A Direct Approach to the Evaluation of the Variance of the Wave Aberration W. B. King Mar6chal's treatment of tolerance theory shows that in designing high quality systems one should aim at minimizing the variance E of the wave aberration. Since the value of E is essentially positive, a useful criterion for the whole field is the sum of the suitably weighted values of E for a typical set of image points. It is shown here that the variance E (for both the axial and extraaxial images) may be calculated very simply by means of a set of universal coefficients [P(ij), Q(ij), and R(ij)] once the wave aberrations of selected rays are known. The values of these coefficients are uniquely determined by the form of polynomial assumed for the wave aberration and by the pattern of rays traced. Tables of P(ij), Q(ij), and R(ij) are presented for the different cases that can arise in practice. Introduction For highly corrected optical systems, that is, those substantially satisfying the Rayleigh X/4 limit, the im- age criterion based on the Strehl intensity ratio (Defini- tionshelligkeit) gives one a single index which measures the loss of intensity at the center of the image of a point source owing to the presence of aberration. The Strehl ratio I is defined as the ratio of the image intensity at the diffraction focus, with the aberration present, to that without aberration. It is given by: J= fA exp [ikW(x,y)] dA , (1) where W(x,y) is the wavefront aberration at the point (xy) of the pupil; k = 27r/X; A denotes the region of pupil, and dA = dxd/ ff dxdy is the element of area expressed as a fraction of the total area of the pupil. Mar6chall has shown that when the wave aberrations are small, such that I > 0.80,the Strehl intensity ratio may be calculated from: I = [1 - (2r,2/X2) E] 2 , (2) where E is the variance of the wave aberration given by E = fIA [W(x,y)] 2 dA - LffA W(x,y)dA]. (3) The author was in the Physics Department, Imperial College, London when this work was done; he is now with the Perkin- Elmer Corporation, Norwalk, Connecticut 06852. Received 8 September 1967. In designing high quality systems, it is important that a diffraction based criterion of image quality should be employed. Since the value of E is essentially posi- tive, a useful criterion for the whole field is the sum of the suitably weighted values of E for a typical set of image points. Optimization of a design then requires this sum to be minimized with respect to the different design parameters. For this purpose, both the values of E and their parameter derivatives are required at different design stages, and these usually incur exten- sive computations. A technique based on the applica- tion of Hopkins' canonical pupil coordinates has re- cently been developed, 23 in which the types of aberra- tion polynomial associated with different classes of op- tical system, as well as the ray trace patterns in the equivalent circular pupil are standardized. The result- ing advantage is that the variance E (for both the axial and extraaxial images) may be calculated very simply by means of a set of universal coefficients P (ij), Q (ij), and R (ij) once the wave aberrations of the selected rays are known. The computation of these coefficients are now described, and the results are tabulated for different classes of optical systems. The Equivalent Circular Pupil for Extraaxial Imagery It has been found that, for extraaxial imagery, the vignetted pupil shape may be represented to a good approximation, by a best fitting ellipse: (x 8 /a 8 ) 2 + (y 8 /b) 2 = 1, (4) where (sys) are the entrance pupil coordinates cor- responding to the limiting rays; the entrance pupil sur- face is taken to be the reference sphere passing through the effective entrance pupil point and centered on the March 1968 / Vol. 7, No. 3 / APPLIED OPTICS 489
Transcript
A Direct Approach to the Evaluation of theVariance of the Wave Aberration
W. B. King
Mar6chal's treatment of tolerance theory shows that in designing high quality systems one should aim atminimizing the variance E of the wave aberration. Since the value of E is essentially positive, a usefulcriterion for the whole field is the sum of the suitably weighted values of E for a typical set of imagepoints. It is shown here that the variance E (for both the axial and extraaxial images) may be calculatedvery simply by means of a set of universal coefficients [P(ij), Q(ij), and R(ij)] once the wave aberrationsof selected rays are known. The values of these coefficients are uniquely determined by the form ofpolynomial assumed for the wave aberration and by the pattern of rays traced. Tables of P(ij), Q(ij),and R(ij) are presented for the different cases that can arise in practice.
