Optical Aberrations in Lens and Mirror Systems

. v. .c "

Yol. 9. No. 10 301

'. . OPTICAL ABERRATIONS IN LENS AND MIRROR SYSTEMS

by W. de ,GROOT. 535.317.6

Whèn dealing with technical-optical problems. such as X-ray screen photography and television projection. the need is felt of an optical system with large aperture • .Apart from the special lens systems developed for this purpose. Schmidt's mirror system, con- sisting of an aspherical correction plate, demands attention. In t1lls article the principal aberrations (so-called third order aberrations) of optical systems in general and of the spherical mirror in particular are discussed, and it is shown how with one exception (the curvature of field) all the third order aberrations of a spherical mirror can be eliminated with the aid of a correction plate. '

Intr6ductión In the field of illumination, photography, picture diaphragm' expressly :fitted for the purpose. As a ' projection, etc. innumerable, technical problems rule the aberrations due tó the aspherical shape of arise where the need of an optical system with large the wave surfaces become greater' the larger the aperture is felt. Certain special applications, as .for diameter of the limiting, apertures. . , instance X-ray screen photography and television , On the other hand, as a result of this limitation projection, have emphasized the need of such a so-called diffraction phenomena arise which also system in the Philips laboratory. cause lack of definition in the image. The effect of Complicated lens systems with large relative these phenomena is the 'greater, the smaller the aperture have already been developed for this diameter of the limiting aperture relative to the purpose, but now a second solution, the application wavelength of the light used. It therefore depends' of mirror systems, is demanding more and more upon the circumstances whether the first or the attention. second cause predominates in the resulting lack of Following the invention by Schmidt, who for definition. With photographic lenses and with the astronomical purposes ·fitted a. spherical mirror mirror 'systems that will be dismissed i_n ,this article with an aspherical correction plate eliminating the one may, as a rule, 'ignore the diffraction effects. most important aberrations, Philips have developed It is thex{ permissible to substitute for the concep- a mirror system with a correction plate which can tion of light waves that of light rays, that is' to. say be applied for the purpose mentioned above and very narrow pencils of light which pass thro~gh, which is simple to construct. the system mdependently of each other and are The object of this' article is to examine more everywhere at right-angles to the wave surfaces: closely the aberrations 'of optical systems' in general ',.In other. words one is satisfied with the approxi- and of spherical mirrors' in partienlar. The proper- mation offered by ge ome t r ic al optics. ties and construction of the correction plate will he First-order aberrations discussed in a subsequent article. ' Fig.lis a representation .of the well-known Causes of the aberrations elementary construction of the passage .of rays It is already known that, with the aid of a lens through an optical system with rotational symmetry. or system of lenses whose limiting surfaces are The rays emitted from point P at a distance Xl in centred spherical surfaces, and also with aspherical front of the first focus F'i and a distance Yl below . concave' mirrors, waves of light emitted from a po~t the axis converge at a point Q at a distance x; source can he approximately éoncentrated at an- behind the second focus F2 and a distance Y2 above other point (the image point], The fact that 'this the axis, where the equa~ons imaging is imperfect is due to two causes. Y2 :Yl Xl = X2 sT= v. In the, first place after refraction or reflection =r: from a convex surface the spherical light waves apply. Thus the plane through P at right-angles emitted from the point source will generally be no to the axis is uniformly projected with a constant longer truly spherical and consequently will not linear magnification 11 upon a plane through Q at converge into -one point. right-angles to the .axis, This is only correct to a The wave surfaces are always limited' by thè sufficient approximation when all the. rays pass so. edges of the lenses or mirrors and often also by a close to the axis ("paraxial") that the angles formed, 302 PHILIPS TECHNICAL REVIEW 1947/1948

