Theory of Concentric Circular Grid M. V. R. K. Murty and A. H. Shoemaker Any image-forming optical system is usually tested for aberrations by looking at the image plane. A point source is placed at some distance from the system (this may be a very large or small distance de- pending on the particular system) and its image by the optical system is studied by any of many methods available. The present paper presents the theory of a method in which a grating containing a series of concentric circles is placed at the image of the point source. Characteristic patterns are derived for the usual aberrations of optical systems and some typical photographs of these patterns are presented. Finally, this method is compared to the Ronchi test in which a series of straight lines are used for the grating. Introduction There are many methods available for testing image- forming optical systems. A most convenient method is to look at the image of a point source of light. This image may be observed directly by means of a high- power microscope, and one can decide how closely this image conforms to the ideal Airy diffraction image. This is the so-called star test and is a very quick and use- ful method. The Ronchi test is one in which a grid con- taining equally spaced straight lines is placed in the focal region, and different aberrations give rise to different characteristic patterns. A recent review article on the history of this test has been published by Ronchi. 12 In- stead of using a series of straight lines for the grating, a grating containing a series of concentric circles is placed in the focal region of the optical system. The patterns obtained by this method differ from those obtained in the Ronchi test in a characteristic manner. Theory of the Test We shall assume that the optical system is represented by the circular exit pupil plane and that the maximum radius of the exit pupil is r. At a distance R from the exit pupil plane is the Gaussian image plane. We use (x, y) coordinates in the exit pupil plane and (x, y) in the image plane. If the grating is placed slightly outside the Gaussian image plane, this longitudinal displacement is denoted by 2. Figure 1 indicates these quantities in relation to each other. If the wavefront The authors were with the Institute of Optics, University of Rochester, Rochester, New York. The present address of M.V.R.K. Murty is Madras Institute of Technology, Chromepet, Madras, India. The present address of A. H. Shoemaker is NOTS, China Lake, California. Received 28 July 1965. aberration is denoted by W(x, y), the transverse aberra- tion on the image plane is given by TA, = = R- ; TA, = = R W ax by (1) Now the system of concentric circles can be represented by (X- )2 + ( - 7)2 = n l1 when the circles are equally spaced and by (X- )2 + (g7 - = 7lp 2 (3) when the circles form a Fresnel zone plate. Here Q,,q) is the center of the concentric circles, pi is the radius of the first circle, and n is the number of the circle. In the discussion that follows we use equally spaced circles, and the theory can be used for the Fresnel zone plate by replacing n 2 by n. Thus we can eliminate x and y from Eqs. (1) and (2) and obtain (R - )2+ (R y (4) Equation (4) will be the basis for the derivation of the characteristic patterns for various aberrations. vet ~~~~~~~CONCENTR CIPCULAR GRID. Fig. 1. Schematic diagram showing the relative positions of the exit pupil and the concentric circular grid. February 1965/ Vol. 5, No. 2 / APPLIED OPTICS 323 (2) (3)
Transcript
Theory of Concentric Circular Grid
M. V. R. K. Murty and A. H. Shoemaker
Any image-forming optical system is usually tested for aberrations by looking at the image plane. A
point source is placed at some distance from the system (this may be a very large or small distance de-
pending on the particular system) and its image by the optical system is studied by any of many methods
available. The present paper presents the theory of a method in which a grating containing a series of
concentric circles is placed at the image of the point source. Characteristic patterns are derived for the
usual aberrations of optical systems and some typical photographs of these patterns are presented.
Finally, this method is compared to the Ronchi test in which a series of straight lines are used for the
grating.
Introduction
There are many methods available for testing image-forming optical systems. A most convenient methodis to look at the image of a point source of light. This
image may be observed directly by means of a high-
power microscope, and one can decide how closely thisimage conforms to the ideal Airy diffraction image.
This is the so-called star test and is a very quick and use-ful method. The Ronchi test is one in which a grid con-
taining equally spaced straight lines is placed in the focalregion, and different aberrations give rise to different
characteristic patterns. A recent review article on thehistory of this test has been published by Ronchi. 1 2 In-
stead of using a series of straight lines for the grating, agrating containing a series of concentric circles is placed
in the focal region of the optical system. The patternsobtained by this method differ from those obtained in the
Ronchi test in a characteristic manner.
