Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. A 1529

Primary aberrations of systems of toroidal Fresnel surfaces

Erwin Delano

The Institute of Optics, University of Rochester, Rochester, New York 14627

Received September 22, 1992; revised manuscript received February 8, 1993; accepted February 19, 1993

By means of a different method from that of Sands [J. Opt. Soc. Am. 63, 425 (1973)], the theory of primaryaberrations for systems of regular (non-Fresnel) surfaces with symmetry about two mutually orthogonal planesis generalized to toric Fresnel surfaces. Computationally convenient expressions are derived for the resulting20 monochromatic and 4 chromatic aberrations. Also given are simple relations between the aberrations of thepupil and those of the object and for the changes in the aberrations with either stop shift or object shift.

1. INTRODUCTION

The primary monochromatic aberrations of regular (non-Fresnel) systems with symmetry about two mutuallyorthogonal planes have been studied at various levels ofgenerality by Smith,' Burfoot,2 Wynne' (for cylinders only),Buchdahl, 4 and Sands.5 6 Sands 5 gives formulas for calcu-lating the monochromatic aberrations and for determininghow they change with stop shift. In the present paper adifferent method from his is used to derive more generalresults that are applicable to toric Fresnel surfaces too.The paper also includes a discussion of the change in aber-rations with object shift and of the primary chromaticaberrations.

The method and the notation used in this paper is simi-lar to that used by Delano7 in his treatment of the primaryaberrations of systems of Fresnel surfaces of revolution.

Consider a Fresnel surface for which both the substrate(base surface) and the Fresnel facets (grooves) possesssymmetry about both the xy and xz planes, so that thex axis is regarded as the optic axis (Fig. 1). The inter-section of that axis with the surface will be referred to asthe latter's vertex. Let (xo, yo, zo) be the coordinates of apoint on the substrate, and assume that the equation ofthe substrate is -

xo = '/2coyy02 + 1/2co Z0

2 + eoyyo4

+ 2eoyzyo2zo2 + ezo 4 + 0(6) (1)

with respect to an origin at the vertex, where coy and co,are the curvatures of the two principal sections of the sub-strate and yo and zo are regarded as quantities of the firstorder. With appropriate choices of coy, co,, eoy, eoy, andeo,, Eq. (1) will approximate any sufficiently smooth sur-face with the assumed symmetry, e.g., a surface of revolu-tion, a torus, and a general toroid.

The Fresnel facets are assumed to be large enough sothat diffraction is negligible but small enough so that therefracting surface of each facet may be approximated by aplane. The surface of the facet at the point (yo, zo) on thesubstrate has a normal parallel to that (at the correspond-ing point) on an associated surface that will be referred toas the effective surface for the given Fresnel surface.Let (x, yo, zo) be the coordinates of any point on the effec-

tive surface, and assume that

X = 1/2Cyyo2 + /2CZZo2 + eyyo 4 + 2eyoyo2 ZO2 + eZ 04 + 0(6)

(2)

with respect to an origin at the vertex.Let ,u, v, and A be the direction cosines of the unit nor-

mal to the effective surface at the point (yo, zo). Calculat-ing the gradient for that surface and normalizing it give

v = -YoCy + 1/2yoCy(cy2 yo2 + czo 2)

- 4yO(eyyo2 + ezo 2) + 0(5),

A = -zoCz + 1/2 zC(Cy2yo2 + Cz2Z02 )

- 4zo(ey2yo2 + eZ02) + 0(5). (3)

Let u and v be the slopes of the projections of an arbitraryskew ray on the xy and xz planes, respectively, and let Nbe the refractive index that precedes the surface. Use aprime to denote quantities after refraction, and definen = N/N'.

Now Eqs. (2) in Ref. 7 state that

U= 1/2U(U2 + V 2) + nu - /2nu(U 2 + V2)

+ (1 - n)v{1 + /2n[(u - V)2 + (V - A)2]} + 0(5),

V = 1/2V(U2 + V 2) + nv - /2nv(U2 + V2)

+ (1 - n)A{1 + /2n[(u - V)2 + (V - A)2]} + 0(5),

which may be rewritten as

u' = nu + (1 - n)v + 1/2U(U,2

+ Vs2) - /2nU(U2 + 2)

+ /2(1 - n)nv[(u - V)2 + (v - A)2] + 0(5),

v = nv + (1 - n)A + 1/2V'(U'2

+ V2) - /2nv(u2+ V2)