by all parts of the rays 'with the axis are so small W!_}may now cally-y 0 and z-zo (= z, since Zo= 0) that their sines and tangents may be interchanged .. t~e aberrations for the ray determined by Ju, Zu The quantity v is called the paraxial enlargement. and Q. These aberrations result from the choice of the plane W. In practice they may occur, for instance, with a photographic camera when the distance between the frosted glass plate and the lens is incorrectly adjusted. As a matter of fact the aberrations described by equations (la, 1b) are sometimes called first-order aberrations, because they are described by an expression of the first Fig. 1. Elementary construction of the image Q of a point degree in Hand Yo' There are two kinds of first- ~ource P in the case of paraxial passage of the rays. Ht and H2 . repreaent. the principal planes of the system; the focal dis- order aberrations. The term with J J still remains .tance is f. even when the magnitude of H ~s zero. It then results in an increased distance between the image. Even in the simple case of the paraxial passage point and the axis, the increase being in the propor- of rays represented above one may speak of aberration of (1 + a2) : L This may be called the first- tions. Let us suppose (fig. 2) that a narrow pencil order enlargement aberration. The terms governed of rays converges at a point Q in the plane V and by H increase ~th H, thus with the opening of the a distance Yo from the axis, If the rays are collected cone of rays. These may be called first-order aper- in another plane W not coinciding with V then ture aberrations. Obviously both enlargement aber- instead of a point of light a small patch is observed. rations and the aperture aberrations are propor- in this plane, displaced with respect to the point Q. tional to the distance d between V and Wand they In particular, a ray intersecting a plane U at a disappe~r when W is coincident with V (d = 0) .. point D having as coordinates J = H cos rp, Z = Higher-order aberrations H sin (/J, will strike the plane Wat a point having the coordinates If (fig. 3) the point source P is chosen on the axis of the optical system, but the condition that the Y = Jo + al H cosç + aaro' (la) .ray ~ust make a very small angle with the à:ds is , z = al H sin (/J, • • • • •• • (lb)

in which ~ and a2 are proportional to the distance d between V and W. . p Let (/J assume all values between. 0 and 2 :n, then . the point Yu, Zu describes a circle with radius H in

.the plane U. The ray DQ thus describes a conic 51012 surface. Next let H assume all the values between Fig. 3. A ray throughP in the plane of the diagram (cp = 0), making a large angle with the axis, intersects the axis not at zero and a maximum value Hl' Then we get a cone the paraxial image point Q but at Ql and the plane V at S, . entirely filled with rays. One mayalso imagine this where to a first approximation QS is' proportional to H3. His· .the distance from the point of intersection of the ray with the plane U to the axis.

dropped, then owing to the rotation-symmetry the ray will lie entirely in a plane through the axis. After refraction, however, it will no longer interseet the axis in. the paraxial' image point Q but in a

51011 point Ql and upon extension meet the plane V at a

Fig. 2. Construction in the xy and yz projections of a cone of point S. if, to determine the ray, we again choose a .rays with apex Q (projections Q'. Q") intersecting the plane U in plane U and a point of intersection therein having a circle with radius H. This. cone intersects the plane W in a circle with respect to which the yz projection is' displaced the coordinates Ju = H cos (/J, Zu . H 'sin (/J, then from Q to Q" (adjustment aberration). the coordinates of S will be

y = A cos (/J, z = A sin (/J, (2) having been brought about by the introduction Ui the plane U of a circular diaphragm with radius Hl' where A is a function of H, which may be devel- We shall therefore call the plane U the diàphragm oped in a series of the form' plane. . (3) Vol.. 9, No. 10 OP,TICAL ABERRATION 303

