JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Primary aberrations of Fresnel lenses*Erwin Delano

Physics Department, St. Jo/hn F/siser College, Rochester, New York 14618(Received 18 August 1973)

Mathematical expressions derived for the primary aberrations of a thin, flat Frcsnel lens (grooved on bothsurfaces, in general) as functions of the constructional parameters are convenient for the analytic design ofoptical systems containing one or more Fresnel lenses. In general. such systems have five primarymonochromatic aberrations of the Seidel type plus a sixth that is identically zero for ordinary lenses. Thenew aberration (called line coma to distinguish it from ordinary circular coma) bears the same relation tosagittal and tangential coma that Petzval curvature bears to sagittal and tangential astigmatism. Moreover.line coma is independent of stop position, whereas Petzval curvature varies with stop position unless theline coma is corrected. The primary chromatic aberrations and the aberration contributions of Fresnelaspherics are the same as for ordinary lenses. Specific applications of the theory are discussed.

Index Headings: Lens design: Aberrations.

A series of articles on Fresnel lenses, by Hofmann'- 6

and his associates, appears to have ushered in a newsurge of interest in such lenses. As is well known, thin-sheet, plastic Fresnel lenses are ideal for some appli-cations in which low cost, light weight and compactnessare important, but where high resolution is unneces-sary.7 However, recently it has been pointed out thatFresnel lenses are theoretically capable of resolutioncomparable to that of ordinary diffraction-limitedlenses of the same aperture.8 This suggests the possi-bility of using high-precision Fresnel lenses to replaceordinary lenses in optical instruments of high quality.It is therefore interesting to compare the aberrationalproperties of the two types of lenses from the viewpointof the optical designer. The primary aberrations of thinlenses in air are of special interest in this regard, because

FIG. 1. Bending a thin, flat Fresnel lens with grooves on bothsurfaces. The lens may be regarded as having a shape determinedby the shape of the ordinary lens from which it is derived, andshown directly above it. The power is redistributed among the twoFresnel surfaces, while keeping the power of the lens constant.

they lend themselves nicely to analytic methods ofoptical design. Such methods provide valuable insightto the designer during the feasibility study and pre-liminary-design stage, which precedes automatic cor-rection of the aberrations by computer.

The main subject of this paper is the derivation ofmathematical expressions for the primary aberrationsof a Fresnel lens which is flat, infinitely thin, androtationally symmetric about the x axis. For generality,both surfaces of the lens are assumed to be grooved,to permit us to analyze the aberrational effects ofbending the lens. See Fig. 1. The bending of the Fresnellens does not refer to the physical deformation of thelens as such-it remains flat-but to the shift of powerfrom one surface to the other. Note that the effectivenormal (i.e., the one used in Snell's law) at any heighton a Fresnel surface is parallel to the normal at thesame height on the ordinary surface from which theFresnel surface is derived. For mathematical con-venience, the grooves are treated as if they wereinfinitely narrow, and the effects of diffraction areignored. Therefore the treatment will be valid if thewidth of the grooves is very small compared to thelens aperture, but still large compared to the wave-length of light. Moreover, individual grooves may beconsidered to have flat instead of curved profiles,because the optical difference between the two de-creases with groove width and can therefore be neg-lected. Finally, the shading effect of a groove onadjacent grooves and other problems of a practicalnature are not discussed here. Two of the articlesalready referred to treat such topics in detail." 7

The notation used is generally similar to that ofHopkins.9 Light is assumed to travel from left to rightthrough the optical system. A prime (') on a symbolmeans that the associated quantity is in the imagespace of the surface, lens, or system as the case may be.A bar (-) over a symbol means that the associatedquantity refers to the principal ray. The analytic-geometry sign convention is used for all angles and for

459

VOLUME 64, NUMBER 4 APRIL 1974

4 ERWIN DELANO

aberrations (i.e., overcorrected aberrations are treatedas positive).

REFRACTION OF A SKEW RAY

Let ii be the unit vector normal to a surface, and aand a' be the unit vectors parallel to the incident andrefracted rays, respectively. Snell's law in vector formis NaXa=N'a'Xfi, where aXfi denotes the vectorcross product of a and fi, etc., and N and N' are therefractive indices of the two media. Using standardvector techniques (see Appendix A), we can show that

wherea'= (N/N')a+r(,

r = [1- (N/N')2(aXn)2]&- (JVN')a- h,(1)

and a- i denotes the dot (or scalar) product of a and ni.Let i, j, and k be the unit vectors in the x, y, and

z directions of a right-handed cartesian coordinatesystem. The vectors a and ii can then be written in

component form a=ac1i+j+yk and n= ,u1+Ai+Xk,where a<= (1 - 2--y2)i and /-= (1-v2-X2)i since a andn are unit vectors. It follows from the definitions of thedot and cross products that

a- n=IAg+/3v+YX=1- [ (,3 2 + (,y- )2] +0 (4) , (2

(aX n)2 (OX -,v) 2 + (yui-aX) 2 + (av-#/3()2

= 8- V)2+(7-y-X)2+0 (4),

where the symbol 0(m) denotes all terms of degree notless than m in the independent variables, assumed small.If we substitute Eqs. (2) into the expression for r, andsimplify, we get

r= [1- (N/N')]{l1+2 (N/N')X[E3-V)2 + (y-X)2 ]}+0(4). (3)

At the two surfaces of the Fresnel lens, which is ofrefractive index n and immersed in air, by Eqs. (1)

a,'= (Ni 1 /N 1 )a1 +rFni1, a 2 '= (N 2 /N 2 ')a 2 +r2n 2 ,

where the subscript refers to the particular surface.Now a2 = a 1', N. = N 2

1 = 1, and N1'= N2 = n. Thereforecombining these into a single equation for a2', we obtain

a2 ' = ai+nrini+r 2 n2 . (4)

