Testing of Curved Surfaces and or Lenses

8. Testing of Curved Surfaces and/or Lenses8.1 Radius of Curvature

8.1.1 Spherometer8.1.2 Autostigmatic Measurement8.1.3 Newton’s Rings8.1.4 Interferometer and Radius Slide

8.2 Surface Figure8.2.1 Test Plate8.2.2 Twyman-Green Interferometer (LUPI)8.2.3 Fizeau (Laser source)8.2.4 Spherical Wave Multiple Beam Interferometer

(SWIM)8.2.5 Shack Cube Interferometer8.2.6 Scatterplate Interferometer8.2.7 Smartt Point Diffraction Interferometer8.2.8 Sommargren Diffraction Interferometer8.2.9 Measurement of Cylindrical Surfaces8.2.10 Star Test8.2.11 Hartmann Test8.2.12 Foucault Test8.2.13 Wire Test8.2.14 Ronchi Test8.2.15 Lateral Shear Test8.2.16 Radial Shear Test

8.0 Curved Surfaces and/or Lenses

8.1 Radius of Curvature

8.1.1 Spherometer

A spherometer is an instrument that measures the sag of a surface with great precision. Acommon spherometer is the Aldis spherometer in which three small balls are arranged toform an equilateral triangle. In the center of the triangle there is a probe mounted on amicrometer. In use, the surface to be measured is placed on the balls, and the probe isbrought into contact with the surface. The sag of the surface is measured using amicrometer. If we denote the sag of the surface by h, the distance between the center ofthe balls d, and the radii of the balls by r, then the radius of curvature of the unknownsurface is given by

Where the positive sign is taken when the unknown surface is concave, and the negativesign when convex.

If the balls are arranged to form a triangle of sides d1, d2, and d3, the radius of curvature isgiven by

If a ring spherometer of radius r is used, the radius of curvature is given by

The major source of inaccuracy in using a spherometer is determining the exact point ofcontact between the probe and the surface being tested.

In some spherometers the point of contact is determined by observing the Newton’s ringinterference pattern formed between the test surface and an optical surface mounted onthe end of the probe. As the probe is brought up to the surface, the ring pattern expands,but when the point of contact is reached, no further motion occurs.

,26

2

rh

h

dR ±+=

( ).

254

2321 r

h

h

dddR ±+++=

.22

2 h

h

rR +=

8.1.2 Autostigmatic Measurement

A travelling microscope with a vertical illuminator is often used to measure the curvatureof a surface directly, as shown in Fig. 8.1.2-1. The instrument consists of a normalmicroscope configuration with a small beamsplitter placed behind the objective. Atungsten bulb is placed off to the side of the beamsplitter by a distance equal to theseparation of the eyepiece reticle from the beamsplitter. If the microscope is focused on asurface, an image of the filament of the flashlight bulb will be projected on the surface,and reimaged in the plane of the reticle by the microscope objective.

To measure the radius of curvature of a surface, the microscope is first focused on thesurface, and then on the center of curvature of the surface as shown in Fig. 8.1.2-1. Theseparation of the two focus positions is equal to the radius of curvature of the surface. Aconvex surface can be measured using an auxiliary positive lens as shown in Fig. 8.1.2-2.

Fig. 8.1.2-1 Autostigmatic test for measuring surface curvature.

Fig. 8.1.2-2 Autostigmatic test for measuring surface curvature of convex surfaces.

8.1.3 Newton’s Rings

A Fizeau interferometer can also be used to measure radius of curvature of a surface. Asurface having a long radius of curvature can be compared interferometrically with a flatsurface to yield Newton’s rings as shown in Fig. 8.1.3-1. When viewed from above wesee a series of concentric rings around a central dark spot. The radius ρm of the mth darkring from the center is given by

where R is the radius of curvature of the surface being measured.

Fig. 8.1.3-1 Radius of curvature measurement using Newton’s rings.