The series necessarily has odd terms only,.because when H The fact that the equations for third-order aberrations are is replaced by -H and rp by rp + n, which does not cause y" . indeed such as given in (4a) and (4b) can most easily be and Zu to change in value, y and z must also remain unchanged, "deduced by a reasoning that originated with Conrady 1). Let us consider for this purposc (fi8. 5) refraction at a The aberrations (2) are known as spherical single couvexboundary plane (centre M, radius r, index of refraction left: "t, right: n2)' We' are not only interested, as aberration. The terms with the coefficient" Cl are '0 called third-order spherical aberration, those with the coefficient el fifth-order spherical aberration, and so on. Let us now consider a point source P off the axis but in the same xy plane (fig.4). If the laws of , paraxial imaging were stili to hold, the image point would be in Q at a distance Yo from the axis.,.An Fig 5. Refraction at a' spherical boundary plane (radius r). arbitrary ray determined by its point of intersec- The point P is imaged by rays paraxial with respect to PM tion Yu . H cos tp, Zu = H sin cp on a plane U will at the point Q. Displacement of P to Pl causes the image point Q to shift to Ql" now' strike the paraxial image plane V at a point S, the xy projection (S') and the yz projection (S") we were above, in all points of light situaied in a plane at' of which are given in fig. 4. right-angles to the axis of the system, but rather in those lying on a sphere with radius Rl and centre M. Let one of these points be P, so that MP = Rl' It is clear that in this special case, owing to the spherical symmetry, the line PM. may equally well be considered as an axis as any other. If we confine ourselves to rays which are paraxial with respect tó the new axis PM then the image point Q'appertaining to it will be at a distance -R from M, in which case it follows -_ 2 x v v from the elementary theory that' 1 1 1 1 1 1 (1) nlR - naRs = -;:- na) =c--: -;:-L1 ;; • pV .310tJ l ("t - f , The quantities Rl and R are to be taken as positive or nega- Fig 4: Construction in the xy projection of a ray through P 2 tive according to whether the concave side of the lens faces to intersecting the plane U at the point y" = H cos rp, Zu = " H sin rp. P lies in the plane of the figure which at the same the right or to the left.' If P is situated anywhere on the , time is the plane of symmetry, but the ray in question does sphere Rl then the locus of the point, Q lies on the sphere R2• , not lie in that plane. This ray Intersecte the plane V at a Ifwe then consider the ray PA (fi8. 6), whichisnot paraxial point S the xy projection S' and the yz projection S" of which are indicated and which deviates from the paraxial with respect to the axis PM, we find that ünder the influence image Q (projections Q' and Q") of P (seeformulae 4:aand 4b). of spherical aberration this ray does not cut the sphere at Q The distance from Q' and Q" to the axis is yo. but at a point S having as coordinates s' = kaS cós''P and z' = kaS sin 'P, Calculation shows that the coordinates y and z of S are given by , in which y' ill'the distance from fhe y projection to the axis PM and z' = z is'the distance to the plane of delineation, y = Yo + C~H3 cos cp + c2H2yo (2 + cos 2cp) + , whilst a represents the distance from A to the axis and 'P is the angle between the plane PAM and the plane of C HYo2 cos cp C4Yo3 (4a) + a + "* ... delineation.

z = clH3 sin cp +' c2H2yo sin 2cp + + Ca'HYo2 sin cp + ... (4b)

in which the constants Cl • • • C4 depend not only upon the nature of the optical system' but 'alSOY'\!_ upon the-position of P and the choice ofthe plane U. h. ~'_, , Since we shall here confine our considerations to a IJl third-o~der aberrations the series expansion has pil --'Z been broken off at the terms of the third degree. Fig. 6. Explanation of the third-order aberrations arising from If one wishes to' find the point of intersection of spherical aberration according to Conrady. The ray from P the same rayon a plane W parallel to V at a dis- to A, of which points the x'y' projections P'A' and the y'z' projections P"A" are indicated, strikes the sphere through tance d, then the right-hand terms of equations Q (projections Q', Q") at S (projections S', S"). (4a) and (4b) are increased by terms of the form (la) and (lb), whilst if d is small enough the coeffi- 1) A. E. Conrady, Thë· five aberrations of lens-systems, cients C may be regarded a~ remaining unchanged: ~onthly Not. Roy. Astron. Soc. 79, 60·66, 1918. 304 PHILlPS TECHNIcAL REVIEW 1947/1948

With the aid of the ;)'z' projection drawn on the left-hand with radius Rp, for which onè finds: side of fig. 6 we easily find that 1 i-Ill . a sin 'lp == h sin rp and a cos 'Ijl = ao + h cos rp. Lli nl:rlRp == i~l Ti (~). The meanings of aa, hand rp are given in the diagram. Since, Here I relates ,to' the Iast refracting surface; Rp is the radius further of the Petzval surface of the system. Here again the ratio 3 :1 exists between the. coefficients of H;) 02 but it is lost again we find for S: when the corrections are made which are necessary for the

