative is abandoned at the vertex, surface types similar to those shown in fig. 2c are possible. A simple special case is a shallow cone of an axicon, which images an axial point as an axial line. Surface types that have curved meridian section representations, as shown in fig. 2c, necessarily result if a single lens is to bring a large aperture bundle of axial-parallel rays to the same sharp focus for two wavelengths (SCHULZ[ 19831). Surface types similar to those shown in fig. 2c at larger distances from the axis are also proposed or used for telescope mirrors for oblique incidence and for concentrators (see § 5.6). Surfaces with steplike discontinuities in Fresnel lenses are not considered in this article.

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[IV, 0 2

The shape parameters of an aspheric according to fig. 2b are essentially the z values of the nodes. These parameters are localized, each of them practically influencing only a limited part of the total surface. In contrast to this, the shape parameters of fig. 2a, for example, the coefficients of eq. (2.4), are of global character. If global parameters in a suitable series representation can be used, often a few parameters are sufficient. However, in cases where this is not possible, up to hundreds of local parameters can sometimes be determined recursively (SCHULZ[ 19841).

2.2. OTHER TYPES OF ASPHERICS

Surface types of a different character are used if the optical problem to be solved is not rotationally symmetrical. Nevertheless, in this case the surfaces themselves may have a rotation axis, but this is not always the optical axis or an axis running through the surface center. An example is a certain type of ophthalmic lens with a continuous run of the refractive power from the reading to the distance area (see 8 5.3). In this example the rotational symmetry of the surface facilitates their manufacture. Such surfaces may have complicated shapes. Surfaces of a simpler form, also with a rotation axis, are toric surfaces, which are generated by the rotation of a circle or a circular arc about an axis running in the plane of the circle but not through its center (see fig. 3a). A toric surface has two shape parameters. Of course, only a small part of the surface is used optically (in the case of a lens surface, for example, the neighborhood of the coordinate origin 0 of fig. 3). A

Fig. 3. Toric surface,generated by the rotation of a circle (radius R , ) about an axis A-A, which lies in the plane of the circle. R , and R, are the shape parameters of the surface. In the case of R , < R,, as shown, a bicycle-tire type is obtained.

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351

simple limiting case of a toric surface is a cylinder (R,+ co in fig. 3). Toric mirrors are also used, particularly for grazing incidence. A toric surface in the neighborhood of point 0 in fig. 3 is a special case of a surface of the more general form z=

2 a,x”y’J. LJ

(2.6)

A surface described by eq. (2.6) has, in general, no rotation axis but only two planes of symmetry,namely, the ( x , z ) and the ( y , z ) planes. Forms like eq. (2.6) can be used in optimization programs (HUGUES,BABOLATand BACCHUS [ 19831). The shape parameters ai, of an aspheric (2.6) are of global character, as are those of eq. (2.4) (cf. fig. 2a). Local parameters can be used as well not only for axisymmetrical surfaces (fig. 2b) but also for surfaces without rotational symmetry. In particular, spline surfaces of this kind have been used (VOGL, RICLERand CANTY[ 19711, STACY[ 19841). As is well known, toric surfaces have been applied for a long time in spectacle lenses for the correction of astigmatism. Such spectacle lens surfaces are not spherical, of course, but their form is comparatively simple and their departures from a sphere matter primarily in the paraxial region; they are generally not regarded as “aspherics”. This term is used mostly for surfaces of a more complicated form (cf. BORNand WOLF[ 1964,p. 197]), which are emphasized in this article.

8 3. Design of Aspherics Since there are a variety of methods for designing aspheric surfaces that require several different survey approaches, a classification of these methods is difficult and of limited value. Anyhow, the design is conditioned by the goal to be reached, and therefore $ 3.1 gives a survey of general optical goals of designing aspherics. In 3.2 examples of simple solutions are given, and for more complicated cases a near-axis region is often determined first and separately, as described in Q 3.3. For further calculations, ray tracing (see $ 3.4) is a frequently used technique. The final shape and position of the optical surfaces are often determined by optimization methods (see $ 3.3, proceeding from results in a near-axis region and using ray tracing methods. In other cases, however, differential equations for the aspheric surfaces are set up and solved (see $ 3.6). In this case the elements of a surface are determined successively.

358

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ASPHERIC SURFACES

Other possibilities for their determination in succession are described in 5 3.7. Optical problems and the principles of designing aspherics are emphasized here. Of the many special methods, only a few examples can be outlined. (Properly speaking, each special goal necessitates its own particular method or design variant.) Required computer programs and special questions of Computation will not be discussed in this article.

3.1. DESIGN GOALS

Two main categories can be distinguished in considering the optical goals of designing aspherics: first, improvement of the imaging properties and, second, proper guiding and shaping of the illuminating beams or power flows. The first category applies to an imaging system that should meet certain requirements of imaging performance. In particular, aberrations will be made equal to zero or will be minimized. This applies either to certain aberration types or to a global or mean aberration. Aberration types of rotationally symmetrical systems can be classified if the wave aberration W is expanded in a series with the variables r, e, and 0 according to fig. 4;for example, in the form (HOPKINS [ 1950, p. 481):

W = W ( r , ~ , e ) = , c 4 , @ 4 +,C3,re3cose

+ 2 C 2 2 r 2 ~ 2 ~ o+s 22C2,r2~2 0 + ,C,,r3ecose + 0c60e6 + , c , , ~ ~ ~ c o. s. - ~. +

(3- 1)

Here the coefficients C are functions of the system parameters including the parameters of the aspherics (for the form of these functions see 3 3.3). Arranging the terms in expansion (3.1) in ascending powers of r or e results in the form (3.2) or (3.3), respectively:

w,(e)+ Wde, e)r + w2(e,6)r2 +. W(r, e, 0) = w,(r, e)e i- w2(r, 8)g2 +. . . . W ,e, 0) =

a

,

(3.2) (3.3)

The functions W, in expansion (3.2) and w , in expansion (3.3) are known expressions (SCHULZ[ 1982, 19851) containing the coefficients C of (3.1). The connection between the power series expansions (3.1), (3.2), and (3.3) is illustrated in fig. 21 (see 3 6.2). By means of aspheric surfaces a limited number of the coefficients C can be made equal to zero (see 3 3.3); these coefficients, for example, may be some or all of the five Seidel aberration coefficients oC40

359

DESIGN OF ASPHERICS

P

OPTICAL

SYSTEM

iQ

t OBJECT PLANE

p’

IMAGE PLANE

PUPIL PLANE (ENTRANCE OR EXIT PUPIL )

Fig. 4. Rotationally symmetrical imaging system (schematic) with coordinates r, e, and 0 in the object plane and the pupil plane. Because of the rotational symmetry, an arbitrary object point P can be assumed to lie in the meridional plane; r and/or e may be normalized.

for spherical aberration to 3Cll for distortion. This is also possible with spherical surfaces; however, using aspherics generally reduces the number of surfaces required or makes possible a freer choice of the paraxial system parameters. Moreover, not only single coefficients C but also total functions W, or w, (each function having an infinite number of coefficients C) can be made equal to zero. Examples include the following (see 8 3.2, 3.6, and 3.7): W,(d I 0 ,

axial stigmatism, i.e., absence of spherical aberration of all orders,

W,,(Q)I W,(Q,0) = 0 ,

axial stigmatism and fulfillment of the sine condition,

wl(r, e) = 0 ,

absence of distortion in the total field,

wl(r, 0) E w2(r, 0) = 0 , absence of distortion, field curvature, and astigmatism in the total field.

Spherical surfaces generally cannot meet these requirements (cf. 3 6.2). Requirements containing To(@) = 0 can be useful in cases of small fields and wide pencils, and wl(r, 0) = 0 can be useful in cases of large fields and thin pencils. In other cases, however, where neither the field angle nor the pencil diameter is small, instead of special aberration types a global function (merit function) containing a number of suitably weighted aberrations is often used. This function has to be minimized by varying the system parameters (see 5 3.5). The imaging also may be wavelength selective, especially in spectroscopic

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ASPHERIC SURFACES

[IVY§ 3

instruments. Corresponding imaging requirements influence or determine the surface shapes of diffraction grating substrates, which may be aspheric. The second category of goals does not relate to imaging performance but to beam shaping and similar problems, for example, for the purpose of uniform distribution or concentration of illuminating energy. Here a correspondence between every point P and its image P’ within a field (fig. 4) is generally not of interest. Rather, other requirements have to be met, such as higher or more uniform energy density. In addition to earlier applications (e.g., condenser optics) a number of new applications have emerged, concerning, for example, the focusing and shaping of laser beams. For such purposes nonspherical surfaces have turned out to be particularly useful or indispensable, and for their design a number of special methods have been developed. This article will mainly consider methods concerned with surfaces that have higher accuracy requirements. Achromatism or similar requirements can also be included in the goals of designing aspherics.

3.2. SIMPLE SOLUTIONS USING CONIC SECTIONS

Very simple solutions with conic sections are shown in fig. 1. In addition, nonspherical (mostly reflecting) surfaces in a number of cases are designed on the basis of conic sections. In a combined refractive-reflective system the focusing properties of two reflecting ellipsoid parts have been used for an approximately uniform illumination of a small spherical target from nearly all sides by two laser beams (THOMAS [ 19751, BRUECKNER and HOWARD[ 19751). Many other reflector surfaces that are used for a variety of purposes have very low accuracy and are generally not regarded as “optical surfaces” and “asphencs”. Conic sections also play an important part in their design (ELMER[ 19781). Certain nonimaging concentrators (for optimal collection of radiation energy) are also designed by using conic sections (WELFORDand WINSTON[ 1978, p. 481). The geometrical basis of a compound parabolic concentrator is formed by a part of a parabola whose axis makes an angle a with the optical axis (see fig. 5). A concentrator surface is generated by the rotation of this parabola part about the optical axis. In other cases as well, conic sections whose axis does not coincide with the optical axis have been suggested or used for the design of reflecting systems (BAKKEN[ 19741, VARGADY[ 19751, DOHERTY [ 19831).

36 1

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DESIGN OF ASPHERICS

PARAB'"~ U LA HPART

/

F' I

'/

OPTICAL AXIS

/

ti'<

I

0

I

/\ P -. APERTURE

p'

ENTRY APERTURE

Fig. 5. Construction of a compound parabolic concentrator. The meridional rays incident at the maximum angle a from the optical axis are required to be rellected in such a way that they just pass through the edge (F) of the exit aperture. Thus the intersection of the concentrator surface with the meridional plane is a part of a parabola with the axis direction angle a and the focus F. The parabola is then completely determined by the condition that the symmetrical point F' is a point of the parabola. The meridional rays with angles < a pass through the exit aperture.

The rneridional curves of surfaces calculated on the basis of conic sections need not be conic sections themselves, as can be seen, for example, in the [ 1979, 19811). This design following geometrical design procedure (MERTZ principle is shown in fig. 6, where the spherical aberration of a spherical primary mirror is corrected by a smaller aspheric secondary mirror. Any axial-parallel ray incident from the left should pass exactly through the focus F. Its optical path length [ABCDF] is, according to Fermat's principle, a SPHERICAL PRIMARY I

INCIDENT RAY

:A I

I I I I

+ I I I I I I

I I

/

I

Fig. 6. A geometrical design principle for removing the spherical aberration of a spherical mirror by an aspheric secondary mirror. The figure shows the determination of an arbitrary point D of the secondary mirror in the meridional section.

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ASPHERIC SURFACES

[IV,§ 3

constant independent of the incidence height; it can immediately be determined from the vertex positions of the two surfaces. If the radius of the sphere is also known, points B and C can be easily determined from A by ray tracing, and hence the optical path lengths [ABC] and [CDF] = [ABCDF] - [ABC] are known as well. Thus the dashed ellipse with foci C and F and major axis length [CDF] is determined, and the intersection D of the ellipse and the ray BC can be calculated. D is a point of the aspheric mirror. The normal to the ellipse at D can be regarded as a normal to the mirror. According to this principle, any number of surface points and normals can be determined. The rays used for the determination are aberration-free. A reflecting quadric surface of revolution according to fig. 1 images an axial point at another axial point stigmatically but does not fulfill the sine condition. Improved imaging properties can be reached by combiningtwo quadric surfaces in such a manner that the axial image point is again stigmatic and the sine condition is fulfilled in a considerably better approximation. Such mirror combinations for grazing incidence (WOLTER[ 19521) are the basis of a number of X-ray telescopes, also with modified surface forms (see 5 3.6 and 5 5.4). Their design is then more complicated.

