JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Computation of Spherical-Mirror Systems by Means of Finite-Difference EquationsLouis P. RAITIERE

General Precision Laboratory Incorporated, Pleasantville, New York(Received July 17, 1952)

Computation of large-aperture mirror systems by conventional methods tends to yield solutions whichare no more than approximately exact. This paper develops and demonstrates a new procedure involvinga finite difference equation. The method offers several advantages, such as a direct analytical approach toand exact solutions of problems of spherical aberration and coma. A numerical example is given. Althoughthe two-mirror case is demonstrated, the method is equally applicable to three mirrors as will, it is hoped,be demonstrated by another article to follow.

1. INTRODUCTION

LIMITED aperture, coma, large central obstructionand other defects have long been regarded as

necessary evils attendant on the use of spherical-reflector optical systems. Some spherical-mirror opticaldesigns have been produced, but the design process haslargely been an empirical one because of computationaldifficulties."2 The classical method of the evaluation ofterms of an infinite series breaks down when applied tolarge-aperture spherical-mirror systems.

This paper demonstrates that the application of thecalculus of finite differences to spherical-mirror systemsyields simple, analytical design methods. An applicationof this method to a particular design is given.

2. BASIC RELATIONS

Let us consider in Fig. 1 a spherical mirror M with itscenter of curvature coinciding with the origin of coordi-nates 0. The radius of curvature OI is the internalbisector of the angle AIA' between the incident ray AIand the reflected ray A'I. The external bisector of theangle AIA' is the line PQ, perpendicular to the radiusOI at the incident point I. The four lines IA, IO, IA', PQform a harmonic pencil.

On the x axis, for any c, we have

x OP-xxi OP-x'

withOA= x and OA'= x'.

Since OP coscw=OI and 1/p=OI, where p is the cur-vature, we have

l/x'= 2p cosw -1/x. (la)

On the y axis we have

y OQ-y

y OQ-Y'

'A bibliography, Reflective Optics, obtainable from AssociateExecutive Secretary of the Armed Forces-N.R.C. Vision Com-mittee, Dr. H. Richard Blackwell, 304 West Medical Building,University of Michigan, Ann Arbor, Michigan, includes all sig-nificant references on this topic through 1949.

2 Norris, Seeds, and Wilkins, J. Opt. Soc. Am. 49, 111 (1951).

withOB=y and OB'= y'.

Since OQ sinc= OI, we have

l/y'= 2p sinco- I/y.

For w =0, we employ a and h instead of xrewrite the basic relations (la) and (lb) as

1/a'= 2 p- I/a,

It'= -It.

We shall alsorelations:

(lb)

and y, and

(2b)

have to deal with the following derived

smiwtana=

cos)- px

smiwtanw'=

cosw- px'

- x' sin' = x sine,

- y' Cosa' = y cosa,2w = a'+ a.

Now we define

Ax=x-a, x'=x'-a'.With (la), (2a), and (4a), we obtain

- (x'/a') sina'

(3a)

(3a')

(4a)

(4b)

= 2xp(l -cosw) since- (6x/a) sinae. (6)

Between the surface t and the surface (- 1) in a com-plex system we have the following transfer relations:

These yield with (6), after summation,

(_ /in/ ll(_ )'- sinan'3n =- sinaetlxi

an' a,n

+, 2(- )nhnXnPn( - COSwn) sinan. (6a)

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VOLUME 42, NUMBER 12 DECEMBER, 1952

an=an-i', bXn=bXn _1 , h./a.=h._j'/an-1'-

SPHERICAL-MIRROR SYSTEMS

3. TWO-MIRROR SYSTEMS

Assuming the object point at infinite distance, therelation (6a) for two mirrors becomes

(h2'/a 2') sina2'ax 2'= 2h2x2p2( 1-cosco 2)+ 2hiylpl(l -coswi), (6b)

sincexisinai=-yi and a2=a= 2wi.

Introducing e, the separation of centers, let us now write

2= xi'- e or x2= x1 (1-e/x').

For an object point at infinite distance, the relation (la)becomes 1/xi'=- 2 pi coscol, and we obtain

X2 = Xl'(1-2ep 2 cosCw0.

But we also have

a2 =al1 -e or a2=ai'(1-e/ai%

or, with (2a), l/a 1 '=2p1 and the transferh2 /a2= hi//ai 1,

hl+h 2 = 2hiple.

Combining (a) and (b) above, we obtain

h 1 x2 = YX1',

wherep= h1 -(h11+h2) COS(1.

Combining now 1/xl'= 2p, cosw1 with

we obtainyp= = sinw1,

Y = Xi' sin2w i.