If we substitute the expressions for ri and r2, accordingto Eq. (3), and take y and z components of both sidesof the resulting vector equation, we get

3' =f3+ (n-1) (V1-V2)

+2 (J1-1-)[{ O-V )2+ (7_X1)2} V1

- {11-v+n(V1-V2)] 2

+[LY-X 1 +11(X 1-X 2 ) ]2} V21+0(5), (5)

ly'Y=+ (n-1) ()1-X2)+1 (J1-n-)[{(O-V1)2+()tyX1)2}X1

-({E-vl+n(v1-V2)]2

+E'Y-X 1+1(X 1-X2 )]2}X2 J+0(5),

FIG. 2. An arbitrary skew ray from the point P, incident upona Fresnel lens lying in the y,z plane. The distance I and the angleU are negative as shown.

where 3,By and O',-y' refer to the object and image spaces,respectively, of the whole lens, their subscripts havingbeen dropped for convenience.

TRANSVERSE ABERRATIONS OF AFRESNEL LENS

Assume the Fresnel lens to be rotationally symmetricabout the x axis, with center at the origin 0 of thecoordinate system. Consider an object point P lyingin the meridional plane (the x,y plane) at a height h,assumed small, above the x axis. Let I be the distancefrom the plane of the lens (the y,z plane) to the point P,measured in the positive x direction. See Fig. 2. Let Abe the point at which an arbitrary skew ray from Pintersects the y,z plane. Let PB be the projection ofthis ray onto the x,y plane and let U be the angle(measured CCW) from the x axis to the line PB.Similarly, let V be the angle from the x axis to theprojection PoC of the ray onto the x,z plane. If wedefine u= tanU and v=tanV, then by inspection

u=-(y-h)/l, v=-z/l.

Also, the distance D along the ray from A to P is

D= [12+ (y-h)2+z 2 I= [i1+-22+Iv2]+0 (4).

Since a, ,3, y are the direction cosines of the ray,

3=-(y-h)/D, -y=-z/D,

and combining with the expression for D above, weobtain

8=U[-1-2 (U2+V2)]+0(5), (6)

vl 2 (-I8=JV2)]+0 (5).

Equations (6) are also valid in the image space, i.e., ifall of the variables in the equation are primed. If wesubstitute the expressions for #, -y, f', and -y' from Eqs.

B

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PRIMARY ABERRATIONS OF FRESNEL LENSES

(6) into Eqs. (5), we can put the latter into the form

'= U+ (n-1) (vP - P2) +2[u' (U2+V2) -U(U 2+VI)]

-V2{ [u--v+n (v1-V2)]2

+Ev-Xl+n(X-X2) ]2)+0(5), (7)v'= v+ (n-1) (X\-I\ 2)+ 2'[v' (U"2+V2) -V(U 2

+V2)]

+1 (1-n-1)[X I{(U-vl)l+ (V-X)2)

-X 2{ [u-v+n (v,-V2)]2+[v-X1+n(X1- X)]2 } 1+0(5).

Now assume that the Fresnel surfaces of the lens arederived from paraboloids of revolution. For suchsurfaces, it is shown (in Appendix B) that

v= -yc+2c y (y'+z)+0(5), (8)X =-zc+ cCz(y 2

+z 2)±0(5),

where c is the vertex curvature of the paraboloid. If cland c2 are the curvatures of the Fresnel-lens surfaces,then since Y =Y2 and z1 = Z2 for an infinitely thinFresnel lens, we obtain

(n-1)G(V-V 2 ) = -Ky+' (n-1)X (C1'-C23)y(y

2+z2)+0(5), (9)

(n-1) (A]-A\2) =-Kz+l (n-1)X (CIa-6C,)Z(y2+Z2)+0(5),

where K= (n- 1)(c,-C2) is the paraxial power of thelens, and where the common values of y and z are used.Equations (7) can therefore be written in the form

u' ==u-Ky+5u', v'=v-Kz+5v', (10)

where the approximations u'=u-Ky and v'=v-Kzare just the paraxial ray-tracing equations and the6u' and Wv' include all the terms of the third order orhigher, i.e., they represent the angular aberrations ofthe skew ray. Let X'=1', Y'=h', and Z'=0 be thecoordinates of the ideal image point P', which lies inthe gaussian image plane. Further, let bY' and AZ' bethe coordinates in the gaussian image plane of theintersection point of the skew ray with this plane,referred to P' as origin. They eSY' and AZ' are transverseaberrations and satisfy the relations 6Y'=l'6u' and5Z'=I'bv'. If we use Eqs. (9) and (10) in Eqs. (7), weeasily obtain

ay,/I,=,2 (n- 1)(C13-C2 3)y(y2+Z2)

+M[u'(u' 2+vI) -U(U2+V2)]

+-(1_1-n-+)vlt(UV12+ V-XI)2)

- P2{[Uvl+fl(Pl-P)]2

+[v-Xi+n(Xl-X2 )]]2}]+0(5),6Z 1 = 2(n_1) (C13_C23)Z(y2+e) (1

+ 12 v' (u12+v12)-v (U2+v2)]

+2 (1-n-l) [X1{ (UV1)I+ (V-X3)2}

-X 2 {[U-V1+f(V1-v 2 )]2

+[v-Xi+n (X1-X 2 )]2} 1+0(5),

where the u' and v' are to be replaced by their paraxialvalues.