3.3. DETERMINATIONS IN NEAR-AXIS REGIONS

It is often useful to begin the calculations in a “near-axis region”, which corresponds to small field angles and pencil diameters. This means (apart from paraxial precalculations not specific for aspherics) that certain higher powers of the field and pupil variables are neglected. In the Seidel region, for example, all the terms of eq. (3.1) with the sum of the power exponents of r and Q greater than 4 are neglected. In this approximation the aberrations of the system to be determined can be calculated if the five first-order aberration coefficients oC40, C, , 2C22,zC2, and 3C, are known. Therefore these coefficients have to be determined, in particular as functions of the system parameters including the parameters of the aspherics. These so-called Seidel aberration coefficients are also denoted by S,, SII,SIII,(SIII+ S,,) and S, or by B, F, C, D and E, apart from different normalizations and references to the object or image space of the field and pupil variables. The determination of these first-order aberration coefficientsfor an arbitrary number of aspherics was already known at the time of SCHWARZSCHILD

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363

[1905]. Since the form of these coefficients as functions of the system parameters is rather complex, even for systems with a small number of spherical surfaces, it is usually given in several steps rather than directly. A description of the procedure can be found in a number of books. As an example, the simplest of these functions, that is the coefficient B of the spherical aberration, is here given according to BORN and WOLF [ 1964, p. 2201:

This formula applies to a system of refracting surfaces with the numbers i (i = 1,2, . ..); the quantities sj and sl! indicate the position of the object plane and of its paraxial images and can be obtained in turn from the equations

and for example, from s1 (distance from the first surface to the object plane) in the succession s;, s2, s;, sj, ..., if the vertex curvatures l/ri, the refractive indices ni and n,- I , and the vertex distances di (between surfaces i and i + 1) are given. The auxiliary quantities hi (paraxial ray heights) can then be obtained successively from

where hl is normalized by h, = first surface,

-

1 if the aperture stop is at the vertex of the

The quantities bi are asphericity parameters of the surfaces; they are the quantities b in the representation (2.2) or (2.4) with (2.5) of the respective

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surfaces. Equations (3.4) to (3.8) define the aberration coefficient B as a function of the system parameters s1,d,, l/r,, ni, and b, for a given stop position. The other quantities have been eliminated. As can be seen from (3.4), B depends linearly on the asphericity parameters b,. This holds also for the other Seidel aberration coefficients C, D, E and F. However, the difference

which determines the Petzval curvature of the field, is independent of the quantities 6,. From the first-order aberration coefficients, values of the wave or the ray aberration can be determined in the Seidel approximation, which is based on the assumption that the wave normals or rays form small angles not only with the optical axis but also (for surfaces corresponding to fig. 2a or b) with the surface normals. The latter assumption is not fulfilled for grazing-incidence mirror systems. Here the rays form small angles with the surfaces themselves. If in this case the rays also form small angles with the optical axis, corresponding low-order aberrations for grazing incidence can be investigated and calculated (WOLTER[ 19711, WINKLERand KORSCH[ 19771; cf. also WERNER[ 19771). The investigations are generally restricted to the most important cases, in particular two-mirror systems. In addition to the first-order aberration coefficients oC,, to 3C1I in eq. (3. l), higher-order aberrations are also considered; these correspond to the coefficients oC,o, Csl, . . . . The coefficients of the next-higher order have received considerable attention. (There are nine coefficients of this kind which are independent of each other; cf. fig. 21 in $ 6.2.) In principle, such coefficients can also be expressed as functions of independent system parameters including the parameters of the aspherics. Publications on higher-order aberration theory [ 19651 (see also GAJ[ 19711, for example). The have been reviewed by FOCKE expressions obtained are still more complex than the expressions for the first-order aberration coefficients. Naturally, there is also a greater complexity in cases where aberration coefficients for nonrotational-symmetrical systems are determined. The case where the system comprises, apart from planes and spherical surfaces, only cylindrical or toric surfaces is comparatively simple. Here aberrations have been determined in particular for anamorphic imaging (WYNNE[1954], BRUDER[ 19601, HACKENBERGER and KLEBE[ 19811). Furthermore, there are investigations on aberration coefficients for systems containing surfaces of the

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form exx2

+ eyy2 = 22 - @Z2

(KLEBE[ 19831)with the z axis being the optical axis. For surfaces of revolution with ex = Q, = l/R, this equation takes the form of eq. (2.1). The knowledgeof aberration coefficientsis one of the bases for the determination of a near-axis region of an optical system. If aspherics are used, however, additional questions emerge. For example, how many and which of the surfaces should be aspherics? Where (within the total system) should the aspherics be located? These questions arise especially if a greater number of surfaces, particularly for an objective, are considered. The answers are of practical interest, since aspherics generally require a greater manufacturing expenditure than spherical surfaces, so that their number and optimal use are important. For example, one effectively applied aspheric can often improve the imaging performance considerably or can make several spherical surfaces redundant. In a first approximation such questions can be investigated in the Seidel region, together with a determination of the paraxial system parameters, for example, of the quantities d,, l/ri, nj, and the field and stop positions. In a tentatively fixed paraxial system, for example, one wants to determine in a simple way which surface or surfaces should become aspheric for an effective control of one or several of the Seidel aberrations. In this regard it is well known that a spherical aberration should be corrected by a surface in the pupil and distortion should be corrected by a surface near the object plane or an image of it. More comprehensive results can be obtained, for example, by [ 19781). For this diagram two means of the Delano diagram (BESENMATTER paraxial rays are traced: One ray starts from the axial object point and has the paraxial height hi, and the other ray passes through the axial pupil point (principalray) and has the paraxial height H,on the surface i. In the (H, h) plane each surface i is then characterized by a point H = Hi, h = hi.The normalized ratio Hi/hi quantitativelydetermines the weight by which an aspheric deformation (bi# 0)of the surface i influences the individual Seidel aberrations. If, for example, I Hi/hi/is small, primarily the spherical aberration can be controlled, and if I Hi/h,I is large, primarily the distortion. If I HJh, I has a value in a middle range, the asphericity parameter b, influences all five Seidel aberrations by nearly equal weights. In this way the surface or surfaces can be found that will control a certain combination of Seidel aberrations most effectively. If the points (Hi, hi)in the (H, h) plane have such a position that the required control is not sufficiently effective, values of paraxial system parameters have to be changed and another trial has to be made.

366

ASPHERIC SURFACES

[IV,5 3

Similar results can be obtained if the aspheric deformations of refracting surfaces are regarded in the same optical space, in particular in the image space (SCHUSTER [ 1977, 19781). In this model the object plane, the stop, and each optical surface (preliminarily imagined as spherical) are paraxially imaged (by the surfaces following it) into the image space of the total system. A possible aspheric deformation of a surface appears in this space as an additional thin lens: one surface of this lens is the image of the spherical surface mentioned, and the additional (positive or negative) optical thickness of the thin lens is equal to the optical thickness variation due to the aspheric deformation. In this paraxial image space model the light rays run undeflected. Thus, without ray tracing one can see or calculate where arbitrary rays meet a surface, and the influence of aspheric deformations on optical path length and direction variations of the rays can be approximately determined. Higher-order aberrations can also be considered. If the number of variable system parameters is less than the number of the aberrations to be corrected, these aberrations can generally not be made completely equal to zero. Then, strictly speaking, an optimization problem should be solved that presupposes a merit function assessing the aberrations (see 5 3.5). If the values of the paraxial system parameters and the detailed form of the merit function have already been fixed, a pre-optimization in the Seidel region concerning the asphericity parameters b, can be carried out. For example, I out of the k surfaces may be allowed to become aspheric. Then the question arises concerning which selection of 1 aspheric surfaces results in the optimal solution. To find the answer, ( f )systems of I linear equations with 1 unknowns need to be solved (KROSS and SCHUHMANN [ 1984, 19851). For any imaging system a surface number as small as possible and an imaging performance as good as necessary will be the goal. The highest imaging performance in the Seidel region is the removal of all five Seidel aberration coefficients:

B=C=D=E=F=O. Here the question arises concerning the smallest number of surfaces that will enable this removal to be attained (for a real imaging with an arbitrary magnification). This question can be answered for refracting surfaces in the following way (SCHULZ[ 19801): One single lens with two surfaces is not sufficient,but by using three surfaces, of which two are aspherics, all five Seidel aberrations can be made equal to zero. This is possible in particular by using a lens in air with two aspherics as outside surfaces and one spherical surface between them. The refractive indices on both sides of the spherical surface, the

367

DESIGN OF ASPHERICS

lateral magnification, and a fourth parameter, can be arbitrarily chosen (within certain limits). For any such parameter combination a solution can be found. Figure 7 shows an example of this system. The strong curvature of the second surface can be regarded as a consequence of the Petzval condition C - D = 0 (cf. eq. 3.9). The fulfillment of this condition, which is necessary here, cannot be influenced by asphericities. This condition imposes a strong restriction on the radii of curvature, ri, particularly in the case of a small number of surfaces. If, for an infinite object position, the refractive index is not less than a critical value ( a1.602) and distortion is permitted, the other four Seidel aberrations can be made equal to zero by a single lens with two aspherics (MARTIN[ 19441, based on BURCH[ 19431). The lens is thick because the Petzval condition here requires the equality of the two vertex curvatures so that the lens power is proportional to the thickness. In addition, more than two surfaces can be made aspheric in order to make certain Seidel aberrations equal to zero (see, for example, KORSCH[ 19731 and ROBB[ 19781).

3.4. RAY TRACING THROUGH ASPHERICS

Ray tracing is a procedure required for some of the design methods described in 3.5 to 3.7. In this procedure real (possibly skew) rays with finite heights are traced by applyingthe law of refraction or reflection, respectively. However, paraxial ray tracing near the optical axis and differential ray tracing near a known principal ray (see, for example, WELFORD[ 1974, p. 1651) are not described in this chapter. The former implies no difference and the latter no essential difference between spherical and aspheric surfaces. Ray tracing starts from an initial ray given by a point and a direction (e.g., in the object space). The ray tracing procedure then comprises the following steps: tl,=1.7

Q-1.5

r, -11182 r,! 0.184 d--0.430 b,Bf- 1.357 4-0 b,Bf-0.890 Fig. 7. Example of an optical system with three rekacting surfaces that is free from all five Seidel aberrations. The lateral magnification is

-4.

368

ASPHERIC SURFACES

[IV, I 3

1. Determination of the intersection point of the ray with the following surface. 2. Determination of the surface normal at the intersection point and of the ray direction after refraction or reflection. 3. Transition to the next surface, reiterating the described steps up to the last surface, which may be the image plane. The determination of the intersectionpoint of a ray with a surface proceeds as follows: In a Cartesian ( x , y, z ) coordinate system with the z axis being the optical axis and with the origin at the surface vertex, let the ray be given by x=A+Bz,

(3.10)

y=C+Dz

and the surface by f(X,Y,

4 =0

(3.11)

*

Then x and y from eq. (3.10) need to be substituted into eq. (3.1 l), and the resulting equation f ( A + Bz, C + Dz, Z) = 0

(3.12)

needs to be solved for z. This yields the z value of the intersection point to be determined, and by substituting this value into eqs. (3.10) the other two coordinates are obtained.* Equation (3.12) has generally the same degree as the surface equation (3.1 1). If the degree is higher than four, a closed-form solution is generally impossible,as is well known; this denotes a characteristicdifference between spheres and general aspherics. The intersection point of a ray with a sphere is simply determined by the solution of a quadratic equation, whereas for the intersection with a more general aspheric (order of the surface higher than four) no closed expressions can be given. In the latter case iterative procedures are appljed. For special aspherics, however, namely for quadrics of revolution, the intersection point is usually determined by solving a quadratic equation (HERZBERGER [ 1958, p. 401, WELFORD[ 1974, p. 561). In this case eq. (3.11) has the special form (2.1), so that the z value of the intersection point is the solution of the equation (A + Bz)'

+ (C + 02)'

=

2Rz - (1

+ b)z2.

* Instead of z , the quantity z/Nrwhere N , is the ray direction cosine with respect to z is often determined first. This fact, of course, makes no essential difference. z/N,is the path length of the ray from the vertex plane to the intersection point.

IV, 3 31

369

DESIGN OF ASPHERICS

For quartic surfaces, closed expressions for the intersection point can be used as well; however, the solution of the corresponding equation of fourth degree, including the selection of the proper solution, is rather laborious. Thus, rapidly converging iterative procedures similar to those that will be mentioned later are preferred, as CHENand HOPKINS[ 19781 have done for ray tracing through concentrators (fig. 5; see 5 5.6) with axisymmetrical quartic surfaces. For general aspherics that are determined by polynomials of (2n)th degree (n > 2) according to eq. (2.4), expression (3.12) takes the form

+ Bz)’ + (C + Dz)’]+ u,[(A + B z ) +~ (C+ D z ) J2~ + - ’ + [ ( A + Bz)2 + (C+ D 2 ) 2 ] n

z = u,[(A *

I

Known iterative procedures are used for the solution of this equation, for example, Newton’s method or variants of it. Some of these procedures are based on simple geometrical principles, such as shown in fig. 8 or fig. 9 (see, for example, HERZBERGER and HOADLEY[ 19461, WELFORD[ 1974, p. 571). For an aspheric defined by a number of curve sections (fig. Zb), the intersection point is determined correspondingly by using the equation that is valid for the proper curve or surface section. The intersection point of a skew ray with an aspheric defined by a cubic spline function (RIGLERand VOGL[ 197 11) has to be determined iteratively, for example, according to fig. 9. An iterative procedure of this kind is not necessary if the curve sections are parts of conic sections (ellipses or hyperbolas) with an axis that coincides with the optical axis (SCHULZ[1985]), because in this case the equation for the corresponding surface section (3.11) is only a quadratic equation,

(a)

I b)

Fig. 8. Iterative determination ofthe intersectionpoint ofa ray with an aspheric:(a) in a stair-like or (b) in a spiral-like succession, depending on the direction of the ray. The coordinates of the points Po, P , , P2,... are successively determined;the points converge to the intersection point P. P2,+, has the same x and y coordinates as Pzk; P2k+zhas the same z coordinate as Pzk+, ( k = 0, 1,2,. ..). The points P,, lie on the ray, and the points P,,+, lie on the aspheric. The ray may be a skew ray.

370

ASPHERIC SURFACES

RAY

z

Fig. 9. Iterative determination of the intersection point of a ray with an aspheric, using tangential planes. The coordinates of the points Po, P,, P2,... are successively determined; the points converge to the intersection point P. P2,+ has the same x and y coordinates as P2k; P2*+,is the intersection point of the ray with the tangential plane of the aspheric at the point P,,, (k = 0, 1,2 ,...). The ray may be a skew ray.