Theory of the Test
We shall assume that the optical system is representedby the circular exit pupil plane and that the maximumradius of the exit pupil is r. At a distance R from the
exit pupil plane is the Gaussian image plane. We use(x, y) coordinates in the exit pupil plane and (x, y)in the image plane. If the grating is placed slightly
outside the Gaussian image plane, this longitudinaldisplacement is denoted by 2. Figure 1 indicates thesequantities in relation to each other. If the wavefront
The authors were with the Institute of Optics, University of
Rochester, Rochester, New York. The present address of
M.V.R.K. Murty is Madras Institute of Technology, Chromepet,
Madras, India. The present address of A. H. Shoemaker is
NOTS, China Lake, California.Received 28 July 1965.
aberration is denoted by W(x, y), the transverse aberra-tion on the image plane is given by
TA, = = R- ; TA, = = R Wax by
(1)
Now the system of concentric circles can be representedby
(X- )2 + ( - 7)2 = n l1
when the circles are equally spaced and by
(X- )2 + (g7 - = 7lp 2 (3)
when the circles form a Fresnel zone plate. HereQ,,q) is the center of the concentric circles, pi is the radiusof the first circle, and n is the number of the circle. Inthe discussion that follows we use equally spaced circles,and the theory can be used for the Fresnel zone plate byreplacing n2 by n. Thus we can eliminate x and yfrom Eqs. (1) and (2) and obtain
(R - )2+ (R y (4)
Equation (4) will be the basis for the derivation of thecharacteristic patterns for various aberrations.
vet ~~~~~~~CONCENTRICCIPCULAR GRID.
Fig. 1. Schematic diagram showing the relative positions ofthe exit pupil and the concentric circular grid.
Primary Spherical AberrationThe primary spherical aberration may be represented
as a wavefront aberration by
W(x,y) = B(x2 + y2)2 + ( + yI),2R2
Thus in the most general case, the fringe pattern is asystem of curves of sixth degree. When the lateraldisplacement () is eliminated, we have a system ofconcentric circular fringes. These circular fringes,however, are not equally spaced in the presence ofprimary spherical aberration. When the primaryspherical aberration is zero, we have a system of equallyspaced concentric circles even when the lateral dis-placement () is present. Figure 2 shows the fringepattern taken with a good lens. Figures 3 and 4 showthe fringe patterns for a lens having primary sphericalaberration.
Primary ComaThe wavefront aberration for primary coma, includ-
ing defocusing, may be represented as
W(X,Y) = Fy(x + 2) + (X + Y2)- (9)
After going through a similar process of normalizationas in the case of primary spherical aberration, we maywrite for the set of fringes seen as,
(2FoF#)2 (u + OU2V
2+ 9
4) + 6o (u + 2
)V
- 4FoF/{2uv + (u2 + 32) + I (U2+ 2)
- (t + V) + (t2
+ 2) = n2p,2
where F =F .The equation (10) gives a system of fringes which are,
in general, not symmetrical about any axis. However,when there is no lateral displacement (go) or when the
(5)
where B is the coefficient of primary spherical aberra-tion and the second term is due to shift-of-focus fromthe Gaussian plane. Thus, we can get
61VR - = 4BRx (x2+ 2 + Z }
R - = 4BR (X + y2) + |
(6)
It is usually convenient to write Eq. (6) in normalizedvariables u = x/1r and v = y/r and the f-number F#= (R/2r). Then Eq. (6) becomes
Fig. 5. Characteristic fringes obtained with primary coma.
The center of the grid is displaced in the meridional direction.
curves and in the general case they are elliptical. Inthe special case when there is no lateral displacement(Q,,) the fringes are centered ellipses. When = 0(at the middle of the two foci) the curves degenerateinto circles. When is chosen so that only eitheru2or v2 term remains the fringes degenerate into straightlines. Figures 7 and 8 show two typical fringe patternsobtained with primary astigmatism.