+ /2( - n)nA[(u - V)2 + (V - A)2] + 0(5). (4)

2. RAY INTERSECTION WITH SURFACE

Before refraction the ray passes through some point A,with coordinates (0, y, z), on the plane tangent to the sur-face at the vertex 0 (Fig. 2). After refraction it passesthrough the corresponding point A', with coordinates

0740-3232/93/071529-06$06.00 © 1993 Optical Society of America

Erwin Delano

f530 J. Opt. Soc. Am. A/Vol. 10, No. 7/July 1993

f acetsla) - ---- - -- - ----- --- ---

regular toroidal esurface

Fig. 1. Comparison in the xy plane (a) and the xz plane (b) of atoroidal Fresnel surface (right-hand side) with the regulartoroidal surface (left-hand side) from which it is derived. Thenormal to each facet (groove) on the Fresnel surface is parallel tothat at the corresponding point on the regular surface. Thewidth of the facets is greatly exaggerated. For generality, thesubstrate is shown as being toroidal, but it need not be.

to obtain

u = us + /2u'(u" + vt2) - /2nu(U2 + V2)

+ 1/2yCy(- 1)n(i2 + j2) - y 2c 2- Z2 2]

+ 2cy (n - 1)u(y 2coy + Z2Coz)

+ 4y(n - 1)(eyy2 + e z 2) + 0(5), (7)

where

i=ycy+u, up=nu+ycy(n-1),J=zcz + , = no + zc(n-1). (8)

In Eq. (7) i and j are the paraxial angles of incidence onthe Fresnel facet in the xy and xz planes, respectively,while up and vp are the paraxial values of u' and v', respec-tively. A similar equation for v' follows from symmetry[exchange y's and z's, u's and v's, i's and j's, etc., every-where in Eq. (7)]. From Eqs. (5),

= yo - Xou', Z' = ZO - XOv'. (9)

Substituting for x, yo, and z from Eqs. (1) and (6) intoEqs. (9) and using up and vp for u' and v', respectively, give

Y = y + /2( - ')(y2coy + z2Co,) + 0(5). (10)

A similar expression for z' follows from symmetry.

A

0

-hz v=tan V

p 2 --

AZ

3. TRANSVERSE ABERRATIONS

Let iy and denote the paraxial image distances thatcorrespond to generally different axial object points in the

'X. X2

substrate

(b)LaxIS

Fig. 2. Projections of a skew ray in object space onto thexy plane (a) and the xz plane (b). Points A and AO are the pointsof intersection of the ray with the vertex plane and the substrate,respectively. P and Pz represent the two mutually perpendicularlines that correspond to a paraxial astigmatic object. If 1, = ly,the object degenerates into a point.

(a)

(0, y', z'), on that plane (Fig. 3). Then

yO = y + xOu = y' + xou',

zO = + xzv = z' + zoo', (5)

where (xo, yo, zo) are the coordinates of the point Ao ofintersection with the substrate. Substituting for x fromEq. (1) into Eqs. (5) gives

YO = y + 1/2u(y2cO + z2CO) + 0(5),

zo = z + /2V(y2 coy + z2 coz) + 0(5). (6)

We substitute for yo and zo from Eqs. (6) into Eqs. (3)and use the resulting expressions for v and A in Eqs. (4)

A'z

(b)0

v'=tan V'l |*

X, x-axis

\substrate

1 2

T SZ

h'

Fig. 3. Projections of a skew ray in image space onto thexy plane (a) and the xz plane (b). Points A and Ao are the pointsof intersection of the ray with the vertex plane and the substrate,respectively. P and Pz represent the two mutually perpendicu-lar paraxial image lines that correspond to Py and P2 in Fig. 2,respectively. If l = i, the image degenerates into a point.

(a)

l / -l

Erwin Delano

I,-"

Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. A 1531

xy and xz planes, respectively, and let h' and h' be theparaxial ray heights at those respective distances. Then

h;=y+l u , h' = z + v. (11)

If Hy and H. are the corresponding exact ray heights, then

H = y' +l$u', H' = z' + v', (12)

and the corresponding transverse aberrations are

BY = Hy - h, SZ' = H - h' (13)

(see Figs. 2 and 3). Using Eqs. (7) and (10)-(12) inEqs. (13), we obtain

By, = /2{(y2COy + z2co,)[u - u' + cyu(n - 1)]

+ lyc(n - 1)[n(i2 + j 2) - 2Cy2 - Z2C22]

+ 81;y(n - 1)(eyy2 + eyZz2)

+ I[u'(u'2 + v2) - nu(u2 + 2)]} + 0(5), (14)

where paraxial values are used on the right-hand side.Using symmetry, one can obtain a similar expressionfor 8Z.