EFFECT OF LENS SHAPE AND CONJUGATES

As in the case of ordinary lenses,' 0 the aberrations ofFresnel lenses can be expressed in a very convenientform by introducing two symmetric variables thatdefine the shape of the lens and the conjugates atwhich it is used. The shape variable S=(cI+c2)/Kdefines the distribution of power between the twosurfaces of the lens. Since K=(n-1)(cI-C2 ), we cansolve for c, and c2 between these two equations to get

Cl= 4K[S+ (n-1)-'], C2= -K[S-(n-1)-1]. (12)

The conjugate variable T= (v+v')/(v-v'), where vand v' are paraxial values, defines the magnification atwhich the lens is used. Since v'=v-Kz paraxially, wecan solve for v and v' between these two equations to get

v=-Kz(T+1), v'= Kz(T-1). (13)Now

Nw-(y-h)/l= -y/l+h/l,

and let U be the angle (measured CCW) from the xaxis to the principal ray P0, assuming the stop incontact with the lens. If we then define u=tanU, itfollows that u=-y/l+27. But v=-z/l, thereforeu= (y/z)v+ft. Similarly, u'= (y/z)v'+'& since '= a forstop in contact. If we substitute for v and v' from Eqs.(13) we get

u= 1Ky(T+i)+?Z, u'= !Ky(T-1)+u, (14)

where paraxial values are used throughout.The next task is to substitute the expressions for cl,

c2, u, v, u', and v' from Eqs. (12)-(14) into the right-hand side of Eqs. (11). The algebra is fairly straight-forward, but tedious, and is outlined in Appendix C.The results are

6Y'11'= =-KlKy(y2+z2 C(- 1)S2

+2(1+w)ST+ (3+w)T2 + (1-cJ)-']-!K 2y2 iE(1+w)S+ (2+w.)T]-1K 2 (y 2+z2)fzT-4KyfZ2 (3+w)+O (5),

SZ'/1'= -lK'z(y 2 +z2)E(co- 1)82+2 (1+w)ST+ (3+co)T2+ (1 -w)-']

- 4K2yzE(1 +w)S+ (2+w)T]-KZt2 (I +C) +0(),

(15)

where co= n-. It is convenient to replace the variablesy, z, and i7 by the fractional variables

7q=y/rm, =z/rm, o-fz/?2m,

where rm is the maximum value of r= (y'+z2)i (i.e., theradius of the pupil) and un is the maximum value of fz(i.e., its value at the edge of the field). Equations (15)

April 1974 461

462ERW1N DELANO

can then be written in the form

a ye= ajq(q12+r2) +a2 -q2.+a3 (7,q2+g;2)0

+a 4sia 2+0(5), (16)

AZ'= al (r/2+ 2) +a21lq<o+asPcT2+0(5),

where

a, =-8l'K'rm '(co-1)S 2 +2(1+w)ST

+ (3+co)T'+ (1 -c ')-],

a2 = - l'K 2 rmkim[(1+co)S+ (2+w)T],

a 3 = -2lK2rm2iumT,

a 4 = - Ul'Krmg 2 (3+w),

a5 = - 1Krmrm2 (1 ±W).

These ai are primary aberration coefficients for thelens. Since the primary aberrations are usually describedin polar coordinates, set -=p cosO and =p sinO, wherep=r/rm and r and 0 are polar coordinates in the planeof the pupil. Equations (16) can then be rewritten as

3Y'=blp' cosG+b2,p'Oa( 2+cos 20)+b21p2'O

+ (3b3+b4)pa 2 cos0+O(5),

AZ' = bip' sin0+b2 ,p2a- sin20+ (b3+b4) pa2 sin0+O (5),

wherebi = aj b2c = a2, b21 =a3-1a2,

(17)

b3=2(a.,-as), b4=4-(3a5 -a 4 ).

Comparing each of the terms in Eqs. (17) with thewell-known properties of the Seidel aberrations," wecan easily identify b6, b2, b3, and b4 as the coefficientsof spherical aberration, coma, sagittal astigmatism,and Petzval curvature, respectively. We might alsoexpect an additional term bsa3 in the expression foray', where b65 is the coefficient of distortion, but thisterm is missing because the distortion is zero for a thinlens with stop in contact. Finally, the coefficient b2l

represents a new type of coma that does not afflictordinary lenses. Some of the properties of this newaberration will now be discussed.

If bait 0, but bi = b2c = b3 = b, = bs = 0, then 6Y' b2lp2 o-

and aZ'= 0, so that all rays passing through an annularzone of fractional radius p, for a given value of a, willcome to a sharp focus on the gaussian image plane.The focus will be displaced from the principal ray in adirection away from the x axis if b21>0 (towards, ifb2l<0). The displacement is a function of p; therefore,the foci for different zones will fall along a straight-linesegment having one end at the principal ray (for p=0)and a length proportional to a. It is easy to show thatthe resulting focal line is uniformly bright along itsentire length. This aberration is therefore appropriatelycalled line coma, to distinguish it from ordinary(circular) coma. An interesting relation connects linecoma to sagittal and tangential coma, similar to thatwhich connects Petzval curvature to sagittal and

P"FIG. 3. The relation between Comas,, ComaT,, and ComaL'

measured in the gaussian image plane. Point P' is the ideal imagepoint; points S and T are the sagittal and tangential foci, respec-tively. All other aberrations are assumed to be zero.

tangential astigmatism. To show this, assume that onlyb2c and b2i are not zero, and set 0=0 in Eqs. (17) toget the tangential coma

ComaT'= (3b 2 c+b2l)p2';

set 0=900 to get the sagittal coma

Comas = (b2 ±+b21)p2 af,

and note that the line coma is

ComaL' = b2 lp2',

the amount of coma if b2, = 0. It follows that

ComaT, - ComaL' = 3 (Comas' - ComaL').