Shorter radius of curvatures can be measured with a Fizeau interferometer if a referencesurface of approximately the same radius of curvature as the surface being measured isavailable. For example, Fig. 8.1.3-2 shows the result of comparing a reference surfacehaving radius of curvature R with a test surface having approximately the same radius ofcurvature. Naturally, one surface is concave, while the other is convex. If d is thediameter of the piece under test, and m is the number of fringe spacings the fringes departfrom straightness, the difference in radius of curvature of the two surfaces is given by

,Rmm λρ =

.4

2

2

d

RmR

λ=∆

m = order #λρm

R2

=

R

ρ

Fig. 8.1.3-2. Radius of curvature measurement using a spherical test plate.

Helium Lamps

Beamsplitter

Part beingtested

TestPlate

Eye

m

d

2

24

d

RmR

λ=∆

8.1.4 Interferometer and Radius Slide

1998 - James C. Wyant Part 3 Page 25 of 28

Measuring Radius of Curvature

Radius

Two positions which give nullfringe for spherical mirror.

8.2 Surface Figure

Two categories1) Techniques that measure the surface or wavefront directly.2) Techniques that measure surface or wavefront slope.

8.2.1 Test Plate

We have previously described the Fizeau interferometer for measuring surface flatness.The exact same procedure can be used to measure the surface figure of a curved surface,even an aspheric surface, if a reference surface is available. The interferograms asanalyzed in the same manner as described above. It should be noted that the two surfacesmust have nearly the same radius of curvature if the resulting interferogram is to have asufficiently small number of fringes to be analyzed.

8.2.2 Twyman-Green Interferometer

The basic Twyman-Green interferometer or LUPI (Laser Unequal Path Interferometer)described above can be used to test concave or convex spherical mirrors. In the test setupthe center of curvature of the surface under test is placed at the focus of the diverger lensso the wavefront is retroreflected back onto itself. In a like manner a lens can be tested.

The good features of the LUPI are (1) any size concave optics can be tested; (2) the twopaths in the interferometer do not have to be matched as long as the conditions forcoherence are satisfied; (3) surface contours are obtained just as for a test plate; and (4)the test is a noncontact test. The disadvantages are (1) the test is sensitive to vibrationand turbulence; and (2) expensive.

Mounting the reference mirror on the surface being tested can reduce bad effects ofvibrations. In this case the vibrations are coupled and the interference pattern willbecome much more stable. If the mirror is close to the interferometer a pencil or meterstick can be placed on top of the mirror and the interferometer to couple vibrations.


Classical Fizeau Interferometer

Helium Lamps

Ground Glass With Ground Side Toward Lamps

Part to be Tested

Test Glass

Eye


Twyman-Green Interferometer(Spherical Surfaces)

Laser

Reference Mirror

Test Mirror

Interferogram

Diverger Lens

Imaging LensBeamsplitter

Incorrect Spacing Correct Spacing

8.2.3 Fizeau (Laser Source)


Fizeau Interferometer-Laser Source(Spherical Surfaces)

Laser

Beam Expander

Reference Surface

Test Mirror

Interferogram

Diverger Lens

Imaging Lens

8.2.4 Spherical Wave Multiple Beam Interferometry (SWIM)

Ref: Heintze, Polster, and Vrabel, Appl. Opt. 6 1924 (1967)Polster, Appl. Opt. 8 522 (1969)

Two surfaces being compared in a Fizeau interferometer can be given a high reflectancecoating to yield sharp, high-finesse multiple-beam interference fringes. The surfacesbeing compared can be either flat or spherical, the only restriction being that the twosurfaces under test must have essentially the same radius of curvature and the reflectancesurface must be at least as large in diameter as the surface being tested. Theserestrictions can be eliminated for the testing of concave surfaces by using the setupshown in Fig. 8.2.4-1, where a single reference surface can be used for the testing of anydiameter and radius of curvature concave surface.

Fig. 8.2.4-1 Spherical wave interferometer, multiple-beam (SWIM).

For the SWIM shown in Fig. 8.2.4-1, both the reference surface and the surface beingtested must be coated. At the common center of curvature of the two surfaces a lens isplaced that images one surface on the other. This lens is very important since, without it,any surface deformation in the mirror being tested or any displacement of the two centersof curvature will cause beam walkoff, which greatly reduces the fringe finesse as shownin Figure 8.2.4-2. A laser having a single longitudinal mode should be used with theSWIM.