4. SURFACE CONTRIBUTIONS

Let the subscript a refer to the marginal ray at full aper-ture in the xy and xz planes simultaneously, and let thesubscript b refer to the chief ray at full field in thoseplanes. One may then write

Ey, = -N'U{coyYa2[lcyua( - 1) + ua - u]+ lyCyYa(n - 1)(ni2 - Cy2ya2)

+ 1[8eyya3(n - 1) + U fl nUa 3]},

Ey2= -N'u{cozza 2[lcyua(n-1) + Ua - Ua]+ lcyya(n - 1)(nja2 _ za2c.2)

+ 1y[8eyzYaza2(n - 1) + u'vo2 - nUaVa 2 1},

Ey3 = -Nuau{Coyya[lcy(n - 1)(2UaYb + UbYa)

+ 2Yb(Ua - Ua) + Ya(Ub - Ub)]

+ ycy(n - 1)[nia(iaYb + 2 ibYa) - 3Cy2Ya2Yb]

+ 31[8eyya2yb(n - 1) + U 2 U, - nUa 2Ub]},

Ey4 = -2Nua'{COzZaZb[lyCyUa(n - 1) + Ua - Ua]

+ lycyya(n - 1)(njajb - Cz2ZaZb)

(18a)

(18b)

(18c)

+ y[8eyzYaZaZb(n - 1) + Uavavb nUaVa~b]},(18d)

Eys= -N'ua'{coZa[lycyub(n - 1) + Ub U']

+ lcyyb(n - 1)(nja2 - C 2 za2 )

+ l1[8eyZYbza2(n-1) + ua 2 - nUba 2]}, (18e)

Ey6 = -Nua'{coyyb[lcy(n - l)(UaYb + 2 UbYa)

+ Yb(Ua - U) + 2Ya(Ub - U)]

+ y'cy(n - 1)[nib( 2iaYb + ibYa) - 3cy2YaYb ]

+ 31;[8eyYayb2(n - 1) + uu' 2 - nUaUb2 ]}

Ey7 = -2N'ua{CozzaZb[lcyUb(n - 1) + Ub - Ub]

+ 1'cyyb(n - 1)(njajb - Cz2ZaZb)

+ 1j8eyZybzazb(n - 1) + U'vavb - nUbVaVbl}

(18f)

(18g)

Z = U'zZa + TzZb

V = OzVa + Tb,

i = O-zja + Tzib

V' = oZ a + ; ia b , (15)

where oy and a, denote the fractional apertures in thexy and xz planes, respectively, and Ty and i, denote thecorresponding fractional fields. Instead of using BY' andBZ' themselves, we find it more convenient to define thequantities

Ey = -2o-yN'ua8Y', E = -2crzNova8Z', (16)

which are directly additive and therefore act as surfacecontributions. Now we substitute Eqs. (15) into Eq. (14)and use the result to calculate Ey, then sort terms accord-ing to powers of o-y, oZ, Ty, and rz to obtain

Ey8 -N'uI{cozzb 2 [lcyu(n-1) + Ua - Ua]

+ lcy ya(n -)(njb 2 - 2Zb2)

+ 1[8eyzYazb 2(n - 1) + UV, 2 - Ua Vb2]},

Ey= -Nua{c0yyb 2 [Ycyub(n - 1) + Ub - Ub]

+ lcyyb(n - 1)(nib2 - cy2Yb2)

+ 1;[8eyYb3(n-1) + U,3 - nUb3]},

Eylo = -N u2{c1zb [lcyub(n - 1) + ub Ub]

+ lycyyb(n - 1)(njb2 - cz2Zb2)

+ 11[8eyzYbZb 2 (n - 1) + UV, 2 - nUbVb 2]}

(18h)

(18i)

(18j)

Similar expressions are obtained for EZ and the Ez2(i = 1, ... ,10) by symmetry.

