Algorithm for determination of the diffraction focusin the presence of small aberrations
Helene Safa
The diffraction focus (or best focus) is calculated for an optical system that contains small aberrations. Byassuming that the wave aberration function is known with respect to an arbitrary point in the imageregion, we obtained simple analytical expressions of the diffraction focus coordinates for any geometry ofthe pupil. The method can be easily worked out on a computer by ray tracing. A comparison is done with
the method that is based on the displacement theorem.Key words: Diffraction focus, wave aberration function, primary aberrations, displacement theorem,
Strehl ratio.
Introduction
When small aberrations are present in an opticalsystem the diffraction image of a point source lookslike a distorted Airy pattern; the maximum intensityis lower than when the aberrations are absent and isno longer located at the Gaussian focus (the focusthat is given by paraxial optics) but at a point calledthe diffraction focus"2 (the Fresnel number of theaperture is assumed to be large'). Knowledge of thediffraction focus is important in optical analysisbecause the fall of the peak intensity, i.e., the Strehlratio (SR), is a simple criterion that is widely used toquantify the aberrations in an overall view, and it iscustomary to consider that a system is limited prima-rily by diffraction when the SR is > 0.8 at the peakintensity. For small aberrations, i.e., when the SR islarge enough (SR 2 0.6) at the diffraction focus,"4 it isknown that the diffraction focus is the center of thereference sphere, which minimizes the variance of thewave aberration. By making use' of the displacementtheorem and of the properties of Zernike polynomials,the diffraction focus and the tolerances for the pri-mary aberrations have been obtained for a circularpupil. Pietraszkiewicz' generalized the use of thedisplacement theorem for an arbitrary wave aberra-tion function in a system with a small aperture. In thesame way Mahajan6'7 established the coordinates of
The author is with the Centre d'Etudes Spatiales des Rayonne-ments, 9 Avenue du colonel Roche BP 4346, Toulouse 31029,France.
the diffraction focus and the tolerances for primaryaberrations for an annular pupil. Here we develop asimple algorithm for calculating the diffraction focuscoordinates for an arbitrary pupil function when thewave aberration is known with respect to a referencesphere centered at any point P in the image region.The SR need not be large at this point, but it is stillassumed to be large enough at the diffraction focus(small aberrations). The method is iterative and isbased on the Newton algorithm. It is shown that theformulas that are obtained with the displacementtheorem for small apertures (large f-number) corre-spond to the first iteration of the algorithm. The needfor going through a second iteration is discussed.Once the diffraction focus is known, the SR is easilycalculated, and the tolerance conditions on the waveaberration can be derived.
In another paper it will be shown that the methodhas been applied successfully for the evaluation of theimage quality of the PRONAOS submillimeter seg-mented telescope (Centre National d'Etudes Spatiales/Centre National de la Recherche Scientifique, France).
I. Aberration Function
Before we begin let us define the notations that areused in what follows. The wave aberration function isdefined within the exit pupil of the system whosecurrent point is denoted Q. In order to specify explic-itly the reference sphere, we denote by A(Q, P) thewave aberration at Q with a reference sphere cen-tered at P. The standard deviation of A(Q, P) aredenoted u(P) and is equal to
SE is the exit pupil area, l; is the integration region,and dQ is the surface element. For an apodized pupil,all the following formulas remain valid provided thatthe average value X is replaced by
X [fXX(Q)P(Q)dQJ/ff p(Q)dQ), 0 p(Q) 1,
where p(Q) is the apodization function. As usual' theSR at point P [which is also called normalized inten-sity i(P)] is the ratio of the intensity I(P) divided bythe intensity that would be obtained at the samepoint if no aberrations were present.
We first derive a convenient expression of the waveaberration A(Q, P). We consider a centered opticalsystem with a point source M. We take a Cartesiansystem of axes with the origin at the center of the exitpupil Q and the X direction along the axis of thesystem.
The optical wave front W and the reference sphereS both pass through the center of the exit pupil Q0.M' is a point of the image region; let Qw and Qs be thepoints at which the ray QM' intersects wave front Wand the reference sphere S, respectively (Fig. 1).
Then' A(Q, P) is the optical path length (QwQs).