,

( z - 7)’ = a

+

( x 2 + y 2 ),

where a, b, and y are the parameters of the surface section. The determination of the su@ace normal at the intersection point is based on differentiations in the surface equation (3.11). The direction cosines (L, N) of the normal are, in vector representation,

a,

(3.13)

The other steps of the ray tracing are the same as for spherical surfaces. When the ray tracing has been carried out, aberrations (ray or wave) can be calculated in the usual manner. An intermediate level between the ray tracing just described and paraxial ray tracing has been called “proximate ray tracing” (HOPKINS [ 19761). Whereas paraxial ray tracing uses only the paraxial system parameters, proximate ray tracing uses aspheric deformation parameters up to a certain order. The intersection point of the ray with a surface is here determined approximately by an inversion of the corresponding series (to appropriate order). In this way an approximate ray path of an accuracy between that of the paraxial and that of the real ray path is obtained. Such methods can also be used for calculating aberration coefficients.

IV, I 31

DESIGN OF ASPHERICS

371

3.5. SYSTEM OPTIMIZATION USING A MERIT FUNCTION

Optimization methods using a merit function are often applied in the final stage of designing spherical and aspheric systems. However, since the optimization methods for systems containing aspherics differ little from those for spherical systems (see, for example, PECK[ 1980]), they can be discussed relatively briefly here. (For some practical rules concerning the inclusion of aspherics, see SHANNON[ 1980, p. 751.) Typical examples of application are imaging systems with medium or larger field and aperture angles (see, for example, 0 5.2), and their optimization is outlined in this section. At first a starting system should be available from which the optimization proceeds. This system can be obtained by calculations in a near-axis region (see 4 3.3). Anather possibility is to proceed from an already known system of which certain properties need to be improved or changed. In any case the optimization concerns the enhancement of the imaging performance of the starting system, the imaging performance being characterized by an appropriately defined merit function. Usually this function consists of the sum of the squares of weighted aberrations and should be minimized. The determination of which aberrations are selected for this purpose and by which individual amounts they should be weighted requires care and experience concerning the particular optical problem to be solved. The merit function, expressed by means of the aberrations, depends on the system parameters, for example on the lens thicknesses and the vertex curvatures. The mathematical problem to be solved is then the variation of the system parameters in such a way that the value of the merit function is minimized and thus the imaging performance optimized. This is generally an optimization with boundary conditions and constraints; in any case the system parameters mentioned must not exceed certain limits. In general, the number of independent parameters is less than the number of aberrations to be corrected by parameter variations, so that the merit function cannot be made equal to zero. The problem is the determination of the minimum of the merit function and of the parameter values by which this minimum is reached. Mathematical methods and programming principles for solving this problem weredescribed in the 1970s (JAMIESON[ 19711, HIMMELBLAU [ 19721, MURRAY[ 19721, GILLand MURRAY[ 19741). The minimum of the merit function and the corresponding parameter values are reached in a number of steps, each step intended to diminish the value of the merit function. When a relative minimum (with respect to neighboring parameter values) has been found, often the question arises concerning whether this is the desired absolute minimum.

312

ASPHERIC SURFACES

[IV, 8 3

If some or aU of the surfaces are aspherics, the system parameters need to be complemented by the aspheric shape parameters, which are mostly global shape parameters, for example, the coefficients u4, u6, .. . in eq. (2.4). Sometimes, however, local shape parameters (cubic spline parameters, see 8 2) are also used. Aspheric parameters add to the number of parameters of a surface that are varied. On the other hand, the application of aspherics often reduces the number of optical surfaces. The aberrations in the merit function, which depend on the system parameters, are generally calculated by ray tracing (see Q 3.4). Because the computationaleffort for optimization is often very high, not only when aspherics are used, computer programs and computational improvements or formulas for program modules that may reduce the expenditure are also important (see, for example, WORMWELL[ 19781, HOPKINS[ 19811, FAGGIANO, GADDAand MORO[ 19831, HUBER[ 19851).

3.6. ATTAINMENT OF AXIAL STIGMATISM AND DIFFERENTIAL EQUATIONS FOR VARIOUS PURPOSES

The design methods described in § 3.5 are typical for systems with larger field and aperture angles. In such systems an arbitrary point of an optical surface is generally passed through by light rays of very different directions. All these rays should be controlled independently as far as is possible. Because of the limitation on the number of surfaces, however, this can be achieved only to a limited degree. Therefore, compromises in the form of merit functions are made, and with small numbers of surfaces the imaging performance cannot be expected to be high. If, however, either the field angle or the beam diameter is very small, often other design methods are appropriate that are different from those described in Q 3.5 but are sometimes combines with them. They are discussed in Q 3.6 and 3.7, which are concerned with axial symmetry systems. An example of their application are systems for the sharp focusing of large-aperture laser beams, in which the field angles are very small. Thus every point of an optical surface is passed through by a very narrow cone of rays or, in the limiting case, by only one ray. These rays can be controlled more locally, considering their immediate vicinity, which is partially achieved by using differential equations for the aspheric surface or surfaces. In spite of the small numbers of surfaces, high requirements concerning small aberrations can be met. In particular, the removal of certain aberrations can be required.

IV, 8 31

313

DESIGN OF ASPHERICS

In this respect, cases have been investigated (without and with differential equations) in which the field angle is so small that in eq. (3.2) r = 0 can be assumed. Then, for the absence of the wave aberration, the only requirement to be met is W&) = 0, which implies axial stigmatism (see 3 3.1). The problem concerning how to attain this requirement is very old. As mentioned in 3 1 and § 2.1, solutions have already been given by Descartes; fig. 1 shows some simple examples and another case is shown in fig. 6 (see 3 3.2). A more general case is the determination of a surface of an otherwise arbitrary system in such a manner that axial stigmatism is attained. In this case the optical system is generally assumed as given, except for the shape of the surface that is to be determined. This problem has been considered by a number of authors some decades ago (for descriptions and references see, for example, NAUNDORF [ 19661 and JUREK [ 19771). If the surface to be determined is the fist or the last surface of the system, the solution can be obtained in a simple way (WOLF and PREDDY[ 19471, BORNand WOLF[ 1964, p. 1971). After a usual ray tracing any point of the unknown aspheric surface can be determined by solving only a quadratic equation. In fig. 10 the system is to image sharply the axial object point 0 at the image point O f ,that is, with a constant and known optical path length [ 0 * - 0’1. The system is known up to the vertex plane of the last surface (the back-focal distance L‘ being known). Thus a ray starting from 0 with the direction angle t can be traced to point V, which enables a determination of the followingthree quantities: the optical path length [ 0 * * V 1, the direction angle U, and the height H of V.Then the coordinates (y, z ) of point A of the aspheric can be determined. Here the following equations hold: [ A O ’ ] = [ O * * . O ’-] [ O * * . V -] [VA], nz

, V ] -cos u’ y =H-ztanu,

,

(3.14)

LAST SURFACE (ASPHERICI

Fig. 10. Attainment of axial stigmatism by one asphenc that is the last surface of the optical system. n and n ‘ are the refractive indices on both sides of the aspheric.

314

[IV,8 3

ASPHERIC SURFACES

n’’{(HztanU)’+ (L‘ - z)’}

=

[ O * * * O -’ ][ O * . . V ]- (3.15)

Equation (3.15) is a quadratic equation for z, and from z, y can be obtained by (3.14). With this calculation the coordinates of a point A of the aspheric have been determined depending on the value of the starting parameter t of the ray tracing. When starting with other t values, any number of points of the aspheric can be determined. A more complicated case occurs when the aspheric to be determined is in the interior of the optical system. The following method describes the solution of this problem (MIYAMOTO [ 1961]), which can be regarded as a differential form of an earlier method (WOLF[ 19481). In fig. 11 again any number of pairs of values (U,H) can be obtained by means of ray tracing, so that the function U = U ( H ) can be regarded as known. Correspondingly the function U ’= U’(H‘) is also regarded as known, since the pairs ( U ’ , H ’ ) can be determined by backward ray tracing from 0’ into the space on the right of the aspheric. The derivatives U ( H ) = d U(H)/dH and U (H’) = dU‘ (H’)/dH’ can be obtained from U ( H ) and U‘(H‘)and are also regarded as known. However, the correct correspondence between t and t’, that is, between H and H ’ ,is still unknown. The function H‘ = H ’ (H)describing this correspondence can be determined, together with the unknown function z = z(H), from the following two differential equations :

-dz- dH

cos’ U ( H )

1 i tanU(H)g(H,H’)

OPTICAL ...

g(H, H ’ 1

(3.16)

3

i

‘

SYSTEM ...

...

Fig. 1 1 . Attainment of axial stigmatism by one aspheric that is in the interior of the optical system. Notations correspond to those in fig. 10; V’ is the (virtual) intersection point of the ray AO’ with the vertex plane of the aspheric, the intersection point being regarded in the space on the right of the aspheric.

IV, 4 31

315

DESIGN OF ASPHERICS A"

dH' -dH

cos' U(H)

-

-

{tan U(H) - tan U'(H')} "" dH

1-2

where g ( K H')

=

U'(H') cos2 U ' ( H ' )

(3.171 .

I

n' sin U'(H') - n sin U(H) n' cos U'(H') - n cos U ( H ) '

The differential equations (3.16) and (3.17) can be numerically solved by standard methods, beginning in the vicinity of the axis and proceeding to larger off-axis distances. From any value of z = z(H) obtained in this way, the corresponding y value can be determined again by (3.14). Another differential method (JUREK [ 1967/68]) for solving the same problem proceeds from an approximate shape (assumed as known) of the surface to be determined and results in a corrected shape. If the correction is still unsatisfactory, the procedure can be iterated. As described so far, the requirement Wo(e)= 0 in eq. (3.2) can be met by using one aspheric, and thus the spherical aberration of all orders can be removed.* A further requirement is Wo(e)= W1(e,6 ) 3 0 (3 3.1), which means the additional fulfillment of the sine condition. Such aplanatic imaging system can be designed with the use of two aspherics. The first methods for designing such systems occurred at the beginning of this century, in the design of mirror [ 1905, No. 21) and lenses (LINNEMANN [ 19051). telescopes (SCHWARZSCHILD Later several other authors described and reviewed solutions of this problem [ 19661 and JUREK [ 19771). (see NAUNDORF For an aplanatic two-mirror system the following differential equation can [ 1905, No. 21): be stated (SCHWARZSCHILD (3.18)

1 da

(notations according to fig. 12, system focal length = 1). The solution of this equation follows: 1 - sin2&)

I

+ c{d - sin2(?ja)}-'/(d-1){cos2(4a)}d'(d-1).

(3.19)

d

* This holds for monochromatic light (one wavelength). The same requirement for two wavelengths can be met by two aspherics, applying a different design method (see 4 3.7).

316

ASPHERIC SURFACES

Fig. 12. System of two aspheric mirrors S and S’, which brings any axial-parallel incident ray exactly to the focal point F, fulfilling the sine condition.

This is the equation of the meridional curve of the mirror S in polar coordinates (r, a) with the integration constant c appropriately chosen. For the mirror S‘ the following parameter representation in Cartesian coordinates is obtained: y’

=

sina,

(3.20)

(3.21)

Thus the general solution (3.19) to (3.21) for the aplanatic two-mirror system appears in a closed form. In principle, this solution also contains all aplanatic two-mirror systems for grazing incidence (in particular for X-ray imaging) if complex integration constants are allowed (WOLTER[ 1952]), that is, it also contains systems of the types I, 11, and I11 according to fig. 13. For systems of type I the integration constant c has a nonzero imaginary part; in this case eqs. (3.19)-(3.21) need to be rewritten if they are to contain only real quantities. Recently Wolter’s classification with three types according to fig. 13 has been extended to a total of eight types (SAHA[ 19871; for further references see ibid.). The extended classification includes types with a real intermediate image in the [ 19631, HEITRICKand BOWER spacebetween the two mirrors (KIRKPATRICK

lo)

b)

(GI

Fig. 13. Two-mirror systems of the types I, 11, and 111 (WOLTER[1952]) for grazing incidence, in particular for X-ray imaging. Type I systems (a) only contain concave mirrors, whereas type I1 systems (b) and type I11 systems (c) also contain a convex mirror. In type I1 systems the mirror near the axial point F is convex; in type 111 systems this mirror is concave. In telescopes point F is the axial image point, and in microscopes (WOLTER[ 19521) point F is the axial object point. The mirror surfaces are parts of confocal quadric surfaces of revolution (see 5 3.2), for example, or their shapes are such that the system is aplanatic according to Schwarzschild (see 8 3.6).