Hence (6b) becomes, with (7),

h1(h2'/a2 ')6X2' sina2'= 2y,[Ah2p2(1-cosW 2)+h 2 pl(1 -Cosw)].

(a)

relation

(b) FIG. 1. Basic spherical-mirror geometry.

"A stigmatic two-mirror system is aplanatic when the(7) mirrors are concentric."

With (7), a2=2w1 and x2'=a 2' (stigmatism for the(7a) zone considered), the relation (4a) becomes

-a 2 l sina2' = x,' sin2coi.

(7b) But xi' sin2w1=y 1 (7c). Hence, with the sine condition- y= sina2 ' and the scale equation (9'), we obtain

(7c)A =-h2, (11)

or, with (7a),

(8) 1+h2= (1+h2 ) cosC0. (11')

Now, with (2a), (2b), and h2 /a 2= hl'/ai', we obtain

-h2'/a2' = 2h2p2 + 2hipl. (9)

This last relation (9) can be taken as the scale equationif we write hi= + 1. We shall write this scale equation as

-0=2h 2p 2+2pi or h2'/a 2'=0. (9')

Using the scale equation in (8), we then obtain

6X2' = 2 [h 2 p2 (1 - COS02 )

+pi(1-coswo)]y,/sina2 '. (10)

The absolute value of the ratio y1/sina2 ' is theaplanatic focal length. This ratio is always finite underthe present conditions. Thus the spherical aberrationis cancelled for a given zone, when

uh2P2 (I- C0Sw2)±P1(1 -cos 1 ) = 0. (10')

4. SINE CONDITION

We can write the zonal sine condition - 0yi= sina2'.We shall now demonstrate the following statement:

If w 0, we must have h2 = -1, which means that themirrors must be concentric. Since , = 1, when h2= - 1,(10') becomes

P2(1 - cosW2) - P1(1 - cosS1) = 0. (12)

Besides, for any number of concentric mirrors, theaplanatic condition is

Instead of considering the exact sine condition, weshall consider the sine relation

- y= sina2'. (13)

With this generalized relation we obtain, instead of (11),

(14)

Hence, the offence against the sine condition is givenby -i1.

961December 1952

F, (- 1) np.(l - cosw ") = 0.

,ua = - 2.

LOUIS P. RAITIERE

FIG. 2. Divergent two-mirror system with minimum obstruction.

Combining (7a) and (14) in order to eliminate At, weobtain

1-cosw 1h2= a-,

I-a cosco1

where co refers to the stigmatic zone.

5. RESOLUTION OF EQUATION (10')

A/12p2(1 -CosW 2)+ P1 (1 - COSCO) = 0. (10')

We eliminate ,u with (14), we eliminate 2 with thescale Eq. (9'), and we eliminate Pi with the aperture re-lation (7b). Thus, we obtain

(y4+ 2 sinwi)(1-cosC02)=2 sinw 1-2 sincol coswi. (16)

We now eliminate W2 with the relation (5), which yieldsin a two-mirror system,

(17)W2=- W+ ca2,

where a2' is given by (13). Let us putI*1P=cos2a2, q=sinoa2'.

We havecosW2= P coscoI-q sin 1.

Thus (16) becomes

2q sin~ci+ykq sincoj+y4= [2(p-a) sinwi+y p] cosco. (A)

Now we deal with a general relation between W2 andW which derives from the transfer relation a,= a,'.

We have (3a and 3a')

sinW2tana2=

COSW2 -P2X2

since,tanal'=

COsW1 -pIX

Hence with (la) and (7),

sin(2w 1-&J2)=Xpusinwi with X=P2/ P1- (18)

If we write An~ sinw = sini2, we obtain

2 = 2wl-i2. (18')

This last relation is convenient as a ray tracing checkformula. But for our present purpose we have to use (18)to eliminate cosw, from (A). We can write, from (17),

2wl-w2= xl- 2.Now with (14) and (7b), (18) becomes

-Y1112P2= (p sinc 1-q cosw ,),

and again we eliminate 2 with the scale equation (9')and P, with the aperture relation (7b). Thus we obtain

2(1-up) sinwl+2uq coswj+y0=0. (B)

Combining (A) and (B) above, we obtain, afterreduction,

(a 2+ 1-2up) sin2WI+4yi(1- up) sinw- 2q4 = 0. (19)

The discriminant is always positive. Furthermore, bychanging the sign of , one changes the sign of the rootsbut not the value. Therefore, we shall take 4= + 1. Theforegoing equation can be solved for a given numericalaperture yi, and a given offence against the sine condi-tion a-1.

6. APPLICATIONS

a. Aplanatic Systems (u = + 1)

Substituting A= + 1, and u= + 1, Eq. (19) becomes

2(1-p) sin2 wi+y(-p) sinwi-q'=0.