A(Q, P) = (QWQS)
= OP(QS) - OP(QW), (2)
where OP(Qs) and OP(Qw) are the optical paths in Qsand Qw. As Qw and QO are on the same wave front, wehave OP(Qw) = OP(QO) and
A(Q, P) = OP(Qs) - OP(Qo)
= OP(Q) + (QQs) - OP(Q). (3)
Let Qs* be the intersection of the line joining Q and Pwith the reference sphere and take the approximation
The diffraction focus will be the point P that mini-mizes the variance of A(Q, P).
II. Calculation of the Diffraction FocusFrom Eqs. (4), the average value of A(Q, P) is
(Q, P) = OP(Q) - OP(Qo) + qP- Q0P, (5)
so that
A(Q, P) - Y(Q, P) = OP(Q) - OP(Q) + QP - - (6)
Thus if we consider two points F and Fo in the imageregion and note, for simplicity, that A(Q, F0 ) = A0 andA(Q, F) = A, we get
A - 2 = Ao - Yo + [(QF - QF0 ) - (F - QF0)]. (7)
Therefore
(A -K)2 = (A. -(,)2 - 20(A, - A) + , (8)
where
0 = [(QFo-QF)-(QF0 - P)]. (9)
Now we assume that A0 is known either analyticallyor numerically by ray tracing. Several numericalmethods can be used to minimize a2 with respect to F(0 depends on F). The simplest method is the Newtoniterative method. Since F and F are supposed to beclose to each other we can develop QF to first order:
QF = QFo + LX + MY + NZ, (10)
where X, Y, and Z are the coordinates of point F withthe origin at F and L, M, and N are the cosines of thethe direction QFO, with
L = (QF/aX)Fo = -XQIQFO,
M = (QFIaY),F = -YQIQF0,
Q1
WI I \SImageplane
Exitpupil
Fig. 1. Wave aberration in the exit pupil.
N = (QFI8Z)Fo = -ZQIQF0 . (11)
So 0 can be rewritten as
0= -(L-L)X- (M-M)Y- (N-N)Z. (12)
By annulling theX derivative of Eq. (8) we first obtainx
-0aO~X)(A0 - Y) + (aolaX)0 = 0; (13)
then, according to Eq. (12), a0/aX = -(L - L).Therefore the coordinates of the diffraction focus(X, Y, Z) satisfy the equation
Similarly by annulling the Y and Z derivatives of Eq.(8), we obtain
X(L - L)(M - M) + Y(M - M)2 + Z(M - M)(N - N)
= - (M-M)(A,-Y), (14b)
X(L - f)(M - M) + Y(M - M)(N - ) + Z(N - N)2
= - (N - T( - Y). (14c)
The coordinates X, Y, and Z are readily calculatedand give the first approximation F 1for the diffractionfocus. The method is then iterated by developing QFin the neighborhood of Fj:
QF = QF1+ LX, + MY, + NZj,
where
L = (QFIOX)F, = -XQIQF,,
and X 1, Y, and Z, are the coordinates of the newdiffraction focus in the repair that is centered at Fl. A0remains unchanged for all the interations. Thuswhen the wave aberration is calculated numericallyonly one ray tracing is necessary. Once the diffractionfocus is known with the desired approximation, thevalue of 3r is obtained from Eq. (8). To reduce thenumber of iterations, FO must be chosen as close aspossible to the diffraction focus. Because we aredealing with small aberrations we can take the focusgiven by paraxial optics for F0. If the calculation iscarried out numerically with a ray tracing, it isusually better first to find the transverse plane wherethe spot diagram is the most confined and then tochoose F 0at the centroid of the image in this plane.
111. Comparison with the Displacement Theorem
To compare the method that is developed in Section IIwith that based on the displacement theorem, con-sider again Eq. (7), where A and A0 are wave aberra-tions with reference spheres centered at F and FO,respectively. Now if we assume that the two followingconditions are satisfied: (1) that F is close enough toFo (how close is discussed below, and (2) that thef-number of the system is large or, equivalently,aid << 1 (Fig. 2), then condition (1) allows us to writeEq. (10) and condition (2) allows us to develop thecosines L, M, and N of the direction QFo.
