JUNE, 1940

Exact and Approximate Computation of Schmidt Cameras

I. The Classical Arrangement

FRANK ALLEN LucyOnyx Oil and Chemical Company, Jersey City, New Jersey

(Received April 13, 1940)

The Schmidt camera is particularly advantageous in certain astronomical and spectrographicstudies. This paper gives an exact derivation for the equation of the correcting surface, togetherwith some approximations rather more precise than those previously available. Performance ofactual instruments made according to the old and new equations will differ significantly if thesurfaces are figured with exceptional care.

FOR a restricted range of optical work, it isdesirable to have a telescopic camera of low

focal ratio and high resolving power over acomparatively broad angular field. In astronomy,these requirements hold wherever faint extendedlight sources must be used, as with gaseousnebulae and comets, and also when rapid surveyphotographs of large star fields are desired. Inspectrography, such a camera is particularlyindicated for short spectral ranges with lowintensity sources, as in Raman and fluorescencestudies.

Schmidt' pointed out that a spherical mirrorwith a stop at the center of curvature gives animage free of astigmatism, and designed a cor-recting plate to remove axial spherical aberration.This plate also corrects coma almost completelyover a wide field; so that the system acts as apractically aplanatic telescope for its whole focalsurface. The advantages and disadvantages ofthis construction for astronomical and spectrumphotography have been discussed previously.'-3

The correcting plate required is practicallyplane-parallel, only slight figuring on one sidebeing necessary. To enable the use of the smallestdeviations and most accurate corrections, theplate should be formed as a weakly positive lensnear the center, with gradual reduction of thepower along a radius outward; so that finally theplate acts as a negative tens near the edge. Forrays of a selected intermediate optical height h,

'B. Schmidt, Cent.-Zeitg. f. Optik u. Mechanik 52,Heft 2 (1931), or Mitt. Hamburger Sternwarte Bergedorf 7,Nr. 36 (1932).

2 B. Strbmgren, Vierteljahrss. d. Astronom. Ges. 70, 65(1935).

3 0. Struve, Astrophys. J. 86, 613 (1937).

the power is zero, and the axial spherical aber-ration is uncorrected. The equivalent focal lengthof the system is increased for rays of smalleroptical height and reduced for those of greater.If the corrections are properly made, all rays willfocus together with those of optical height h.

Apparently Schmidt himself never publishedthe equation of the figured surface. Stromgren2

gave an approximation to the third order, in thesense of Seidel's theory of aberrations. That is,he expressed the excavation required, x, as afunction of the optical height of incident rays, h,in the form

x = ah 2-Oh4+.. . (1)

and gave instructions for evaluating a and .The problem, however, may be solved exactly,

and, if desired, power series approximations maybe derived from the exact solution. For telescopesof ordinary dimensions, the constants obtaineddiffer only slightly from Str6mgren's, but fortelescopes of large aperture and relatively shortfocus, the more exact formulations may bedesirable. Differences in notation between thispaper and that of Strdmgren are in part arbitraryand in part arise from the use here of the positivex direction for the incident light.

It is required to express the depth of excavationin terms of the optical height of an incident ray.The depth of excavation is also a function of therefractive index, n, of the plate, and the radius ofcurvature of the mirror. In the present develop-ment, this radius is suitably taken as the unitmeasure of distance: it is thus the scale factor ofthe system. A convenient parameter is the angle,

251

J. O. S. A. VOLUME 30

FRANK ALLEN LUCY

A, between the axis and the radius to the point ofincidence of a ray on the mirror.

From a plane wave front outside the instru-ment the optical length must be the same for allrays reaching the focus. Three rays should beconsidered; namely, a paraxial ray, the selectedundeviated ray, and a general ray parallel to theaxis before incidence but at a moderate distance.The necessary values for finding their opticallengths may be determined from Fig. 1. Allquantities referring to the paraxial ray have thesubscript zero; all referring to the selected rayhave the subscript s; while those referring to thegeneral ray are without subscript.

The second principal point of the instrumentmay be located by dropping a perpendicular tothe axis from the intersection of projections of aparaxial incident ray and the corresponding rayfrom the focus. To locate the intersection, onemay solve simultaneously the equations ofthese rays. Recall that in the paraxial regionarc = tan = sin. Referring to Fig. 1, we see that70 = 00+ (61+a 2 ) 0 = )o+6o-20o = Oo- is the equa-tion for the projection of the incident ray, whereflo is the ordinate. For the ray through the focus,X = t tan 0, no= o00o, where is the abcissa of apoint on the ray. Thus the focal length f = H

=(6o4-o)/6o=1-o/6o, tH being the abcissa ofthe second principal point measured from F.