It may be worthwhile to point out that, in the case ofRonchi test, astigmatism gives rise to straight fringes andone detects its presence by noting the change of thedirection of fringes. In the test described here withconcentric circles, the astigmatism is detected by thepresence of elliptical fringes.
Brief Comparison of Ronchi Testwith the Present Test
In the case of the Ronchi test, the equation to thefringe pattern may be written as
1:3W 'A)VR - cosa + R- sina = nd,
a)x by(13)
Fig. 6. Characteristic fringes obtained with primary coma when
the center of the grid is displaced laterally in the sagittal direction.
displacement is in direction only, the fringe patternshows symmetry about the y axis, i.e., meridional di-rection. If there is no longitudinal or lateral dis-placement, the expression in eq. (10) contains fourth-degree terms only and hence the system of fringes issymmetrical about the x and y axes. Figures 5 and 6show typical patterns obtained with a lens havingprimary coma.
Primary AstigmatismThe wavefront aberration for primary astigmatism,
including defocusing, may be represented by
W(x,y) = C(x - 2) + (X2 + y2).2R2
Fig. 7. Characteristic fringes obtained with primary astigma-tism. The center of the grid is at one of the foci and thus
straight fringes are obtained.
(11)
After going through the normalization, we may writefor the fringes the following expression:
(4CFh)2 + >2 (U2 + V2) -\o F# / --8oFII(tu - --
- 4Ci (u' -v) )(tu + 7v) I
+ (2 + 712) = n2p,2)
(12)
where CO = Cr2 . Thus the fringes are second-degree
Fig. 8. Elliptical fringes obtained with primary astigmatismwhen the center of the grid is between the foci but not at the
where ce is the angle of inclination of the grid and d isthe spacing of the grid. Thus the order of the fringes is(m-1), where m is the order of the wavefront aberration.In the case with concentric circles, the order of the fringepattern is 2(m- 1). Because of this, if the order of thewavefront aberration is 2, the order of the fringes ob-tained is also 2. Thus defocusing and astigmatismgive rise to second-degree fringes.
Conclusion
The concentric circular grid test is very simple toperform and gives characteristic fringes for different
aberrations. The fringes resemble closely those ob-tained in a Twyman-Green interferometer. Thesensitivity is comparable to the coarse Ronchi test.Presence of astigmatism is detected in a better mannerthan the Ronchi test.
This work was supported by a National Aeronauticsand Space Administration contract.
References1. V. Ronchi, Atti Fond. G. Ronchi 17, 93 (1962).2. V. Ronchi, J. Opt. Soc. Am. 3, 437 (1964).
Meetings Calendar continued from page 292
20-24 Internatl. Cong. on Crystal Growth K. J. Button,MIT, Natl. Magnet Lab., Cambridge, Mass.
21-23 Conf. on Precision Electromagnetic Measurements,Boulder J. B. Brockman, NBS, Boulder Labs.,Boulder, Colo. 80301
22-24 2nd Rochester Conf. on Coherence and QuantumOptics, University of Rochester E. Wolf, Dept. ofPhysics & Astronomy, U. of Rochester, Rochester, N.Y.14627
26-July 1 ASTM 69th Ann. Mtg. and 17th Materials TestingExhibit, Chalfonte-Haddon Hall, Atlantic City,N.J.
July5-8 Internatl. Conf. on Lens Design with Large Compu
ters, Sheraton Hotel Rochester, Inst. of OpticsUniv. of Rochester, Rochester, N.Y. 14627
12-19 Internatl. Union of Crystallography, 7th Gen.Assembly and Internatl. Cong., Moscow N. V.Belov, c/o Inst. of Crystallography, Acad. of Sciences,Leninskii Prospekt 59, Moscow B-333, U.S.S.R.
20-21 Symp. on Crystal Growth, Moscow N. V. Belov,c/o Inst. of Crystallography, Acad. of Sciences,Leninskii Prospekt 9, Moscow B-833, U.S.S.R.
August14-19 InternatI. Cong. of Ophthalmology, Munich B.
