surface is obtained. After computation of this normal vec-tor, the procedure is the same as in aspherical surfaces.

To find the normal to the spherocylindrical surface we firstneed to find the gradient of the following function:

CsX2 + CTYl

1 + V1 - (CSX2 + CTY2)2 /(X 2 + y2)-, (3)

thus finding

-[2C5 X(1 + 1 - 1/S2 ) + 2(CsX2/S4)(2CS - F)F]

(1 + 1 - F/S2 )2

-[2CTY(1 + 1 - P/S 2 ) + 2(CTY21S4)(2CT - F)F]

(1 + 1-F 2 /S2 )2

where F = CSX2 + CTY2,S2 = X 2 + Y2, and

U,V,W are defined as in Ref. 2.

Fig. 2. Spherocylindrical surface.

Position coordinateson the tangent sphere

X - Y plans (Z * O) tangent tothe J" aspheric surface at coordinateorigin.

Fig. 3. Tracing of rays through an aspherical surface.

These expressions replace Eqs. (22)-(24) in Ref. 2, andthen the standard procedure is followed.

Summarizing: it has been shown that toroidal and sphero-cylindrical surfaces are mathematically different, but forpractical optical purposes they may be considered almostequivalent. The procedure of tracing rays through sphero-cylindrical surfaces has been described.

This work was done with a grant by Conacyt.

References1. D. Malacara and J. Z. Malacara, "Diamond Tool Generation of

Toroidal Surfaces," Appl. Opt. 10, 975 (1981).2. R. E. Hopkins and R. Hanau, Military Standardization Hand-

book, MIL-HDBK-141 (U.S. Defense Supply Agency, 1962),

Chap. 5.

Ronchi test with daylight illumination: 2

Krzysztof PatorskiWarsaw Technical University, Institute of Design of Pre-cise & Optical Instruments, 02-525 Warsaw, Poland.Received 9 June 1986.0003-6935/86/183009-02$02.00/0.© 1986 Optical Society of America.

In a recent letter7 a simplification of the Ronchi test usedfor testing optical mirrors has been reported. The lightsource intended for interferometer use has been completelyabandoned and replaced by the sun or room light present inthe laboratory or optical workshop. One-half of the binary-type amplitude grating placed close to the center of curva-ture of the mirror under test has been made to serve as theextended periodic source. This has been achieved by plac-ing a strip of alluminum foil behind it. Conducting, as usual,the observation of the mirror surface through the second halfof the grating, the Ronchigram fringes giving the informationabout the mirror quality can be monitored.

The aim of this Letter is to propose another modificationof the grating test configuration using day or room lighting.It is even simpler and more convenient for fast test imple-mentation and results in brighter interference patterns.This fact is important when testing uncoated mirrors andlenses.

The common principle of the extended source Ronchi testis to employ a periodic spatially incoherent light source. Asmentioned above this is done by diffusely illuminating theupper or lower part of the grating. Such off-axis illumina-tion and detection mode is widely used for testing relativelyslow mirrors.

The faster the mirror is the smaller the lateral dimensionsof the grating are. This requirement is dictated by comatic-type aberrations arising due to the slightly off-axis test con-figuration. Consequently, the number of slits in the sourceand detecting parts of the ruling might become too small toconduct the test. One solution is to introduce the beamsplitter directing the light from the separate diffusely illumi-nated grating. (The normal of this source grating is perpen-dicular to the axis of the mirror under test.) The light

15 September 1986 / Vol. 25, No. 18 / APPLIED OPTICS 3009

-f U-a = =ax W

-A V-o = =ay W

(4)

(5)

reflected from the mirror and transmitted by the beam split-ter is analyzed by the second (detecting) grating identical tothe source grating. The light source (a small bulb withdiffuser), the beam splitter, and both gratings form usually acompact interferometric head placed near the center of cur-vature of the mirror under test.

A much simpler solution to the on-axis Ronchi test andusing a room or daylight illumination is schematically shownin Fig. 1. The diffuse illumination of the binary amplituderuling of proper lateral dimensions is simply provided by apiece of white paper held by the observer in front of his head.The observation is conducted through an opening in thediffuser generated in this way. It is necessary to emphasizethat in such a configuration the whole grating surface actssimultaneously as the source and detecting ruling. Conse-quently, the interferogram brightness and observation con-ditions are improved, not mentioning the unusual simplicityof the test configuration. The only requirement is to centerthe grating on the optical axis and to provide the possibilityof its axial displacement with respect to the center of curva-ture of the mirror under test. Depending on the lightingconditions and reflectivity of the surface being examined thewhite paper can be replaced by other diffusers or even com-pletely abandoned. In the latter case, for example, the re-flectivity of the human face or white dress (shirt, apron, etc.)might be satisfactory.

During the experiments it has been found that the bright-

or

6 D

Fig. 1. Schematic representation of the Ronchi test with room ordaylight illumination. The light L illuminates the test setup com-posed of the optics under test OT, diffraction grating G, and diffuserD. The tested surface is observed through the grating G and the

opening in the diffuser.

est interference patterns are obtained with side-type illumi-nation. It is provided by ceiling lamps or daylight passingthrough the window. In the latter case the window should beabout parallel to the optical axis of the test configuration.

Two special cases of window-type illumination are worth ashort comment. The setup with the mirror surface facingthe window is not recommended because the observer seesthe window reflection itself given by the mirror. In such acase the traditional bulb lighting of half of the grating isrequired.2 3 On the other hand, when the grating and ob-server face the window the white paper diffuser gives smallimprovement to the interferogram brightness. It has beenfound experimentally that in such a case the best solutionsare to hold in hand a small plane mirror near the head behindthe grating and direct the light into the optical system or touse the grating composed of reflective stripes replacing thedark lines in the binary-type amplitude grating.

Although the descriptions given above considered the test-ing of mirror surfaces the concept can be extended in astraightforward way to lens systems. For example, in thecase of testing at unity magnification, two identical ampli-tude diffraction gratings are placed at the double focal lengthon the opposite sides of the lens under test. The lines of bothrulings must be set mutually parallel.4 The white paper orscreen is placed behind one of the gratings, and the observa-tion is conducted through the other ruling. The shearing-type interferogram fringes are visible immediately. Again,the astonishing test simplicity is to be emphasized.

All the observations stated above follow from the experi-mental work. However, because the Ronchigrams obtainedwhen testing optical systems are widely published, we willnot repeat them here.

The author wishes the acknowledge the participance in theexperimental work and the suggestions of J. Kozlowski.

References

1. K. Patorski and A. Cornejo-Rodriguez, "Ronchi Test with Day-light Illumination," Appl. Opt. 25, 2031 (1986).

2. J. A. Anderson and R. W. Porter, "Ronchi's Method of OpticalTesting," Astrophys. J. 70, 175 (1929).

3. M. V. R. K. Murty and A. Cornejo, "Sharpening the Fringes in theRonchi Test," Appl. Opt. 12, 2230 (1973).

4. K. Patorski, "Heuristic Explanation of Grating Shearing Inter-ferometry using Incoherent Illumination," Opt. Acta 31, 33(1984) and references therein.

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3010 APPLIED OPTICS / Vol. 25, No. 18 / 15 September 1986