Jacques Lewandowski, Bernard Mongeau, Maurice Cormier, and Jean Lapierre
Infrared (10.6-,um) dynamic holograms are obtained on a recording medium consisting of a thin oil film on aglass substrate. Reconstruction with visible light (0.63 urm) permits copying the initial hologram on aphotographic plate. Any subsequent displacement of the object modifies the corresponding reconstructedwave at the recording medium, thereby creating a characteristic visible interferogram. Analysis of the doubleimage in copy holograms together with fringe localization and visibility did not show stringent setupconditions. Experimental results are well related to various objects and displacements while the longwavelength (a 10 jm) used offers the advantage of large displacements or wavefronts deformation measur-ability and considerably decreases the setup sensitivity to vibrations.
1. Introduction
Holographic interferometry has become a usefultechnique in the measurement of small deformationsand displacements: this technique has been usedmostly in the visible spectrums 2 for high resolutionmetrology. However, due to the use of these shortwavelengths, this technique is highly sensitive to airturbulence and mechanical vibrations.3 Moreover inmost cases of real-time holographic interferometry onemust take additional precautions in developing thefilm4'5 and placing it back in its original position, aswell as obtaining good fringe contrast. 6 7 As a result,the localization interpretation of the fringes is highlycomplex in most cases.",6 8
Having previously developed techniques for infra-red (IR) holography,9"10 we have used the longer wave-lengths which facilitate fringe interpretation for largeobject displacements and minimize experimentalhandicaps of visible holographic interferometry. Pre-vious work" demonstrated real-time interferometryusing IR holography of low-diffusing surfaces and thispaper reports some results obtained by adapting thesomewhat standard techniques of holographic inter-ferometry to the use of IR wavelengths (_ 10.6 Mim) ininspecting surfaces that may be diffusing.
To that effect a copy of the IR hologram was made,using visible (0.6 3 2 8 -Am) reconstructed waves on aphotographic plate similar to the standard methods ofduplication.12 When this copy is put back in place, itsreconstruction interferes with the visible image recon-structed in real time from the IR hologram.
Jean Lapierre is with Ecole Polytechnique, Department de geniephysique C.P. 6079, Succursale A, Montreal, Quebec H3C 3A7; theother authors are with College Militaire Royal, Laboratoire Laser(LLCMR), Saint-Jean, Quebec JOJ RO.
A laser beam of 10.6-,um wavelength, in its funda-mental mode, is used with a 50% beam splitter (BS)and a plane mirror M1 to obtain the reference andobject beams shown in Fig. 1. These two beams havean approximate diameter of 2 cm and they are incidenton the holographic recording medium (RM) consistingof a thin oil film on a glass substrate.9 Due to theirlong coherence length (several meters), the IR beamsinterfere on this dynamic recording medium and cre-ate a phase relief hologram with a resolution of 20cycles/mm, a diffraction efficiency between 5% and20%, and a response time from 5 to 200 ms when thetotal incident power is below 0.5 W/cm. 2
The He-Ne laser beam, incident normally on the oilat RM in Fig. 1, is used to reconstruct the IR waves inthe visible spectrum (0.6328 ,m) as well as to produce acopy of the initial hologram on holographic plate CP(Fig. 1): the use of a He-Ne laser ensures good coher-ency of the reconstructed waves at CP. Once devel-oped, this plate is put back in its original position CP ata distance d behind the oil film.
When the IR beams interfere again on the oil filmresulting from a displaced or distorted object, the He-Ne beam reconstructs simultaneously the displaced/distorted image from the oil film at RM and the initialimage from the copy at CP. These waves give visibleinterference patterns corresponding to the displace-ment or distortion of the object.
An object transparent to IR is placed directly intothe object beam (from mirror Ml) whereas mirrorM2 inFig. 1 is used to illuminate an object which is opaque tothe IR wavelength. In all cases the reconstructedimages from the oil and from the copy superimposequite well: the misalignment due to distance d be-tween the copy and the IR medium will be shown asnegligible in specified conditions. To optimize fringevisibility, fringe contrast will be considered in terms ofthe transmittance of the copy plate, while the problemof fringe localization will be discussed for each experi-ment presented in this paper.