For a given aberration, the shape of the interference fringes obtained using the SWIM isthe same as for the LUPI; the difference being the increased finesse of the fringes

obtained using the SWIM. This increased finesse is useful if the interferogram is to beinterpreted by eye; however, if the interferogram is to be scanned with a comparator, thegain in accuracy achieved by using multiple beam interferometry to give high-finessefringes is minimal. The greatest disadvantage of the SWIM is that the surface beingtested must be coated to have a high reflectance, which is a problem except for the finaltesting where the mirror would generally already be coated.

Fig. 8.2.4-2 Reduced finesse fringes due to beam walkoff. (ref: K. Kinosita, “Numericalevaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8,219-225 (1953), J. R. Rogers, “Fringe shifts in multiple-beam Fizeau interferometry”, J.Opt. Soc. Am., 72, 638-643 (1982).)

638 J. Opt. Soc. Am./Vol. 72, No. 5/May 1982 John R. Rogers

Fringe shifts in multiple-beam Fizeau interferometry

John R. Rogers

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Received August 7, 1981; revised manuscript received January 7, 1982

A computer program was developed to compute the profile of multiple-beam Fizeau fringes and was tested againstlaboratory observations. By using the program as a tool for studying the behavior of the fringes, a simplified modelof the fringe formation process was developed. The shifts of the fringes from their expected locations are causedprimarily by a reduction in spatial frequency that is due to the oblique projection of the fringes onto the detector.The angle at which the fringes are inclined to the input beam of the interferometer may be calculated by using aphase law developed by Brossel [Proc. Phys. Soc. 59,224 (1947)] and is independent of the reflectivity of the cavity.Once the orientations of the fringe planes and the detector are known, the fringe shifts may be calculated by usingsimple trigonometry. Although the absolute shift of a particular fringe may be an appreciable fraction of a fringespacing, neighboring fringes have nearly identical shifts, making the relative fringe shift inconsequential. The fi-nesse of the fringes may be increased by either tilting the Fizeau cavity or defocusing the detector optics. Unlikethe tilt angle required to minimize the fringe shift, the tilt angle that maximizes the finesse is a strong function ofthe cavity reflectivity.

INTRODUCTIONIt has long been known that multiple-beam Fizeau fringes donot exhibit the symmetry and degree of finesse characteristicof Fabry-Perot fringes because of the tendency of thehigher-order beams to walk in the direction of the wide endof the wedge. An expression governing the relative phases ofthe beams was developed by Brossel,1 but computing thefringe profile still required the summation of a great numberof beams. Kinosita,2 in 1952, laboriously performed this sum,obtaining a fringe profile for one particular wedge arrange-ment. More recently, Hall3 constructed a computer modelfor performing the summation.

It is the modern high-speed computer that permits theshape and location of a multiple-beam fringe to be computedand the dependence of these on the various parameters of thesystem to be studied. In this paper, a model for computingthe profile of a transmission Fizeau fringe is described, andits output is verified quantitatively against experimental re-sults. The computer model is then applied as a tool forstudying the shifts of the fringes from the traditional expec-tation that they be located at the positions where the Fizeauwedge is an integral number of half-wavelengths thick. Asimplified model of the fringe formation process is developed,and it is shown to predict accurately the locations of thefringes. The concept of fringe sharpening, introduced byLangenbeck,4 is also studied.

COMPUTATIONAL MODELThe intensity profile of the fringe is formed point by point inthe image plane by adding the complex amplitudes of thebeams as they emerge from the wedge, as shown in Fig. 1. Thejth-order beam, having experienced two transmissions and2j reflections, has a magnitude (relative to an input beam ofunit amplitude) given by

(1)

where T and R represent the intensity transmission and re-flection coefficients, respectively.