Let o- be the axial spherical aberration for theselected ray, and let L be the projection on theaxis of a ray from the focus to the mirror. ThenLo= 2-o, and 4)o/Lo=6o, or ONo/ o= -. There-fore f =+ a. The optical length of a paraxial rayis

7ro = nto+ 2-f (2)

measured from the first contact of the plate andwave-front to the focus. In terms of the chosenparameter 0, f may be evaluated as follows:

L ro PC=5. alt. int. L's=.X FPC= 4),. Z incidence = z reflection.

Therefore ACFPS is isosceles, and FP,=f. Also

06,= 24),. ext. X =sum of opp. int. z 's.

Thereforef =L,/cos 24).,. (3)

Subtracting 1 -cos 4 (the sagitta) and -+oa fromthe mirror radius, we find L =cos 4-f. Thus (3)

FIG. 1. Portion of meridian section through a Schmidtcamera. Left shaded region is the transparent correctingplate, right shaded region is the first-surface sphericalmirror. For focal ratios of f: 1 to f: 3, the curvatures ofmirror and correcting surfaces will be very much less.

becomes f= (cos 4,-f)/cos 24), which readilyreduces to

f = (2 COS (4)

Also 0 may be evaluated in terms of 4). Theoptical height of a ray at the mirror is sin 4. Thensin O/L = tan 0; whence

0 = tan- sin O/ (cos (A-f)].

Further, FP=L/cos 6, or

FP= (cos 4)-f )/cos tan-' Esin 4)/(cos 4)-f )].Using the Pythagorean theorem, one may reducethis to

FP = (1-2f cos 4 +f 2) . (5)

From Fig. 1 and Eq. (5), the general optical pathis

7r=x+nt/cos 31+cos 4)/cos ( 1+62 )+(1-2f cos 4)+f2)12;(6)

where 31 and 2 are the deviations produced bythe first and second faces of the correcting plate.Equating (2) and (6) and collecting,

x(n/cos 6 -1) =cos 4/cos ( 1+82)+(1-2f cos 4+f2 )

+f-2-no(l-1/cos b1), (7)where io is the central thickness of the plate.

Equation (7) gives exactly the depth of figuringfor the correcting plate. Since it is possible tocompute 61 and 2, as detailed below, (7) could beused as it stands. This is not really convenient,however. Various less exact but simpler solutionsare obtainable. For instance, differentiating (1),

dx = 2abdh - 4ih 3dh+ - -. (8)

252

COMPUTATION OF SCHMIDT CAMERAS

At the incidence point of a selected ray,dx8/dh = 0; whence a = 20h2, or = a/2h, 2 . Sub-stituting in (1),

x8,ah, 2- ahs2/2a-hs2/2.Then

x-2xsh 2 /h 2 -xh 4/h 4. (9)

From (7), xs may be determined exactly; sincefor the selected ray

x5= (cos k8+sec Uk-2)/(n- 1). (10)

This is to be evaluated and substituted in (9). Atrigonometric table may be used, but if none ofsufficient precision is available, series expansionswill serve, yielding

For most purposes, the 4, term alone will beenough, and insertion of terms higher than thesecond of (11) in (9) would be meaningless. Ifgreater precision is required, more points shouldbe computed from (7) or (12) below, and a powerseries fitted in the usual manner.

Another equation of intermediate precision,suitable for any telescope of size and proportionslikely to be constructed, is

(n -)x (1-2f cos O+f 2)X[1+1/(1-f cos 20/cos k)]+f-2, (12)

obtained from (7) by substitutions for a, and 62,

and cancellation of terms affecting only theeighth or further significant figures for an f 2telescope.

Certain details of the derivation of (12) from(7) may be interesting. The reader will noticethat cos 3, occurs in (7) only as a divisor of n. Itis very close to unity, and equating it to oneamounts approximately to correcting the ray fora wave-length very slightly different from that forwhich n is measured. In white light, imageblurring from this source will be indistinguishablefrom that produced by the chromaticity of theplate. Only for a large spectrograph camera couldit be worth while to consider 100. The term in61+32 is on a different footing. It is apparentfrom the construction of Fig. 1 that 31+62 = 0- 2.Putting in terms of and collecting yieldscos (1+32) = (cos k -f cos 2k)/(1 -2f cos O+f 2) .Substitution of this in (7), together with cos 31 = 1,gives (12). Tracing a ray from the focus shows

that cos a,= [1-sin (-24)/n 2 ]1; where

0=tan-1 [sin 4/(cos k-f)],

and, as noted above, this value may be substi-tuted in (7) if it is ever important to have anexact solution.

The parameter enables the derivation offairly compact expressions for x, but it must beremembered that sin = y, the optical height atthe mirror, and therefore is not generally equal tothe plate radius, , for the intersection of thefigured surface by the same ray. In fact, sin 4,should be corrected by 31+a2 in radian measureto get a sufficiently close approximation to h. Analgebraic rather than trigonometric form of (12)results from the substitutions y=sin , (1 -y 2)

= cos 4,. For coarser approximations, y and h maybe used interchangeably.