The IR object and reference beams incident on therecording medium RM (Fig. 1) are represented, re-spectively, by 0 exp(joo) and R exp(j4R), where 0, R,ko, and OR are real functions of the spatial coordinates
(x,y) on the RM plane. These waves give rise to thefollowing interference pattern of intensity I:
The reference beam used has a constant amplitude Rover RM; for the case where the object is a diffusescatterer, 02 is also essentially constant over RM.
It has been verified that for small intensity modula-tion the oil film thickness at RM13 is proportional tothe last two terms of Eq. (1) leading to a phase modula-tion Ark:
A = RO{expU( 0 0 - OR)] + exp[-(o - OR)]1' (2)
where A is a constant characteristic of the oil film.The amplitude transmittance tRM of the phase holo-
gram at RM (for AO << 1) is then
tRM = exp(jAO) 1 + j V'n[a(xy) + a*(x,y)], (3)
where XIj = ORO is the amplitude diffraction efficiencyof the phase hologram, a(x,y) = expU(O0 - OR)I, and *denotes the complex conjugate.
B. Reconstructed Waves
The He-Ne plane waves of wavelength X,, incidentnormally on the recording medium (RM), can be de-scribed by a real amplitude disturbance A, over thehologram plane. For a diffraction efficiency of RMsmaller than 10%, consistent with our experimentalwork, the light diffracted has a complex amplitudeARM well approximated by the first-order waves:
ARM(XY) = tRMAC A, + jA,1n[a(x,y) + a*(x,y)]. (4)
The zero-order planar wave front A, acts as the refer-ence and reconstruction beams for the copy hologramat CP.
At a small distance d from RM, the replica emulsionCP is exposed to the diffracted waves Acp which can berepresented by
Acp(x',y') = A, + AcIn[AV(x',y') + AR(x',Y)], (5)
where AV (x',y') and AR (x',y') are the Fresnel trans-forms of ja(x,y) and ja*(x,y), respectively, and (x',y')are the spatial coordinates at the CP. Both a and Avrepresent complex amplitudes (in different planes) ofthe same wave diverging from the virtual image.
Due to their various inclinations through the glasssubstrate at RM, each wave of Eq. (4) travels a differ-ent path length: this difference in path length issmaller than X,/65 for the 1-mm thick substrate and isconsidered negligible. Hence the glass substrate doesnot alter the phase relationship of the waves from RMbut simply adds a constant phase to all three waves ofEq. (5). As seen from Eq. (4), plane-wave light illumi-
nating the phase hologram maintains its unmodulatedintensity at the surface of the hologram (since IRM =ARMA*RM = A' forq << 1), and only when light propa-gates a sufficient distance from the surface can inter-ference and intensity modulation take place [since AVand AR are no longer complex conjugates in Eq. (5)].
For the sake of clarity in the equations, future refer-ences to the amplitudes A in this paper will not showexplicit dependence on the coordinates (xy) or (x',y')unless judged necessary.
From Eq. (5), the intensity Icp of the waves incidenton plate CP is, in first approximation ( << 1), equal to
ICp(x',y')-; A2 + jjA'(A, + AR + AV + AR). (6)
The photographic plate at CP once processed has anamplitude transmittance tp given by the relation8
tcp = to - kIcp TO- kV A'(A, + A + A + , (7)
with
To = to - kAc, (8)
where k is a constant characteristic of the photograph-ic plate.