In the case of a circular pupil let (r, 0) be the polarcoordinates of a current point in the exit pupil, let abe the radius of the pupil, and let d be the distancebetween the pupil and FO (Fig. 2). Point Q of the exitpupil has (-d, r cos 0, r sin 0) as coordinates. Sinceaid < 1, Eqs. (11) can be approximated to
L = (1 - M2 - N2)1/2= 1 - r2/2d2,
M = -r cos old,
N = -r sin old. (15)
a
zI
IY
F0
x
z
Fig. 2. Focus F, lies on the axis of revolution. (r, 0) are the polarcoordinates in the exit pupil.
Thus for A(Q, F) we get
A(Q, F) = A(Q, FO) + Ar2 + Br cos 0 + Cr sin 0 + D,
where A = -XI2d 2 , B = -Y/d, C = -Zid, and D =X + QoFo - QoF.
This is the exact expression of the displacementtheorem.' Therefore the method developed in SectionII is in complete agreement with that based on thedisplacement theorem, which appears to correspondto the first approximation of the Newton algorithmunder condition (2).
When either condition (1) or condition (2) is notfulfilled, the displacement theorem may fail for thedetermination of the diffraction focus. Condition (1)is self-consistent. However, one must bear in mindthat, for large apertures, the scalar theory of diffrac-tion, which is implicitly assumed in this paper, mayfail. A vectorial propagation of the electric field is thennecessary for describing correctly the electric fieldnear the focus.
To make condition (1) exact we note that theneglected term in the Taylor development of Eq. (10)can be written as
3 3
e = (1/2) 2 2 [a2QFI(aXiaXj)]XiX2,i-1 j=l
(16)
where (Xi) are the X, Y, and Z coordinates. Thesecond derivatives, i.e., the derivatives of the cosinesL, M, and N, are easily calculated at Fo:
aLlaX = (1 - L')IQFQ,
aLlaY = LMIQFO
aL/aZ = LNIQFo, (17)
and similar expressions for the derivatives of M and Ncan be calculated. Clearly the neglected term willhave no incidence on the calculation of the diffrac-tion focus if its contribution to the wave aberrationA(Q, F) - A(Q, F) is much smaller than the wave-length I (e -e) I << A.
When condition (2) is satisfied, (M << 1, N << 1,QF0 d), we get
(e - )/X = [(M2 + N2) - (' N)]X2- (M2 _ )y2
-(N2 _ Th)Z2 + 2(M - M)XY + 2(N - Y)XZ
+ 2(MN - MN)YZj/Xd. (19)
Because MI << /2f, INI << /2f, where f is thef-number, the conditions on the diffraction focuscoordinates X, Y, and Z that are obtained with thefirst iteration with ( - e) I<< X can be written as
X 2 /2f2 Xd << 1, y 2/2f2 Xd << 1, Zl2f2rXd << 1,
XY/fXd << 1, YZ/fXd << 1. (20)
These are the conditions that must be satisfied ifwe do not go through a second iteration and make useof the displacement theorem method under condition(2). We have assumed, for simplification, that F0 is onaxis, but inequalities (20) are still valid within Gaus-sian optics. Usually point F0 is chosen at the Gaussianfocus. Then the practical cases in which the diffrac-tion focus strongly departs from F correspond tooptical systems that contain a certain amount ofdistortion or field curvature. We consider here, as asimple illustrative example, the f/10 Cassegrain PRO-NAOS telescope (Fig. 3), which is intended for submil-limeter observations. The optical combination is givenin Table I. The telescope is rigorously stigmatic for aplane wave that is parallel to the optical axis butexhibits a field curvature, as is often the case for theCassegrain combination.
For a plane wave with an incidence 0 = 0.02 rad andfor = 150 ,um, Table II shows that two iterations arenecessary for obtaining a precise diffraction focus. Arough application of the displacement theorem wouldhave led to a SR of 0.55 at the diffraction focus, whilethe exact value is 0.71. For this example inequalities
M
Table I. Optical Parameters of the Pronaos Telescope
(20) are not satisfied. The SR's are calculated numer-ically from the Huygens-Fresnel diffraction formula.
