Any two-surfaced optical system, either refracting or reflecting, can be made stigmatic (aplanatic and anas-tigmatic) for object at infinity by using curved Fresnel surfaces instead of the usual conics. Such systemswill also have a flat field under certain conditions and may even be distortion free. It is also shown that sys-tems of flat Fresnel miirrors have incurable line coma.
Introduction
This paper deals with the correction of the mono-chromatic primary aberrations of an axially symmetricoptical system of two curved Fresnel surfaces-eithera thick lens in air or two mirrors-with object at infinity.Starting with any first-order layout, we show that twocurved Fresnel surfaces suffice to correct astigmatismas well as spherical aberration and the two kinds of comaand thereby achieve stigmatic correction. We alsodescribe a family of flat-field stigmats, one member ofwhich is distortion free. Finally, a limitation of systemsof flat Fresnel mirrors is demonstrated.
First-Order Layout
Let u and U be the slope angles of the paraxial mar-ginal and paraxial principal rays, respectively, in anylens space. Define a = Nu and a- = NU, where N is therefractive index in that space. Consider a normalizedsystem with power P = 11, ray heights Yi = 1, and Yi= 0 for the marginal and principal rays, respectively, onthe first surface, object at infinity, and LaGrange in-variant Q 1. Then
ai=° a =al =1 = 1
N 1 =1 N = n
at' = -Pa2 = b
N'2 = 1(1)
where a and b are arbitrary, n is the refractive index ofthe lens in the refracting case, and n = -1 in the case oftwo mirrors. These parameters completely determineall the first-order properties of the system, including thelens thickness d (which is negative in the case of twomirrors), the back focal length, surface powers P1 andP2, and the paths of the two rays.
The author is with St. John Fisher College, Physics Department,Rochester, New York 14618.
The primary monochromatic aberration contribu-tions of a curved Fresnel surface are given in Eqs. (21)in a previous paper by the author.1 Those equationsare
S = yNiAu 2 - 8ey4tN + Coy2A(NU2)
SIIc = yNiA(uU) - 8ey3 57AN + coy5iA(Nu2 )
XIIL = Qy(co - c)Au
Nii = yNiA(uU) - ey2y 2 AN + coyy/A(NuU) I
I = QycAiU - QcoAu
WVv = yNiAiU2 - 8eyy 3AN + Coy2A(Nuii)
(2)
where i and i are the angles of incidence of the marginaland principal rays, respectively, c is the vertex curvatureof the equivalent surface (which determines the surfacepower), e is the coefficient of y 4 in the expression for thesag x of the equivalent surface, and co is the vertexcurvature of the substrate on which the Fresnel facetsare cut (Fig. 1). The symbol AA denotes the change inany quantity A upon refraction or reflection.
Equations (2) can be transformed into a more con-venient form by using first-order identities. The al-gebra is lengthy and is outlined in Appendix A. Theresult is
SI y(Ni)2 A(u/N) + ay2A(Nu 2) - Ky4 c3 AN
= (NI)S + (J)f + (VI)a
X = (&1 )(i/i) + (SI)j(y/y) + (eVI)a(Y/Y)
= (SINC)S + (&IIC)f + (SIIC)a
SIIL = QYAU
SVIII = (IIC)S (/i) + [(SIIC)j + &IIL](Y/Y)
+ ('IIC)a( /Y)
= (III). + (III)f + (VIII)a
cVIV = Q2 CA(1/N) - IIL(YY) = (IV)s + (Iv)f
SI = (,SIIC))S(I/i) + [IIC)f + cSIILI(Y/Y I+ (SIIC) (Y/Y)
x= cy' +ey'Fig. 1. The slope of a facet F on the Fresnel surface is the same asthe slope at the corresponding point E with coordinates (x,y) on theequivalent surface. The Fresnel facets are cut on a flat or curvedsubstrate of vertex curvature co = 1/ro. The size of the facets is shown
greatly enlarged.
where cr = co - c measures the Fresnel roughness of thesurface, and K = -1 + 8e/c 3 measures the departurefrom sphericity of the equivalent surface. For an or-dinary spherical surface, = K = 0; for an ordinary pa-raboloid, o. = 0 and K = -1; for any Fresnel surface, a d
0; for a flat Fresnel surface a = -c. Each aberrationcontribution is expressed as the sum of three terms, i.e.,the contribution of an ordinary spherical surface (sub-script s), the Fresnel contribution (subscript f), and thecontribution due to asphericity (subscript a).
To calculate the primary aberrations of the system,we need to express the quantities appearing in Eqs. (3)in terms of the independent first-order parameters.Using Eqs. (1) and first-order identities, we get
2 = -G2 (yY) 2 , M 2 = (H2 + K 2 )(i/i) 2,N 2 = I2(Y/Y)2
02 = J2 (YIY) 2 -
, (6b)
A derivation of Eqs. (6a) and (6b) is outlined in Ap-pendix B.