S 2 transition to ~ plane image field. ;)'= k[ao + 3a0 hcos tp + aohl (1 + 2 coss rp) + h3 cos.ç] (Sa) z' = k[a02 hsin rp + 2aoha sin tp cos rp + h3 sin rp] •.• " (Sb) Further considerations of third-order aberrations Since it is only a question of small quantities and we break A hetter insight into the significance of the terms off the expansion at terms of the third degree, the right-hand of the equations (4a) and (4b) is obtained by study- ,expressions represent at the same time the aberrations j--:>o ing the intersection figures formed when, with H and Z-Zo of the point S in the system of coordinates X;)Z. (In the transformation from y' to y, one may take cos {}.= 1 constant, cp is made to assume all possible values. because higher terms only affect aberrations of the fifth and higher orders.) Bearing in mind, further, ..that to a sufficient It must then he borne in mind that although the analytical approximation (and again the factors ignored affect only the, treatment proposed gives a complete insight into the geome- higher order aberrations) try, of the refracted-pencils of rays this is no longer the case if, as we shall do presently, each of the terms is viewed separately. ao = ayo and h = PH, The fact that this has often led to incorrect conclusions being drawn in the literature on the subject has been pointed where j 0' Hand rp have the same meaning as before, Then we find . out already by Gullstrand 2). Even though one may have an absolutely true picture of ;)-jó == hH3 cos rp+ YaHsYo (2 + cos 2rp) + the course of the pencils of rays after refraction, one may not 3Y3HyoS cos rp Yd03, • . • (6a) + + Immediately draw conclusions therefrom about the distribu- (6b) tion of intensity in the beams'. Here the approximation of geometrical optics fails more or less, since it does not allow for' by which 'ecjUati~ns (4) are obtained, but with thi~ difference the coherence between the rays. This has been pointed out by .that the co~fficie~ts of H;>asin (4) are not, as here, in the ratio' Picht 3) and Zernike ') among others. S: l.This is bècause in this case the coordinates are determined from the point óf intersection of the' refracted rayon the sphere For the sake óf, simplicity we shall now examine Ra;whiIst the image point is chosen on the sphere Rl' If P is moved along MP to PI 'on' a sphere with' radius R'l whichi s equations (4a) and (4b) one term at a time. . ' tangent to the sphere with radius Rl' then the point Q'is displaced to Ql' Equations (6) now repreaent to no less good an . Spherical aberration approximation (for the term with Y03 only is this not abso- , In the terms with JI3 one recognises the spherical lutely correct) the coordinates on a sphere with radius Ra' aberr~tion previously discussed. This therefore re- \ tangent to the sphere with radius Ra, for which we again have: mains unaltered for points outside the axis'. When . 1,"- 1 'Ill 1'1 -,--,= -(---) =-~LI (c-). this defect occurs alone then as cp varies the point ~Rl naRa T ~ na 'T n S describes a circle around the point Q, the radius A flat object plane at right-angles to the principal axis of the of which is proportionalto H3 (fig. 7.). system is obtained by taking RI' == 00. Ra' thereby' assumes a certain val~,~ Rp, for which the expression Coma : , ...~ The terms with lJ2y 0 all come under the name of co m a. If these terms alone were present, then with applies. a variation of cp and constant H, owing to the term This is the so-called Petz'valsurface. The aberration in the 2 cp, the point of intersection S would' describe a paraxial image plane, i,e. the plane surface at right-angles to circle in such a way that if the zone in the dia- the main axis tangent to the sphere Ra, differs from the expres- / , phragm plane is passed through once then that circle sion'{6)by an amount proportional to the distance from Ql to that plane (== :> 02/2Rp)and also proportional to the aperture would be described twice. The centre of this circle parameter H. When this correction is made an 'equal amount lies at a distance 2c2H2yo from the paraxial image is added to the two coefficients 3ys and Ya and the simple point. Owing to the proportionality with Yo one may ratio of 3 : 1 is lost. here speak of an enlargement aherration, which, . When dealing with a series of refracting surfaces (a system of lenses) instead of one such surface, one may proceed further and consider the spherical surface with radius 'Rp as the 2) See for instance A. Gullstrand, Naturwiss. 14, 653- 664, 1926. object plane for the next refraction. One then still arrives at 3) J. Pi~ht, Optische Abhildung, Braunschweig, 1931. an equation of the form of (6), which then applies to a sphere ') See B. R. A. Nijb oer, thesis, Groningen, }942.