IntroductionFor highly corrected optical systems, that is, those
substantially satisfying the Rayleigh X/4 limit, the im-age criterion based on the Strehl intensity ratio (Defini-tionshelligkeit) gives one a single index which measuresthe loss of intensity at the center of the image of a pointsource owing to the presence of aberration. The Strehlratio I is defined as the ratio of the image intensity atthe diffraction focus, with the aberration present, to thatwithout aberration. It is given by:
J= fA exp [ikW(x,y)] dA , (1)
where W(x,y) is the wavefront aberration at the point(xy) of the pupil; k = 27r/X; A denotes the region ofpupil, and
dA = dxd/ ff dxdy
is the element of area expressed as a fraction of thetotal area of the pupil.
Mar6chall has shown that when the wave aberrationsare small, such that I > 0.80, the Strehl intensity ratiomay be calculated from:
I = [1 - (2r,2/X2) E]2, (2)
where E is the variance of the wave aberration given by
E = fIA [W(x,y)]2dA - LffA W(x,y)dA]. (3)
The author was in the Physics Department, Imperial College,London when this work was done; he is now with the Perkin-Elmer Corporation, Norwalk, Connecticut 06852.
Received 8 September 1967.
In designing high quality systems, it is importantthat a diffraction based criterion of image quality shouldbe employed. Since the value of E is essentially posi-tive, a useful criterion for the whole field is the sum ofthe suitably weighted values of E for a typical set ofimage points. Optimization of a design then requiresthis sum to be minimized with respect to the differentdesign parameters. For this purpose, both the valuesof E and their parameter derivatives are required atdifferent design stages, and these usually incur exten-sive computations. A technique based on the applica-tion of Hopkins' canonical pupil coordinates has re-cently been developed,2 3 in which the types of aberra-tion polynomial associated with different classes of op-tical system, as well as the ray trace patterns in theequivalent circular pupil are standardized. The result-ing advantage is that the variance E (for both the axialand extraaxial images) may be calculated very simplyby means of a set of universal coefficients P (ij), Q (ij),and R (ij) once the wave aberrations of the selectedrays are known. The computation of these coefficientsare now described, and the results are tabulated fordifferent classes of optical systems.
The Equivalent Circular Pupil forExtraaxial Imagery
It has been found that, for extraaxial imagery, thevignetted pupil shape may be represented to a goodapproximation, by a best fitting ellipse:
(x8/a8 )2 + (y8/b) 2 = 1, (4)
where (sys) are the entrance pupil coordinates cor-responding to the limiting rays; the entrance pupil sur-face is taken to be the reference sphere passing throughthe effective entrance pupil point and centered on the
March 1968 / Vol. 7, No. 3 / APPLIED OPTICS 489
-0.2
-0.4
Fig. 1. The scaled pupil coordinates (x,/a,, y,/b,) of the rim raysplotted against a unit circle for a 15.24-cm (6-in.), f/2 system at
full field (500).
object point; a and b, are the major and minor semi-axes corresponding to the particular field considered.Next we scale this ellipse into an equivalent circle:
x2+ y2 = 1 (5)
by defining
x = (x,/a.), y = (y.8/b,). (6)
A typical example is illustrated in Fig. 1 that shows howaccurately the scaled coordinates (x,y) of the pupil maybe represented by a unit circle (this is discussed in de-tail in another paper).4
The usefulness of this equivalent circular pupil lies inthe fact that one may adopt a ray trace pattern withprescribed values of (xy) at the entrance pupil, andthis ray pattern can then remain the same for differentimage fields and for different optical systems of thesame class. This facilitates an analytical formulationof the general problem of extraaxial image formation.