The error involved in truncating the infinite summationafter some finite number of beams may be estimated bymaking the pessimistic assumption that all the terms left outof the sum are exactly in phase with one another, therebyensuring that the estimated error exceeds the actual error.The error caused by truncating the series after n terms thenbecomes an infinite geometric series with the summationindex beginning at n + 1. This series may be expressed as thedifference between an infinite series beginning at one and afinite series beginning at one and ending at n. By using theanalytic solutions for these last two sums, the intensity errormay be shown to be

For simplicity, the assumption is made that there is no ab-sorption (a good approximation, if dielectric mirrors are used),so the factor of T2/(1 - R)2 becomes unity. The above ex-pression may be inverted to produce the number of beamsrequired for a given accuracy in the sum

(2)

In this expression, it is convenient to use base-10 logarithms,so that the effect of changing the allowed intensity error AIby an order of magnitude can readily be seen.

With the modulus of each beam given by Eq. (l), and thenumber of beams required given by Eq. (2), only the relativephases of the beams remain to be computed. This will bedone according to the method described by Brossel. As shownin Fig. 2, an input wave front drawn through the apex angleof a wedged pair of mirrors will be imaged into a family of wavefronts fanned out about the apex, the angle between any twowave-front images-being twice the angle of the wedge. Thisis true regardless of the orientation of the wedge relative to

John R. Rogers Vol. 72, No. 5/May 1982/J. Opt. Soc. Am. 639

Order

Fig. 1. Interference of three beams emerging from a Fizeauwedge.

Fig. 2. Fan of wave-front images produced by a wedge.

the input wave front. Because these wave fronts are all im-ages of one another, they have the same relative phase, andso their phases at the detector plane may be computed simplyby propagating them along their normals to the detector.Simple trigonometry shows that the path length from the jthwave front to the detector at position (z, y) is given by

(3)where z, y, and are as seen in Fig. 2. The phase of the jthbeam is computed directly from the path length and makesuse of the fact that the beam angle increases by each timeone beam number j is increased. Considerable computationalsavings are gained by computing the sine and cosine of once and then computing the values for the higher-order an-gles recursively.

R = . 9 8 4 4 F r i nges / Inch

Fig. 3. Computed and observed fringe profiles; normal incidence.

Tilt = - .7mil l iradians

Fig. 4. Computed and observed fringe profiles, wedge tilted -0.7mrad.

EXPERIMENTAL VERIFICATION

The ability of the program to predict fringe profiles waschecked by setting up a Fizeau interferometer and imaging

8.2.5 Shack Cube Interferometer

Only 1 good surface required.

Single longitudinal mode laser required.

Accessory optics required for testing lenses or convex surfaces.

Thin absorber, such as a pellicle, is required for testing coated surfaces.

Lower cost, but not as flexible as Twyman Green or regular laser based Fizeauinterferometer.

1998 - James C. Wyant

Shack Interferometer

Ref: R. V. Shack and G. W. Hopkins, Opt. Eng. 18, p. 226, March-April 1979


Shack Interferometer

8.2.6 Scatterplate Interferometer

The scatterplate interferometer illustrated in Fig. 8.2.6-1 for the testing of a sphericalmirror gives fringes of constant contour just as does the LUPI; however, its operationdoes not depend on the knowledge of the quality of auxiliary optics. Due to the commonpath feature of the interferometer, the light source can be either a laser, or morecommonly, a white light source such as a zirconium arc or tungsten bulb with a Wrattenspectral filter. The light source is focused onto a pinhole, which is then reimaged ontothe surface under test. A scatterplate a few millimeters in diameter is placed at the centerof curvature of the mirror under test. Part of the light illuminating the scatterplate willpass unscattered to focus directly on the mirror surface, while part of the light will bescattered uniformly over the mirror surface. The mirror will reimage the scatterplate backon itself--inverted, of course.

Fig. 8.2.6-1 Scatterplate interferometer for testing concave mirror.

Fig. 8.2.6-2 shows a photo of a scatterplate interferometer where the light source is ahelium neon laser.