Since n is a factor in the shape of the plate, thecamera is corrected for one X only. For other Vs,there will be a confusion disk at the best focus.This may be minimized by proper selection ofqAs. The radius of the plate for the selected rayshould be somewhat less than the edge radius:

hs/h6= Y, (13)

where Y is a proper fraction. Stromgren seems toprefer to choose Y= 0.866; so as to have the leastpossible angular departure from flatness at anypoint of the plate. This means that the greatestchromatic error for any of the rays will be assmall as possible.

Between the axis and selected ray, dx/dhjhas a maximum at inflection, and drops to zeroat h. Then it increases without limit as hincreases. However, for Y= 0.866, dx/dh at theedge equals the value at inflection, as one may seeby differentiating (9) and substituting this valueof Y. Str6mgren, though, states further thatwhen the chromaticity blur becomes roughlyequal in size to the blur produced by diffraction,0.707 would be a better choice. This will be thecase only for wave-lengths rather far from thatselected, especially if glass of low dispersion isused for the correcting plate.

Another choice for Y would be that giving thesmallest average deviation; thus concentratingthe chromaticity blur near its center and gettingthe best possible resolving power. If the always-present effects of diffraction are neglected, it is

253

FRANK ALLEN LUCY

TABLE I.

EQUATION X AT he AT , AT hi

(7) 123.9 138.95 76.8(12) 123.9 138.95 76.8(17) 122.1 137.3 76.3(18) 122.7 138.4 . 76.8

only necessary to minimize the integral of alldeviations over the useful surface of the cor-recting plate. Part of the light through theplate will be intercepted by the film-holder, andin a spectrograph* the rectangular beam from theprism or grating will not fill the outer regions ofthe correcting plate. Thus the integral for anactual instrument should be minimized withconsideration of such unused areas. These effectstend to cancel, and for a spectrograph, a sufficientapproximation may be found by putting h. for ksin (11), evaluating dx/dh, and assuming that thedeviations of refracted light rays are proportionalto the departure of the plate from flatness. Then

haeI | 82h 2- 741 dh = minimum (14)

is the desired condition. Here he is the edge radius.Constant factors outside the integral have beendropped: they do not affect the value of Yrequired for a minimum integral. The pales maybe removed by dividing the range and taking thenegative of the part beyond hs:

h. he,

( ( 2h2- h4 )dh -Jh (h,2h2 -h 4 )dh= min. (15)

o S h8

Integrating, collecting, differentiating, and settingthe derivative equal to zero, gives

Y=() =0.79. (16)

However, the exact value is not critical, andanything in the neighborhood of 0.8 is satisfactory.

* To get full advantage from a Schmidt camera in aspectrograph, one should use a Newtonian telescope ascollimator. If a slit of small height is used, the only aber-rations ahead of the camera will then be those caused bytechnical imperfections and by the unavoidable effects ofdiffraction.

Finally, in order to compute a telescope of theSchmidt type, one should select values forf, theratio of Eq. (13), the mirror radius of curvatureand edge radius, the plate edge radius, andshould determine the thickness and refractiveindex of the plate. Then the equation for theaspherically figured surface may be found ap-proximately from (11) or (10), with (9). Asomewhat better approximation may be reachedby the use of (12), or, if desired, the surface maybe computed exactly from (7).

A numerical example is given in conclusion.Strdmgren 2 gave a table of excavations for atelescope of 2 m radius of mirror curvature,25 cm radius of correcting plate, Y=0.866, andn= 1.5. With distances in mm, and using thenotation of this paper, his equation is

x= 5.86 10-3 h2 -6.25 10- 2h4 . (17)

Use of the first term of Eq. (11) with (9) gives

x= 5.903 10-3 h2-6.302* 10-2 h4 . (18)

Table I shows a comparison of the values of xfor sample values of h, using different equations.The tabulated numbers give the excavation inmicrons for values of h corresponding, respec-tively, to the edge of the plate, the selected(undeviated) ray, and the inflection.

The technical problem of figuring an asphericalsurface to a quarter-wave-length is still the mainobstruction to the use of the full resolving powerof available photographic materials in a cameraof large aperture and small focal ratio.

None of the differences between the values of xas obtained from (7) and (12) is greater than0.004,u for the telescope here computed, andprobably the differences are not appreciable forany telescope likely to be constructed. Thus theexact formula here derived is per se devoid ofpractical significance, but the approximationsare better than the one previously available, andit is conceivable that a correcting plate can befigured accurately enough so that the improvedcalculations will be advantageous.

254