See Fig. 3. This is analogous to the well-known relationthat

XT-XP 3 (Xs5 -Xp,))

where XT', Xs,, and Xp are longitudinal displacementsfrom the gaussian image plane to the tangential,sagittal, and Petzval surfaces, respectively. In thepresence of both ordinary coma and line coma, raysfrom an annular zone of fractional radius p will fall on

Vol. 64462

PRIMARY ABERRATIONS OF FRESNEL LENSES

where

NEW PUPIL

FIG. 4. As the pupil is shifted, its size is changed in such a waythat the marginal ray PoC from the axial object point Po isunchanged. Because the object height It is also unchanged, thevalue of the Lagrange invariant Q for any ray also remains thesame.

a circle of radius b2cp2 a . This circle subtends an angle

2t from the principal ray, where

4, = sin-'Eb2,/ (26~+6~)1.

If no line coma is present, b2l = 0, so that 24' = 600, asit does for ordinary coma.

Another important property of line coma is itsinvariance with respect to change of stop position,which is proved in the next section.

EFFECT OF STOP SHIFT

The effect of a change of stop position on the primaryaberrations of a Fresnel lens is now considered. Supposethat the entrance pupil is shifted from contact with thelens (at distance d= -I from the object plane) to a newposition a't distance d*. Moreover, assume that theradius of the entrance pupil is changed from its originalvalue rm to the new value rm*=rm(d*/d) as in Fig. 4.Let y* and z* be the coordinates of the arbitrary skewray in the plane of the new entrance pupil. By inspectionof Figs. 2 and 4, we see that

(y*-h)/ (y-h) = z*/z = d*/d.

Solving for y and z in terms of y* and z*, and using thecorresponding fractional variables 7, r and s*, ¢*, weobtain

where (18)p= m/?rm, ?m im (d*-d)/d*.

The quantity ?m is the height on the lens of the newprincipal ray coming from the edge of the field. Sub-stituting Eqs. (18) into Eqs. (16) and collecting terms,we obtain

W'=j a*X* (n *2 +v* 2 ) +a,*fl*2 cr+a,* (X7* 2 +.* 2)u

+a 4*?7*o 2+a6*o 3+0 (5),

SZ'= ai*v* (77*2 +t*2) +a2*f7*P*a+as*¢*0I 2 +O (5),

a,* =a1 ,

a2* =a 2+2pai,

a3* =a3+pa,

a 4 = a 4 +2p(a 2+a3)+3p 2 a1 ,(19)

a5 =a5+pa2 +p2a,,

a6* = pa4+p2 (a 2 +a 3 ) +p 3 a,.

The a,* are the aberration coefficients with the stopin the new position. Transforming to polar coordinatesp* and 0*, in the plane of the new pupil [as was done toobtain Eqs. (17)], we obtain the corresponding equa-tions for the bi*,

aY'= bl*p*3 cos0*+b2 c*p*2 a (2+cos20*) +b2l*p* 2.7+ (3b3*+b ,*)p*0i2 cos0*+b5* l3+0(5),

bZ'= b*p*3 sinO*+b 2e*p*2I sin20*+ (b3*+b4*)p*a2 sin0*+O(5),

where

bl* = b1,

b2c* =b2,+ pb ,,

b2l* = b2l, (20)

b3* = b3+P (2b26+6 21) +p 2b1 ,

N4*= b4-pb2l (N.B. a minus sign here!)

b5*=a6* = b5+p(3b3+b4) +p 2 (3b2c+b21)+p 31b,,

where b5 (which is zero in this case) is introduced forthe sake of generality. Equations (20) are a general-ization of the usual equations for stop shift'2 for thecase b21sw0. Note that b21 (line coma) does not changewith stop position, but that b14 (Petzval curvature)does depend on stop position unless b21=0, as in thecase of ordinary lenses.

ADDITION OF ABERRATIONS

Consider an optical system consisting of k Fresnellenses in air. The transverse aberrations 3Y'tt andand 6Z'tto of the complete system are related to thoseof the individual lenses by the relations

6ytot = mJkk1YJ+0(5),i

WZtot'= 2 mJk11Zj+0(5), (21)i

where Mnk' = N/'v1'/Nk'vk' is the paraxial transversemagnification from the image plane of the jth lens tothe final image plane, and the paraxial values of v' areused.

It is convenient to define the aberrations differentlyso as to make them directly additive. To accomplishthis, define the modified aberrations of the jth lens as

E' =-2N/'v/3Y /', F==-2N/'v/'3Z/'. (22)

463April 1-974

ERWIN DELANO

The total aberrations for the complete system are then

Ett'= - 2NL'vk'6YtOtt'= E= + 0(5),j

Ftot= - 2Nk'vk'6Ztot'= F='+0(5),i

so that the E/' and F/' are additive as desired. NowN'= 1 for a lens in air, and

v, = - z/l = (- r./l) (zlr.) = vm T = vm p sinO,

where vm' is the maximum value of v'. Using Eqs. (20)and (22), we obtain the values of E' and F' for a singlelens,

E' = [81*p*3 COS0*+82c*p* 20r(2+cOs20*)

+821*P* 2 f+ (383*+8.1*)p*o.2 cos0*

+8,*a3]p* sin0*+O(5), (23)

F'= [Si*p*3 sin0*+8 26 *p*2a sin20*

+ (83*+S.*)p*a2 sin0*]p* sin0*+O (5),

where Si= -2vm'bi, and the asterisk refers to the generalposition of the stop for the given lens. Each of thevariables p* and 0*, in the plane of the pupil, has thesame value for all of the lenses because all of theindividual pupils are images of each other. The variableoa, in the plane of the image, also has the same valuebecause all of the individual image planes are imagesof each other. Therefore the coefficients 8i* must bedirectly additive, i.e.,

j

In this respect, the Si are analogous to the wave aber-rations of ordinary lenses, as defined by Hopkins,9

except for the sign convention.

