Sandwich hologram interferometry. 5: Measurement ofin-plane displacement and compensation for rigidbody motion
Nils Abramson and Hans Bjelkhagen
Sandwich holography has been used for measurements of in-plane displacement of an object. The sign of
the displacement is found by tilting the sandwich hologram during reconstruction. Fringes caused by in-plane rigid body motion can be compensated for, and local displacements evaluated. It is shown that an in-plane motion of more than 1 mm of the object placed at a distance of about 1 m from the plates can be com-
pensated for and a local tilt of 1.5 X 10-3 degrees evaluated. A comparision between conventional evalua-
tion and evaluation of sandwich holograms by measuring the tilt angle of the plates to calculate an object'sdisplacement is shown with a series of experiments.
Introduction
Sandwich holographyl-21 can be used for solvinga number of measurement problems. In this paper thepossibility of using the method for in-plane measure-ment is considered. Many experiments have beenperformed to ascertain the reliability of the sandwichmethod for practical applications, and in-plane dis-placements as well as tilt were studied. For conven-tional evaluation of in-plane motions, formulas in apaper presented by Liu and Kurtz22 have been used.
Theory
According to Ref. 4 the following formula is given forthe tilt of an object between exposures when the sand-wich method is used:
1 ds = - arctan2 rI n \2 lliJ
L - 1]J
where s = tilt angle of the object between exposuresd = distance between the emulsions in the
sandwich hologram,L = distance between the plates (observing
point) and the object,n = refractive index of the glass plates, and
= tilt angle of the sandwich hologram forcompensating the tilt angle of the object.
For translation the definitions presented in Fig. 1 areused. The following three equations are combined forobtaining the formula for translation:
Ax = (t x S)/(S + L),
tan kB tid,
sin = n x sinfB,
(I)
The authors are with Royal Institute of Technology, Division ofProduction Engineering, S-100 44 Stockholm 70, Sweden.
Fig. 1. Conditions concerning translation of an object:S = distance between point of illumination and object,L = distance between object and holographic plate,
Ax = translation in the x direction between exposures,t = distance between recorded points in the emulsion,d = distance between emulsions in a sandwich hologram,n = refractive index of glass plates,B = refraction angle, and0 = tilt angle of the sandwich hologram for compensation
It can be assumed that one speckle beam from the objectis hitting the back plate in the first exposure at a certainpoint. The speckle beam can be regarded as reflectedfrom a small mirror fixed to the object. In the secondexposure the mirror has been moved the distance Axand the speckle beam hits the front plate at a point thatis displaced in relation to the first recording on the backplate. The distance between the two recordings is t.
Finally we assume that fringe-free reconstructionoccurs when the two recorded points (speckles) from thetwo recordings are aligned along the line toward thepoint of the reconstructed object being studied. Thisis simply done by hand tilting the sandwich hologramduring reconstruction.
For accurate evaluation of a sandwich hologram onehas to take into consideration the change of the positionof the reconstructed image when the plates are tilted' 3
(see Fig. 2). The following formulas are used for thiscompensation:
,Y2sin- cost. 2 2sin- = +
2 cos(lf + )(III)
0corr = - (Q2 - TO, (IV)
where yi = angle between image point P and refer-ence point R prior to rotation of thesandwich hologram,
Y2 = angle between the new image point P 2and reference point R after rotation of theplates,
I = angle between the plates and a planeperpendicular to the bisector to Yi,
= angle that the plate has been rotatedduring reconstruction, and
Ocorr = corrected value of 0 due to the change ofposition of object point P during recon-struction.
Equations (III) and (IV) can be combined with theprevious formulas for tilt [Eq. (I)] and translation [Eq.(II)]. Equations (III) and (IV) are combined to give
Fig. 2. The change in the position of the image when the sandwichhologram is rotated from position H. to H2. Then object point P1becomes P2 . Reference point R plus Pi create a recorded separationof interference fringes of di in the hologram positioned at angle inrelation to the perpendicular distance di between fringes. After ro-tating the angle k of the hologram, the distance between the inter-ference fringes seems to be d2 in relation to the direction of the ref-erence beam. This distance gives the angle 72 to the new position P2
of object point P1.
'Y1 sin- cosil
2 sinkC =sn-cosk[ os('J + /,) ] VsinkPcorr = sin - o 2 arcsin - _Yi (V)
Then the corrected formulas for tilt and translation willbe:
I i ]I2 \1/2
p = d 2L[ l -1-I
I lI COS(' + 0) ]
Ax = (d x S) (S+ L) co |-1
lin - cos 2arcsin sin(,yl/2) cost Yiljsin~ k - cosq5 [2 co s('I + 0)
These complicated equations are, however, only usedif high accuracy is needed. If Eqs. (I) and (II) are usedinstead of (VI) and (VII), the error will be about onefringe when a hundred fringes are evaluated. The errorwill be less when the number of fringes is fewer.
