# Testing Flat Surface Optical Components 1. Mirrors 1.1 Fizeau

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Testing Flat Surface Optical Components 1. Mirrors 1.1 Fizeau Interferometer 1.2 Twyman-Green Interferometer 1.3 Ritchey-Common Test 2. Windows 2.1 Interferometer 2.2 Autocollimator 3. Prisms 3.1 Interferometer 3.2 Goniometer 3.3 Autocollimator 4. Corner Cubes 5. Diffraction Gratings 6. Index inhomogeneity

7.1.1 Fizeau Interferometer

The basic use of a Fizeau interferometer for measuring the surface height differencesbetween two surfaces was described earlier. If the reference surface is flat, then surfaceflatness is being measured. The figure shows several Fizeau interferograms with varioustypes and magnitudes of defects. The drawings were obtained from the Van Keuren Co.

It should be noted that even if two surfaces match for various rotations and translations,the two surfaces need not be flat; the two surfaces could be spherical, where one isconvex and the second is concave. To be sure the surfaces are indeed flat, three surfacesare required, where the interference between any two gives straight equi-spaced fringes.

28 NEWTON, FIZEAU, AND HAIDINGER INTERFEROMETERS

the pinhole. To facilitate preliminary adjustment, the screen is used to projectthe two pinhole images from the two reflecting plane surfaces. This is accom-plished by removing the negative lens between the beam divider and the ground-glass screen. The pinhole image from the reference surface is at the center ofthe screen, whereas the one from the surface under test is somewhere on thescreen; by manipulation of this surface, the two spots of light on the screen canbe brought into coincidence. Then the negative lens is inserted in the path, andthe Fizeau fringes are projected on the screen. These fringes can be furtheradjusted in direction and number as required. By the use of another beam di-vider, it is possible to divert part of the beam to a camera for taking a photo-graph of the fringe pattern. The whole instrument must be mounted on a suitablevibration-isolated platform.

This instrument can be used for various other applications that are normallynot possible with conventional sources of light. We describe some such appli-cations in the sections that follow. In addition, many possibilities exist for otherapplications, depending on the particular situations involved.

Several commercial Fizeau interferometers have been available for severalyears, but probably the two most widely known are the Zygo interferometer(Forman 1979), shown in Fig. 1.23, and the Wyko interferometer, shown inFig. 1.24.

Figure 1.23. Fizeau interferometer manufactured by Zygo Corp. (Courtesy of Zygo, Corp.)

1.2. FIZEAU INTERFEROMETER 29

Figure 1.24. Fizeau interferometer manufactured by Wyko Corp. (Courtesy of Wyko, Corp.)

(1.25)

7.1.2 Twyman-Green Interferometer

The basic use of a Twyman-Green interferometer for measuring surface height variationswas previously described. The Twyman-Green and Fizeau give the same interferogramsfor testing surface flatness; the main advantages of the Twyman-Green are moreversatility and it is a non-contact test, so there is less chance of scratching the surfaceunder test, while the main disadvantage is that more high-quality optical components arerequired.

8.6. RITCHEY-COMMON TEST

According to Ritchey (1904), this test was suggested by a Mr. Common as aWay of testing large optical flats (Shu 1983). For this purpose a good sphericalmirror is used, with the flat acting also as a mirror, to obtain an apparatussimilar to the one used for the knife-edge test. See Fig. 8.32.

Any small spherical concavity or convexity of the surface under test appearsas an astigmatic aberration in the image of the illuminating point source. The

310 FOUCAULT, WIRE, AND PHASE MODULATION TESTS

I L I

Figure 8.32. Geometry of the Ritchey-Common test.

two focal planes associated with the sagittal and tangential foci can be accu-rately found (by employing the knife-edge test as indicated in Section 8. l), andwith this information the radius of curvature of the surface under test can becalculated. Of course, any localized deformation or deviation from flatness canalso be observed.

The advantage of the Ritchey-Common test over other methods is that areference flat is not needed as in the Newton or Fizeau interferometer. Theautocollimation method requires a paraboloid as large as the largest dimensionof the flat under test. On the other hand, the auxiliary mirror used in the Ritchey-Common test has a spherical shape, which is the easiest to make and test. Anadditional advantage is that the spherical mirror needs to be only as large as theminor diameter, if it is elliptical, like many diagonal mirrors used in telescopesor other instruments.

