Talbot interferometry in noncollimated illuminationfor curvature and focal length measurements
Kuppuswamy Venkatesan Sriram, Mahendra Prasad Kothiyal, and Rajpal Singh Sirohi
We describe a simple method for measuring the radius of curvature by using Talbot interferometry in anoncollimated light beam. This scheme can also be used to determine the focal length of the collimatinglens employed in the setup. Results of the measurements are presented. A discussion of achievableaccuracies and the proper choice of parameters is included.
Talbot interferometry has been used recently formeasuring focal lengths." The measurement of focallengths and curvatures by using moir6 deflectometryhas also been reported.5 We report here on a methodof measuring focal lengths and curvatures that isbased on Talbot interferometry in a noncollimatedlight beam by employing a dual-field grating approachused by us in connection with collimation testing.6 7
Measurement Principle
Figure 1 explains the principle of the method that isused for measuring a concave surface. A collimatedlight beam is obtained by placing a point source at thefocal point F of a collimating lens (CL). The colli-mated beam that is reflected from the concave surface(TS) converges. If the point source is displaced from Fto a position S' to produce a beam that is incidentnormally on the measurement surface; that is, if thebeam appears to originate from the center of curva-ture of the surface being tested, the reflected wavefront retraces its path. A grating G1 placed in thebeam forms a self-image on the grating itself if theself-imaging conditions is satisfied. In this situationthe spatial frequency of the self-image is identical tothe grating frequency. The normality of the wavefront on the surface is tested by the moir6 patternbetween the grating and its self-image by using a newtype of grating construction, which has been pro-
posed in earlier papers.6'7 The displacement of thesource from the collimation position is related to thefocal length of the collimating lens as well as thecurvature of the test surface, and hence it can be usedfor the measurement of both. The dual-field gratingwas first introduced by Swift.8 In our gratings thelines are orthogonal to those proposed in Ref. 8. Thishas some practical advantages, and a comparativestudy is being prepared.
If f is the focal length of the lens, D the distancebetween the lens and the test surface, and x thedisplacement of the source, from the ray diagramshown in Fig. 1 the radius of curvature R, of theconcave surface is expressed as
R,, = x'-f + D,
where x'as
(1) .
and x are related through Newton's formula
xx' = If 1. (2)
Hence
f2R - f + D.x (3)
In a similar way from Fig. 2 we can determine theradius of curvature RX of a convex surface as thefollowing:
The authors are with the Department of Physics, Applied OpticsLaboratory, Indian Institute of Technology, Madras 600 036,India.
As is explained below, a cube beam splitter (CBS) isintroduced into the ray path for testing. The passageof light through the beam splitter introduces an
R 1Fig. 1. Optical configuration for the testing of a concave surface.
effective longitudinal shift, which is given by
h = (n - 1)t/n, (5)
where t is the thickness of the beam splitter and n isits material refractive index.
The corrected formulas for the radii of curvatureare therefore given as
f2Rv,= -f+D-h , (6)
Ro=(f+f-D+h) * (7)
The focal length of the collimating lens can bedetermined from Eqs. (6) and (7) by repeating theexperiment twice, i.e, for two values of D (D1 and D2)giving two values of x (x1 and x2). Manipulating theseequations yields
D = 'x(x2 11/2
CL
from Eq. (6) and
f D(XI-2 1/2
from Eq. (7), whereD' = D2- D1.By using this set of D and x values we can establish
the following relations:
R +f =D' (x 2 + D1 - h, (10)
Rx - f= D 2 -D +h.(11
Experimental Arrangement
Figures 1 and 2 show the optical arrangements thatare used to test the radii of curvature of concave andconvex surfaces by using this technique. As explainedbefore, we need to determine whether the light isincident normally on the surface. Thus we use agrating G of the type shown in Fig. 3 that has been
C BS
S . F...........
Is a
Fig. 2. Optical configuration for the testing of a convex surface.
Fig. 3. Schematic of the modified gratings G1 and G2.
(a) (b)
Fig. 4. Moir6 fringe patterns obtained (a) at the self-image planeand (b) away from the self-image plane. The orientation of thefringes reverses as we go from one side of the self-image plane tothe other.
proposed for collimation testing.6 7 A CBS deviates thelight beam reflected from the test surface. A secondgrating G2, which is a replica of G1, is placed in thebeam reflected from the beam splitter so that G, andG, are equidistant from the beam splitting surface.Ideally the gratings and the beam splitter should bemounted on a common platform. If the beam incidenton the test surface is normal, a self-image of G, withthe same spatial frequency is formed on the plane ofG, as well as on G2 provided that the self-imagingcondition is satisfied. To satisfy the self-imagingcondition the beam splitter-gratings assembly isshifted along the optic axis until moir6 fringes with agood contrast are obtained on G2. If the G, self-imageand the grating G2 are oriented as in Fig. 3, the moir6fringes formed in one half of the grating G2 areparallel to those formed in the other half as in Fig.4(a). When the normal condition is not satisfied, thetwo fringe patterns make an angle with each other asin Fig. 4(b). The normal condition can also be testedby other techniques by using the Talbot effect and themoir6 technique.9
Initially the lens is set for its collimation positionby using a plane mirror in place of the spherical
surface. In practice it is found to be more convenientto shift the lens for various settings. The sphericalsurface is then placed in the collimated beam, and thelens is shifted so that a spherical wave front falls onthe surface. The lens is adjusted until a fringe patternas shown in Fig. 4(a) is obtained, in which case therays on the test surface are normal. The lens ismounted on a translation stage whose traverse ismeasured with the help of a 0.01-mm resolutiongauge.