With y=+0.50 we obtain

0.254333 sin 2 O 1+2X0.0317917 sinw-0.0167468=0.

The roots are

-0.4104316 and +0.1604316.

The negative one (-0.4104316) gives a combinationwith the first mirror divergent. The positive one(+0.1604316) gives a combination with the first mirrorconvergent.

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SPHERICAL-MIRROR SYSTEMS

With the scale equation and the aperture equationwe obtain

(1) sinwl=-0.4104316P =-0.8208632 P2=-0.3208632;

(2) sinw 1= +0.1604316Pi= +0.3208632 p2= +0.8208632.

The second combination is the first in reverse. Inci-dentally, we note that the third-order theory gives thefollowing equation and roots:

4pi2 +2pi- 1 = 0

cr = 0.96824

- 0.809017P1=

+0.309017

In the systems under discussion, where we haveo-= 1, it is easy to determine the value com of co, corre-sponding to the maximum residual spherical aberrationbxm' by using relation (18). We obtain

3(1+X2 ) 2 cos2w3= (1-2X 2) 2, X= p2/Pl,

which gives

(a) (sinwi= -0.4104316);

ordinate of the maximum aberration y mi= 0.36020,maximum aberration ax'= -0.00044.

(b) (sincwl= +0.1604316);

ordinate of the maximum aberration y l = 0.36020,maximum aberration aXm'= -0.00044.

However, there is a very important difference be-tween the two systems. In the second one the imagefocal point is virtual. In Fig. 2, we show how it ispossible to design such a combination, in order to havevirtually no obstruction at all.

b. Systems Stigmatic Only (, 0 1)

The maximum residual spherical aberration Axm' is afunction of a-. This poses the following problem: Whatvalue of a. gives the best correction of the sphericalaberration?

The study of the variation of the residual sphericalaberration as a function of a- in the first system issurprising. We have previously seen that axm'=-0.00044 when a= 1. Now, when a-= 0.96824 (thereciprocal of 1.0328-we shall see the significance of

TABLE I. Spherical aberration and offense against sine conditionfor stigmatic two-mirror system.

y ax' -I

0.1366608 +0.0000049 -0.00128040.2041586 +0.0000082 -0.00286450.2706618 +0.0000079 -0.00507490.3358462 +0.0000025 -0.00789030.3993944 -0.0000068 -0.01128450.4609970 -0.0000111 -0.01521980.5000000 -0.0000014 -0.01804850.5203535 +0.0000089 -0.0196424

=1

Sx'XIo3

-1.0 .5 ; .5 1.0+

FIG. 3. Spherical aberration in a two-mirror system with variousoffenses against the sine condition.

this figure later), the residual aberration becomespositive (Fig. 3) and the ray-tracing process givesbxl'= + 0.00074. By continuously varying a- from 1 to0.96824, we must pass without discontinuity from thecurve ( 1) to the curve (o-=O.96824), since we cansee no reason for discontinuity. On the other hand, it isimpossible to admit that a given curve coincides withthe y axis for any one value of a-. But there is no con-tradiction in these two statements, if we assume thatsome curves cut the y axis in two points P and Q forsome values of a. The point P, corresponding to a givenvalue of yi, is steady, while the point Q runs along they axis.

For o= 1, the root (in the first system) is smallerin absolute value than the corresponding absolute valueof sina2' For a-0. 9 6 8 2 4 , the root is larger in absolutevalue than the absolute value of sina2'. We expect twopoints of coincidence between the curve of aberrationand the y axis for the value of a- that corresponds with

)1i= a2'. With the condition Cw1= a2', Eq. (18) becomes

4 sin 2 cw ±sinc 1 = -2 sin w1 ,

which gives i = - 30.610. The corresponding value ofa is 0.9819486. From these data we obtain

ri= -0.9819486

r2=- 1.704243512=-0.8834512 e=-0.0572225.

The relations (13) and (7a) give ri= - a-.

In Table I and Fig. 4 we have the curve of sphericalaberration and its coordinates, and also the offenceagainst the sine condition a--1.

The spherical aberration is very small (r' is 40times smaller than in the aplanatic combination, and itis not necessarily the best correction) for only two

963December 1952

LOUIS P. RAITIERE

0.6

0.3

0.2

0.1

Y

/ 8X 1X I05

- 1.0 ( 1.0 +

FIG. 4. Spherical aberration for final design of two-mirror system.

mirrors and a N.A. of 0.50. However, the column (a-1)shows that we are far from the sine condition. Although,from that point of view, this last combination is muchbetter than the parabolic mirror, it is unfortunately nota good reference.