Specification of Pupil Coordinates in Ray TraceIt is assumed that the optical system has axial sym-
metry and zero obscuration. The rays to be tracedare specified by giving the values of the two quanti-ties A and S: A is the number of aperture raystraced in the azimuth = 0 of the pupil and S denotesthe number of skew rays to be traced (S being always anodd number). The total number of rays traced fromeach extra-axial object point will be
J = 2 + S, (7)
of which there are A in the azimuth f = 0, A in theazimuth 7 = ir, and a single ray in each of the skewazimuths
pi = i[vr/(S + 1)], i = 1, 2, ... , S. (8)
The major and minor semiaxes (a, and b) of theelliptical pupil are given, and the scaled pupil coordi-nates (x,,y,) j for ray tracing are found from
(xs)j = axi; (y,)j = b, (9)
where j denotes the jth ray; (x,y)j are given as for a cir-cular pupil, that is, with a, = b = 1.
In the azimuth = 0, the rays are numbered j =1,2,. . .,A and have coordinates
xj = O, yj = + [(A + 1 - j)/A]I. (10)
The forms of the wave aberration polynomials to bedescribed later argue against having a uniform distri-bution of rays over the whole pupil area. With theabove convention, the rays are equally spaced in theaperture radius squared andj = 1 is the marginal ray.
For the axial object point, S is set to zero and Aaperture rays are traced in the azimuth = 0 of thepupil.
For an extraaxial object point, S is set to the speci-fied value, and A aperture rays are traced, using Eq.(10). Inaddition,theraysj = A + 1,A + 2,. . ., [A +(S + 1)/2] are traced, respectively, in the azimuth 0i[see Eq. (8)] with i = 1, 2,. . ,(S + 1)/2. The polarradii r of these rays are
S =1: i = 1, -1 = 1
S 3 3 i = 1, 2' 11 2 = +[(A - 1)/A]2 (11)~i =2, r2 = 1,
and the ray coordinates for these are
xi = ri sinoi, (12)yi = r cosoi.
The raysj= ,, ...,A, A + 1,. . ., N, with N = A +(S + 1)/2, are thus traced in the upper half of the pupilwith the j = Nth ray in the azimuth = 7r/2. Therays in the lower half of the pupil, with j = N + 1,N + 2,. . ., N + A + (S - 1)/2, are traced with
Xj = Xj-N (13)Yj = -j-N)
so that for j < N, the rays j and j + N are symmetricalabout the x axis leading to simplified calculations of oddand even aberrations from the ray trace. With theabove conventions, it is possible to specify the rays to betraced for any system by means of the two numbers Aand S. An example of the ray pattern correspondingto A = 3, S = 3 is illustrated in Fig. 2.
Wave Aberration PolynomialsThe aberration polynomial is written as a function of
the equivalent circular entrance pupil coordinates (xy).There will be (A - 1) orders of spherical aberration,(A - 1) orders of circular coma, a transverse focal shift,a sagittal curvature coefficient, and an astigmatism term.This gives a total of (2A + 1) coefficients. The totalnumber of rays traced (2A + S) equals the number ofaberration coefficients required. In any practical case,the values of A and S are kept as small as is consistentwith an accurate representation of the wave aberration.
490 APPLIED OPTICS / Vol. 7, No. 3 / March 1968
From Eq. (14), Vj may be written as
V = WV1yj + W31(X' + Y2)Yj
+ W51(X2 + Y')Y2 Yi + W33Yy3 . (18)
Thus forA = 3, S = 3, N = A + (S + 1)/2 = 5, wehave five equations for U, and four equations for V;these can be solved for the nine aberration coefficients.
(=N) By substituting the values of (xj,yj) as specified by Eqs.(10) and (11) into Eqs. (16) and (18), it can be shownthat the coordinates of the ray pattern lead to well con-ditioned equations for the aberration coefficients.
For the axial object point, the aberration polynomialfor A = 3, S = 0 reduces to
A=3S=3
< - H~~~~=A 2~=5
Fig. 2. The specification of rays is determined by the values ofA and S. An example of the ray pattern for A = 3, S = 3 is
shown.