To simplify the notation, we now discard the a subscript(marginal ray) and replace the b subscript (chief ray) by abar over the variable. We may therefore write

Iy= - Y/U', =- Z/v'.

Now defineEy = EY1,cY4 + Ey2 cr0y2o2 + Eys¢y 3

Ty + E 4 LOT2LT2T2

+ Ey5Osyr~y22 + Ey6cr2'r 2 + E 7o-TzcrTz

+ EY80uY2T2 + Ey9o-'rT 3 + Eylocryr 2,

Qy = N(yu - yu),

(17) fy = cy -Cy,

QZ = N(zv - zv),

f = Coz-cZ, (19)

where

Y = OYyYa + TyYb,

U = O'yUa + TyUb,

i = Oyia + Tyib,

U = I + Tlx

Erwin Delano

1532 J. Opt. Soc. Am. A/Vol. 10, No. 7/July 1993

where Q and Q, are the Lagrange invariants in the xyand xz planes, respectively. Also, define the operator Aby AX = X' - X for any quantity X. Using well-knownparaxial identities, after extensive simplification oneobtains

Eyl = Y2 coyA(NU2 ) - 8 4eyAN + yNiA(U 2),

Ey2 = z 2 coA(NU 2 ) - 8y2 Z2eYZAN + yNiA(V 2 ),

Ey3 = 3yycoyA(NU 2) + QyyfyAu - 24y 3 yeyAN

+ 3yNiA(uu),

Ey4 = 2zzco0 A(NU2 ) - 16y 2 zzey, 2 AN + 2yNiA(vv),

Ey5 = z2coA(Nuu) - 8yyZ2eyzAN + yNIA(V2),

Ey6= 3yycOyA(Nuu) - QyycoyAu - 24y2y 2eyAN

+ 3yNtA(uu) + QyycyAu,

Ey7 = 2ZZcozA(Nuu) - 16yyzzey, 2 AN + 2yNzA(vv),

Ey8 = Z2 c oA(NU2 ) - 8y 2 z 2 eYzAN + YNiA(W2 ),

Ey =y 2coyA(Nuu) - 8yy 3eyAN + yN A(U2),

Eylo = Z2coZA(Nuu) - 8yyz 2eyzAN + YNZA(V2), (20)

and similar expressions for the Ezi. The coefficients Eyi,Ey2,Ezl, and Ez2 represent four types of spherical aberra-tion; Ey3, Ey4, Ey5, Ez3, E 4, and E25 represent six types ofcoma; Ey6, EY7, EY8, Ez6, Ez7, and E28 represent six types ofastigmatism/curvature; and Ey9, Eyl0, Ez9, and E2 lo repre-sent four types of distortion.

It follows from Eqs. (16) that the values of 6Y' and 8Z' inthe final image space for a complete system of k surfacesare given by

k k= gy E (Ey)r, 5Zk = g> E (Ez),,

r-1 r=1

For a system of k surfaces the totals for the same sixpupil aberrations are

k

Ey3 = 3 [Ey -QA(U 2 )] + 2QY (fyAU)r,r-1

Ey5 = E0 _QYA(V2),

Ey6 = Ey6 - 3QyA(uu) + Qy> [fy(YAU + YAU)]r,r=1

Ey7 = Ey7 - 2QyA(vv),

Ey9 = Ey3 _ QyA(U2) + 2 Qy(YfyAU)r,3 ~3 r-1

Eyl = EY5 QYA(V2), (23)

where quantities with no r subscript refer to the completesystem. Similar expressions are valid for the aberrationtotals Ez 3, Ez 5 , Ez 6 , Ez7, Ez 9, and E_o.

6. EFFECT OF STOP SHIFT

Let an asterisk denote the value of a quantity after a stopshift is performed, and define

py= (Y* - Y)/Y = (U* -u)/u,

Pz = (z* - z)/z = (*- vv, (24)

where py and pz are each invariant throughout the system.It follows that

Y* =Y + PY,*= U + PyU,

u*=z + py,(21)

z*= z + pZz,

v*= v + PzO,

j*=j + Pzi,where

g = -(2oryNk U k) ', g2 = (2o-zNk vk ) X

and Nk, Uk, and vk are values in the final image space.