Fig. 8.2.6-2 Scatterplate Interferometer

Source

Scatterplate(near center of curvatureof mirror being tested)

InterferogramMirror beingTested

The light transmitted through the scatterplate the second time can be divided into fourcategories dependent upon the influence of the scatterplate on the transmitted light: (1)unscattered-unscattered, (2) unscattered-scattered, (3)scattered-unscattered, and (4)scattered-scattered. The unscattered-unscattered light will produce a bright or hot spot inthe interferogram, while the scattered-scattered beam will reduce the fringe contrast. If alaser source is used the scattered-scattered beam will give a high-frequency specklepattern across the interferogram. The interference of the unscattered-scattered andscattered-unscattered beams gives fringes of constant contour that are just the same asthose produced by a LUPI. Tilt in the interference fringes is introduced by lateraltranslation of the scatterplate, while longitudinal translation controls the amount ofdefocus. Fig. 8.2.6-3 shows 3 typical interferograms obtained using a scatterplateinterferometer. The scatterplate interferometer was moved slightly between recording thethree interferograms to change the amount of tilt and defocus.

Fig. 8.2.6-3 Scatterplate interferograms of parabolic mirror.

The scatterplate interferometer is a very simple device, having a minimum of qualitycomponents. The most critical part of the instrument is the scatterplate itself, which caneasily be made. The basic procedure for making a scatterplate is to expose aphotographic plate to a speckle pattern produced by illuminating a piece of ground glasswith a laser beam. Since the scatterplate must have inversion symmetry, twosuperimposed exposures to the speckle pattern must be made, where the plate is rotated180o between the exposures. To ensure that the scatterplate illuminates the surface undertest as uniformly as possible, during the making of the scatterplate the solid anglesubtended by the illuminated piece of ground glass, as viewed from the photographicplate, should be at least as large as the solid angle of the surface under test, as viewedfrom the scatterplate during the test. After development, the photographic plate shouldbe bleached to yield-a phase scatterplate. The exposure, development, and bleachingshould be controlled so that the scatterplate scatters 10 to 20% of the incident light.

Fig. 8.2.6-4 shows a high magnification photograph of a scatterplate. This scatterplatewas made for operation at a wavelength of 10.6 microns so the detail making up thescatterplate was large enough to easily observe through a microscope. Note the inversionsymmetry of the structure making up the scatterplate.


Lateral Displacement Introduces Tilt

Á

A•

••

LateralDisplacement


Longitudinal Displacement Introduces Defocus

Á

A•

••

Center of curvature

Image of A

Fig. 8.2.6-4 High magnification photograph of a scatterplate.

The advantages of the scatterplate interferometer are that the instrument is simple andinexpensive, requires no accessory optics, and because the interferometer has acommon path, it is less sensitive to vibration and turbulence. If an incoherent source isused as the light source, the coherent noise (extraneous fringes), normally associated withusing a laser as the light source, is absent. The disadvantages are that the hot spot cancause a loss of fringes for a small portion of the interferogram and the twice-scatteredbeam will cause the interferogram to have somewhat lower contrast than can be obtainedusing a LUPI.

References

J.M. Burch, Nature (London) 171, 889 (1953).

J.M. Burch, “Interferometry in Scattered Light,” in Optical Instruments and Techniques,J.H. Dickson, Ed. (Oriel, London, 1970), p. 220.

J. M. Burch, “Scatter Plate Interferometry,” J. Opt. Soc. Am. 52, 600 (1962). (Abstractonly).

R. M. Scott, “Scatter Plate Interferometry,” Appl. Opt. 8, 8531 (1969).

John Strong, “Concepts of Classical Optics”, Appendix B (J. Dyson), (W.H. Freeman,San Francisco 1958), p. 377.

L. Rubin and J.C. Wyant, “Energy distribution in a scatterplate interferometer,” J. Opt.Soc. Am. 69, 1305 (1979).

L. Rubin and O. Kwon, “Infrared scatterplate interferometry,” Appl. Opt. 19, 3219(1980).

J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatter plateinterferometer using polarized light,” Optics Comm. 68, 235 (1988).

D. Su and L. Shyu, “Phase shifting scatter plate interferometer using a polarizationtechnique,” J. Mod. Opt. 38, 951 (1991).