IV,I 31

DESIGN OF ASPHERICS

311

[ 19841) and types with F being virtual, and distinguishes three classes, according to the specific imaging requirements. General surface equations are given, where the sign and magnitude of the design parameters determine the type and class. Other authors have considered additional aspects of designing aplanatic two-mirror systems. Surface equations describing a continuous transition from systems with near-normal incidence to systems With grazing incidence can be used for studying the performance of such systems as a function of the angle of incidence(KORSCH[ 1980al).Another possibility is the numerical evaluation of the differential equations for the surfaces (KORSCHand WARNER[ 19801). Differential equationsfor three-mirror telescopes with the same property (axial stigmatism and fulfillment of the sine condition) can also be set up and solved, the shape parameters of one of the three surfaces being used to minimize residual aberrations (KORSCH[ 1980bl). Aplanatic two-mirror systems were also the starting point for designing reflecting microscopes for spectral regions ranging from the near UV to the near IR (BURCH [1961], with further [ 19721). references; BURCHand MURGATROYD For simple aplanatic mirror systems, as described above, surface equations can be given in a closed form, for example, in the form (3.19)-(3.21). For corresponding lenses forming real images, however, this is not the case, and their differential equations are solved numerically. The following differential equations (3.22) and (3.23) hold for two neighboring surfaces of an optical system, making the system aplanatic (WASSERMANN and WOLF[ 19491, BORN and WOLF [ 1964, p. 200]), see fig. 14. The system is specified, except for the shapes of these two surfaces. Thus the quantities U and H can be determined by tracing a ray with the startingparameter t = sin U, from 0 to V. On the other side, the quantities U' and H' belonging to the same ray can be determined by backward ray tracing with the parameter t' = sin U& from 0' to V', where t' is known because of the sine condition sin U& = sin U, x constant. Thus, the quantities U,H , U' and H' can be regarded as known functions of the starting parameter t. Likewise, after calculation the derivatives d(tan U)/dt, dH/dt, d(tan U')/dt and dH'/dt are known. Then the quantities z and z' be determined from the following two simultaneous differential

378

ASPHERIC SURFACES

FIRST

OPTICAL SYSTEM

Fig. 14. Optical system with two aspherics, which brings any ray that starts from 0 with the direction parameter t = sinU, exactly to the image point 0 ' , fulfilling the sine condition sin V& = sin U, x constant. The two aspherics separate the three media of the refractive indices n, no, and n', respectively.

-={ dz'

n'

dt

n'

d

m cos U' - n D .,tan,'}

,/mU' sin

O

- n,Dy

-'{ dH'

y doan dt

u')l

(3.23)

where Dy=y-y',

D,=dO+z' - z ,

y = H - ZtanU,

y'

=

H'

- Z'

tanU'.

.(3.24)

(3.25) (3.26)

The last two equations also provide the required values of y (belonging to z ) and of y' (belonging to z ' ) . The case of two neighboring aspherics can be generalized in the following way: Between the two aspherics to be determined there may lie a number K of known surfaces (VASKAS[1957]). Then, instead of the two first-order simultaneous differential equations (3.22) and (3.23) with the two unknowns z and z' { y and y' being eliminated by (3.25) and (3.26)), K + 2 first-order simultaneous differential equations in K + 2 unknowns hold. Two of these differential equations correspond to the light path through the two aspherics, and the other K differential equations correspond to the light path through the K intermediate known surfaces. The first-mentioned two differential equations should be solved numerically, whereas the determination of the light path through the K known surfaces is equivalent to the solution of a set of K linear equations, which have to be solved at each integration step (BRAATand GREVE[1979]). The unknowns that are determined by this set of linear

IV, I 31

DESIGN OF ASPHERICS

319

equations are the small increments by which the ray heights at the K known surfaces change when the ray parameter value, changes corresponding to a small integration step. The two differential equations of the aspherics and the K linear equations are solved step by step, which yields the coordinates of the two aspheric surfaces in a succession of points and, together with the other surfaces, an optical system with axial stigmatism and fulfillment of the sine condition. If three aspherics are used, astigmatism can additionally be reduced (BRAAT [ 1983b1) by combination with an optimization procedure (see $ 3.5). The coefficients of the third aspheric are used as variable parameters for the optimization, and at each step of the optimization the other two aspherics are calculated in such a manner that axial stigmatism and the fulfillment of the sine condition are obtained as described earlier. Aspheric surfaces as substrates of diffraction gratings can also be calculated by differential equations (KASTNERand WADE [ 19781). Whereas without diffraction (only by reflection or refraction) one aspheric can be used to image a point at another one stigmatically, by a diffraction grating on a substrate of an appropriate shape a point can be imaged stigmatically at severalpoints, each for a different wavelength (under certain restrictions). If a holographically recorded grating on an aspheric substrate is assumed, the weakest restrictions concerning the choice of points and wavelengths can be obtained (GUTHERand POLZE[ 19821). If the substrate and the grating are symmetrical about the z axis, the following simultaneous differential equations hold for the groove separation g = g(z) and the substrate shape h = h(z), where h is the distance from the z axis: dh dh cos arctan- - arctan- + sin arctan- h dz Z dz

(

i = 1,2,.. .

')

(

(3.27)

Here the axial point z = 0 should be imaged stigmatically at the axial point z = ei for the wavelength li,and k is the order of the spectrum. For two stigmatic points (i = 1,2) g can be eliminated and the remaining differential equation can be solved numerically. For three stigmatic points (i = 1,2,3) with e2 = 0 the solution is conditioned by the restriction

(3.28)

380

ASPHERIC SURFACES

[IV,B 3

The cases examined so far concern systems with imaging properties, although these systems are often applied for other purposes than imaging in the usual sense. Other systems are used to guide an illuminating or energy flux in such a way that a desired distribution is obtained (e.g., a uniform distribution on a certain surface, cf. 8 3.1). In such cases differential equations can also play a part in the calculations. This is the case, for example, if the radiation of a discrete or continuous line source should be reflected by an aspheric surface in such a way that a specified irradiance on a receiver surface (symmetrical about the line) is obtained (BURKHARD and SHEALY[ 19751). A lens with one aspheric can also be used to obtain a specifled energy distribution on a given receiver surface (BURKHARD and SHEALY[ 19761). However, if for the output radiation not only a certain energy distribution but also a certain direction or distribution of directions is required, two aspheric surfaces are necessary. The problem is partly similar to that shown in fig. 14. An example appears in fig. 15. Here a parallel laser beam with a given axisymmetrical intensity distribution is to be converted into a parallel beam with uniform intensity distribution (KREUZER[ 19651). For this case the following differential equation (3.29a) has been given, together with eqs. (3.30) and (3.31) (RHODESand SHEALY[ 19801; notations according to fig. 15):

+

+

(Fy (5

- 1) ( ( y ’ -y)2

(3 -

{ -2(y‘ - y ) ( Z ’

- ( Y ‘ - YI2

-

+ ( Z ’ - Z)’)

Z)}

=o. *

(3.29a)

* It should be noted that eq. (3.29a) is here equivalent to eq. (3.22) with U = 0 and H = y if the parameter t is identified with y ; that is, eq. (3.29a) is equivalent to _ dZ no(Y‘ - Y ) (3.29b) dy n J ( y ’ - Y ) +~(2’- Z)2 - no(Z‘ - Z )

DESIGN OF ASPHERICS

38 1

Fig. 15. System of two plano-aspheric lenses, which converts a parallel bundle with a given axisymmetrical intensity distribution into a parallel bundle with uniform intensity distribution over a given diameter. The two aspherics separate the three media of the refractive indices n, no, and n ’ , respectively.

The relation between y and y’ is determined by the conservation of energy (neglecting reflection and absorption losses): d 8 Joy a(u)u du = JO2= d 8 Joy’ u du x const. , that is,

y‘

=Ja/

FYa(u)udu

JO

const.

9

(3.30)

where a(u) is the energy density of the input radiation. The dependence of Z ’ on y, 2, and y’ is determined by the requirement of a constant optical path length between the two plane surfaces of the lenses. This results in 2’ =

(nn’ - ng)Z + n ’ K n f 2- n t

This can be shown in the followingway: The problem is to determine the inclination ofthe surface element of the first aspheric in such a manner that an incident ray is refracted into a certain direction. This problem is solved by eq. (3.22). where, in this case, we have to put U = 0 and H = y and can put r = y. Then eq. (3.22) yields the form (3.29b) if the notations of fig. 15 are used. Equation (3.29a) is fulfilled by (3.29b), as can be verified. However, (3.29a) is an equation of the fourth degree and contains not only the correct solution, corresponding to (3.29b), but also three wrong solutions. Thus, eq. (3.29a) and the selection of the right solution are not necessary if one and WOLF[1949]). proceeds from eq. (3.22) (WASSERMANN

382

ASPHERIC SURFACES

where K

=

n'(d

+ do) - nd - nodo.

Equation (3.29a) (or 3.29b, see the footnote following eq. 3.29a) together with eqs. (3.30) and (3.31) mean a differential equation of the general form F(dZ/dy, Z, y ) = 0 for the first aspheric, which can be solved numerically. Then, for the second aspheric in case of n = n', a simpler mathematical form results. Other differential equations can be used for designing reflectors in illumination engineering (WEISS[ 19781). Here graphical methods of solution are also applied (ELMER[ 19801).

3.7. POINT-BY-POINT COMPUTATION BY RAY CONSTRUCTIONS WITHOUT

DIFFERENTIAL EQUATIONS

According to a number of methods described in 8 3.6, the surface points or elements of aspherics are calculated successively,changing the off-axis distances step by step. This process is also possible under similar conditions and goals (see second paragraph of 5 3.6) without deriving and solving differential equations as we will now discuss. Simple examples were shown in figs. 6 and 10. In either case an axial point should be imaged stigmatically, which can be achieved with one aspheric. However, if in systems similar to that shown in fig. 6 the sine condition should also be fulfilled (coma correction), a ray leaving the spherical primary mirror not only should pass through a specifled point (F) but also should have a pre-assigned direction at this point. Thus a one-parameter set of meridional rays leaving the primary mirror should be transformed into another set of completely specified meridional rays, the total optical path length being constant. This transformation is accomplished by using two mirrors (following the primary mirror); the reflecting surface elements and their succession can be obtained by geometrical considerations (MERTZ [ 1979, 19811). Thus the aplanatic three-mirror system is designed point by point but without using differential equations. The methods mentioned so far use a succession of ray constructions that are of the same kind for all the rays that construct the system. New possibilities emerge if dflerent types of rays are considered and their constructions connected with one another, which permits the design of optical systems with additional or other desired properties, especially systems that up to now have

IV,$31

DESIGN OF ASPHERICS

383

Fig. 16. The design of a stigmatically and achromatically imaging single lens with an alternating construction of rays of two different types. One type of rays belong to the wavelength L = L* (continuous lines in air, broken lines in the lens); the other type of rays belong to I = A** (continuous lines everywhere).Both types of rays run from point 0 to point 0’. The refractive index n of the lens is wavelength dependent.

not been designed by other methods (e.g., by differential equations). An example is the calculation of an achromatic single lens forming a stigmatic real image of an axial point for two wavelengths at one axial point (SCHULZ[ 19831). Such a lens must have surfaces of the type shown in fig. 2c. Its design can be described on the basis of that in fig. 16, in which the axial point 0 is imaged sharply at 0’ for the two wavelengths A* and A**. Thus the optical path length from 0 to 0’ must have a constant value for all A* rays (belonging to the wavelength A*) and another constant value for all A** rays (belonging to A**). If these two constants and one surface element, for example, the one at P,, are known, the other surface elements can be determined as follows. The P, surface element refracts the A* ray emanating from 0 and so determines the path of this ray within the lens. Point Pi lies on this ray. Only the distance P,P; is still unknown, but it can be easily obtained from the optical path length constant for A*, for example, similar to the position of A in fig. 10. Thus the position of Pi is known. The normal of the Pi surface element then results from the requirement that the refracted ray must run through 0‘.Therefore the Pi surface element (i.e., its position and normal) has been determined, proceeding from the PI surface element. The determination has been carried out together with the construction of a A* ray. In the same way, proceeding from the Pi surface element, the P, element is calculated, but now by constructing a 1**ray in the opposite direction. From the P, surface element, again using a A* ray, the P; element is determined. In this way the two aspheric surfaces are obtained point by point in the succession P I ,P i , P,, Pi,.. . , By appropriate interpolation between these surface elements, the lens obtains the required imaging property also between those points.

3 84

ASPHERIC SURFACES

u

Fig. 17. Result of the computation of a single lens, which brings a bundle of axial-parallel rays to the same sharp focus 0’for two wavelengths A* and A**, the corresponding refractive indices of the lens being n( L*) = 1.78 and n( A**) = 1.84. The central part of the second surface is not met by light rays and is therefore arbitrary.

Figure 17 shows the result of such a lens design, where point 0 lies at infinity. This design principle can be easily extended to the more general case of predetermined spherical surfaces lying between the two aspherics to be calculated (SCHULZ [ 19841). In this design the discontinuity of the surface normal at the axis can be avoided. Figure 18 shows the result of such a design. It is an achromatic doublet, where not only the back focus difference but also the spherical aberration of all orders is equal to zero for two laser wavelengths. In the way just described, by constructing two kinds of rays, two aspherics can be calculated step by step. In addition, by using other types of rays, more than two aspherics can be designed. The general design principle is the same as that for fig. 16, in which rays of different kinds are constructed by turn and in an appropriate connection. These rays meet surface elements that are either already known (from preceding determinations) or still unknown. If a ray meets a known surface element, its path through this element is determined by ray tracing. If the ray meets an unknown surface element, this element is chosen and determined in such a manner that the ray traced through becomes aberration free or fulfills another desired condition.

INDICES:

1.595 1.608

{1.619 1624

REFRACT.

(,I1060nm *= ) (A*: 530nm)

Fig. 18. Result of a point-by-point computation of an achromatic doublet, where the spherical aberration of all orders is equal to zero for the two wavelengths A* and A**.

IV, B 31

385

DESIGN OF ASPHERICS

Fig. 19. Designing three aspherics (2,3,4) of an optical system free from distortion, field curvature, and astigmatism. The near-axis region (bounded by the dotted line) and the spherical surface 1 have been determined by precalculations. The system consists of two single lenses (1.2) and (3,4) in air. The distance between the principal ray and the neighboring skew ray has been exaggerated in the figure. The points P lie in the meridional plane and are nodes according to fig.2b (see 5 2.1).