One way to explain the manipulation of the fringesin the sandwich hologram is the use of "Young's fringesof observation" 9 (see Figs. 3 and 4).
The two recordings in a double-exposed hologram canbe regarded point by point as created by a speckle beamreflected at the object from a small mirror fixed to theobject that has been displaced distance Ax due to thein-plane motion Ax of the whole object. (This is ap-proximately true when the distance from plates and il-lumination to the object is large in comparison to thesize of the object.)
The two recordings in the emulsions of the plates arethen displaced distance t. If these points were twocoherent points of illumination, they would create aninterference pattern (Young's fringes) projected ontothe object.
By assuming two points of coherent observation,exactly the same interference pattern would be seen onthe object (observation through many double slits dis-placed exactly the same distance t). The distance be-tween Young's fringes is
X A XL
.a t t2 sin-
22 X-
2Lwhere L is the distance from the holographic plates andthe object and t is the distance between the displacedrecorded points in the emulsion. According to theformula
X/[(/S) + (1/L)] Ax
for the fringe separation on the object given by Liu andKurtz 2 2 for translation of an object, it can be shown thatthis expression is exactly the same as Young's fringesof observation. If in the Young's fringe-separationexpression (XL)/t, t is replaced with t = [(S + L) Ax]/Sobtained from Fig. and Eq. (1), the fringe separationwill be
X XLS X
(S + L) Ax [(11S) + (1/L)] Ax
which is the same as the Liu and Kurtz expression.22
Creation of the interference fringes seen on the imageof the object depends on distance t, no matter how it isobtained, i.e., either by translation of the object (a smallmirror reflecting the speckle beam displaced the dis-tance Ax) or by rotation (a small mirror rotated an angles° which rotates the speckle beam 2%). Therefore itappears that the hologram is more sensitive to tilt thanto translation.
The advantage of using a sandwich hologram insteadof a conventional double-exposed single-plate hologramis that the distance t can be affected at reconstructionby tilting the sandwich hologram. Then t can be set tozero, and accordingly no fringes are visible on the re-constructed image of the object, i.e., compensation for
the rotation or translation of the object between expo-sures has been performed with a fringe-free recon-struction of the displaced object. Tilt and translationcan be distinguished because the information distri-bution over each sandwich plate is different. When thefringes caused by tilt are set to zero, the object will be-come alternately dark and bright as the observer moveshis head behind the plate. The fringes caused bytranslations will not produce this effect. When they areset to zero, the object surface will look the same whenstudied through different parts of the sandwichplates.
Let us compare sandwich holography with conven-tional holography for the separation of out-of-planemotion (e.g., tilt) from in-plane motion. In conven-tional double-exposure holography one can find thein-plane component by moving the eye behind the ho-logram plate and there observe the motion of thefringes.23 From this observation it is possible to eval-uate the in-plane motion and to calculate the numberof fringes it would have caused. By substracting thesefringes from the original fringe system it is finally pos-sible to find the fringes caused by tilt alone.
Using sandwich holography the evaluation is lesscomplicated. The hologram is tilted until the fringesappear fixed to the object as the eye is moved behind theplate. During this tilt the fringes caused by in-planemotion are automatically eliminated so that the re-maining stationary fringes represent the tilt alone.
The resolution of the method described is, however,limited by the fact that the hologram tilt resulting instationary fringes is not so well defined as the corre-sponding tilt resulting in zero fringes.
Fig. 5. The test object, a steel plate that can be translated in itsplane. In the center is a door that can be tilted around a vertical axlefixed to the plate. This means that the door can be both translated
The experiments were made by using a test objectwhich was a plate (150 mm X 150 mm X 10 mm) fixedto an x -y table. In the middle of the plate there was asquare hole to one edge of which a door was hinged.This construction made it possible to combine a tilt ofthe door with an in-plane motion of the whole test object(see Fig. 5).
Experiments with pure translation as well as combi-nations of tilt and translation were performed. Theholographic setup used is shown in Figs. 6 and 7. Astable reference surface of similar size to the test objectwas placed at the side of the test object. This referenceobject was unchanged between exposures and used forindicating a fringe-free reconstruction of this objectwhen the plates were positioned as they were re-corded.
