Circular fringe carrier technique in holographicinterferometry
Pin Long, Da-Hsiung Hsu, and Ben Wang
An improved fringe carrier technique can eliminate the ambiguity in holographic fringe interpretation. Thispaper presents a new fringe carrier technique-a circular fringe carrier technique that not only eliminates theambiguity in holographic fringe interpretation but also determines the direction of displacements and obtainsthe required fringe spacings without precisely adjusting the object beam.
1. Introduction
Double-exposure holographic interferometry hasbeen successfully demonstrated in nondestructivetesting and metrology.' There are still some unsolvedproblems which hamper the wide use of holographicinterferometry in the industrial sector and more workis needed.
One common problem in applying holographic inter-ferometry to deformation studies is the difficulty inthe unambiguous determination of fringe orders anddirections.2 In that paper Kupper and VanDijk pre-sented the concept of a reference fringe (here called thefringe carrier) to study the transient phase object inholographic interferometry. But they left some prob-lems to be solved. First the authors used this methodonly to measure the relative difference between thereference fringe and the shifted fringe in a specificfringe order. Second, the circular reference fringe wasonly used in the cylindrical symmetric variation of theindex of refraction of the object. For the usual case,the linear reference fringe must be used. Third, theproblem of the difficult determination of fringe ordersand directions remains.
Recently, Plotkowski and Hung presented an im-proved fringe carrier technique that eliminates theambiguity in the proper sequencing of fringes pro-duced in holographic interferometry.3 This techniquecan also determine the direction of deformations with-out calculation.4 In this technique a set of highlyprecise angle controls is required to obtain the expect-ed fringe spacings.
The authors are with Beijing Institute of Posts & Telecommunica-tions, P.O. Box 27, Beijing, China.
We now present a new fringe carrier technique-thecircular fringe carrier technique. The fringe carrier isa set of circles with the same center and differentspacing. The circular fringe carrier technique hassome advantages over the linear fringe carrier tech-nique. The spacings of the circular fringe carrier de-pend on the movement of the object beam not being assensitive as the linear fringe carrier. Therefore theprecise requirements of the mechanism decrease.
II. Description of the Technique
Let us assume that along the optical axis there aretwo point sources at a distance d and that their inter-ference pattern is a set of circles with the same centerand different fringe spacings.5 Let us further assumethat the waveform produced by the point source is
S = a, exp[-jk(x2 + y2+ z
2)1/
2], (1)
after the point source is removed a distance d, thewaveform becomes
S2 = a2 expl-jk[x 2 + y2 + (z d)2l/2i (2)
The S and 2 were recorded on holographic platessequently through interference with the same refer-ence beam. After development, the hologram was re-produced by the reference beam:
I = s1 + 212
= a2 + a2 + 2ala2 coskt(x2 + y2 + z2)1/2
- [X2 + y2 + (z d)2]1/2
Now let us assume that a1 = a2 = a:
(3)
I = 2a(1 + coskh(x2 + y2
+ Z2) - [x2 + y2
+ (z - d)2 ]12 1), (4)
because z >> x, z >> y. So
(x2 + y2 + Z2)1/2 = z[1 + (x2 + y2)/2z], (5)
[x2 + y2 + (z - d)2 ]1/2 = (z - d)[1 + (X2 +-y2)/2(z - d)]. (6)
Fig. 1. Schematic diagram of the technique for generating a circu-lar fringe carrier.
I = 2al1 + cosk[d - d(x2 + y2)/2z(z - d)]1.
The optical path difference is
d - d(x 2 + y 2)/2z(z - d).
The bright fringe locations are
X2 +y 2 = 2z(z -d)(d-mX)/d.
(7)
(8)
(9)
From formula (9), we know that the interferogram isa group of circles with different spacings. Figure 1shows a schematic diagram of the technique for gener-ating a circular fringe carrier. A parallel beam wasfocused, by a lens and created a point source. Thispoint source is used to illuminate the object. After thefirst exposure, the lens is moved a distance d along theoptical axis (this means the point source is moved adistance d). Meanwhile, the object is loaded, and thiscreates a deformation; then the second exposure ismade.
After the object is deformed, the deformationalfunction is P(x,y). So the function of the optical pathdifference D(xy) is
D(x,y) = d - d(x2 + y 2)/2z(z - d) + P(xy)
= M(xy) + P(xy), (10)
(1!in )
Fig. 3. Interferometric fringe pattern.