According to this general principle, by using an alternating construction of a principal ray and a neighboring skew ray, for example, systems with three aspherics can be designed that are free from distortion, field curvature, and astigmatism, also for large fields (SCHULZ [1985]). Thus w,(r, 0) = w2(r, 0) 3 0 in the wave aberration equation (3.3) (see 5 3.1). The principle of the step-by-step design of such a system is shown in fig. 19 and can be described as follows: Let the aspheric surfaces and their slopes be known up Then they are determined up to the neighto the dashed line Pi(2)-Pi3)-Pi4). as follows. boring line P~~)l-P~:)l-P~~), A new principal ray is constructed, first between the stop center and Pi3), where the ray meets only known surface elements. Then the principal ray can be traced on the left to the object plane and on the right, by the law of refraction,

\

IMAGE PLANE

Fig. 20. Determination of the free parameter of the surface element above

4") of fig. 19.

386

ASPHERIC SURFACES

“, 0 3

into the second lens, where the ray then meets an unknown surface element above PI:4).The free parameter of this surface element is now varied and is determined according to fig. 20 in such a manner that the ray traced through this surface element passes through the Gaussian image point and so gets zero distortion. The determination can be carried out, for example, by Newton’s method of determining a zero of a function (i.e., of the distortion as a function of the free parameter mentioned). Then, as shown in fig. 19, a new skew ray is constructed. The ray is traced through the known object point of the principal ray constructed immediately before, the known stop point Q (Q has been appropriately chosen before), and the known spherical surface 1. On the surfaces 2 and 3 the ray meets unknown surface elements, each of which has one unknown free parameter. These two parameters are now varied and are determined in such a manner that the skew ray through these surface elements and through the element above P!*), which is already known, passes through the Gaussian image point. Thus, altogether, three new surface elements have been determined. Their end points (P!:),, P$t)l, and Pf:),) in the meridional plane can be fixed by the condition that they get the same distances from the optical axis as the intersection points of the skew ray with the respective surfaces. Now i is increased by 1 and the procedure described is iterated a sufficient number of times. In this way, proceeding from i = 1 in the near-axis region, the total system is designed in the succession i = 1,2,3,. .. . If ea is chosen to be very small, the system becomes free from distortion, field curvature, and astigmatism. Its computation in a succession of small pieces or surface elements does not use differential equations, but it corresponds to a numerical solution of such equations (see § 3.6): In both cases the exact solution is the limiting case if the lengths of the elements converge to zero. Futhermore, a principle of designingfour aspherics is known according to which rays of three different kinds (two different meridional rays and one skew ray) are constructed by turn and in such a way that the resulting optical system fulfills the condition W&) = Wl(@,0) = W,(e, 8)= 0 in eq. (3.2) ( S c ~ u t z [ 19821). Such systems are aplanatic systems, whose wave aberration function does not contain any terms depending on the zeroth and the first power; nor does it contain any terms which depend on the second power of the object coordinate r (“second-order aplanatic systems”). These systems are examples of higher-order aplanatic systems (see 6.2). Another construction principle for designing a second-order aplanatic system has been discussed (GUTHER[ 19861). This principle also uses the three kinds of rays just mentioned, but the system is assumed to consist of two aspherics being substrates of diffraction gratings whose groove separations become suitable functions of the height.

IV,8 41

FABRICATION AND TESTING METHODS

387

Computational results according to $ 3.6 or 5 3.7 can be used directly, or the methods can be conbined with optimization methods (see $ 3.5).

4 4. Fabrication and Testing Methods

4.1. FABRICATION METHODS

The fabrication of aspherics is much more complicated than that of spherical surfaces because of the lack of spherical symmetry. A spherical surface has the same curvature 1/R everywhere and in all directions so that it can be worked everywhere in the same manner; the tool also has a spherical surface (of the curvature r - l/R), and the workpiece and tool are in continuous contact and guided in appropriate relative movements about the common center of curvature. These movements cover all surface parts statistically in the same manner, but this is not possible if the optical surface to be made has only one axis of symmetry (its rotation axis). A brief survey of fabrication methods for aspherics follows, in particular for aspherics with a rotation axis (STONECYPHER [ 19811, WALTER[ 19831, HEYNACHER[ 19841, MARIOGE [ 19841, SHANNON[ 1980, p. 771). There are several kinds of fabrication methods, including working by tools (the workpiece is in principle regarded as a rigid body); utilization of elastic deformations; and casting, molding, and related techniques using a negative master. In the fabrication methods that involve working by tools, from a geometrical viewpoint three cases can be distinguished: The workpiece and the tool contact each other on a surface, along a line, or at a point (KUMANINA[1962], RIEKHER[ 19641, MINKWITZ[ 1965, 19661, RUSINOV[ 1973, p. 1931). (1) A surface-like contact occurs in the oldest method of making aspherics that has been applied, which is locally variable polishing (see, for example, HORNE[ 19821). This is carried out, for example, by means of a polishing tool with an appropriately formed pitch layer. A flexible lap can also be used. Here the tool has no rigid form (in contrast to the surface-like contact in grinding spherical surfaces). In most cases the workpiece is of glass and has a spherical form initially. In order to obtain the desired aspheric shape, the material is partially and gradually polished away by small and locally differing amounts. The polishing process can be controlled by the distribution of pitch over the surface of the lap and by variations in tool size and form. A flexible tool can

388

ASPHERIC SURFACES

[IV, 8 4

also be used for grinding before polishing if the desired departures from sphericity are more than a few wavelengths. The h a l shape can be obtained with a high degree of accuracy, but this requires great experience and only small departures from the initial form can be achieved. (2) For greater departures from a spherical form the grinding tool has a rigid surface which, generally, is at most in a line-like contact with the workpiece. Such a contact occurs in grinding machines according to a principle by [ 1973, p. 2181, HEYNACHER [ 19841). Mackensen (see, for example, RUSINOV In this method the grinding tool (a wheel) and the workpiece (a lens) rotate about their respective axes, which intersect each other at right angles. The outer surfaceof the grinding wheel is not cylindrical but by means of a dressingprocess has obtained the negative profile of the meridional curve of the aspheric, so that both rotating surfaces contact each other along a line which obtains the form of the meridional curve mentioned and lies in the plane of the two rotation axes. One grinding tool works several lenses simultaneously. (3) For grinding with apoint-like contact a number of principles are known (see the references mentioned above). In general, the workpiece and the tool rotate; the tool can be pot shaped, for example, and have a grinding brim. The mutual position of the two rotation axes is varied in the course of grinding to reach all points of the meridional curve of the aspheric successively. These variations in position consist of pivoting and translatory motions of the axes, which usually are controlled electronically. There also is a point-like contact in the turning process. In the last decade diamond turning has gained greater [ 19831, SANGER importance (see, for example, SAITO[ 19781, LANGENBECK [ 19831). Unfortunately, not every material is suited to be machined in this way. In particular glass, the material most frequently used, is practically unusable for this purpose. However, a number of metals, for example, aluminium and electroless nickel, are used successfully for the manufacture of mirrors, and germanium is used for infrared lenses (for further materials, see SANGER [ 19811). Aspherics can be turned with a high accuracy and surface quality so that subsequent polishing becomes unnecessary or is shortened. Single-point turned surfaces can also be used as master molds for aspherics of plastics (see later). Methods utilizing elastic deformations for making aspheric surfaces have [ 19841. The oldest and best known method applies been reviewed by MARIOGE to the famous Schmidt plate, which is located near the center of curvature of a spherical telescope mirror and corrects its spherical aberration (see $ 5.1). After the first publication about this plate (SCHMIDT [ 1931]), its fabrication has repeatedly been described (see, for example, RIEKHER[ 1957, p. 3721). A thin,

iv, o 41

FABRICATION AND TESTING METHODS

389

plane-parallel glass plate is supported vacuum tight along its circular rim. Then the air pressure under the plate is decreased and the atmospheric pressure from above causes the plate to sag. In this state the upper plate surface is ground and polished to a spherical form of weak curvature. Then the underpressure is removed, and the plate in its untensioned state has the desired aspheric profile. This method has been varied in several ways. For example, the vacuum-tight circular support can be at a specified distance from the plate rim, and the underpressure on the two sides of the support can be different (LEMA~TRE [ 19721). Aspheric mirrors can also be worked in a state of tension, with an appropriately formed back surface of the mirror, or the mirrors already have their desired shape from bending and are used in the state of tension, as is the case for nonrotationally symmetrical mirrors in X-ray optics (see, for example, HOWELLS[ 1981, sess. 21). Several groups of methods can be distinguished that use a negative master forming the aspheric by surface contact (PARKS[198l]). These methods include casting of plastics, injection molding of plastics, thin-film replication (aspheric layer on a substrate), and electroforming of metals. Casting and molding techniques are chiefly applied in large-scale production. The material used in these techniques is often acrylic (for other materials and their properties see GREISand KIRCHHOF[ 19831 and WOLPERT[ 19831). In plastic casting the liquid material is poured into a master mold, in which the material cures. The problem of shrinkage during polymerization is handled in several ways, for example, by a precompensation in the master shape and by timing and temperature control. One application of this technique is the manufacture of ophthalmic lenses. The injection molding process (see, for example, GREISand KIRCHHOF [ 19831) uses heated, plasticated material, which is forced into the heated mold cavity at high pressures. Shrinkage must also be minimized in this technique, and a number of parameters have to be controlled, including temperature, injection velocity, and the pressures. Aspherics manufactured in this way are applied in photographic lenses.* Thin-filmreplication techniques (e.g., epoxy replication) are used not only for making diffraction gratings but also for the fabrication of aspheric lenses and mirrors (see also WEISSMAN[1981] and LOEWEN[1983]). The aspheric element has a substrate, for example, which is made of glass and has a spherical

* Low-accuracy condenser lenses are often made by molding of glass. For a description of precision-molded glass aspherics see MASCHMEYER, ANDRYSICK, GEYER,MEISSNER, PARKER [ 19831 and MASCHMEYER, HUJAR,CARPENTER, NICHOLSON and VOZENILEK and SANFORD [1983].

390

ASPHERIC SURFACES

[IV, 8 4

surface. The substrate surface is coated with a thin epoxy layer whose outer surface obtains its accurate aspheric shape from the properly aligned master surface. In electrofoming the master is treated with a conducting release agent and then, by means of electroplating, is coated with a sufficiently thick layer of metals, which is removed from the master when the plating operation is finished. One application of this technique is the production of reflectors for high-power light sources. Electroforminghas also been used to produce X-ray LOUGHLIN and KOWALSKI [ 19841). mirrors, for example (ULMER,PURCELL, In addition to these fabrication methods there are special methods with limited applications, for example, vacuum evaporation for small departures [ 19841, ANG~NIEUX, MASSONand ROUCHOUSE from sphericity (MARIOGE [ 19831) and a mosaiclike composition of mirror segments for large-aperture and BARR[ 19821, ULRICHand KJAR [ 19841, telescope mirrors (BURBIDGE MEINEL, MEINELand TULL[ 19841, MAST,NELSON, GABOR and BUDIANSKI [ 19841).

4.2. REFERENCES TO THE LITERATURE FOR TESTING METHODS

A large number of papers have appeared that examine the testing of aspherics. However, since they have already been reviewed or referred to in literature compilations, the following description is primarily confined to those references. For precise testing of aspherics, interferometric tests are widely used. These tests, without and with computer-generated holograms, have already been and SCHWIDER [ 19761). In addition,noninterdescribed in this series (SCHULZ ferometric tests of aspherics (e.g., screen tests) have been described in a review edited by MALACARA [ 19781. Testing methods for aspherics can also be found CORNEJO and in comprehensive bibliographies on optical testing (MALACARA, MURTY[ 19751, CAULFIELD and FRIDAY[ 19801, RODRIGUEZ, CAULFIELD and FRIDAY[ 1982]), and in parts of conference proceedings (BAKERand ROSENBRUCH [ 1981, sess. 41, STONECYPHER [ 1981, sess. 41, WYANT[ 1983, sess. 31). Applications and further developments of such methods for testing aspherics have also been described in recent journal articles including interferometric methods such as shearing and scatter-plate methods (MURTYand SHUKLA[ 19831, v. BIEREN[ 19831, HARIHARAN, OREBand ZHOUWANZHI [ 19841, OHYAMA, YAMAGUCHI, ICHIMURA,HONDAand TSUJIUCHI [ 19851, Su, OHYAMA,HONDAand TSUJIUCHI [ 19861); interferometry using holo-

IV, B 51

FIELDS OF APPLICATION

39 1

grams that are computer generated or obtained from a material master surface (LUKIN, RAFIKOVand TOPORKOVA [ 19811, FANTONE [ 19831, D ~ R B A Nand D TIZIANI[ 19851,ONo and WYANT[ 19851); null tests (RODGERSand PARKS [ 19841); and the Hartmann test (MORALES and MALACARA [ 19831).

8 5. Fields of Application Aspheric surfaces have been used or suggested for application in a variety of fields. Classifications of this kind, because of the various aspects to be considered and the need for simplification, are bound to be arbitrary to some extent, and overlapping and omissions are unavoidable. Of the extensive literature only examples can be given here. Since the manufacture of aspherics is much more complicated than that of sphericalsurfaces, aspherics are also much more expensive, at least in the initial stage of development. Thus, in cases where aspherics are used, their application must be necessary or advantageous for other reasons: Certain optical problems cannot be solved without aspherics, for example, the attainment of a real aplanatic imaging as for telescopes, the shaping of ophthalmic lenses with a continuous transition from the distance to the reading area, short-wave UV imaging by grazing-incidence mirrors, and problems of concentration and uniform distribution of laser radiation. A number of other problems can be solved by using spherical surfaces exclusively,but in this case one or a few aspherics often reduce the total number of optical surfaces considerably, which means a reduction of light loss and possibly disturbing reflections, diminution of weight, and saving of space. Such advantages are, on the one hand, important for a number of special applictions, whereas on the other hand they may become important in fields where a large-scale production of aspherics is getting, or may get, suflicientlyprofitable, for example, for photolenses. However, since such profit is often uncertain, most of the imaging surfaces produced are still spherical, and the large-scale manufacture of aspherics is growing only slowly. Further development in this area will depend primarily on technological developments and improvements of manufacturing procedures and on the solution of optical problems, some of which were mentioned in 8 3.3. In this regard, a comparison of the imaging performance between spherical and aspheric surfaces is also of interest, for example,the question concerninghow many spherical surfaces can be replaced by an aspheric that gives the same imaging performance. This number varies considerably in different cases (see 8 6.4).