Stigmatic SystemsFor a system of two ordinary conic surfaces with given
first-order parameters, only two degrees of freedom areavailable for aberration correction, namely, Ki and K2.These are commonly used to make the system aplanatic.Thus if we set cj = NIIC = a = 0'2 = 0 in Eqs. (6a) andsolve for K and K2 we get
The subscripts refer to the two surfaces of the system. -Moreover, if S = d/n is the air-equivalent thickness ofthe lens, and I is the back focal length;
S = (1 - b)(ab + P)-', I = (1 + Pa)(ab + P)-', (5a)so that
a = (Pl-1)S- 1 , b = (P-S)l- 1 . (5b)
Substituting Eqs. (4) into Eqs. (3) results in the fol-lowing expressions for the total primary monochromaticaberrations of the system:
XI = Al + A2 + aIiBI + o2B2 + KlCl + K2C2
Slc = Di + D2 + G2E2 + K2F2
ePI1L = oGI + 0 2G2
Xiii = Hi + H 2 + Or2I2 + K2 J2 ,
Siv = KI + K2 + r2L2
v = MI + M 2 + 2N2 + K202
and the coefficients are:
K2 = - (Di + D2 )F2 , xi = -(A 1 + A 2 + K2C2 )CI 1. (7)
An example of an aplanatic two-mirror system is theRitchey-Chretien telescope, a variation of the Casse-grain telescope.
In general, neither &III nor Siv are zero, although oneor the other may be controlled by proper choice of thefirst-order parameters. The sagittal image surface hasa curvature proportional to (II + IV). Substitutingthe expression for K2 from Eqs. (7) into the expressionsfor &111 and Siv in Eqs. (6a) we get
SIII + Siv = H1 + H2 + K1 + K 2 - (D1 + D 2)(Y/Y) 2 - (8)
Two new degrees of freedom, al and 2, become(6a) available if we replace the two conics by curved Fresnel
surfaces. One of these can be used to correct astigma-tism, the other must be used to correct the line comaordinarily afflicting systems of Fresnel surfaces. Set-ting S = IIC = IIL = Sil = 0 in Eqs. (6a) and solvingfor K, K2, al, and 2, we get
In general, SIIv 0 for these stigmats, but designs withSIV = 0 exist under the condition stated in the nextsection. We now show that the curvature of the sagittalimage surface for the stigmat is the same as that for theaplanat having the same first-order layout. To provethis, observe that substitution of the expressions for U2
and K2 from Eqs. (9) into the expressions for S111 and Sivin Eqs. (6A) yields Eq. (8) again. Moreover, since YI'I= 0 for stigmats, it follows that for them
SI = H1 + H2 + K1 + K2 - (D1 + D2 )(y/Y) 2 , (10)
so that the astigmatism in the aplanat is converted intoadditional Petzval curvature in the derived stigmat.
Flat-Field Stigmats
The condition for a flat-field stigmat is obtained bysetting &IV = 0 in Eq. (10). Using Eqs. (6b) and (4) toexpress the right-hand side of Eq. (10) in terms of a, b,n, and P, and simplifying, yields
Equation (11) states the necessary condition that atwo-surface system must satisfy to be a flat-field stig-mat. Substituting the expressions for a and b from Eqs.(5b) into Eq. (11) results in the following cubic equationfor 1:
This equation has the root 1 = P - S, which implies thatb = 1 and produces a singularity. 2 Removing that root,Eq. (12) reduces to the following quadratic:
12 (n2 + 1) + l[Sn(n + 1)-2P(n2 + 1)]
+S 2 n3 -SPn 2(n+ 1) +n2 + 1 = O. (13)
When S is specified, one can solve to get two, one, or novalues of 1 which result in a flat-field stigmat. In par-ticular, the discriminant of Eq. (13) is negative when P= -1 and n > 1, so that no real solution exists in the caseof a thick lens with negative power.
Equation (13) is also quadratic in S so that, alterna-tively, one can specify and solve for S from the equa-tion
In either case, having both I and S one can get a and bfrom Eqs. (5b), and ci, c2, U1, U2, K1, K2 from Eqs. (4),(6b), and (9), so that the design of the flat-field stigmatis completed. For a given value of n it is possible to finda value of S for which the flat-field stigmat is also freeof distortion. An example is given in the next sec-tion.
For two-mirror systems, we set n = -1 in Eq. (14) toget the two solutions S = I21/2(1 - P), from which
= P S/A/2. (15)
From Eqs. (4) and (5b) one obtains a = P/V/2 = 2c1.Consider the two cases P = 1 and P = -1 separately:
(a) If P = +1,1 = 1 S/V\2, ci + 1 21/2/4. The minussign corresponds to a concave primary mirror and aCassegrain type of solution. The plus sign correspondsto a convex primary and an inverse-Cassegrain solution.Since S must be positive, Cassegrain solutions have areal image ( > 0) for any S < \/2, whereas inverse-Cassegrain solutions always have a real image.3
(b) If P = -1, 1 = -1 S/V\2, c = 21/2/4. Theupper sign corresponds to a concave primary and aGregorian type of solution. The lower sign correspondsto a convex primary and an inverse-Gregorian solution.Gregorian solutions have a real image for any S > /2,whereas inverse-Gregorian solutions always have avirtual image.