V ( X. '
&, X(mrn)
0 2() 3() 4 () 'I
t { ) W ~~~~~~~~~T' ( . )
I i) 2 ) :3 ( 4 0 5ix ";)
Fig. 4. Plot of fringe order vs location for the reference fringecarrier M(xy), a plot of the perturbed fringe carrier D(x,y), and a
plot of the deformational fringe order P(x,y).
where M(xy) = d - d(x2 + y2)/[2z(z - d)] is thecircular fringe carrier, P(x,y) is the deformationalfunction, d is the moving distance of the point source,and z is the distance between the point source andobject.
From formula (10) we know that the circular fringecarrier M(x,y) is modulated by the deformationalfunction P(xy).
111. Experiments
To simplify the calculation, we put the center of thecircles on one corner of the object, as shown in Fig. 2.To set up a basic standard, we first make a referencecircular fringe carrier. 4
The interferometric fringe pattern produced by thecircular fringe carrier technique is shown in Fig. 3.From the calculation, a plot of fringe order vs locationof the reference fringe carrier M(x,y) is shown in Fig. 4.Also shown is a plot of the perturbed fringe carrier(modulated fringe carrier) D(xy). A plot of the defor-mational fringe order is obtained by subtracting thereference circular fringe carrier from the perturbedfringe pattern. This curve is shown in Fig. 4.
Fig. 5. Schematic diagram of the tested beam and the loadingsituation.
Figure 5 shows a schematic diagram of the testedbeam and the loading situation.
IV. Discussion
From the analysis we believe that the circular fringecarrier technique has some advantages over the linearfringe carrier technique as follows:
(1) In generating the linear fringe carrier, the linearfringe spacings are sensitive to the deflection of theobject beam between the two holographic exposures.A precise adjusting deflection of the object beam isneeded to obtain proper spacings of the linear fringe.
Fig. 6. Perturbed fringe pattern produced by the linear fringecarrier technique.
Fig. 7. Perturbed fringe pattern produced by the circular fringecarrier technique.
In the circular fringe carrier technique, the movementof the object beam affects the spacings of the circularfringe less sensitively than in the linear fringe carriertechnique. Therefore we do not need precise adjust-ment of the moving distance d but can still obtain therequired spacings of the circular fringe.
(2) In the linear fringe carrier, the spacings betweenthe fringes are uniform, the spacings of the perturbedfringe carrier (modulated fringe) are unequal, asshown in Fig. 6. It is difficult to count the fringe orderin the places where fringe spacings are very narrow.
In the circular fringe carrier technique, the spacingsof the circular fringe are unequal, but the spacings ofthe perturbed fringe are almost uniform if you makethe proper location of the circular fringe center. Thenthe narrow spacings of the fringe carrier are in theplaces where the deformations are large, and the widespacings of the fringe carrier are in the places where thedeformations are small.
This is shown clearly in Figs. 2-4. The deformationin the edge of the beam, where the fringe carrier centeris located, is very small; but in the loaded place of thebeam, where the fringe carrier spacings are narrow, thedeformation is large. The results of comparing thelinear fringe carrier and the circular fringe carrier tech-nique are shown in Figs. 6 and 7. Figure 6 is themodulated pattern of the linear fringe carrier: Fig. 7 isthe modulated pattern of the circular fringe carrier.The loading amount on the object is the same in bothcases. We can see that the fringe order in the per-turbed fringe pattern in Fig. 7 can be more easilycounted.
(3) In experiments we have found that the circularfringe carrier is sensitive to the direction of deforma-tion. If the deformation is positive, the center of thefringe carrier will move toward the center of the defor-mation. The center of the fringe carrier will leave thecenter of the deformation if the deformation is nega-tive.
The circular fringe carrier technique has some dis-advantages. For example, the fringe spacing near thecenter of the circular fringe is large making it difficultto determine the fringe centers.
We expect that the circular fringe carrier techniqueof holographic interferometry will find use in nonde-structive testing and metrology.
References
1. R. K. Erf, Ed., Holographic Nondestructive Testing (Academic,New York, 1974).
2. F. P. Kuper and C. A. VanDijk, "Reference Fringes in Holograph-ic Interferometry," Opt. Laser Technol. 5, 69 (1973).
3. P. D. Plotkowski and Y. Y. Hung, "Improved Fringe CarrierTechnique for Unambiguous Determination of HolographicallyRecorded Displacements," Opt. Eng. 24, 754 (1985).
4. Long Pin, Hsu Da-Hsuing, and Wang Ben, "The Further Consid-erations of the Linear Fringe Carrier Technique," presented atOptical Information Processing Conference, Hefei, China (1987);submitted for publication.
5. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1980).
e_/< ~I
S. E. Harris of Stanford University at the Seattle 1986 OSA AnnualMeeting. Photo; W. J. Tomlinoon Bilcore.