392

ASPHERIC SURFACES

5.1. MIRROR TELESCOPES

The first application of aspherics was in telescopes (cf. 8 1). In the course of time the increasing demands for resolution and especially for image brightness resulted in the apertures becoming increasingly large, particularly in astronomical telescopes. The correspondingly increasing expenditure for manufacture required that they be as versatile as possible. Thus the mirror telescopes gained greater importance, an essential reason being that their performance is independent of the wavelength (no chromatic aberrations). Moreover, lenses with diameters of more than about 1 m cannot be used because of elastic deformations, whereas mirrors can be provided with special relief systems at the back. The great majority of telescope mirrors, particularly those with larger diameters, are nonspherical. In cases where the primary mirror is spherical (cf. fig. 6), its aberrations are usually corrected by one or several aspherics. All cases considered, a comparatively great part of the optical surfaces of mirror telescopes are aspherics, for a number of reasons. Because of the large aperture and the resolution required, including for weak stars, the imaging must be accomplished by a minimum number of surfaces to keep the manufacturing expenditure, obscuration, and stray light as low as possible. However, the minimum number of surfaces generally can be attained only if aspherics are used. Their application in telescope mirrors is favored by the fact that they can be manufactured individually and do not require mass-production techniques. Moreover, the field angles are comparatively small and the apertures are large, which makes the performance of aspherics especially effective(see 8 6.4). Thus, an exhaustive description of the application of aspherics in mirror telescopes would necessitate a discussion of almost the total field of mirror telescope optics, which is impossible within the scope of this article. Therefore, this section will describe only some essential aspects to give a general impression. Special features of imaging with nonvisible radiation are examined in 5.4. Important basic types of mirror telescopes presently in operation are (a) the arrangement using the prime focus (directly or as Newtonian focus), (b) the Cassegrain, (c) the Ritchey-Chrktien, and (d) the Schmidt system (see, for [ 19821). Most of today’s mirror example, BAHNER[ 19671 and SCHIELICKE telescopes are variants and combinations of these types. (a) If the primary mirror is the only optical element in the telescope, it is a concave paraboloid. At its focus (the prime focus) a receiver, such as a photographic plate, can be positioned. In a Newtonian arrangement, this focus is displaced to one side of the telescope tube by a small plane mirror so that

IV, 8 51

FIELDS OF APPLICATION

393

a star image can be viewed through an eyepiece. Because of the coma, the field angles in such arrangements are extremely small (e.g., of the order of a minute) if no correctors are used. (b) In a Cussegruinsystem a convex hyperboloidal secondary mirror is added to the concave paraboloidal primary mirror. The secondary mirror, reversing the light direction, shifts the focus back to an axial point near the vertex of the primary mirror, where the latter has an opening. The axial image point is again stigmatic because the paraboloid focus coincides with one of the hyperboloid foci. The field angles are again very small. (c) Since the development of Schwarzschilds design of a strictly aplanatic two-mirror system (see 8 3.6), a number of solutions for larger fields have been found, an example of which is the Ritchey-Chrktien system. This is similar to the Cassegrain system, but the surface shapes have been modified in such a way that larger fields are obtained, with a field angle, for example, of about 1 degree. (d) The Schmidt system consists of a spherical mirror and a thin corrector plate, one side being plane and the other one aspherically figured (see 4.1). The center of the plate is positioned at the center of curvature of the mirror. The incident light at first traverses the plate, whose rim acts as the aperture stop; the plate corrects the strong spherical aberration of the mirror. This correction of the incident plane wave fronts is nearly independent of their direction because of the position of the plate, which makes still larger, although curved, fields possible, with a field angle, for example, of 5 degrees. Schmidt’s principle has also been modified and applied for other purposes. The wavelength dependence of the original Schmidt plate can be eliminated by replacing the plate with an aspheric mirror that is nearly plane and acts in a manner similar to the plate. This is the case, for example, in UV wide-field cameras (COURTBS, CRUVELLIER, DETAILLE and SAYssE [ 19831). &rangements with Schmidt mirrors can be off-center sections of rotationally symmetrical systems (EISENBERGER, LEWISand MEIER[ 19831). The Schmidt plate principle has also stimulated a number of modifications for non rotationally symmetrical and catadioptric systems (see, for example, SHAFER[ 1978, 19811).

The Schmidt corrector plate lies in the parallel ray path and therefore has rather a large diameter. Essentially smaller diameters are possible in field correctors positioned in the converging ray path of an aspherical-mirror telescope (WYNNE[ 19721). Such correctors can also contain one or several [ 19841). aspheric surfaces (see also LIANZHEN The largest telescope mirrors with a coherent surface presently used are the Hale telescope of the Palomar Observatory in California (BOWEN[ 19601) and

394

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[IV,§ 5

the Telescope BTA of the Astrophysical Observatory in the North Caucasus [ 19791). Both telescopes have a parabolic primary mirror, the (MICHELSON former with a diameter of 5.08 m and a focal length of about 17 m and the latter with a diameter of 6 m and a focal length of 24 m. In both telescopes the prime foci can be used (field angles in the order of about 2 minutes), or convex hyperboloidal secondary mirrors can be added with which larger focal lengths can be obtained, partly with the aid of plane path-folding mirrors. Field correctors increase the usable field and vary the focal length. For even larger telescope primary mirrors a composition of smaller parts is increasingly discussed and investigated (multiple, mosaic, or segmented mirrors, see 4.1). If these mirrors are spherical, and thus easier to make and test, their aberrations are corrected or partly corrected by smaller aspheric mirrors or plates (SHAFER [ 19791, ROBBand MERTZ[ 19791, MEINELand MEINEL[ 1981, 19821). Ground-based astronomical telescopes with apertures of more than 10 or 20 cm cannot directly give diffraction-limited images, because of image degradations by the atmosphere, as is well known. Apart from special highresolution techniques, further possibilities have emerged with the development CRUVELLIER, of space optics (see, for example, WYMAN[ 19791, COURT~S, DETAILLEand SA~SSE[ 19831, BOKSENBERGand CRAWFORD[ 1984, sess. 3-51), where no atmospheric limitations of image quality and of wavelength ranges exist for astronomical objects. Special requirements for space optics include spatial compactness and low weight. The first large-aperture optical and UV telescope that is planned to be in space over a longer period is “Space Telescope” (HALL [1982]). Its basic arrangement is a Ritchey-Chretien system with a 2.4 m aperture and an F number of 24. The total field has an angular diameter of 28 minutes and is divided into eight segments of different forms and with different resolutions for several purposes. The resolution requirement for the inner parts of the field is about 0.1 arcseconds (diameter of the circle of least confusion at an angular distance of 4 or 5 minutes from the axis, taking account of astigmatism).

5.2. PHOTOLENSES, WIDE-ANGLE SYSTEMS, AND ZOOM LENSES

Many proposals have been made (mainly in the form of patents) for the use of aspheric surfaces in photolenses, wide-angle systems, zoom lenses, and similar systems. A great number of these proposals concern photolenses of various kinds, for example, objectives with an aperture or a field that would not be attainable by an equal number of surfaces if all of them were spherical. Other

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wide-angle systems, for example, those with reflecting surfaces, often include aspherics in their specification as well; similarly with zoom lenses (e.g., in TV cameras) and with projection lenses. In these systems the number of aspherics is generally small in comparison with the total number of surfaces. In most cases there are one or two aspherics, sometimes three, but seldom more, and the material is frequently plastic. The optical effect of such an aspheric consists in a Correction of spherical aberration, a control of distortion, or more generally, an improvement of the imaging performance. On the other hand, if a certain imaging performance is given, aspherics can substantially reduce the total number of optical surfaces needed. Despite the advantages of such a reduction, the application of aspherics still plays a comparatively minor role in this field, as described in the introduction of $ 5 . If one of the surfaces of a photolens is aspherically figured, the number of optical elements can often be reduced by one element (one single lens). Such a photographic objective with one aspheric can consist of three or four (RUBEN [ 19851) or more single elements and can also substantially enhance the relative [ 19761). Aspherics can also reduce the number of optical aperture (GLATZEL surfaces in zoom lenses. For example, a variator (the variable part of the zoom lens) consisting of three single lenses with six spherical surfaces can be replaced by a variator of two single lenses with two aspheric and two spherical surfaces (BESENMATTER [ 19801).

5.3. OPHTHALMIC LENSES

Normal ophthalmic lenses determined for one focal power are spherical, with the exception of toric lenses for the correction of astigmatism. Such toric lenses have been known for a long time and will not be considered here. Presbyopes need lenses with different powers for different distances. They often use multifocal lenses with two or three spherically shaped areas of different focal powers (the reading area below and the distance area, that is, the area for an infinite distance, above). However, these multifocal lenses have some disadvantages. They form disturbing double images if the light bundle passing through the eye pupil traverses the border between areas of different power. Moreover, objects at intermediate distances appear unsharp. To avoid these problems, a number of lens types with special aspheric shapes have been developed, since an aspheric has a locally varying curvature and thus permits a continuous run of the lens power between the reading area and the distance area.

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For example, an ophthalmic lens with a rotational-symmetrical aspheric surface has been developed. The rotation axis of this surface does not pass through the center of the lens but coincides with the average line of sight for the reading area (LAu, JAECKEL and RIEKHER[ 19541, LAU [ 19561, RIEKHER [ 19591). At this line the meridional focal power has its maximum; from there it decreases continuously downward and upward until it remains constant in the distance area. The locally variable part of this run of the meridional focal power is generated, for example, by a part of a quadric surface of revolution at whose axis the curvature has its maximum. A problem with all ophthalmic lenses that have a locally continuous power variation is their astigmatism. This characteristic is caused primarily by the differences of the two principal curvatures of the aspheric surface, the “surface astigmatism”. Only on a sphere does surface astigmatism vanish everywhere. Therefore its identical removal cannot be attained, and, one therefore tries to minimize the astigmatism of the lens. Ophthalmic lenses with aspheric surfaces that have an umbilical line have been developed (MAITENAZ[ 1954, 19591). At every point of such a line the surface astigmatism is equal to zero. The umbilical line runs nearly vertically through the center of the lens. Upward along this line the power decreases continuously from the upper end of the reading area to the beginning of the distance area. The parts of the aspheric that are not in the vicinity of the umbilical line can be controlled by further parameters (MAITENAZ[ 19701, GUILINOand BARTH[ 19781). These parts are not free from astigmatism. Compromises need to be made between the desired power variation and the unwanted astigmatism, which is necessarily connected with the power variation. The possibilities of such compromises are restricted by certain conditions, [ 19611) and for which have been derived for surfaces of revolution (MINKWITZ surfaces with a plane of symmetry that intersects the surface along an umbilical [ 19631). In the latter case the surface astigmatism varies in the line (MINKWITZ direction perpendicular to that line in its vicinity twice as fast as the curvature along the line. This curvature variation is desired and necessary for generating the focal power difference between the reading and the distance area; however, this variation inevitably leads to a certain unwanted astigmatism on either side and MINKWITZ [ 19621). The continuous of the umbilical line (cf. also RIEKHER power variations cause not only astigmatism but also continuous variations of the lateral magnification. The latter variations can also disturb the user of such a lens ( h s s o w and ZANDER[ 19721). Nevertheless, ophthalmic lenses with aspherics have increasingly been appearing on the market. For their design a number of different requirements

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(including physiological requirements) need to be considered and balanced because they partly counteract each other; for the corresponding optimization, spline surfaces are also used (FURTER[ 19841). Individuals with aphakic eyes (as a consequence of a cataract operation) need lenses of a very high power (e.g., + 12.5 diopters, WARRENBLAKER [ 19831). For these lenses, which can be worn in spectacles, correction requirements arise that cannot easily be met. Aspheric ‘lenses have been used for a longer time for this condition. They are optimized, for example, to balance oblique astigmatic and power errors of these lenses; an inclusion of the distortion in the minimization has also been investigated (KATZ [ 19821; for further references see this reference). The optimal fulfillment of all conditions for aphakic lenses is a difficult problem, even if aspherics are used.

5.4. SYSTEMS FOR NONVISIBLE SPECTRAL REGIONS

Observations using nonvisible radiation have a number of special features or peculiarities, some of which also apply to aspheric surfaces. For imaging in the extreme UV and soft X-ray region, grazing incidence mirror systems are generally used.* Their surfaces need to be aspheric. Microscope mirror systems according to fig. 13, for example, have been known for and GALE a long time (WOLTER[ 19521; for further developments see FRANKS [ 19841). Such arrangements have also been considered for laser-pellet diagnostics (CHASEand SILK[ 19751). The central obscuration can be avoided in rotational-symmetrical systems of more than two mirrors (RIESENBERG [ 19631).