The first part of the experiment series was tensandwich holograms with different displacements Axbetween exposures. The next in the series wastwenty-two sandwich holograms with displacements Ax50 /um, combined with a tilt of the door of 1.5 X 10-3 deg.One experiment with displacement Ax = 0.5 mm andone with Ax = 1.0 mm were also performed where thedistance between the object to plate was increased bya glass plate 8 mm thick that was used both at the re-cording stage and for the reconstruction.
In all other experiments the separation between thetwo emulsions was equal to the glass plate thickness.This thickness was individually measured for each ex-periment, because this value is important in the calcu-lation of the object's displacement. In some casesconventional double-exposed holograms were made.The displacements of the object were also measuredwith the aid of mechanical and electronic measuringinstruments.
Results and Evaluation
The results from the experiments are double-exposedholograms, sandwich holograms, and readings from theinstruments. The following treatment of the materialhas been performed:
(A) Conventional evaluation of the hologram by useof the fringe information.
(B) Evaluation of the sandwich holograms by mea-suring the tilt angle of the holograms needed for com-pensation of the tilt or translation of the object. Thisangle was then used for the calculation.
(C) Comparision between the two methods as well aschecking with the information from the measuring in-struments.
A. Conventional Evaluation of the Holograms
Both real and virtual images were used and thefringes counted. Knowing the geometrical setup andusing the formulas given in Liu and Kurtz,2 2 we havecalculated the translations. The method for in-planedisplacements, where the eye is moved a certain dis-tance along the plate while observing the fringes thatmove through a special point of the object, was alsoused.
B. Evaluation of the Sandwich Holograms
The sandwich holograms were put in a special eval-uating instrument in which their tilt angle could bemeasured. Both virtual (identical setup as at the re-cording stage) and real images were used. The averageof the two measured angles was used for this calculation.The results are shown in three tables and three di-agrams (Figs. 8-10). In Table I (Fig. 8) the translationof the object with different translations is shown. InTable II (Fig. 9) the translations around 50 Ium areshown. In Table III (Fig. 10) the tilt angle of the door
= measurement of the angle 0 when using thevirtual image,
= measurement of the angle 0 when using the realimage,
= average of 0 Virtual and 0 Real,= corrected value of 0 due to the change of posi-
tion of the reconstructed image when thesandwich hologram is tilted,
= distance between emulsions in the sandwichhologram,
= calculated value of translation Ax using sand-wich formula f (,/),
= conventional evaluation by using the informa-tion from the fringe pattern on the objectwithout tilting the hologram,
= evaluation of translation by using the methodof moving the eye along the plate and observingthe fringes passing through a special objectpoint; this method has an error of - +10 Mm inthis case,pure translation with no influence of unwantedtilt measured by instruments.
pm
130
120-
110I
100-
90.
is shown where the whole object also has been trans-lated.
In these diagrams (Figs. 8-10) the uncertainties havebeen indicated. For conventional evaluation the fringeerror equals 0.25 fringe. Influence of the error inmeasuring geometrical distances and angles has alsobeen taken into consideration. For sandwich hologramsthe error is mainly caused where it is necessary to de-termine when the part of the object being studied isfringe-free. For the experiment this error was mea-sured to +0.40 of the tilt angle 0.
Photographs of some of the holograms are presentedin Figs. 11 to -22.
Discussion
Sandwich holography works well for practical in-vestigations and can be used both for qualitative andquantitative evaluations of tilt and translation. Com-pared with conventional evaluations (underlined in thetables) the agreement is good for sandwich holography.By using the formulas [Ax = f(0) for translation and ,o= f(0) for rotation] evaluation of object translation andtilt can be measured with an error often less than 5%depending on the holographic setup. When there is a
80-
70-
60-
50-
40
30
20
10.
DIAGRAM 1
TRAN SLAT ION
-I
I._, _
I.
PURETRANSLATIONI1/
I
FRINGEEVALUATION
SANDWICHEVALUATION
1 2 3 4 5 6 7 8 9 10 HOLOGRAM NO
Fig. 8. Comparison between conventional evaluation and sandwichevaluation for translations of object with different translations.
measurement of the angle when using the vir-tual image,measurement of the angle 0 when using the realimage,
= average of 0 Virtual and / Real,= corrected value of X due to the change of position
of the reconstructed image when the sandwichhologram is tilted,
= distance between emulsions in the sandwich ho-logram,
= calculated value of translation Ax using sandwichformula f(0),
= conventional evaluation by using the informationfrom the fringe pattern on the objectwithout tilting the hologram,
= evaluation of translation by using the method ofmoving the eye along the plate and observingthe fringes passing through a special objectpoint; this method has an error of
+10 ,jm in this case,= pure translation with no influence of unwanted
tilt measured by instruments.