Scatterplate Interferograms

647.1 nm

568.2 nm, rotatingground glass

476.2 nm All λ’s except red

Absorptivescatterplate



520.8 nm

Scatterplate InterferometerScatterplate InterferometerJ. M. Burch, Nature 171, p. 889, 1953.

A.H. Shoemaker and M.V.R.K. Murty, Appl. Opt., 5, p. 603, 1966.

MirrorUnder Test

ImagingLens

FocusingLens

Scatterplate(Near center of curvature of mirror being tested)Interferogram

Source

Mirror


Phase-Shifting Scatterplate Interferometer

Source

Scatterplate(near center of curvatureof mirror being tested)

Mirror beingTested



• Direct-scattered beampasses through a 1/4wave-plate

• Phase shift producedby applying voltage toE/O Cell

• Must correct for phaseerror produced by 1/4wave-plate

Ref: Huang, J., Honda, T., Ohyama, N., and Tsujiuchi, J., 1988, Opt. Commun., 68, 235.



• Scattered-direct beampasses through apolarizer

• 1/2 wave-plate isrotated to producephase shift

• Must correct for phaseerror produced by thepolarizer

Ref: Su, Der-Chin and Shyu, Lih-Horng, 1991, J. Modern Optics., 38, 5, 951.


Common Path InterferometersCommon Path Interferometers• Advantages

– Less Sensitive to Vibration– Less Sensitive to Air Turbulence

• Challenge– Difficult to Separate Test and Reference Beams

for Phase-Shifting

Scatterplate Interferometer

• Scatterplate has inversion symmetry• Scatterplate is located at center of curvature of test mirror• Direct-scattered and scattered-direct beams produce fringes

Source

(Near center of curvature ofmirror being tested)Interferogram Mirror Being

Tested

Four Terms:Scattered-ScatteredScattered-DirectDirect-ScatteredDirect-Direct

.

Scatterplate

High magnification imageof a scatterplate

Separating Test and Reference Beams

• Cannot Separate Spatially– Both Test and Reference Traverse the Same Path

• If Test and Reference Have Orthogonal Polarization then Phase-Shifting is Possible

Birefringent Scatterplate Makes Phase Shifting Possible

• Scatterplate is Made of Calcite

• Oil Matches Ordinary Index of Calcite

• Scattering is Polarization Dependent

Calcite

Glass Slide

Index Matching Oil

Scatterplate Production

Generated using six step process

1. Clean substrate2. Spin coat with Photoresist3. Expose

» Holographic» Photomask

4. Develop5. Etch with 37% HCl

Diluted 5000/1 in DI6. Remove Photoresist

1

6

5

4

3

2

Photomask Exposure

• Desired pattern is designed in computer • Pattern is written into chrome on a quartz substrate using an

electron beam• UV source illuminates photoresist only where chrome has been

removed

{}

Incident UV Radiation

PhotomaskSample

Phase Shifting Scatterplate Interferometer

InterferogramMirror Being

Tested

SourceRotating

Ground Glass

Polarizer (45°)Liquid CrystalRetarder (0°)

λ/4 Plate (45°)

Analyzer (45°) Scatterplate (0°)

Phase-Shifted Fringe Pattern

Frame 3Frame 3

FrameFrame 11

Frame 4Frame 4

Frame 2Frame 2

Surface Measurement Using Phase-ShiftingScatterplate Interferometer

Measurement Comparison

RMS = 0.008738 Waves

PV = 0.03750 Waves

RMS = 0.008738 Waves

PV = 0.03405 Waves

ScatterplateScatterplate WYKO 6000WYKO 6000

Summary Phase-Shifting Scatterplate

•• Birefringent scatterplate makes phaseBirefringent scatterplate makes phase--shifting possibleshifting possible

•• For this exampleFor this example–– Accuracy Accuracy ≈≈ 0.035 waves peak0.035 waves peak--toto--valleyvalley–– Repeatability Repeatability ≈≈ 0.003 waves RMS0.003 waves RMS

•• Low sensitivity to vibrationLow sensitivity to vibration

Testing of Curved Surfaces and or Lenses

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