Wolter (or Wolter-Schwarzschild)-type systems according to fig. 13 have been discussed by many authors (see also 3.6). Such systems have gained particular importance in X-ray astronomy (ROCCHIA[ 1984]), for example, type I with two collecting mirrors, which has been investigated and modified in various ways (see, for example, WERNER[ 19771, KASSIMand SHEALY [ 19841, BARSTROW, WILLINGALE,KENT and WELLS [ 19851; for further references see ibid). For example, a diverging mirror can be added to the two collecting mirrors of a Wolter-Schwarzschild type4 telescope as a third aspheric mirror in such a way that a telescope with two coaxial channels results. The outer channel is a Wolter-Schwarzschild type-I telescope whose second-

* However, for such imaging normal incidence systems with multilayer coatings have also been investigated and proposed (SPILLER[1982]).

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ary mirror is the primary mirror of the inner channel so that either channel comprises two of the three mirrors and directs an incident beam to the same image point, apart from aberrations. In this system one can try to keep the image blurring in both channels sufficiently low and approximately equal. Another modification is the nesting of, for example, three Wolter-Schwarzschild type-I mirror pairs as designed for the X-ray astronomy satellite ROSAT. Type 11, with a collecting and a diverging mirror, has also been the subject of many papers. Nonspherical surfaces for grazing incidence are applied not only before but also within spectroscopic instruments. For example, for the study of synchrotron radiation (KOCH [ 19831) off-axis sections of ellipsoids and paraboloids of revolution and toric surfaces are investigated and used for focusing purposes and as substrates of diffraction gratings (see, for example, HOWELLS [1981, sess. 1 and 41, MALVEZZIand TONDELLO[1983]). Other surface shapes have also been investigated (see also, ASPNESand KELSO[ 19811). Mirror systems for the infrared region (see, for example, Zimmermann and Wolfe [ 19801) can be used with smaller angles of reflection. Therefore their surface shapes do not depart from those for the visible region in the same way as do the shapes described earlier. For example, the telescope of the infrared astronomical satellite IRAS (IRACEand ROSING[ 1983]), which has surveyed the sky at wavelengths from about 10 to 100 pm, is a Ritchey-ChrCtien system with a primary mirror diameter of 60 cm. In addition, for earth remote sensing purposes, for example, for earth resource surveys from a satellite, systems operating in the infrared region are of importance. In such systems nonspherical mirrors are also used (see, for example, BRECKINGRIDGE,PAGE,SHANNON and RODGERS[ 19831). Among other requirements, wide fields are essential in these systems. Aspheric lenses are also used in the infrared region, for example, for correction of spherical aberration (KUTTNER[ 19811). However, only a few materials are available, for use in limited portions of the infrared region. A material frequently used is germanium (refractive index m 4). Reflection and absorption losses and costs require the use of a minimum number of surfaces, which in principle favors the application of aspherics, since a given small number of surfaces can be better utilized by aspheric than by spherical shapes. Moreover, because of the greater wavelengths, there are no very close manufacturingtolerances, which partly reduces the higher manufacturing expenditure of aspherics. Some aspheric systems in the infrared region have been compared with corresponding spherical systems (ROGERS[ 19781, ROGERSand NORRIE [ 19811).

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5.5. FOCUSING OF LASER BEAMS AND IRRADIANCE REDISTRIBUTION

The focusing of high-power laser beams is needed for a number of applications, including the processing of materials and laser fusion experiments. In the latter case often a large number of laser beams are needed to illuminate a small spherical target as symmetrically and uniformly as possible. For example, an incident, almost plane wave of such a beam may be required to be converted into a converging spherical wave (corresponding to a sharp focusing). If small adjustment errors are taken into account, such a focusing should also work if there are small angles between the arriving parallel bundle and the axis of the focusing system (approximate fulfillment of the sine condition). Such requirements can be met in principle by some of the methods described in $ 3.6, but there are often essential additional requirements to be met. For example, the focusing may not only be necessary for one but for several wavelengths, either successively or simultaneously. Too high energy concentrations within lenses should be avoided. The optical material must be sufficiently resistant to the radiation. The higher the energy is, the more critical is the selection of the glass for lenses, which complicates the design of achromatic systems. The term “imaging”, as far as it can be used here, refers to extremely small fields, combined with large apertures. The application of aspherics in these cases is especially effective and necessary (cf. $ 6 . 2 and 6.4), the more so because the number of surfaces needs to be as small as possible. Some solutions of corresponding problems were mentioned in $ 3, for example, in $ 3.2 a solution using a combination of refracting and reflecting [ 19751, BRUECKNERand HOWARD[ 19751) and in $ 3.7 a surfaces (THOMAS solution using two single lenses with two aspheric and two spherical surfaces (fig. 18, SCHULZ[ 19841). In the latter case, for two laser wavelengths a stigmatic focus at the same axial point is obtained (without separation changes). Other requirements have been met by using two single lenses with one PATAKYand WELFORD aspheric and three spherical surfaces (NICHOLAS, [ 19781, ELLISand WELFORD[ 19811). Furthermore, systems of one single lens with one aspheric have been designed in such a way that the spherochromatic [ 19841). aberration is minimized over acertain range of wavelengths(NICHOLAS In this case the separation between the lens and the focus changes with the wavelength. A sufficiently uniform illumination of the target is important. If the target is a small sphere, a number of illuminating beams are required. On the other hand, often the intensity over the cross-section of a single beam should also be made sufficiently uniform. A similar uniformity may be necessary for the illumination

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of flat targets and for other purposes where often only one laser beam is used. Since originally the intensity of a laser beam is not uniformly distributed over its cross-section, it is correspondingly redistributed, described by several different terms, including intensity redistribution or irradiance redistribution (CORNWELL [ 19791). Some papers and methods for solving such problems were mentioned in f 3.2 (DOHERTY [ 19831) and in $ 3.6 (see, for example, fig. 15, KREUZER[ 19651, and RHODESand SHEALY[ 19801).

5.6. ILLUMINATING SYSTEMS AND CONCENTRATORS

In illuminating systems and concentrators, as well as in systems described in f 5.5, imaging is not the primary purpose. The goal is, rather, to obtain a desired light distribution or an energy concentration. When compared with the systems described in f 5.5, however, the accuracy requirements for the surfaces are often greatly reduced here. This feature facilitates the fabrication of the surfaces,for which mostly nonspherical shapes are advantageousor necessary. If the accuracy is too low, however, one can no longer speak of “optical surfaces” and even less of “aspherics”. Illuminating systemsof various kinds with nonsperical surfaces have already been used for a long time. In simple systems the surfaces are often quadrics of revolution, particularly in reflectors (ELMER[ 19801) of simpler types, but often reflectors also have much more complicated shapes (see also, WEISS [ 19781). Illuminating systems with refracting surfaces are in use as condensers of various kinds, for example, in slide and motion-picture projectors. The accuracy requirements for aspheric condensers of microscopes with large fields and high numerical apertures (RIESENBERG [ 19771) and of interferometers for testing spherical surfaces (HERRIOTT [ 19673, SCHULZand SCHWIDER [ 19761) are comparatively high. Aspheric surface shapes in illuminating systems reduce the number of surfaces needed. For condenser-lenssystems this characteristic means a reduction of weight and of reflection losses, for example. In reflectors, restrictions in the number of surfaces are inevitable due to obscuration problems. The more the number of surfaces is reduced, the more important are their shapes, which often depart considerably from spherical forms. Nonimaging concentrators (WELFORD and WINSTON [ 19781, RUDA [1984]) became more widely known about two decades ago, and their importance and the literature on this subject are increasing. Their purpose is the concentration of radiation energy, for example, from the sun, onto a receiver

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surface. This energy is then converted into other kinds of energy, for example, in a power plant, or it is analyzed in physical or other investigations. The property of a concentrator that is of primary interest is its concentration ratio; it can be defined by the factor by which the concentrator increases the power density from the entry to the exit aperture. Conventional image-fonning systems often do not give the optimum solution for such a concentration, and nonimaging specular surfaces are used, whose chapes are quite different from spherical forms (cf. fig. 5 in § 3.2). Nevertheless,for such concentrator surfaces the term “aspherics” is not appropriate or not common; their shapes are generally uncomplicated, and the tolerances for their fabrication can be very large in comparison with those of optical surfaces of good quality. However, it is not always possible to make a clear distinction between concentrator surfaces and some other surfaces mentioned in this article. A very simple and well-known concentrator is the cone concentrator. Its (inside reflecting) surface is the lateral area of a truncated cone, at whose smaller aperture the receiver surface lies. Much more effective is the compound parabolic concentrator (CPC) shown in fig. 5. If the parabola part of this figure rotates about the optical axis, a rotational symmetrical CPC (“3D CPC”) is generated. On the other hand, if the parabola part moves perpendicularly to the plane of fig. 5, a cylindrical CPC (“2D CPC”) is obtained. Cylindrical concentrators are important for the collection of solar energy; if they are properly orientated, they need not be guided to follow the daily movement of the sun. A number of variants of the concentrator types mentioned and other solutions have been suggested and investigated in numerous papers. The most widespread application of nonimaging concentrators is the collection of solar energy; however, there are also a number of other possibilities and applications, such as the collection of Cerenkov radiation, detection and investigation of infrared radiation, optical pumping, and use in a star-sensor.

5.7. FURTHER APPLICATIONS

There are a large number of further applications for aspherics and it is impossible to mention all of them here. In one factory, for example, within a period of 10 years aspherics were made for more than a hundred applications, with one- to five-digit numbers of pieces (HEYNACHER [ 19841). Millions of pieces of some kinds of aspheric camera lenses have been produced (see, for example, RUBEN[ 19851).

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Examples of further applications include night-vision systems, ophthalmological instruments, geodesic lenses in integrated optics, nonaxisymmetrical lenses for manufacturing color television picture tubes, read-out lenses for optical disk systems, and aspherics as substrates of diffraction gratings for nongrazing incidence. The latter two examples are briefly described in this subsection. Optical disk systems are gaining increasing importance for the storage of information of various kinds. Large production quantities are expected for entertainment electronics. For video or audio disk systems the information is stored on spiral-like tracks, in a way similar to that on a gramophone record, but the rotational speed of the disk and the information density are essentially higher (track spacing is about 1.6 pm). The information is stored in the form of very small depressions (pits about 0.5 prn wide and 1 to 3 pm long). It is read out using the reflection of a laser beam focused on the disk surface. The focusing of the incident beam and the redirecting of the reflected laser light are performed by a reflected-light micro-objective, which in this case can consist of three or four spherical single lenses. For the latter purpose, the application of aspheric surfaces has been investigated and aspheric read-out objectives have been designed and made (HAISMA,HUGUESand BABOLAT[ 19791, BRAAT[ 1983b], MASCHMEYER, HUJAR,CARPENTER,NICHOLSONand VOZENILEK [ 19831, OERTMANN [ 19851). The objective can consist of just one single lens with two aspheric surfaces or, for lower requirements, with one aspheric surface. Diffractionlimited or nearly diffraction-limited imaging can be attained for numerical apertures of about 0.4 or 0.5 and fields of 0.2 to 0.6mm in diameter, for example. There are several reasons for the application of aspheric lenses in optical disk systems. The reduction of the number of single lenses from 4 to 1 not only decreases the space required and the light loss but also the weight of the read-out element, which is advantageous for automatic focusing. Moreover, for the mounting of the objective a mutual alignment of several single lenses is no longer necessary. Aspheric surfaces as substrates for diffraction gratings with grazing incidence were mentioned in 5 5.4. Nongrazing incidence gratings on substrates being neither plane nor spherical, for visible and UV spectral regions, are also made or proposed for application. The disadvantage of gratings with such shapes is their greater manufacturing expenditure, and their advantage is the greater number of parameters available for their optimization. Before holographic gratings were known, the optical properties of ruled ellipsoidal concave gratings

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were investigated (NAMIOKA\ 1961]), and later, toroidal holographic gratings were investigated, designed, and made (MASUDA,NODAand NAMIOKA [ 19781). Similar or other surface forms of diffraction gratings for astronomical purposes have also been proposed and made (see, for example, LEMAiTRE [ 19811, RICHARDSON[ 19821, DUBAN[ 19851).

# 6. Limits of the Imaging Performance of Aspherics Using aspheric surface shapes leads to a number of new possibilities as has been described in the preceding sections, but the possibilities are not unlimited. There are limits that may be shifted in the course of time and limits that are of a fundamental nature. The movable limits depend on the state of development, for example, of the computers available for designing complicated systems and of the fabrication methods practicable for making the designed aspherics. The following discussion examines not the movable but the fundamental or final limits, which are general performance limits of optical surfaces. Sections 6.1 to 6.3 describe the absolute imaging performance of optical surfaces (generally aspherics), and 8 6.4 describes the imaging performance of aspherics when compared with spherical surfaces. Discussion is confined to rotational-symmetrical systems, and the effects of dispersion are disregarded. 6.1. SHARP AND PERFECT IMAGING

The imaging performance of optical systems is not solely limited by diffraction. Already within the scope of geometrical optics, fundamental restrictions of sharp and of perfect imaging exist that were investigated some decades ago (for references see BORNand WOLF[ 1964, ch. 4.21). In particular, if the object space and image space are homogeneous and degenerate and trivial cases (e.g., the perfect imaging by a plane mirror) are excluded, not more than two secondorder surfaces may be imaged sharply (see also HERZBERGER [ 1958, p. 2421). Moreover, ifa plane is to be imaged perfectly, it is generally impossible to image an off-axis point of another plane (with another lateral magnification) perfectly, that is, without any aberration (see also WALTHER[ 19701). In investigations of such problems, characteristic functions or eikonal functions often play a role.* However, even if such a function with a desired property, for example,

* Without characteristicfunctions severe restrictions of freedom from aberrations can also be deduced by considering the imaging of an interference pattern by an optical system (SCHULZ [ 1974, Secs. B6 and C61).

