Null Hartmann and Ronchi-Hartmann tests

Alberto Cordero-Davila, Alejandro Cornejo-Rodriguez, and Octavio Cardona-Nunez

Assuming the Ronchi and the Hartmann tests to be null tests, we were able to design special screens for eachtest that produce aligned straight fringes and a square array for the observed patterns. It also became clearthat the screen filter and observing planes for both tests can be interchanged.

1. Introduction

There are many methods of testing optical surfacesand they have been used, reviewed, and classified."2

Among these techniques are the so-called Ronchi3 andHartmann4 tests. A close relationship between bothtests will be described.

An important point to note in both tests is that theyhave two reference planes. The pattern is observed orrecorded on one plane; the ruling or screen is located onthe other. In the Ronchi test, the Ronchigram is locat-ed in a plane near the vertex of the surface under test;the screen, in this case a Ronchi ruling, is placed nearthe paraxial center of curvature. For the Hartmanntest the screen (the so-called Hartmann screen) andthe observation plane are located near the vertex andthe paraxial center of curvature, respectively. On thebasis of this analysis, we conclude in principle that ifwe interchange the screen and pattern planes in anyone of the tests we will obtain the other test. Thephysical basis for this rests in the geometrical opticspropagation of light between both planes.

Accordingly, we propose two modifications to theHartmann test. The main changes are in the conceptand design of the screens and in the type of observedpattern. The problem was originally solved as de-scribed in Sec. III; however, the presentation is ordereddifferently to aid understanding. The idea of produc-ing null tests comes from the fact that the interpreta-tion and information about the surface shape is simpli-fied, and the visual analysis turns out to be morepowerful.

II. The Ronchi-Hartmann Null Test

Sherwood5 and Malacara 6 have independentlyshown by two different methods that, when the ar-rangement of Fig. 1 is used, the transverse aberrationTA at the Ronchi ruling plane is given

( + L -2Z _ (dZ\21 + 2dZ S_ (1-z)(L-Z)]

TA(S) = dS, dS l S(l- Z) [ _ (dZ)2] + 2 dZ

(1)

where S is the distance from the optical axis to a pointon the surface of the mirror. From Fig. 1 we can alsosee that

TA = mdSinO'

(2)

A simulated pattern for the Ronchigram can be con-structed by assigning values to S and then calculatingo for different values of m. In general, one wishes toobtain the Ronchigram over a flat surface parallel andclose to the vertex of the surface since this is what isobtained when a photograph is taken of the Ronchi-gram. In many cases the error introduced by consider-ing the Ronchigram fringes over the mirror surface isminimal, but when the radius of curvature of the mir-ror is small compared to its diameter, the error be-comes significant. We can compensate for that errorby using the following formula instead of the calculat-ed S, as is shown in Fig. 1:

SP = S{ -[ Z(S) [ S

Alberto Cordero-Davila is with EFCM, UAP, Apdo. Postal 1152,Puebla, Puebla-72000, Mexico; the other authors are with INAOE,Apdo. Postal 216, Puebla, Puebla-72000, Mexico.

Received 2 December 1989.0003-6935/90/314618-04$02.00/0.

© 1990 Optical Society of America.

(3)

When the simulated Ronchigram is used as a Hart-mann screen covering the mirror, one must eliminatealternative zones situated between two consecutivesorders. Based on the rectilinear propagation of light,we should observe a pattern with straight and parallelbands with constant separation between them at theplane where the Ronchi ruling is usually located. Con-sequently, we can say that we have a null test. In Fig.

4618 APPLIED OPTICS / Vol. 29, No. 31 / 1 November 1990

PLANE

7/RONCHI RULIN(

PLANE

;LIGHT II SOURCEII II I

Il

I m m

L I0

Fig. 1. Parameters used for the simulation of Ronchigrams.

2(a) is shown the Ronchi-Hartmann screen for a pa:bolic mirror (dimater of 14.9 cm, r = 229.5 cm). Figi2(b) shows the observed pattern in a plane located228.5 cm from the vertex of the surface. As canseen, the pattern becomes a null one except for sopossible errors in the surface. We would like to poout that a normal test in some optical shops is to the simulated Ronchigram for drawing some linesthe surface, and to compare such lines with the frinproduced by observing the surface through the Ronruling. Figure 2(c) shows the observed pattern of Eother unfinished parabolic mirror.

An important consideration about the proposed tis the fact that if visual observations are to be maithe simulated Ronchigram used as the Ronchi-Hamann screen can be calculated to produce an obseable pattern of adequate size for the particular surfEunder test. On the other hand, if it is desiredobserve a pattern with a certain number of fringes aseparations between them, we must pay attentionthe adequate width of the bands in the Ronchi-Hamann screen. To do so, we must understand how tholes in a common Hartmann screen are usually calm

lated. Hence, we recommend that the Ronchi-Hart-mann screen be designed and constructed with bandsof a width equal to the optimum diameter of the holesof the standard Hartmann screen.4 The bands shouldbe centered around the curves generated for each order

(s) in the simulated Ronchigram.

Ill. The Hartmann Null Test

For the Hartmann test there are several types ofscreen showing different hole patterns: the holes canbe distributed radially, helically, and squarely. How-ever, no report known to us describes a distribution ofthe holes on the screen that produces aligned dots inthe observing plane in the form of a square array. If adesign of the screen can be achieved for producing sucha square array, this means that a null test for theHartmann set up has been obtained.