7. RELATIONSHIP BETWEEN THE SECONDDERIVATIVES AND 2

The spherical aberration and coma are even or sym-metrical functions. This means that the first derivativesof these two functions equal zero at the origin ofcoordinates.

Now, the formula giving the radius of the osculatingcircle at a given point of a curve f(u) is

ro= [1+f'()]'/f"(u)Hence, for the aberration functions considered, thecurvature of the osculating circle at the origin equalsthe second derivative.

Furthermore, the Maclaurin series expansion ofthese functions,

f(h)= f(O)+2 f (0) l2+. .

shows that the second derivative equals the coefficientof the third order multiplied by 2, since we have, forinstance for the spherical aberration,

x= a+Ahz2,

where A is the coefficient of the third order. As animmediate application, if we again consider Fig. 3 andthe reasoning about the point Q, we see that this pointQ coincides with the origin 0 when A or f"(0) equalszero.

Now, starting from the relation (4a) -x' sina'- x sina, it is easy to arrive at

We can consider X2' as a spherical aberration functionand. a as a coma function, and both as functions of theindependent variable w. Thus we have

dx21 = yuda+ adpz,

butdt= (1+h2) sincodw, d= a'dcw,

and, if we write d 2'= X'dw in order to simplify thenotation, we obtain

X'= 0a(1+112) sinco+tta'.

Hence, we have for the second derivatives

X"=a(1+h 2 ) cosw+2o'(1+h 2 ) sinw+,vu".

At the origin of coordinates we have

a=1, a'=0, =-h2 , &J=O;hence,

Xi/ = (l+h2)-h2a11. (21)

If the system is concentric ( 2 = - 1), we have

X"1 = a.

We have already seen that "u is 1, when 2 is -1.Under these conditions, the relation (20) becomesx2'= a. Since we have a2'= 1, 5x 2'= a.- 1. In other words,in a concentric combination, the residual sphericalaberration equals the offence against the sine conditionfor any zone. From this remark it was possible to con-clude that X"= a". On the other hand, from (21) we canconclude that only concentric systems are aplanatic.

We believe this last development justifies our choiceof origin of coordinates at the center of curvature of thefirst mirror. With these coordinates, the transfer coeffi-cients I are also coefficients of eccentricity. However,if we have to investigate only concentric systems, anysystems of coordinates are good, since we can obtainas the aplanatic condition a relation between anglesonly. For example, if we take, after W. H. Steel,3 theangles of reflection i as basic argument, we havewith (16)

P1 sini 2 = P2 smnw,or

sini 2 = YP2.

But we also have

Hence,yPl = sin 1= - sinil.

sini2 + sinil= y(p 2 - P1);

or, with the scale equation (8') and the sine condi-tion (13),

sini 2+ sini1 = - sina2',

which yields the solution with the general relation

i2+ il =--2 2 -

(20) -W. H. Steel, Australian J. Sci Res. (A) 4, (1951).X2'= ,.0

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SPHERICAL-MIRROR SYSTEMS

8. A FIRST MIRROR CONVERGENT COMBINATIONWITH A REAL FOCAL POINT

We have seen that the aplanatic positive-root system(o-1, sinwi=-0.1604316) has its image focal pointvirtual, since the image produced by the first mirror islocated in front of the second one. Under these condi-tions, 2 is negative. But 2 increases with a. Is itpossible, then, to give to a some acceptable values, largeenough to have W2 positive and consequently to havethe image focal point real?

In this problem, there is a discontinuity correspondingto 2=0. That discontinuity corresponds to o-p=1 inEq. (19). For o-p= 1, that equation degenerates into abinomial one. The roots are iy,/ 2 . They give h2 = °°.With yl=0. 5 0, ao1.0328 (the reciprocal of 0.96824).Theoretically, we have to give to a values higher than1.0328 in order to obtain a real image focal point.Practically, this value of departure is too high, sincethe residual spherical aberration increases very rapidlywith a. With yi= 0.25, the situation is better. The valueof a corresponding to the solution of discontinuityis 1.0079. From this datum, for instance, we can take

a= 1.025 in order to have a real focal point. With thislast figure we obtain

ri= +0.796704

r2= + 1.101783h2=-1.933819 e=-0.371989.

The maximum residual spherical aberration is Ax'=-0.00072, while we have aX4/'=-0.00044 in theaplanatic combination having 0.50 for N. A. There isanother important drawback in this kind of system. Theobstruction by the second mirror is always considerable.

9. CONCLUSION

Far more work has been done, using the methoddescribed, than is indicated by the limited content ofthis one paper. The computation process has beenfound to be relatively simple. Indeed, the computationof multiple spherical-mirror systems has proved to beeasier with this method than that required for a com-parable lens system using the classical method. Resultshave been found to be in startlingly good agreementwith theory.

December 1952 965