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perfect imaging of a plane, has been found, we cannot deduce from this that in principle an optical system with that property exists. The existence of an eikonal function is a necessary but possibly not a sufficient condition for the existence of a corresponding optical system. Furthermore, if systems with inhomogeneous media, for example, Luneburg lenses, are excluded, in general not even a single off-axis point can be imaged sharply by a finite number of optical surfaces (WELFORDand WINSTON[ 19791). The consideration of limits of sharp and perfect imaging is just one aspect; other aspects are dealt with in the followingsections. These aspects are justified by the experience that, in fact, perfect imaging of extended fields is not attained but that the corresponding aberrations can be reduced the more optical surfaces are used.

6.2. ABERRATION-FREEIMAGING WITH SMALL FIELD ANGLES OR PENCIL DIAMETERS

The imaging process described in this section (SCHULZ[ 1982, 19851) is based on series expansions of the wave aberration {seeeqs. (3-1)-(3.3)}. Since aberration-freeimaging of large fields with wide pencils cannot be attained, the question arises as to whether aberration-free imaging is possible if the field angle or the pencil diameter is small. The term “small” here means that if the field angle is small, in eq. (3.2)all powers r p of the object coordinate r in which p is greater than a certain value m can be neglected; if the pencil diameter is small, in eq. (3.3) all powers e”of the pupil coordinate p in which v is greater than a certain value n can be neglected. This raises the following question: Is aberration-free imaging under such conditions of neglect possible, and if so, how many surfaces are necessary for such imaging if m or n is given? The answer is that such aberration-free imaging is possible and that the number of necessary surfaces is A& according to eq. (6.1)in the case of small field angles and A; according to eq. (6.2)in the case of small pencil diameters. These cases are illustrated in fig. 21 and can be compared with other cases. A small field angle means that all terms above a line parallel to the v axis are neglected, and a small pencil diameter means that all terms on the right of a line parallel to the p axis are neglected. The number of the remaining terms, which are not neglected, is still infinite.*Thus freedom from aberrations here signifies that all

* If, however, only a finite number of terms (e.g., the seven terms up to and including the dashed oblique line) are taken into account, the field angle and the pencil diameter must be small. These cases are considered in 3 3.3 and 3 6.4.

LIMITS OF IMAGING PERFORMANCE OF ASPHERICS

405

POWER EXPONENTu

(OF OBJ CT COORDiNATEr)

8

W&9)

~

0

0

0

-

0

POWER EXPONENTV

[OF PUPifCOORDINATE9) 1st

2nd

3rd

ORDER ABERRATIONS

and so on

Fig. 21. Graphical representation of the terms of the wave aberration and connection between its power series expansions (3.1), (3.2), and (3.3). Each number in a circle is the number of those aberration coefficients C in (3.1) that belong to the respective r, p power combination. This number is equal to the number of powers of cos 0 that belong to this combination. The two small squares represent the two paraxial terms with their coefficients oC,o and ,C,,, which have zero values if the image plane and the lateral magnification (or the focal length, respectively) are appropriately chosen. The four circles connected by the dashed oblique line represent the five usual first-order or Seidel aberrations. On the other hand, each row represents one of the functions W,,, Wl, ... of expansion (3.2), and each column represents one of the functions w l r w2,... of expansion (3.3). Each of these functions W,, and w , comprises an infinite number of aberration coefficients C.

aberration coefficients of the terms that are not neglected are made equal to zero by a suitable choice of optical surfaces. In particular, freedom from aberrations in the case of small field angles means that the first (m + 1) functions W, ( p = 0,1, ...,rn) vanish identically. This kind of freedom from aberrations can be called mth order aplanatism; m = 0 denotes axial stigmatism, and n? = 1 denotes the usual, or first-order, aplanatism. Aplanatic systems of any order cannot be constructed by spherical surfaces but require A:, aspherics (rn being the order of aplanatism)*: A'm

={

(1 + im)', a(l

+ m)(3 + m),

m

=

0,2,4, ... ,

m

=

1,3, 5, ... .

(6.1)

* This fact holds except for special cases (e.g., object and image point at the center of a reflecting sphere; virtual imaging of aplanatic points by a refracting sphere).

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On the other hand, freedom from aberrations in the case of small pencil diameters implies that the first n functions w , in fig. 21 ( v = 1,2 ,.. ,n) vanish identically. Systems with this property can be called thin-pencil aberration-free systems of nth order; n = 1 implies freedom from distortion, and n = 2 implies freedom from distortion, field curvature, and astigmatism (also for oblique pencils). Such systems also cannot be constructed by spherical surfaces; they require A: aspherics:*

6.3. PROBLEMS WITH THE NUMBER OF RESOLVABLE POINTS

Diffraction effects have not yet been considered in this chapter. These effects cause the image of an object point to be extended, which also occurs in the case of sharp or aberration-free imaging, where the point image is an Airy disk. In cases of nonsharp imaging, aberrations cause further extensions of the point image. In any case, the total number N of the points of a field that can be resolved by an optical system is finite. It can be defined by N=

area of the image field mean area of a point image

(6.3)

Fig. 22. Optical system with pupil and field.

* This fact again holds except for special cases; for example, a reflecting system with n = 1 can be constructed by using a small part of a reflecting sphere in the aperture stop, but in general, the case of n = 1 requires one aspheric.

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N is a measure of the imaging performance of the system and is equal to the and POHL[ 19791). One may ask which optical transfer measure (HOFMANN values of N can be attained by optical systems. For a given numerator in eq. (6.3) N approximates its upper limit Ndl if the imaging is diffraction limited. This is the case if the transverse aberrations are small compared with the radius 0.61 ADIR, of the Airy disk (A = wavelength, D and R , according to fig. 22). Thus, from eq. (6;3) one obtains

where 2 R,/D x 2 a = angular aperture, 2 RF/D x 241 = field angle. The pupil and the field can be defined either in the image space (exit pupil and image field) or in the object space (entrance pupil and object field), since R p R F / Dhas paraxially the same value in each space. Essentially Nd, is the number of degrees of freedom that can be transmitted by the system, or the number of sampling points that are necessary for a complete description of the distribution in the image (Toraldo di Francia [ 19551; for further references see, for example, SCHULZ[ 1974, Secs. B9-10 and C9-lo]). For the attainment of N x Ndl, that is of a diffraction-limited imaging, a sufficient number of optical surfaces is needed. Thus a fundamental question concerns which number s of surfaces (generally aspherics) is necessary for attaining a diffraction-limited imaging with N x Ndl, and which is the minimum value of s as a function of Ndl and other quantities, for example, of quantities on the right-hand side of eq. (6.4). On the other hand, one may ask how many points can be resolved if the numbers of surfaces is given, that is, what is the maximum value of N as a function of s if diffraction limitation is no longer presupposed? Undoubtedly, general limits exist here, and it would be of great interest to know them, but these problems have not yet been solved. In special cases, however, a number of solutions are known. For example, we know from designs of micro-objectives for optical disk systems (see $ 5.7) that two aspheric surfaces are sufficient for attaining a diffraction-limited imaging of a field of RF = 300 pm with R,/D x 0.45 and I = 0.63 pm. This means that, according to eq. (6.4), N = N ~x, 1.2 x 105;

that is, in this case about lo5 points are resolved by two surfaces.

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6.4. IMAGING PERFORMANCE OF ASPHERICS COMPARED WITH SPHERICAL

SURFACES

The general imaging performance of aspherics can be compared with that of spherical surfaces. As a result, relative values can be obtained in the form of S / A , where S is the number of spherical surfaces and A is the equivalent number of aspheric surfaces that can be expected to give the same imaging performance. This ratio S / A is a measure of the expected efficiency of aspherics in an optical system, On the one hand, there are empirical values according to which S / A sometimes amounts to 2 or 3 (see, for example, ROGERS[ 19781and HOFMANN [ 19801; see also 83 5.2 and 5.7). On the other hand, the systems discussed in 8 6.2 (for example, strictly aplanatic systems) cannot be designed with a finite number of spherical surfaces, which means S / A + 00. Thus, S/A may have very different values. However, all possible values can be estimated, under simplifying assumptions, in a consistent theoretical framework containing the cases mentioned as special cases (SCHULZ [ 19871). This estimation is based on the possibilities of controlling the wave aberration by the parameters of the spherical surfaces, on the one hand, and by those of the equivalent aspherics, on the other hand. For such a control, not only a sufficient number of parameters must be available, but the parameters must also be sufficiently distributed in the depth of the optical system, that is, distributed over a sufficient number of separate optical surfaces. The reason is that in a rotational-symmetrical system the wave aberration to be controlled is a function of three independent variables, whereas each aspheric to be used for the control represents a function of only one variable (for example, of a height) and thus will enable a function of only one variable to be controlled. Therefore, if the wave aberration according to eq. (3.11, W(r, e, 6) =

C C Cp~v7rpevcos7~, U

V

T

is regarded as a terminating series with functions of one variable, for example, with functions fpT(e)in the form W(r, e, 0) =

C I

r

fp T

where fpT(@)

=

1

pCVrev

V

7

.(el r p cos

6,

LIMITS OF IMAGING PERFORMANCE OF ASPHERICS

0

10

0

20

30

409

0

S

Fig. 23. Relative imaging performance of aspherics: efficiency S / A of aspherics, as a function of S; S is the number of spherical surfaces, and A is the equivalent number of (in general) aspheric surfaces that can be expected to give the same imaging performance. This performance is characterized by the assumption that all coefficients ,,Cvrwith p d pMAx and v d bAX have been made sufficiently small. This control of the coefficients is assumed to be effected by variations ofthe surface parameters (a) includingor (b) without variations of the vertex position parameters (e.g., lens thicknesses). If the latter parameters are partly varied, the S/A values can be expected within the hatched regions (or nearby). Similar results are obtained if bAX/pMAX is replaced by its reciprocal, and still higher S/A values are obtained if y ~ ~ ~ / p , , (or , pMAX/bAx)becomes considerablygreater than 3. The low hatched region with bAx/pMAx= 1 in the case of condition (6.5), which is shown here, is similar to the region in the case of (6.6), which is not shown.

then, in principle, it should be possible to control each of these fUnctionsfJe) by a separate aspheric. At least it will be reasonable to assume that the number of the functionsfpT(e)that can be controlled and are to become sufficiently small is equal to the numb:; A of controlling aspherics. On the other hand, regarding the equivalent number S of spherical surfaces, the number of the coefficients pCvTthat belong to the functions f,,(e) and are controlled is assumed to be equal to the number of controlling sphere parameters. Based on such assumptions, the surface numbers A and S, which would give the same imaging performance, can be determined. This performance can be characterized, for example,by the assumption that all coefficientspCv7of a certain region in the p, v plane of fig. 21 are controlled in such a way that they become sufficiently small. This region can, for example, be a rectangle P

VG YMAX

PMAX

(6.5)

or a triangle fl+ v <

gMAX

*

(6.6)

Figure 23 shows some results in the case of (6.5).The abscissa is the number S of spherical surfaces by which all coefficients of such a rectangle region (6.5) with a given side ratio V M A X / P M A ~can be controlled. The figure shows two

410

ASPHERIC SURFACES

[IV

typical features: First, the efficiency S/A of aspherics increases with an increasing number of surfaces. Thus it can be expected that systems consisting of a greafeer number of spherical surfaces can be replaced by aspheric systems with a comparatively high efficiency.* Second, for a given abscissa S the efficiency S / A becomes higher if the ratio k A x / p M A x deviates substantially -P co,for a given rectangle side from the value 1. The limiting case kAX/pMAx pMAx= m = constant, denotes an aplanatic system of mth order (see 8 6.2). The other limiting case, /LMAX/VMAX -P 00, for a given rectangular side vMAX = n = constant, denotes a thin-pencil aberration-free system of nth order. Either limiting case denotes SIA -+ 00. Thus, the efficiency S / A of aspherics will be especially high for wide apertures and comparatively small field angles (v M A X / ~ M A Xlarge), on the one hand, and for wide field angles and comparatively small apertures ( p M A X / k A X large), on the other hand. Acknowledgements The author is greatly indebted to Dr. K.-E. Elssner, Dr. R. GUther, and Dr. J. Schwider for reading the manuscript as well as to Dr. G. Minkwitz and R. Riekher for reading parts of it and would like to thank them for their many valuable comments. References ANG~NIEUX, J., A. MASSONand Y.ROUCHOUSE, 1983, Proc. SPIE 3 9 , 362. ASPNES,D.E., and S.M.KELSO,1981, J. Opt. SOC.Am. 71, 997. BAHNER, K.,1967, Telescope, in: Encyclopedia of Physics, Vol. XXIX,ed. S. FIUgge (Springer, Berlin) p. 259. BAKER,L.R.,and K.J.ROSENBRUCH, eds, 1981, Aspheric Optics: Design, Manufacture, Testing, Proc. SPIE 235. BAKKEN,G.S., 1974, Appl. Opt. 13, 1291. BARSTROW, M.A., R. WILLINGALE, B.J. KENTand A. WELLS,1985, Opt. Acta 32, 197. BESENMATTER, W., 1978, Optik 51, 385. BESENMATTER,W., 1980, Optik 57, 123. BOKSENBERG, A., and D.L. CRAWFORD, eds, 1984,Instrumentation in Astronomy V, Proc. SPIE 445. BORN, M.,and E. WOLF, 1964, Principles of Optics, 2nd Ed. (Pergamon Presss, Oxford).

* Similar or better efficiency values are obtained in cases where a fixed small number of aspherics (e.g., two aspherics) are made to reduce the number of spherical surfaces of a system as far as possible.

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