5. ABERRATIONS OF THE PUPIL

Because of the duality between the object and the stop,one can easily convert the object aberrations Eyi and Ezi(i = 1, ... ,10) into 20 pupil aberrations, denoted by Eyi andEzi, respectively, by merely interchanging barred and un-barred quantities everywhere in Eqs. (20). It is then easyto verify, for the case of a single surface, the followingrelations that connect six of the pupil aberrations withthose of the object:

Ey3 = 3[Ey9 - QYA(U2 )] + 2QYYfYAU,

Ey5= EylO - QyA(V2),

Ey6 = E, 6 - 3QyA(uu) + Qy fy(yAu + yAu),

Ey7 = Ey7 - 2QYA(vv),

Ey = Ey3/3 - QyA(u2) + 2/SQYYfYAU,

Eylo = Ey5 - QyA(v2). (22)

Similar expressions hold for Ez3, Er5, E26, E27, E29 , and E2 10.

(25)

whereas y, z, u, v, i, j, Qy, and Qz are unchanged by a stopshift. If we apply a stop shift to Eqs. (20) and useEqs. (25) in the result, after simplification we obtain

EY* = Eyi,

Ey*2 = Ey2,

Ey*3 = Ey3 + 3pyEyl,

Ey*4 = Ey4+ 2pzEy2 ,

EY*5 = Ey5 +PyEy2,

Ey*6 = E, 6 + 2pyEy3 + 3Py2Eyl,

Ey*7 = Ey7+PyEy4 + 2zEy 5 + 2pypzEy2 ,

E* = Ey8 + PzEy4 + pz2Ey2 ,

EY* = Ey +PyEy6 + py2Ey3 + py3Eyl,

EY*,0 = Ey10 + yEy8 + pzEy7 + pypzEy 4 + Pz2Ey5

+ pypz2Ey2 (

Similar expressions are valid for the E (i = 1,... ,10).The above results and those in Sections 7 and 8, are to becompared with those given by Delano' for Fresnel systemswith rotational symmetry about the x axis.

Erwin Delano

(26)

Vol. 10, No. 7/July 1993/J. Opt. Soc. Am. A 1533

7. EFFECT OF OBJECT SHIFTNow let a dagger denote the value of a quantity after anobject shift is performed, and define

p = (yt - y)/y = (Ut -U)/u,

Pz (zt - z)/z = (vt - )/V, (27)

where py and pz are each invariant throughout the system.It follows that

Yt = + Pyy,

Ut = U + pyU,

Zt= + Pzz,

Vt = V + pzV,

it =i+ yl, jt=j+ p j, (28)whereas y, z, u, v, , j, Qy, and Qz are unchanged by an ob-ject shift. Now an object shift is applied to Eqs. (20), and,using Eqs. (28), after much simplification one obtains

Eyt1 = Eyl + py(Ey3 + Ey9) + p 2(Ey6 + Ey6)

+ p 3(E 9 + Ey3) + Py4Eyl,

Ey2 = Ey2 + py(Ey5 + E 10) + P.Ey4 + py2Ey8

+ pz 2Ey8 + pypz(Ey 7 + E 7) + py2PzEy4

+ p p 2(E 10 + E) + py2pz2Ey2 ,

Ey'3 = Ey3 + p(2Ey6 + E 6) + pY2(3Ey9 + 2Ey3)+ 3p 3Eyi,

Ey'4 = Ey4 + py(Ey7 + E 7) + 2zEys + PyEy4

+ 2pz(Ey 1o + Ey5) + 2 2pzEy2,

Ey5 = Ey5 + pyEy8 + pzEy7 + pypzEy 4 + pz 2 Ey10

+ p p 2Ey2 ,

Et = Ey6 + p(3Eyg + E 3) + 3py2Eyl,

Eyt7 = Ey7 + PyEy4 + 2pzEy1 + 2pzEy 2,

Eyt = Ey8 + py(Ey 10 + Ey5 ) + Py2E

Ey = Ey9 + pyEyl,

EY10 = E 10 + pyEy2 ,

(29a)

plane. Moreover, since Eqs. (30) and (31) are valid forboth Fresnel and regular surfaces, the same is true for theprimary chromatic aberrations. Inspection of the well-known expressions for the primary chromatic aberrationsof regular surfaces of revolution immediately yields thefollowing result:

F, = -yNiA(dN/N),

Fy2 = -yNzA(dN/N), (32)

where F and Fy2 represent the primary axial color andlateral color, respectively, in the xy plane. The quantitydN is the difference in refractive index at the two wave-lengths used (e.g., for a visual system, dN = NF - Nc).Expressions for F., and FZ2 are exactly analogous.