-, Vol. 9, No. 10 OPTICAL ABERRATION 305 .

however.is proportionalto H2. If Hbe given differ- points when Yo varies is a circle in the xy plane ánd ent values in succession then one obtains (fig. 8) a convex surface in' space, the so-called sagittal a system of circles the common tangents of which image surface; likewise the tangenrial image points interseet each other at an angle of 60°. ' lie on' another' convex surface, the tangential image surface. The curvatures of the image surface are given by

1 2ca 1 2ca' (5) R, = ut/d resp. Rs - ai/d' The equation

_2_ . 1/2 (~+~) . . . . (6a) Rm ~s n, represents what is called the average field curv- a ture. The figure of intersection on the sphere "016 "corresponding to this curvature is a circle. The Fig. 7. Intersection figures on the plane V corresponding to the terms with H3 (spherical aberration) for values of H quantity which are as 1 : 2 : 3. A = 1/2 (~ - ~), . . . . (6b) R, Rs • I Astigmatism and field curvature ; determining the distance of the points Ms and Mt Owing to' the proportionality with H, the terms is called the astigmatism. An interesting rela- containing HYo2 are 1:0 he regarded as aperture' aberrations, which; however, depend upon Yo' The intersection of the rays of a .zone (H constant, qJ' variable) with the plane V is an ellipse with' axes caHYo2 and ca'HYo2. When we consider the intersection upon a plane Wat a distance d from V then the terms utH cos qJ + a2Yoand a1H sin rp respectively are added to the aberrations, thereby altering the 51018 relation between the axes of the ellipse. By a suitable , choice of d one can reduce either the coefficient of Fig. 9. Series of cross sections' (H = constant) with a number cos qJ or the coefficient of sin qJ to zero, in conse- of planes W (d negative, increasing in absolute value) corresponding to the terms HY02 (field curvature ànd astigmatism). sequence of which the figure of intersection conver- M. is the sagittal, lilt the tangential or meridional image point. Halfway between' Ms and M, -the cross section' is a circle, At P (the intersection with the Petzval-surface, PM. = 1/3 PMt) the ratio of the axes is 1 : 3.

tion arises between Rs and Rt when these quantities are related to. the quantity Rp, the so-called Petz:val radius of curvature of the system.

As we 'have seen above, this is simply related to the data of the system. Instead of the expression given above one may also write

StOlP Fig. 8. Intersectîon figure on the plane V corresponding to the terms with H2yo (coma) for values of H which are as in which (J)k represents the strength of the lens (placed in a ' 1 : 2 : 3. vacuum) formed by, the two successive boundary surfaces, and nk the index of refraction of the medium between those ges into a smallline at right-angles to the y axis and boundary surfaces, Here (J)k has to be taken for zero thickness 50 that . the z axis respectively (fig. 9). The centres Ms and . . . M, of these degenerate ellipses are called respectively (J)k = (nk-I) (__!__: - _!_) , the sagittal and the tangentialor meridional , . rr.-I rk image points. The locus of the sagittal image whilst I/ra and I/rIH have to be taken equal to nil. . 306 PHILlPS TECHNICAL REVIEW 19.47/1948

The Petzval curvature llRp and the two field though the adjustment errors for a given Ä are nil, curvatures are related thus: for any other wavelength aberrations of the first order will arise. These are called first-order ch r o- 1 1 - 1 1 _. -~2A=--A=--3A. (7) ma t'i c aberrations. The coefficients Cl ••• c4 s; e; s, Re will also depend upon the wavelength t These conditions can be met by requiring that the coefficients Distortion UI • • • é4 for a number (two or three) of values of .Finally the term Y03 represents an enlargemerit Ä must be zero; in the hope that they will not differ aberration dependent upon Yo but independent of much from Ä zero for intermediate wavelengths. , H and thus existing for an infinitely narrow beam. This, however, considerably increases the number It 'results in a non-linear radial distortion of of conditions that have to he satisfied: Moreover, the image. A square lying in the object plane sym- with an objective having a large aperture (large metrically with respect to the axis is distorted into HIJ) and an extensive field (large Yo/f) notonly the the form of a barrel (barrel distOl:tion)' or into the first and third-order aberrations but also those of form of a cushion (pincushîon distortion) accord- higher (fifth and seve~th J order must be considered.