CALCULATION OF THE Si AND THE Si*

Explicit expressions for the Si as functions of S andT are obtained from the definition Si= -2vm'bi aftersubstituting the expressions for the ai from Eqs. (16)into Eqs. (17) for the bi. If we note that vmlT= -rmand put Qm r=mim (the value of the Lagrange invariantQ for the extreme sagittal ray from the edge of thefield"3), we easily show that for a thin, flat Fresnel lenswith stop in contact

81 =- rm4 K'[(w-1)S2+2(1+w)ST

+ (3+w)T2

+ (1 -)-I],

82. = -Qmrm 2K'[(1+w)S+ (2+w)T],

821= +4Qmrm22K

2[(1+w)S+coT]

(N.B. the plug sign),83 =-Qm2 K,

84 = - Qm2

K,

8,5=0.

(24)

These equations assume that the Fresnel surfaces ofthe lens are derived from paraboloids of revolution.The effect of figuring is discussed in the next section.

If the stop is shifted, it is clear from Figs. 2 and 4that

vm* = r.*/d* = rr/d = v,.

Therefore vm'*=vm' so that the Si obey the same Eqs.(20) as the bi do, thus

81* = 81,

82c* = 82c+P8I,

821* = 821,

83* = 83+P (282.+821)+p 2 81,

84* = 84-P821,

85* = 85+P (383+84) +p2

(382,+821) +p38',

(25)

where p=fm/rm as before. Explicit expressions for theSi* are obtained by substituting Eqs. (24) into Eqs.(25) and simplifying. First, however, it is convenientto define the new variable T= (which, like p, also defines the position of the stop,where uz* and ii'* are paraxial values. Now i'* = - Ky,where y is the height of the new principal ray on thelens. We can solve for W* and *'* between these twoequations to get

(26)

which are analogous to Eqs. (13) for v and v'. As isproved in Appendix D, we can then write Qm in theform

Qm= 'Kr. m(rT-T).

Substituting Eqs. (24) into Eqs. (25) and using(27) for Qm where needed to combine terms, weshow that for an arbitrary stop position

8i*= -4rm4 K3[(- 1)S2 + 2 (1+co)ST+ (3+,)T2+ (1- )-1],

82c* = 4-rm3 tmK 3[(w-1)2+ (1+co)S(T+T)

+T 2+ (2+w)TT+ (1-w)-t1],

821* = +2Qmrm 2K2[(1 +&)S+wT1,2

IS3* =-rm2 fn2K3 (co-I)S2+ (I +co)S (T+ P)+T 2 + (2+w)TT+ (1-co)-],

84* = -QmrmJmK2 [ (1 +&)S+WT],

8,*= =-7mrmK 3[ (w-1)32 +2(1+o)ST

+ (3+w)T2 + (1 -C)-I].

(27)

Eq.can

(28)

Equations (28) therefore allow us to calculate theprimary monochromatic aberrations of a thin, flatFresnel lens for any position of the stop or of the object,assuming that the Fresnel surfaces are derived fromparaboloids of revolution.

464 Vol. 64

fz*=1KP(T+1), fz1*=1Kp(T-1),'�Y 'i

PRIMARY ABERRATIONS OF FRESNEL LENSES

EFFECT OF FIGURING

Suppose that we remove the restriction that theFresnel surfaces be derived from paraboloids of revolu-tion. Assume instead that the Fresnel surfaces arederived from surfaces whose equations are of the form

X= ic(y2 +z 2)+e(y2+z 2)2 +0(6),

where e is an arbitrary constant. The changes of theprimary aberrations of the Fresnel lens (if el and e2are not both zero) are identical to those of the equiva-lent ordinary lens having the same values of el and e2.This statement is now proved.

In Appendix B, it is shown that, for a surface of theabove form,

= -yc+± (c 3 -8e)y(y 2 +z 2)+0(5), (2

X= -zc+ (c3 -8e)z(y 2+z 2)+0(5).

For the stop in contact, the case e5,0 produces third-order correction terms to v and X, which will affect onlythe second term in each of Eqs. (7). Consequently, theright-hand sides of Eqs. (9) must be modified by addingthe term -4(n-1)(ei-e2)y(y 2 +z 2 ) to the first equa-tion, and the term -4(n-1)(e]-e2)Z(y2

+Z2

) to thesecond equation. The right-hand sides of Eqs. (11)must be modified in exactly the same way.

Inspection of Eqs. (15)-(17) shows that the modifi-cation will affect only the coefficient b1, increasing itby zAb,=- -'rm3 (n-1)(el-e2 ) and leaving all the otherbi unchanged. Therefore, only spherical aberration isaffected. The value of 5, is increased by the amount

AS 1 =-2vm'Abj =-8rm4 (n-1) (el-e2), (30)

which is the same as it would be for an ordinary lens.If the stop is not in contact, Eqs. (25) show that

AS I* = s,

A821 * =0, (31)

A83* = p2AS1,

which again is the same as for an ordinary lens.14

If a Fresnel surface is derived from a sphere, thene=0c3 for that surface. If both surfaces are derivedfrom spheres, then using Eqs. (12) gives

el= c13= =1 K3ES 3'+3S2(n -1)-i+3S(n- I -I+ (n- j)-3],

=13= 3 _ -e2= 8C23 'lKE S-3S'(n-l)-1

+3S(n-1)2 - (n-1)-3],

and substituting into Eq. (30) gives

AS= -r 4 K 3 [3S2

+cW2

(1 C)-2].