If we measure the astigmatism, we can obtain the magnitude of the concavityor convexity of the flat mirror, but it is assumed that this mirror is spherical,not toroidal, in shape. If the mirror is not circular but is elliptical, the possibilitythat it will develop a toroidal shape while being polished is very high. Thistoroidal shape introduces an additional astigmatism that complicates the anal-ysis. Interested readers can consult the work of Tatian (1967), Silvemail(1973),and Shu (1983).

The mathematical description given here assumes that the mirror under testis not toroidal and follows the approach used by Couder (1932) and Strong(1938), that is, finding the curvature from the Coddington equations for astig-matism. In what follows it is assumed that the pinhole size source and the wiretravel together, in the same plane. Hence, when evaluating the transversal rayaberrations, it is necessary to multiply the value obtained in this way by a factorof 0.5 to get the real value of the transverse my aberrations. However, sincethe aberrations of the wavefront arc duplicated in the Ritchey-Common testbecause of the double reflection on the optical flat under test, we can conve-

8.6. RITCHEY-COMMON TEST 311

niently omit the double reflection factor of 2 and the factor 0.5 when calculatingthe value of the transversal ray aberrations.

Applying the well-known Coddington equations for astigmatism to a reflect-ing surface, we find that the sagittal focus of a converging beam incident on anoptical reflecting surface is given (see Fig. 8.33) by

(8.88)

and the tangential focus by

(8.89)

where is the angle of incidence at the surface under test and r is the radius ofcurvature of the optical surface under test.

If we assume that the incident beam is free from astigmatism, that is, s = t,Eqs. (8.88) and (8.89) can be used to obtain

(8.90)

Figure 8.33. An astigmatic wavefront and its associated focal images. It is to be noted that theorientations of the images are changed when the setup in Fig. 8.32 is used. That is, the sagittalimage is a horizontal line and the tangential image is a vertical one.

312 FOUCAULT, WIRE. AND PHASE MODULATION TESTS

Now, by denoting s = L + L and f = L, and substituting these values inEq. (8.90), we have

(8.91)

Since |L / L | > 1, it follows from Eq. (8.91) that, when the sagittal focus islonger than the tangential focus ( L > 0), the surface under test has a positiveradius of curvature, that is, the surface is convex. On the other hand, when thesagittal focus is shorter than the tangential focus ( L < 0), the surface has anegative radius of curvature and is concave. The sagitta h of a surface with adiameter D and a radius of curvature r can be approximated by

Using Eq. (8.91) in (8.92) we have

(8.92)

For L >> L (|L/ L | >> 1), Eq. (8.93) reduces to

which gives the value for the sagitta of the optical surface under test for anyangle of incidence . For the particular case of = 45o, the sagitta is given by

(8.95)

Couder (1932 in Texereau 1957) has indicated that an optimum angle for testingoptical flats is = 54o45. Of course, in practice it is more convenient to set = 60o. It is also common practice to use any eyepiece, instead of the knifeedge, to determine visually the positions of the sagittal and tangential foci, bylocating the positions of the patterns shown to the right of Fig. 8.33.

8.7 CONCLUSIONS

AS mentioned in the introduction of this chapter, the examples examined dem-onstrate the versatility and usefulness of the Schlieren techniques and the related

REFERENCES 313

phase modulation tests. Of course, the choice of any particular test depends onthe circumstances.

L 2L2r

Sin 2Cos

L 2L2r

Sin 2Cos

James C Wyant

To appreciate the sensitivity of the test, it is of interest to do a sample calculation. Let aflat have a 60-inch diameter, = 60o, and L = 200 inches. If L = 0.01 inch, the sag h =37.5 microinch (approx 1.7 ), and r = 106 ft.

It is important to note that if the distance between the two line foci is measured, it is notnecessary to measure the distance between the reference sphere and the flat, nor is itnecessary to know the radius of curvature of the sphere. However, if only the differencein curvature of the tangential and sagittal fan is measured for a single focal position, it isnecessary to know both the distance between the flat and the spherical mirror, and theradius of curvature of the spherical mirror. Hence, the test is generally performed asdescribed.

Another important item is that if a null test is obtained, the mirror may not be a flat, itmay be a hyperbola. To check to make sure that the mirror is flat, the mirror should berotated 90o and the test repeated.

Although the Ritchey-Common test is great for measuring the radius of curvature of anearly flat surface, it is difficult to measure accurately local irregularities. Neglectinghighe

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