To obtain that kind of screen, let us recall from theprevious section that if the centers of the holes are

ra- along the curves of the simulated Ronchigram, theyire will appear on straight lines in the observing plane.at When one desires a square array of dots in the Hart-be manngram, the positions of the centers of the holes onme the Hartmann screen must be on the crossing points ofint two superimposed simulated Ronchigrams rotated 90°ise to each other. From this conceptual point of view, theon problem for having a null Hartmann test has beenres solved; however, the crossing points for the centers of-hi the holes must be calculated with satisfactory preci-an- sion. The method that we follow to meet this require-

ment is now described.st As we want the dots distributed on a square array

lde, (see Fig. 3), each point will have coordinatesrt-*rv-aceto

ndtort-theIcu-

x = md,

y = nd,

(4)

(5)

where m and n are integers, and d is the distancebetween two consecutives straight lines. From Eqs.(4) and (5) the distance of this point from the origin isgiven by

T = m2 + n2d, (6)

(a) (b) (c)

Fig.2. (a) Simulated Ronchigram for aparabolicmirror. (b) Observed null Ronchigram for the Ronchi-Hartmann screen designed using theRonchigram of (a). (c) Observed null Ronchigram for unfinished parabolic mirror.

1 November 1990 / Vol. 29, No. 31 / APPLIED OPTICS 4619

Y

Fig. 3. Parameters used for the expected squared dots pattern inthe null Hartmann test.

but T corresponds to the transverse aberration of Eq.(1), and it is a function of S. For each set of values of,L, c, K, m, and n, we need to solve Eq. (6) for S; andwith the aid of Eq. (3) we will obtain Sp. Finally, sincethe angle 0 is invariant to the transformations of Eqs.(1), (2), and (3), the coordinates (x,y) of the centers ofthe holes, can be obtained by

xp = Sp(m/Im 2 + n2), (7)

Yp = Sp(n/ m2 + n 2). (8)

The problem is to find the solution of Eq. (6) for S.To do this we use Newton's method to find correctionsAS to the linear approximation

T(S) + dT AS = m2 +n 2d, (9)

and iterate until convergence is obtained. The num-ber of iterations can be lowered when the initial value Sin Eq. (9) corresponds to the solution found before forthe last nearest calculated point. This procedure wasfollowed by increasing to unity one of the integers mand n used in Eqs. (4) and (5) one by one. Because ofthe four axis symmetry, it is only necessary to calculateone eigth of the points on the screen.

Following the algorithm described in the previoussection, we calculate the positions of the centers of theholes of the Hartmann screen for the parabolic mirrormentioned in Sec. II. Such a Hartmann screen wasdesigned to produce a square image of points to adistance of 224.5 cm from the vertex of the surface.Figures 4(a) and (b) show the designed Hartmannscreen and the experimental observed pattern, respec-tively. As was expected, the dots in the image planeare in a square array, equivalent to having a null Hart-mann test. The noncircular image point on the leftside of Fig. 2(b) is caused by a defect on the corre-sponding hole of the Hartmann screen.

(b)

Fig. 4. (a) Designed Hartmann screen for testing a parabolic mirrorin the null method. (b) Observed Hartmanngram in the null Hart-mann test, showing the squared array of image dots points for the

screen of Fig. 4(a).

IV. Conclusions

In this work we have established a relationship be-tween the well-known Ronchi and Hartmann tests.From that relationship we were able to propose twonull tests using the theoretical concepts of the Ronchitest and the experimental scheme of the Hartmanntest. Some advantages of the proposed tests are thatthe observed patterns have straight fringes [Fig. 2(b)]or a square array of points [Fig. 4(b)], corresponding tonull tests. The size and position of the observingplane can be considered for the design of the corre-sponding Ronchi-Hartmann screens. A null test givesopticists an opportunity to have a simple and powerfultechnique to analyze the actual shape of an opticalsurface.

We would like to mention that the elimination of theRonchi ruling, as explained in Sec. II, avoids the prob-lems normally introduced by imperfections in the rul-ing itself. Those effects are due to the position of theruling close to the center of curvature. It is possible tosay that the null Ronchi ruling actually is the Hart-mann pattern observed for a certain plane.

We would like to acknowledge valuable commentsmade by R. H. Noble, Instituto de Astronomia, UNAM(IAUNAM), and the help of S. Zarate-Vazquez, A. M.Zarate and T. Flores in the computer work, and S.

4620 APPLIED OPTICS / Vol. 29, No. 31 / 1 November 1990

(a)

Tecuatl for the testing in the optical shop. One of us(ACR) has at the present a partial academic commis-sion with IAUNAM. A. Cordero-Davila is pursuinghis doctoral degree at INAOE.

Preliminary results of this paper were presented atthe Thirty-Second Annual Meeting of the SociedadMexicana de Fisica (SMF), held in Leon, Guanajuato,23-27 October 1989.

We are grateful to the referees for valuable com-ments on our original manuscript.

References1. D. Malacara, A. Cornejo, and M. V. R. K. Murty, "Bibliography of

Various Optical Testing Methods," Appl. Opt. 14, 1065-1080(1975).

2. A. Cornejo-Rodriguez, H. J. Caulfield, and W. Friday, "Testing ofOptical Surfaces: A bibliography," Appl. Opt. 20, 4148-4148(1981).

3. A. Cornejo-Rodriguez, "Ronchi Test," Optical Shop Testing, D.Malacara, Ed. (Wiley, New York, 1978), Chap. 9.

4. I. Ghozeil, "Hartmann and Other Screen Tests," Optical ShopTesting, D. Malacara, Ed. (Wiley, New York, 1978), Chap. 10.

5. A. A. Sherwood, "Ronchi Test Charts for Parabolic Mirrors," J.Proc. R. Soc. New South Wales 43, 19 (1960).

6. D. Malacara, "Geometrical Ronchi Test of Aspherical Mirrors,"Appl. Opt. 4, 1371-1373 (1965).

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