ing to whéther c4 is negative or positive. , One must always bear in mind that the aberrations The designing of a system of lenses dealt with here and given separate names are equi- It is evident that from a mathematical analysis valent terms in a series expansion and that the true . ' of the aberrations alone it is almost impossible to _ deviation of a point of intersection with 'respect to design a system of lenses that has to satisfy high . - the paraxial image point is obtained by summing .demands. In practice, therefore, one does not pro- all the terms. Consequently the true intersection ceed in this way at all. The designing of optical figure of the rays of a zone upon the plane V or systems is more often than not a matter of practical upon a plane W may be a complicated curve (in experience rather than one of theory' and to' some generalof the fourth degree). Further it must he . extent more of an art than a science. Nevertheless, remembered that the coefficients C depend not only the result will always be put to the test by applying upon the properties of the system in general but an analysis of the aberrations to a solution arrived also upon the positio~, of the object and the choice at empirically. It will then usually be found that of the diaphragm plane. the third-order aberrations are not zero but that Fina11y it is obvious that the third-order aberra- -there isacombination of these together with aberrations discussed here represent only an approxima- tiolis of a higher order, such as to make the result on tion and that to obtain a complete insight it the average as satisfactory as possible. would be necessary to consider also the terms of The difficu1ties caused by chromatic aberrations the fifth "and higher orders. do not arise when mirrors are used for the projee- ...... tion. This is one of the reasons why in some cases . Chromatic aberrations . a mirror system is preferred to one oflenses. Another- One would be inclined to deduce from the fore- reason is that the spherical aberration of,à concave going that in order to obtain a good image formation mirror is less than that of a refracting surface of the in a flat image plane with the desired enlargement same strength. it is only necessary to ensure that the system has The spherical mirror .the right focal distance and that the first and third-

order aberrations, i.e. the coefficients al • • • c4' Let us imagine that we have a spherical mirror are zero. This would involve satisfying a number with circular rim. The axis of symmetry is chosen of conditions by giving .the right values to the curv- as the main axis of the optical system. The paraxial atures and the distances of the lens surfaces arid rays radiating from a point PI) on this axis converge to the indices of refraction of the kinds of glass, upon a point Qo on the same axis. A non-paraxial

since the coefficients al • • -. c4 are functions of the ray from Po after reflection meets the axis at a dif- latter quantities. No allowance would then have ferent point Qo' and reaches the paraxial image

-been made, however, for the fact that the indices of plane at a point S, so that QoS (= c1H3) again

refraction and hence the coefficients ~ . . • C4 represents the spherical aberration. depend on the wavelength Ä. Ifthe focal distance We shall now consider a point' source P outside \ for' a given Ä has one value, for some other Ä it 'will . the axis but at the same distance from the centre be different, Consequently for the same adjustment, M as Po' The line PM for the convex surface of thus for the same position of the image plane, even- which the mirror forms a part will then likewise be Vol. 9, No. 10 . OPTICAL ABERRATION 307

an axis of symmetry, Rays which are paraxial with (radius IJD and that the other rays only display a respect to PM will converge upon PM at a point Q, spherical aberration, where QM = QoM. The rays which are not paraxial We may, however, also - .as was done above for with respect to QM strike this line at another point a .system of lenses - consider' a Hat image field Q'. Thus the aberrations in the imaging of the point through the' focus (paraxial image plane with res- P may be described as spherical aberrations with pect to the main axis PoM) and express the aberra- respect to the seeondary axis PM. tion with respect to this plane by the coefficients'

Cl ••..• C4• It then appears that when y., = ftan f} The situation is now absolutely the same as that in the case (f) = L PoMP) and Zo = 0 the point of intersection described above (see figs, 5 and 6) of refraction by a single of the reflected ray from P on the image plane is convex refracting surface. To this, too, one can apply the theo- given by . ry of Conrady and in this way deduce the values of the. coefficients of the third-order aberrations. .