If we add this to the expression for Si from Eqs. (24),and simplify, we get

81++ /8 = -- rm4K 3[(2+w)S2+2 (1+w)ST+(3+w)T 2 +1+w(1-w)-21 (32)

for the spherical aberration of a Fresnel lens withsurfaces derived from spheres. Expressions for theprimary aberrations of an ordinary lens are given inAppendix E for comparison.

CHROMATIC ABERRATIONS

The primary chromatic aberrations of a Fresnel lensare identical to those of the equivalent ordinary lens.This is so because both longitudinal color and lateralcolor arise from the chromatic variation in the first-order properties of the lens. Since these properties areunaffected by using Fresnel surfaces, the chromaticvariation will also be unaffected.

To a first-order approximation, Eqs. (10) state thatfor a Fresnel lens of power K= (n-1) (C1 -C 2) we have

u':u-Ky, v'=v-Kz.

Let An be the difference of refractive index correspondingto light of two different wavelengths, usually F and C.If 5K is the corresponding increment of K, then5K/K =n/ (n-1) = v-1, where v is the Abbe number ofthe glass. Thus 6K=K/v, so that the primary angularaberrations are

3u'=-y3K=-yK/v, 6v'=-z6K=-zK/v,

and the corresponding primary transverse aberrationsare

aY' =a 7 7, 6Z'=a7i,where

a 7 =-l'rmK/v.

For a shift of stop position, using Eqs. (18), we obtain

aY' = a7*-q*+a 8*o, Z' = a 7 **,

where a7 *=a7 , a8*=a8+pa7, and a 8 =0 since the stopis in contact. Converting to polar coordinates as usual,we obtain

1Y'=b7*p* cosO*+b8*0r, 3Z'=b7*p* sinO*,

where b7 =a7 and b8 =a8 are the coefficients of primarylongitudinal and lateral color, respectively. By Eqs.(22) it follows that

E'= (1*p* cos0*+Ci2*o)p* sin6*,FP= ei*p*2 sin20*,

where ]--2vm'b7 and 62=- 2vmb8. The primarychromatic aberrations for a thin Fresnel lens withstop in contact are therefore

e1l -2r 2K/v, 6 2 =0. (33)

For an arbitrary stop position, the aberrations are

465April 1974

ERWIN DELANO

easily shown to be

ei*= e= -2rm2 K/IP, e2*=e 2+pel=-2rm mK/v.

Note that e1 and e2 differ by a factor of -2 from thecorresponding wave aberrations L and T defined byHopkins.

EXAMPLE NO. 1: ABERRATIONSOF A FIELD LENS

It is intuitively obvious that all the aberrations arezero for a thin, flat Fresnel lens in contact with theobject plane. This is unlike the case of an ordinaryfield lens for which, in general, the Petzval curvatureand the distortion will not be zero.

Consider a Fresnel lens with general stop and con-jugates. Suppose that the position of the pupil, itssize rm*, and the angular radius of the field am* are allheld fixed, whereas the object plane approaches the(fixed) lens. It follows that rm and Q -r *uZ* willalso remain fixed. The aberrations of a Fresnel fieldlens are obtained by letting rm -4 0 in Eqs. (28). Nowrm -> 0 implies that vm'= vm-Krm Vm, so that

T= (Vm+Vm')/(Vm- m') 2 vm(Krm)-l c o

but rmT=2vm/K remains finite. Examination of theright-hand sides of Eqs. (28) shows that, in eachequation, the highest power of T that occurs inside thesquare bracket is less than the power of rm outside thebracket. Since rm -- 0, while rmT remains finite and allother quantities are fixed, it follows that all of theaberrations approach zero for a thin, flat Fresnel fieldlens.

In a similar manner, it is easy to show that Eqs. (24)may be derived from Eqs. (28). Let tm -- 0 whilekeeping rm and Qm=rm fm fixed and use the fact that

PmT 4 2i1r/K, which remains finite.

EXAMPLE NO. 2: DESIGN OF ALOW-POWER TELESCOPE

As another example of the use of the formulas,consider the design of a keplerian telescope, consistingof three Fresnel lenses. See Fig. 5. The stop is in contactwith lens a, which serves as the objective. Lenses b andc together serve as a huygenian eyepiece of unit focallength, and their powers and separations are determinedby the condition that lateral color is corrected. Assumethat the telescope has an angular magnification of 3X,and that the same optical material is used in all threelenses.

One possible first-order layout that satisfies theabove conditions is

Ka=3, Kb=l, K,=2, A=2, B=1, C=AL,

where the K's are lens powers, and A, B, and C aredistances which are defined in Fig. 5. The paraxial data

STOP

14 -A *_ B C.

FIG. 5. A keplerian telescope consisting of three Fresnel lenses.It can be corrected for all the primary aberrations except longi-tudinal color.

for suitable marginal and principal rays are

Ua=O, Ua'= -1, Ub'= -2, u,'=0,

Ya= 3, Yb=l, Yc=-l,

-a= 13 - =1 -3 13 C =- 1,

Ya=0, Yb=32, - 1

which gives Q= 1 and

Ta=-1, Tb=-3, T,=1,T a=oo, Tb=O, T,=-2.

Assuming a given optical material (say polymethylmethacrylate, with nz 1.5 and vz60), each lens hastwo degrees of freedom, viz., its shape factor S andthe spherical aberration AS, contributed by figuringthe surfaces. The problem is to determine the valuesof the six parameters Sa, Sb, S,, (AS1)a, (AS1)b, and(AS,), which will yield specified values, say zero, of allsix primary monochromatic aberrations. Note that ifordinary lenses were being used, the Petzval curvaturewould already be determined and could not be controlledat this stage!