To obtain the effect of a stop shift, the chromatic aber-rations of the pupil, and the effect of an object shift, oneshould use the same procedures as those described inSections 6 and 7 for the monochromatic aberrations.

For the change in chromatic aberrations with stop shift,one obtains

F* = , y2 = Fy2 + pyFyl, (33)

and there is an analogous result for Fz* and Fz*2.As for the pupil aberrations, note that, in general, Fy,

(29b) and F_1 are independent of the chromatic aberrations ofthe object, so that the only pupil aberration to be consid-ered is Fy2, which satisfies the relation

(29c)Fy2= Fy2 + QyA(dN/N), (34)

(29d) with an analogous relation for FZ2.For the change in chromatic aberrations with object

shift, one obtains

(29e)

(29f)

F' = Fyl + Py(Fy2 + Fy2 ) + Py2FyI

Fyt2 = Fy2 + pyFyl, (35)

(29g) with similar relations for Fzt1 and F1.(29h) This section completes the analysis of the primary aber-

rations for a system of Fresnel surfaces with double-plane(29i) symmetry.

(29j)

with similar expressions for the Ezli.

8. CHROMATIC ABERRATIONS

The paraxial refraction equations [see Eqs. (8) and (10)]

u' = nu + Ycy(n - 1),

v' = nv + zcz(n - 1),

Y =Y,

Z' =Z, (30)

and the paraxial equations for transfer from the rth to the(r + 1)th surface at distance dr,

Yr+ = Yr + dr ur,

Zr+1 = Zr + drvri

Ur+1 = Ur

Vr+1 = Vr (31)

together show that paraxially the xy and xz projections ofa skew ray are independent. Since the primary chromaticaberrations arise from the variation of first-order quanti-ties with wavelength, it follows that the chromatic aberra-tions in the xy plane are independent of those in the xz

9. SURFACES OF REVOLUTION

For a surface of revolution, cy = cZ, coy = Coz,eyz = ez. Furthermore, let us assume that y =y = z, and u = v. It follows that Qy = Qz, iz= j, which gives

Eyl = Ey2 = E-A = Ez2 = S,

Ey3 = EZ3 = 3S2C + S2L,

Ey4 = EZ4 = 2 S2C,

Ey5 = Ez5 = S2C + S2L,

Ey6 = Ez6 = 3S3 + S4,

Ey7 = Ez7 = 2S3,

Ey8 = Ez8 = S3 + S4,

Ey= = Ez9 = Ezio = S5,

Fy,= Fzi C1,

Fy2 = FZ2 C 2 ,

and e =Z, U = ,= j, and

(36)

Erwin Delano

1534 J. Opt. Soc. Am. A/Vol. 10, No. 7/July 1993

where the S's and the C's are the aberration coefficientsused in Ref. 8. Since all the coefficients in Eqs. (36) areadditive, these equations are also valid for the aberrationtotals for systems of surfaces of revolution under thestated assumptions.

REFERENCES

1. T. Smith, "On toric lenses," Trans. Opt. Soc. (London) 29, 71-87 (1927-1928).

2. J. C. Burfoot, "Third-order aberrations of doubly symmetricsystems," Proc. Phys. Soc. London Sect. B 67, 523-528 (1954).

3. C. G. Wynne, "The primary aberrations of anamorphotic lenssystems," Proc. Phys. Soc. London Sect. B 67, 529-537 (1954).

4. H. A. Buchdahl, Introduction to Hamiltonian Optics (Cam-bridge U. Press, Cambridge, 1970), Chap. 9.

5. P. J. Sands, 'Aberration coefficients of double-plane-symmetricsystems," J. Opt. Soc. Am. 63, 425-430 (1973).

6. P. J. Sands, "Thin double-plane-symmetric lenses," J. Opt. Soc.Am. 63, 431-434 (1973).

7. E. Delano, "Primary aberration contributions for curvedFresnel surfaces," J. Opt. Soc. Am. 68, 1306-1309 (1978).

8. E. Delano, "Stop and conjugate shift for systems of curvedFresnel surfaces," J. Opt. Soc. Am. 73, 1828-1831 (1983).

Erwin Delano