Diaphr~gni at the centre of curvature

.We shall not go into the details of the general case but discuss the special case in which the diaphragm

plane passes through M. Further we shall choose the , so that c~= lf2R2,Ca = ca'= -2fR2 and C2 = c4 = o. distance PoM = 00 and thus consider the imaging From this we find again that although there is field of an' infinitely distant plane. The paraxial image curvature there is no astigmatism nor any coma or " point of Po then coincides with the focus of the spher- distortion. Further the only defect is the spherical ical mirror. It is easily seen (fig. 10) that owing to aberration.

As may easily be proved, Ilt = djj = 2dlR and the field curvature is thus .

2 Ca 2 IltId = R'

The radius of the curved image plane is thus IRf21 = Ifl. Since astigmatism is zero, this surface is at the same time the Petzval surface. This may be deduced, according to a common method from the general formulae given abo~e for the Petzval curvature, which apply for the case of refraction, by . , putting n prior to reflection = 1and after .reflection = -1, so that Lt (1/n) = 2. .

S1"'fJ Fig. 10. A diaphragm the centre' of which coincides with the' Spherical mirror with correction plate centre of the mirror M brings about a practically sharp image on a sphere with radius IJ I = R/2, Depending on the diameter If when using a spherical mirror one had a means' of the diaphragm, however, spherical aberration occurs. The point sources Po and P are to be imagined as the infinitely of causing the jncident r,ays to change direction distant points of the main and secondary axes respectively. when passing through the diaphragm plane in such a way as to eliminate the spherical aberration, then the particular position of the diaphragm all beams a perfect projection would be obtained on a sphere passing through the diaphragm are subjeet to the with radius IRf21. same conditions (apart from the fact that from the Such a possibility is of great practical importance. point of view of a beam falling obliquely on the The means of attaining this to a high degree of circular aperture of the diaphragm the aperture approximation is to be found in Schmidt's cor- assumes the form of an ellipse having slightly dif- rection p l a te, which is optically Hat on one side ferent axes). Rays which are paraxial with respect and has an aspherical surface on the other side -, to PM converge at a distance Ijl = IRf21from M. . It is clear that with this a system can be devised The other rays have the same spherical aberration which combines the advantages of a ~arge aperature with respect to the sphere with radius Ifl as that with those of a wide image field. This provides a shown by the rays from' Po with respect to that simple way of obtaining a result which would other- sphere. We may therefore say that for rays which wise entail a highly complicated system of lenses. are paraxial with respect to a secondary axis the It is even possible to exceed considerably the light image formation is sharp in a curved imagè plane flux 'of the present lens systems. One must, , I

308 PHILIPS TECHNICAL REVIEW 1947/1948

however, remember that the image field is not flat the sphere. In fact one finds by calculation that but curved. . Cl = 0, IC21= 1/R2, Ical = 8/R2, ICs'1 = 4/Rs, Ic,1 = 4/R2. The construction and applications of the correc- Except for spherical aberration - which is naturally absent- tion plate will be dealt with in subsequent articles. all other aberrations do indeed occur.

Appendix: Note on the parabolic mirror Finally some remarks are to be added regarding the parabolic mirror. ' , Fig. 11 represents the cross section of the plane of the diagram for a parabolic mirror whose radius of curvature at the top is R. If the normal is drawn from a point P on the mirror surface it will interseet th~ axis at a point N, such that the distance from N to the foot Q of the perpendicular from P on N the axis' (the so-called sub-normal) is R, contrary to the case with the sphere with centre M, where the length of the normal itself is R while all normals pass through M. This difference in behaviour of the two curves has the effect that rays striking the axis obliquely show no spherical aberration but . all pass exactly through the focus F, whereby MF = Fa = 1/2 IRI. Without further thought one might suppose that a parabolic mirror with a diaphragm at the centre of curvature Fig. 11. The passage of rays with a parabolic mirror. A ray M would autómatically.be free from aberrations, but this is through P parallel to the axis, after reflection, strikes the axis at F, MF = Fa = 1/2 IRI. Here R is the radius of curvature not so, becnuse with the parabola there is not such a high de- at 0 (broken line). NP is the normal drawn from P on the gree of sym!lletry with respect to the point M as there is with ' . paraboI~; IYQ = MO = IRI. .