By use of Eqs. (24) for the aberrations of lens a,Eqs. (28) for the aberrations of lenses b and c, andEqs. (31) for the effects of figuring, the six aberrationconditions reduce to

(8))tot - ya 4Ka

3 [Ya+23aTa± (3+w)Ta2 ]

TybOKb3EYb+ 2fbTb+ (3+w)Tb2 ]

- 1 W4KEy,,+20J,+ (3+w)T 2] = 0,

(S2 c)tot =-Qya2 Ka 2{Ia+ (2+co))Tal1 3-

-Yb ybKb 3ELb+/b(Tb+Tb)+Tb2

+ (2+w)TbPbJ-4yc3ycKc3 [Yc+Ic (T,+TP)

+TC 2 + (2+w)TTcJ =0,

(821)tot= 2QYa 2Ka2 (a+wTa)+2Qyb 2 Kb2 (13b+coTb)

+ 1Qy 2 K 2 (, +J T) = 0,

(83)tot -Q2Ka-4yb2?b2Kb3EYb+0b(Tb+ Tb) +Tb2

+ (2+Cw) TbTbJ-4 yc2 Yc2Kc3Eyc+0c,(To+Tc)

+PT 2 + (2+w)TcTc] =0,

(8 4)tot =-Q 2 K w-QybibKb (0b+C.)Tb)

- Q 2 (Oc+COT,) = 0,

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PRIMARY ABERRATIONS OF FRESNEL LENSES

(35)tot= -4 ybyb Kb 3[Yb+ 2 0bTb+(3+C0)Pb2]

-4ycy,3Kc3[y.+20J-,+ (3+(o)T 2] = 0,

where the symbols fla and ya are defined by

13a= (1+CO)Sa,

ya= (Co- 1)Sa2 ± (1 -o)-I-4ya- 4KKa3 ( i3)a,

and similarly for lenses b and c.The aberration conditions therefore give six simul-

taneous linear equations in the six unknowns ga, b, #c,

Ya, 'Yb, and -yc. Putting co= and using the data fromthe first-order layout, we can solve to get

fla=-29/3, bb= 1, f3c= 13/6,

,ya =- 1 6 9 /3 , Wb=19 / 3 , -cy=1/ 3 .

The defining equations for the fl's and Iy's then yield

Sa -5.8, Sb=0.6, S,= 1.3,

(Z&;l)a = 36.09, (ASi)b =-0.8633, (zSl)C 4.2067.

The surface curvatures cl and C2 for each Fresnel lensand its figuring constant (el-e 2 ) are then calculatedby use of Eqs. (12) and (30), respectively.

The only uncorrected primary aberration is longi-tudinal color, with (C)t 0 t= -0.2, which correspondsto an angular aberration at the eye of 6u.'= (C=)./2y,=0.1 rad. However, this assumes an objective witha speed of f/0.5, i.e., ua'= -1. If the speed of theobjective is reduced to a reasonable value, 5u,' isreduced proportionately.

APPENDIX A: DERIVATION OF EQS. (1)

Snell's law in vector form is NaXfi=N'a'Xni, wherethe symbols are defined above. Taking the dot productof both sides with a, we get Na- (aXii)=N'a- (a'X i).Now a (aXi) is a triple scalar product with one factorrepeated and so is identically zero. Thereforea * (a' X ii) = 0, so that the vectors a, a', and fi lie inthe same plane, i.e.,

a'=Aa+Fi, (Al)

where A and r are scalart. that are to be determined.Taking the cross product of both sides of Eq. (Al)with i, we get

a'Xi=AaXfi, since iiXi=o0.

Comparing with Snell's law, it follows that A = N/N'.To determine r, we take the dot product of each side

of Eq. (Al) with itself, and noting that a-a=a'-a'=i* = 1 (since a, a', and n are unit vectors), we get

1 = (Aa+fi) * (Aa+ri) =A2+2Ara- n+r2.

Solving this quadratic equation for F yields

r={-A a 2i- (Aa -)2]-(A 2- I)J (A

= at {l-AE[1-( * )2} -A a * . (A2)

From the definition of dot and cross product, it followsthat (a.i*)2 +(aXfi)2=1; thus 1-(a.* )2=(aX i)2.Therefore, Eqs. (Al) and (A2) agree with Eqs. (1)except for an ambiguity of sign, which is easily resolvedby inspection.

APPENDIX B: DERIVATION OF EQS.. (8)AND (29)

Consider a surface of revolution about the x axis,whose equation is

X= 1C(yl+z2) (yl+Z2)2 (B 1)

where c is the curvature at the vertex, and e is anarbitrary constant. If e=O, the surface is a paraboloid.If we write Eq. (Bl) in the form

ck(x,yz) = x- 2c(y 2+z 2) - e(y 2+z 2)2 = 0,

then the direction cosines A, ', X of the unit normal fito the surface are

11=Ka4l/aX, v=Kalo/ay, X=Ka01/aZ,

K= E(a01aX)2+ (.94/dy)2+ (.90/aZ)2]-!y,where (B2)

since t2+v2+X2 = 1. Calculating the partial derivativesof 4, substituting them into Eqs. (B2), and using thebinomial expansion for the square. root, we obtain

'= -yc+l (c3-8e)y(y 2+z 2)+0(5),

X= -zc+ (c3-8e)z(y2+z 2)+0(5).

These equations are the same as Eqs. (29). If thesurface is a paraboloid, e=0 and the equations reduceto Eqs. (8).