IV. Application to Small Apertures
Equations (14) can be simplified for an optical systemof revolution with a small aperture. We give here, asan illustrative example of the way to make thecalculation, an expression of the diffraction focus forprimary Seidel aberrations and for the case of anannular pupil of inner and outer radii a and a,respectively. As in Section III, r and 0 are the polarcoordinates of a point Q in the exit pupil (Fig. 2), butFo is not necessarily located on the X axis. Then
L = [1 - (M2+ N2)]"2
M = (Y - r cos 0)ld,
N = (Z - r sin 0)1d,
where Yo and Z0 are the transverse coordinates of F.Therefore
(L - T) = -(r2 - 7)/2d',
(M - ) = -r cos old,
(N - N) = -r sin old.
Since the system is of revolution, we have
(L - )(M - M) = ,
(N- )(L - ) = 0,
(M- M)(N- N) = 0.
y
x
d
Fig. 3. Geometry of the Pronaos telescope.
Table II. Necessary IteratIons for Obtaining a Precise Diffraction Focusfor a Plane Wave with Incidence 0 = 0.02 rad and X = 150 tim,
aThe Gaussian focus coordinates are X = 2818 mm, Y = 400 mm(origin at the sag of the secondary). The wave-front error at theGaussian focus is 0.77X. DX, DY are the coordinates of F withrespect to F . The pupil stop is on the primary mirror.
with e = 0 for a circular pupil, we rewrite Eqs. (14) as
X = {24d2/[a(1 - E2)2]}[(r2 - r2 )Ao],
Y = {4d1[a2 (1 + e2)]Jr cos OAo,
Z = 4d/[a 2(1 + e2 )llr sin OAo. (24)
These equations give the coordinates of the diffrac-tion focus with respect to F0, provided that conditions(20) are satisfied.
For the standard deviation of the wave aberration,according to Eqs. (8), (12), (21), and (22) we get
a2 = or 2 - (L -)2X2-(M -M)2Y2-(N _ )2Z2,or
(b) Coma aberration: Ao = A(r'Ia')cos 0; we getX = Z = 0 and Y= (2A/3)(d/a)(1 + E2 + E
4 )/(1 + E2 ).
(c) Astigmatism: o = (A/a2 )r2cos2 0. Here Y=Z = 0 andX = A(dla)2 .
(d) Curvature of field: Ao = Ar2 /a2 ; Y = Z =
andX = 2A(d/a)2.(e) Distortion: o = Ar cos 0/a; X = Z = 0 and Y
=A(da).These results are in agreement with those given in
previous papers.-6
Conclusion
It has been shown that the diffraction focus of anoptical system can be worked out for small aberra-tions by minimizing the wave aberration. The methodcan be performed analytically for simple forms of thepupil and the small aperture. However when thepupil is complicated or the aperture of the systembecomes large, this method can be used with acomputed ray tracing only. The method that is basedon the displacement theorem corresponds to the firstiteration of the developed algorithm. In most casesthis approximation is valid but may fail when thediffraction focus is too far from the Gaussian (in thepresence of a strong field curvature). The need forgoing through a second iteration has been discussed.
a2 = - [a4(1 - e2
)2 48d4]X2
- [a2(1 + e2)/4d 2]y 2- [a2 (1 +
2 )/4d2 ]Z2 (25)
Now, let us consider the five primary Seidel aberra-tions:
(a) Spherical aberration: o = Ar4/a4 , where Ais a constant. Ao does not depend on 0 so then we haveY = 0 andZ = 0. ForXwe get
X= {24d2 [a4(1- e2)2]l r 4; - T),
or
X = 2A(d/a)2(1 + e2).
References
1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon,New York, 1980), Chap. 9.
2. A. Marechal and M. Frangon, Diffraction, Structures des Images(Masson, Paris, 1970).
3. Y. Li and E. Wolf, "Focal shifts in diffracted convergingspherical waves," Opt. Commun. 39, 211-215 (1981).
4. V. N. Mahajan, "Strehl ratio for primary aberrations in terms oftheir aberration variance," J. Opt. Soc. Am. 73,860-861 (1983).
5. K. Pietraszkiewicz, "Determination of the optimal referencesphere," J. Opt. Soc. Am. 69, 1045-1046 (1979).
6. V. N. Mahajan, "Zernike annular polynomials for imagingsystems with annular pupils," J. Opt. Soc. Am. 71, 75-85(1981).
7. V. N. Mahajan, "Strehl ratio for primary aberrations: someanalytical results for circular and annular pupils," J. Opt. Soc.Am. 72, 1258-1266 (1982).