Experimental ResultsWe have carried out experiments to measure the radiiof curvature of a pair of spherical surfaces (a convexand a concave) with the proposed scheme. The valuesof R calculated by using Eqs. (6) and (7) for various Dvalues are given in Table I. The value of f used in thecalculation was 420.79 mm (the average of the valuesin Table II) and h = 17 mm. The radii of curvature ofthese surfaces were also measured with a precisionspherometer that measures the sagittal height to anaccuracy of 1 pm. The sagittal height was measuredby using a ring that was 80 mm in diameter. Thefollowing values were obtained: R, = (4967 + 30) mmand R. = (4992 ± 30) mm. For smaller ring diametersthe errors in R, and R, will be accordingly higher. Thedata in Table I show that the present method canprovide much better accuracy. This method may beparticularly useful for measuring a long radius ofcurvature in cases where other methods may bedifficult to use.
The same set of data was also utilized to determinethe focal length of the collimating lens; the results areshown in Table II. The lens's focal length, which wasmeasured with a commercial focometer device, had avalue of 421 mm with ± 0.3% accuracy.
The calculated values from Eqs. (6), (7), (8), and (9)are influenced by errors in the various parameters,namely, 8f in f, 8x in x, and oD in D. An analysis iscarried out for the error in the calculated values of theradius of curvature AR and the focal length Af asexplained in Ref. 10. Error curves (AR versus R andAf versus f ) have been obtained for different parame-ters. Figures 5 and 6 represent typical AR versus Rand Af versus f curves, respectively.
The main contribution to AR comes from the errorBf. Therefore the values of f should be known accu-rately. In Fig. 5 Sf is ±0.3%. (The accuracy of acommercial focometer has been used in the calcula-tion.) The error is 0.5% for R, = 5000 mm and D =
Table I. Calculated Values of the Radii of Curvature
Concave Surface Convex Surface
D (mm) x (mm) (av) R, (mm) D (mm) x (mm) (av) R. (mm)
1000 mm. However, with better accuracy in themeasurement off the error can be reduced.
Figure 6 shows that a smaller error in f is obtainedwhen the spherical surface with a shorter radius ofcurvature is used in the experiment. For a givenspherical surface the error is small if Ix2 - x is large,which is obtained for large D'. In Fig. 6, D' is 800mm. In Table II R 5000 mm and D' varies in the100-350-mm range; the values are not optimum. Thelarge amount of scatter in the calculated values of fmay be a result of this. However, better accuracy canbe obtained by the proper choice of parameters.
We mentioned above that G. and G2 are placed atequal distances from the CBS. If these distances arenot equal, the moir6 fringes may not be parallel evenif normal conditions are satisfied. To estimate hownearly equal the two distances should be, we deter-mine how much G2 can shift from its ideal location sothat the parallelism in the moir6 fringes just begins tobe disturbed. With this process the contrast in themoir6 fringes also deteriorates as we move away fromthe ideal self-image plane. If the grating G2 is shiftedby a distance AZ, the change in the frequency (Lp ofthe self-image is given by
AP. AZ
p - Z' (12)
32
28
1 2420
>16
12
B
4
0
RV (mm) -
Fig.5. ARU versusR. forf= 400 mm,8f=and 8D = 1 mm.
1.2 mm,8x = 0.05 mm,
OR- 500mm* R= 1500 mmV R= 2500mmA R= 5000mm
I G100 200 300 400 500 600 700 800 900 1000
f (mm) -
Fig. 6. Af versus f by using a concave surface for D' = 800 mm,8D = 1 mm, x = 0.05 mm.
where Z is the distance between the grating and themirror and [., the grating frequency. This produces arotation a in the moir6 fringes that is expressed as6
a = tan P' (13)
where 20 is the angle enclosed by the grating lines inthe two halves of the dual gratings. Equation (13)may be rewritten as
tan 0 (14)
For our grating with p. = 5 lines/mm and 0 = 87.7°,we have determined" that the least detectable valueof a. is 0.70, which gives A[l = 0.0049 line/mm.Substituting this into Eq. (12), we obtain AZ = 3.9mm for R = 5000 mm (which is a test radius in thiscase) and Z = 1000 mm. This value of AZ varies inproportion to R. By using a symmetrical CBS and adistance gauge/plate, G and G can be adjusted sothat they are equal within a fraction of a millimeter.If we are not certain of the symmetry of the CBS, wecan use an arrangement as shown in Fig. 7. The CBSand the gratings are mounted on a common plate. By
G., CBS
77vV p
Fig. 7. Experimental arrangement for setting the gratings G1 andG2at equal separations from the CBS.
using a mirror with a shorter radius of curvature wecan adjust G1, G2, and the mirror so that the G, imageis superimposed on G2 with unit magnification. Thisresults in parallel moir6 fringes. This ensures that G,and G2 are equally separated from the beam splittingsurface within a close tolerance. Therefore the effectof this error source on the final result is kept negligi-bly small.
ConclusionWe have presented a simple method based on self-imaging for measuring the radius of curvature of aspherical surface and the focal length of a lens. Bychoosing the experimental parameters properly, goodmeasurement accuracy can be achieved. Our tech-nique uses instrumentation that is obtained easily. Aquasi-monochromatic light source is required.
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