APPENDIX C: DERIVATION OF EQS. (15)

Consider the individual terms in the right-hand sidesof Eqs. (11), one by one. For the first term in 8Y',

using Eqs. (12), we easily show that

(n-i) (Ci 3-C2 3) = 4K{[3S 2 + (n-1)-2]. (C1)

For the second term in WY', using Eqs. (13) and (14)and a moderate amount of algebra, we get

u (u'2+v'

2) -u(u

2+v

2)

= -Ky[FK2y2 (3T2+ 1)+3KyTf+3u2 ]

-WK2z2[2Ky(3T 2 +l)+2Ti7]+0(5). (C2)

The calculation of the third term in 6 Y' is moreinvolved. First, we recall that v =-yc+O (3) andX= -zc+0(3) by Eqs. (8), and using Eqs. (12), (13),and (14) we get

A= v1E(U-V,)'+(v- X)2

1

- -KyES+ (n- 1)-I][{Ky[S+T+n(n- 1)-]+f) 2 +{2KzES+T+n(n- l)-I])2]+0(5), (C3)

B= v2{Eu-v 1+nf(VI-Y 2 )]2+Ev-.l+n(X,-X 2 ) ]2}

W-AKyS- (n- l)-][{ KKyES+T-n(n -)i]

+a}2

+{ 2Kz[S+T-n(n- 1)-]}21+0(5). (C4)

467April 1974

ERWIN DELANO

If we make use of symmetry to reduce the amount ofalgebra, we can show that

A-B =-KyS (I -n-1)'[K(y'+z') (S+T)+Kya] -Ky (n - I)- '{KI(y2+Z2 )E(S+T)l

+ (11-n-1)-2]+Kya(S+T) +f2} +0(5). (C5)

Substituting the expressions given by Eqs. (Cl), (C2),and (C5) into the first of Eqs. (11), and collecting termsappropriately, we obtain the first of Eqs. (15).

Proceed in a similar manner for 5Z'. In Eqs. (11),the second term in 5Z' contains the factor

v' (uI2+-V2) -v(u2-+v 2)

=-KzE4K 2y' (3T2+ 1)+M2KyTfi+72]-'Klz3 (3T 2+ 1) +O(5) . (C6)

For the third term in IZ', the parts C and D corre-sponding to A and B, in Eqs. (C3) and (C4), differonly in having a factor of z in front instead of a factorof y. Therefore, (C-D) differs from (A -B), in Eq.(C5), only in having the same factor z in front insteadof y. The second of Eqs. (15) follows as in the previouscase.

APPENDIX D: DERIVATION OF EQ. (27)

The Lagrange invariant for an arbitrary skew ray isQ=N(zu-yv). Consider the case of an extreme sagittalray from the edge of the field, assuming an arbitrarypupil position. At the first surface of the lens, beforerefraction, we have z=rm, u=u=fm*, y=y=im, andV=Vm, and since N= 1, the value of Q in this case is

Qm =rmUn *rmv.

Substituting for v and ut* from Eqs. (13) andrespectively, we get

(26),

Qm=2rmKrm('T+l)- mKrm (T+1)= -Krm m(T-T),

which is Eq. (27).

APPENDIX E: ABERRATIONSOF AN ORDINARY LENS

The primary aberrations of an ordinary lens withspherical surfaces and stop in contact are

8, = -1rm 4K{E(1+2w)S'+4(1+w)ST

+ (3+2w)T'+ (1-wo)-'],

82c= -2Qntrm2K'[(1+w)S+ (2+w)T],

83 = Qm2K,

84= = Qm2K,

85 = O.

el =-2rm 2KIP,

e2=0. (El)

Equivalent equations are given by Hopkins.' 5 If thestop is not in contact, the aberrations will changeaccording to Eqs. (25). Substituting Eqs. (El) intoEqs. (25) and using Eq. (27) for Qm where needed tocombine terms, we obtain

8,= -Irm 4 K 3 [E(1+2w)S+4(l+ )ST

+ (3+2c)T 2 + (1 -W)-2],

82c* =- rm3 ?mK3[ ( 1 +2w)S'+ (1+w)S(3T+ T)

+ (I +w) T2+ (2+w)7TT+ (1 -4)-],821* = 0,

:1r* =-rm2K3[(1E( +2w) +2 (1 +w)S(T+T)

+2(1+co)TP+T2+ (I -w)-2],

1* = - Q.n2K,

85* =-rmr'K'[(1 . 2KE )S2 + (1 +w)S(T+3T)

+wTT+ (3+w)T2+ (1 -)-2],

(E2)

l*= =- 2r.2KIP,

e2* = - 2 rm'mK/v,

Equations (E2) allow us to calculate the primaryaberrations of an ordinary thin lens with sphericalsurfaces, for any position of the stop or of the object.

REFERENCES

*Paper presented at the Annual Meeting, Optical Society ofAmerica, Rochester, October 1973 [J. Opt. Soc. Am. 63,1295A (1973)].

IC. Hofmann, and R. Tiedeken, Jenaer Jahrb. 1964, 109(1964).

IC. Hofmann, Jenaer Jahrb. 1966, 89 (1966).IC. Hofmann, Jenaer Jahrb. 1967, 7 (1967).4C. Hofmann, C. Z. Jena, and J. Klebe, Optik 25, 389 (1967).'C. Hofmann and J. Neumann, Jenaer Jahrb. 1968, 7 (1968).6C. Hofmann and J. Neumann, Jenaer Jahrb. 1969/70, 69

(1969/70).70. E. Miller, J. H. McLeod, and W. T. Sherwood, J. Opt.

Soc. Am. 41, 807 (1951).8E. Barkan, and R. J. Kapash, J. Opt. Soc. Am. 61, 686A

(1971).9H. H. Hopkins, Wave Tkeory of Aberrations (Oxford U. P.,

London, 1950)."See Ref. 9, Ch. IX."See Ref. 9, Ch. VII."See Ref. 9, p. 134, Eq. (191)."E. Delano, Appl. Opt. 2, 1251 (1963). See Appendix."See Ref. 9, p. 153, Eqs. (216) and (217).'5See Ref. 9, p. 135, Eqs. (192).

468 Vol, 64