Suitable values of A and S for five different classes ofoptical systems may be listed as follows:
Case (1) Photographic objective: A = 3, S = 3;Case (2) Microscope objective (low power): A = 3, S = 1;Case (3) Microscope objective (medium power): A = 4,
S = 1;Case (4) Microscope objective (high power): A = 5, S = 1;Case (5) Eyepiece: A = 2, S = 1.
The value of S = 3 is used for Case (1) owing to itslarger angular field coverage. In this case, the numberof aberration coefficients required is (2A + 3), twomore than that which has been described above. Thesetwo coefficients may conveniently be chosen to beW 33 y 3 , W 42 (X2 + y2 ) y2 .
Thus, the aberration polynomial for A = 3, S = 3[Case (1) ] will be
For the jth ray, W (xy) may be written as W1 . As aconsequence of using the ray coordinates defined by Eq.(13), the even and odd parts of W (xy) may be sepa-rated out. Thus, for the extraaxial case,
(W1).dd = V = (W1 - Wj+ N)/2; j = 1, 2,... N - 1. (17)
Wj = V2OYi' + W40Yj' + W6OYj, (19)
with] = 1, 2, 3 (= A).The aberration polynomials for the different classes
of optical systems considered earlier may now be statedin a general form; the actual permissible values of n andq (introduced below) depend on the values of A and S,and these are tabulated in Table I for reference.
(a) for the even part of the wave abberration (extraaxialimages):
Uj = E E W-n,(X2 + y2)1 (n-)12 !jqn q
(20)
with n, q both even; j =1,2,. . . ,N. These N equationsmay be inverted to give
NWn = Ei K(nq;j)Uj.
j=1
(b) for the odd part of the wave aberration:
Vj = E EWVn (X2 + y2)j(n-q)/2 yjqqn
with n, q bothodd;j = 1,2,,. .. N -1.equations may be inverted to give
N-1Wnq = E K(nq;j)Vj.
j=1
(c) for the axial image (S = 0):
Wj = E W,0Oyjn,n
(21)
(22)
These (N - 1)
(23)
(24)
n being even; j = 1,2,. . . ,A. These A equations may beinverted to give
AWno = E K(n,O;j)Wj.
j=i(25)
Evaluation of the Variance E of theWave Aberration
As pointed out earlier, the aberration polynomialsgiven above are functions of the coordinates (xy) of theequivalent circular pupil. The aberrations Wj,Uj, andV1 calculated in the ray trace depend on the scaledpupil coordinates (xs,ys)j [see Eq. (9) ]. In conse-quence, the aberration coefficients in the polynomialhave no direct significance in analytical aberrationtheory (except for axial images) and are only used as
March 1968 / Vol. 7, No. 3 / APPLIED OPTICS 491
Table I. The Permissible Values of (n, q) are Tabulated forFive Different Classes of Optical Systems
[F EVEN PART OF ODD PART OF AXIAL IMAGESCASE NO. ABERRATION (Uj) ABERRATION (Vj) r =0 (Wj)
intermediate auxiliary quantities for the present pur-pose.
Referring to the expression for the variance E of TV(xy) as given by Eq. (3), if we assume the equivalentpupil to be X
2+y
2= 1, dA = (dxdy) /7r. It is more con-
venient to use polar coordinates to evaluate E, andthe general expression for the wave aberration is then
TV = Z E IV_, 2-, COS70-
7 q
(n,q). The terms in the square brackets in Eq. (30)have the value unity if p = q = 0.