DIAGRAM 2TRANSLATION
pim
70-69
68
67
66
6564
63 -626160.59585756
5554
5352
51
50-49
48
47
46
45
4443
424140.393837 -36
35 -3433 -32
31 -30-
Fig. 9. Comparison between conventional evaluation and sandwichevaluation for translations of object around 50 jm.
Fig. 11. Double-exposed hologram. A tilt of the door between ex-posures causes the vertical fringes seen on the quadratic inner part
of the test object.
Fig. 12. Double-exposed hologram. Translation Ax = 50,um of thewhole test object combined with a tilt of the door that exactly com-pensates the translation. (The door has been moved along an ellipsein the holodiagram and therefore it is fringe-free despite the combined
tilt and translation.)
Fig. 13. Sandwich hologram, with the same translation and tilt ofthe object as in Fig. 12.
Fig. 14. The sandwich hologram in Fig. 13 now tilted so that the50-jim translation has been compensated for and the fringes at the
door reveal only its tilt.
Fig. 15. The sandwich hologram as in Fig. 14. The compensatedtranslation has now been phase-shifted by a slight tilt of the sandwich
Fig. 16. The sandwich hologram as in Figs. 13-15. A tilt of thesandwich hologram produces an identical fringe separation over thewhole test object and on the door despite the fact that it was bothtilted and translated while the surrounding parts were only trans-
lated.
Fig. 17. Double-exposed hologram. Translation Ax = 0.5 mmcombined with a tilt of the door. Because of the large translation, thefringes are so closely spaced that they cannot be resolved on the test
object with its door.
Fig. 18. Sandwich hologram. Translation Ax = 0.5 mm combinedwith a tilt of the door. Fringe-free reference object, i.e., identical to
the double-exposed hologram as in Fig. 17.
Fig. 19. The same sandwich hologram as in Fig. 18, now tilted in sucha way that the 0.5-mm translation has been compensated for. The
fringes at the door are now clearly seen and reveal its tilt.
Fig. 20. Double-exposed hologram. Translation Ax = 1 mm com-bined with a tilt of the door. This time the fringes are still more closly
Fig. 21. Sandwich hologram. Translation Ax = 1 mm combinedwith a tilt of the door. Fringe-free reference object, i.e., identical to
the double-exposed hologram as in Fig. 20.~~~. . b .. 0Fig. 22. The same sandwich hologram as in Fig. 21, now tilted in sucha way that the 1-mm translation has been compensated for. Thefringes at the door reveal a tilt of < 2 jm despite a rigid body trans-
lation of 1 mm.
combination of translation and tilt, and in particular ifthe translation is large compared to the tilt, the sand-wich method is the only possible method for evaluation.Then a larger error can be accepted.
When making the experiments we observed that thex -y table was not producing a pure translation. Therewas an undesired slight tilt of the table, and the regis-tration in the hologram was, of course, the combineddisplacement. Because the sensitivity to tilt is muchgreater than to translation, even a very small tilt willaffect the object's fringe pattern. That is the reasonwhy there is disagreement between the measured valuesobtained from the instrument and those from theevaluated holograms.
The instrument is not affected by a slight tilt but isonly sensitive to the pure translation. The hologram,however, was evaluated as if the combined displacementof the object was a pure translation. The evaluationmethod of observing the fringes that move through apoint on the object while the eye is moved along theholographic plate will give only the in-plane translationof the object.
Due to the size of the plate and its distance to theobject in our experiments, this method was insensitivebecause we made no attempt to resolve fractions offringes that move through an object point. One fringein our experiments corresponds to -10-Am translation.That means that the accuracy, when using this methodin our experiments, will be no better than h10 Am.Therefore the sandwich holography error was evaluatedonly by comparison with the other, conventionalmethod described above.
However, in a comparison of sandwich holographyevaluation with conventional evaluation, both methodsevaluate the combined displacement (the translationcombined with the slight tilt). This combined dis-placement was treated as a pure translation as the un-wanted tilt was very small. That explains the differencebetween the pure translation indicated by the instru-ments and the holographically evaluated translation.
The most useful application of sandwich holographyis to compensate for rigid body motion. The possibilityof measuring translation is also good, especially as thesign of the translation is found by using a sandwichhologram.
The project was sponsored by the Swedish Board forTechnical Development.
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