For (normalized) two-mirror systems, I is also thecentral obscuration diameter ratio and should be assmall as possible for astronomical telescopes. In cases(a) and (b), cannot be small except for Cassegrainianor Gregorian solutions with S 2. Therefore two-mirror flat-field stigmats tend to be too long and theimage too inconveniently placed for practical use asastronomical telescopes. Moreover, since 1 - S < 1 forinverse-Cassegrainian solutions, flat-field stigmats ofthat type are not suitable as microscope objectives.
Numerical Examples
Example 1: Aplanatic Lens
In this example, n = 1.5, d = 0.2, 1 = 0.7, and P = 1.Therefore S = d/n = 0.133333, and using Eqs. (5b) weget a = -2.25 and b = 1.238095. From Eqs. (4), (6b),and (7) one obtains the design given in the first columnof Table I.
Example 2: Stigmatic Lens
This lens has the same first-order layout as Example1, but ol, 2, K, and K2 are obtained from Eqs. (9) in-stead of Eqs. (7). The design is given in the secondcolumn of Table I.
Example 3: Two-Mirror Flat-Field Stigmat
In this example, n = -1, d = -0.5, and P = 1, and Eq.(13) is solved for I to get two roots: 1 = 0.646447 (Cas-segrainian solution), and 1* = 1.353553 (inverse-Cas-segrainian solution). Using Eqs. (5b) to get a and b foreach case and then using Eqs. (4), (6b), and (9), oneobtains the two designs given in the third and fourthcolumns, respectively, of Table I.
Example 4: Distortion-Free Flat-Field Stigmat
For n = 1.5, d = 1.203703,1 = 0.444444, and P = 1, allthe primary monochromatic aberrations are zero. Thedesign is given in the fifth column of Table I (also seeFig. 2).
Fig. 2. An optical schematic of the distortion-free flat-field stigmatgiven in Example 4. The marginal and principal rays are representedby solid and dashed lines, respectively. The size of the Fresnel facets
is shown greatly enlarged.
Systems of Flat Fresnel Mirrors
A system of two flat Fresnel surfaces is cheaper tomake than a system of two curved Fresnel surfaces, buthaving fewer degrees of freedom (since c 0 for eachsurface) cannot be as well corrected. One might expectthat adding one or two more flat Fresnel surfaces wouldpermit the same degree of correction as with two curvedFresnel surfaces, but that is not always true. We nowshow that a system consisting of only flat Fresnel mir-rors cannot be corrected for line coma unless u = u,where the subscript k refers to the last surface of thesystem.
Since = -c for a flat Fresnel surface, by Eqs. (3) thetotal line coma for a system of k surfaces is
(Smjtot = -QjyjCjuj. (16)
But yc = i - u = i' - u', and for reflection, i' =-i;therefore, i = (u - u')/2 and yc = - (u + u')/2. Sub-stituting into Eq. (16) and simplifying give
(GS'IIL)tot = Q(u, - ul)/2. (17)
Therefore, in general, a system of flat Fresnel mirrorshas incurable line coma. However, the line coma of asystem of flat Fresnel lenses can always be corrected.
Appendix A: Derivation of Eqs. (3) from Eqs. (2)
Since = co - c and --1 + 8e/c3, the first of Eqs.(2) can be written,.' = NiAu 2
- c 3(1 + )y 4AN + (c + )y2A(Nu2)
= yNiAu 2 - c3 y4 AN + y2 A(Nu 2 ) + ay 2 A(Nu2 ) - Ky4c3AN.
Using the first-order identity yc = i - u and noting thatNi and yc are invariants on refraction or reflection, weget
yNiAu 2- c 3y 4AN + Cy2A(NU2)
= yAIN[iu2- (i - U)3 + (i -U)U2]
= yAIN[-iU2- i
3 + 3i2u] = yNiA(-u 2 - i2 + 3ui)
= yNiA[ui - (i - U)2
] = yNiA(ui) = y(Ni)2A(u/N),
from Which the first of Eqs. (3) follows.Similarly, the second of Eqs. (2) can be written
by using Eqs. (4) and (6b). A similar expression is ob-tained for the second surface. The total spherical ab-erration of the system is just the sum of the two. Theexpressions for the other aberrations are obtained in alike manner.
References1. E. Delano, J. Opt. Soc. Am. 68, 1306 (1978).2. If b = 1, S = 0 unless a = -P, in which case S and I are both inde-
terminate. Some of the formulas break down in either case.3. Both mirrors are assumed to have a central perforation if neces-