(b) for the odd part of the aberration (n, q both odd):
Eodd = E y Wm,pWnq7n n p q
X J( 2 ) [ 3 (p + q -) (31)+ n + 2 2 X 4... (p + q)
(c) for the axial image case (p, q both zero):
Eaxial = E] T/V WlVOWno[ + n + 2 (m + 2)(n + 2)nt n Ir In(n±/f )
The aberration coefficients from Eqs. (21), (23), and(25) may now be substituted in Eqs. (30), (31), and(32), respectively. The final forms of Eevcn, Eodd) andEaxial are
N NEevel = E E Q(i,j)UiUj,
i=1 j=1
N-1 N-1Eodd = T E R(ij)ViV1,
i=l j=1
A A
E.-i, = E E P(ij)WiWj,i=lj=l
(26)
The even and odd parts of V contribute separately toE, so that
E = E-C + Edd (27)
where
=, f1fl 2" Ur drd - f12 Urdrd42, (28)
7r 0 0o 7r f o
where
Q(i,j) = Fm n q
( 2 )[I X3.. .(p+q- 1)-n + n + 2 2 X 4 ... (p + q)
Edd= -I1 Er V'rdrd4,the integration of the odd function V(r,O) over thepupil being zero. If E is evaluated for the generalaberration function [Eq. (26) ], we have
1 rl 2 _E =- I f I f21 E [ E Wnq(?"' COSqq5) ]rdrdk
7ro J o 71 q
- [-1 X fi V f2 Wnq(1., CoSqo)rdrd ]7r 0 fo 7 q
Evaluating these integrals leads to the following results:(a) for the even part of the aberration (extraaxial images,
with n, q both even):
Eeovon = E Z Z E Wmarpl~n,q j( ( 2 2m n p q in\fl n + n
X [ 2 X .. (p 1)] [1 X 3 (q ])' (30)2[1X3... p I 2X4..q
where the range of values of in and n, p and q in thesummations are the same as those given in Table I for
4 +[X3...p 1)](mn + 2)(n + 2) 2 X 4... p
X [ X .. (q )1) K(m,p;i)K(n,qj), (36)
7n 7 p q { t1+ n+ 2
X [2 X 4. (P + - ) K(-np;i)K(n~q;j), (37)
P(i,j) = (7n + 2)(n + 2)(m + n + 2)J
X K(m,O;i)K(n,O;j). (38)
The values of n, q (also m, p) are those given in Table Ithat correspond to the particular values of A and S. Thecoefficients K(m,p;i), K(n,q;j), as defined earlier, areknown constants once the values of A and S are given,since the ray trace pattern will then always be the samein the equivalent circular pupil. Thus, the above co-efficients P(ij), Q(ij), and R(ij) need be calculatedonce for all for any given values of (A ,S).
The computation of these coefficients has been carriedout on the IBM 7094 computer, and double precisionarithmetics have been specified in a few cases. Theresults, corresponding to the five pairs of (A,S) values,are tabulated below; all these square matrices have di-agonal symmetry [this is expected from Eqs. (36), (37),and (38) ].
When the wave aberrations UjVj, and TV; are knownfrom the ray trace, the values of E__0n, E0,Id and Enxiaimay be calculated at once from Eqs. (33), (34), and(35), respectively, using the tabulated universal coeffi-cients corresponding to the pair of (A,S) values em-ployed in the ray trace. The variance E for an extra-axial image is given by the sum of EK,_e and Eodd.
The image quality merit function (4r) referred to in theintroduction may now be defined as
' = Z (a!Eaxia)x + ZZ (iE0lre + .yEodd)X,
x x a
where a, fi, and Py are weighting factors dependent on
the wavelength X and the image field r. cD is requiredto be minimized with respect to the design parameters,'using a damped least-squares technique. The Strehlintensity ratio of the optimized design must exceed 0.8,this being calculated from Eq. (2) using Eqs. (33), (34),and (35). Optical design programs based on the vari-ance technique are now under development and thesewill be reported at a later date.
The author thanks H. H. Hopkins for suggesting thisproblem and for useful discussions and the British\Iinistry of Aviation who financed this work. It is also
a pleasure to acknowledge the facilities offered by theresearch group of W. Watson and Sons for discussions.They now use these techniques in their regular opticaldesign programs.
References1. A. Mar6chal, Thesis, University of Paris (1948).2. H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).3. H. H. Hopkins, Opt. Acta, 13, 343 (1966).4. W. B. King, Appl. Opt. 7, 197 (1968).