After the object in Fig. 1 has undergone a smalldeformation/displacement, the modified reconstruct-ed waves incident at CP are from Eq. (5):
ACp = AC + Ac6J[AV + AJ, (9)
since the zero-order beam remains insensitive to amodification in the IR object beam. Wave A recon-structed from wave A'Cp incident on the copy plate(CP) is then
A = tcpA6P (10)
From Eqs. (7) and (9) this last equation in firstapproximation becomes
A cTAC + AC4[TOA - kA2(AR + AV)]
+ Ac7[TA, - kA2(Av+A)]- (11)
The first term on the right-hand side of Eq. (11) isthe zero-order beam transmitted through plate CPwhile the second and third terms represent the wavesreconstructed at CP as related, respectively, to the realand virtual images. These last two expressions areinterference patterns generated by two wave frontscorresponding to the standard waves (AR and AR; AVand AV). However, superimposed on these two wavesin each case, there is a disturbing wave (A* or AR)which might considerably modify the observed pat-
terns. This consideration comes from the fact thateven if the equality AV = AR is true in the plane ofhologram RM, this equality may not hold in the planeof the hologram CP at a distance d from RM. Theseundesirable waves are related to the double-imageproblem encountered by many authors while copyingholograms.'4
C. Analysis of the Double-Image Problem
The image doubling effect in copy holograms couldbe avoided for a Kodak 649F emulsion and an illumi-nation at 0.6328 m (Ref. 14) if the angle betweenreference and object beams exceeds 10°. However, inour experiments, because of the wavelength scalingfactor (16.75), we obtain small diffraction angles atRM and consequently a small angle between referenceand object beams at the plate CP, preventing us fromusing this technique in our experiments. Thereforewe must evaluate the phase difference between AR andAV of Eq. (5) and its effect on the final interferencepattern. For simplicity we consider the particularcase where the IR object beam consists of a sphericalwave issued from point P(ro,00) of the IR object wherero is the distance along the line from point P to thecenter of RM and 00 is the angle between this line andthe normal to RM.
Moreover in our experiments, the IR reference wavewith similar coordinates (rROR) and the visible recon-struction wave (rcOc) are planar waves, the latter be-ing perpendicular to RM (Oc = 0). One can show" thatthe first-order Gaussian images from RM have thecoordinates ri and Oi:
ri = iir0 , (12)
i =-(00-R) = i-' (13a)where
AROOR = 00 OR ju =- = 16.75. (13b)Ac
The reconstructed images (real and virtual) are thuslocated symmetrically in reference to the hologram atRM but not in reference to the hologram at CP. Thiscan be illustrated by considering the virtual images Pvand P* corresponding to wave amplitudes AV and ARof Eq. (11), as shown in Fig. 2. It has been shownl 4 thatthe double image PR is closer by a distance 2d to RM,thus introducing a path difference 6 between the wavefronts from Pv and PR interfering on a screen at adistance s from CP:
, x - __x2 dx2 (14)2(s +ri-d) 2(s +ri+d) (s +ri)2
where x is the coordinate of an observation point M onthe screen, as shown in Fig. 2, and the approximateexpression is obtained assuming a small distance d (<<s+ r) between RM and CP. Equation (14) shows thatthe double-image problem does not exist for the unre-alizable case where d = 0 as well as for the trivialsituation where the object waves are planar giving ri -
- from Eq. (12).
Fig. 2. Geometrical analysis of the double-image problem (dis-tances and angles are not to scale).
For the real case of an object at a finite distance rofrom RM the IR spatial frequency fR at RM is
/ 0 R _ 0ifR I -O ,
AR AC(15)
where GOR is defined in Eq. (13b). With the realisticapproximation x (s + r)0i, from Fig. 2, the expressionfor the path difference 6 becomes
6 d R. (16)
From this equation and our experimental value for d(5 mm), it can be calculated that the path difference 6will remain smaller than (c/4) if fR < 8.9 lines/mm,which is equivalent to
(17)OOR = 00 - R < 54-
Consequently the double-image problem will benegligible (6 < Xc/4) whenever the experiments complywith the condition in Eq. (17).
D. Fringe Visibility
The maximum diffraction efficiency for a thin phasehologram (33.9%) like the one at RM is much greaterthan that for a thin amplitude hologram (6.25%) suchas the one at CP; as a consequence, the first-orderreconstructed waves A'V and AR at RM in Fig. 1, even ifattenuated by the plate's average transmittance at CP,will have amplitudes higher than that of the wavesreconstructed from CP thus yielding poor fringe visi-bility.
As a consequence, the diffraction efficiency of thereplica emulsion at CP is made the largest possible byimproving the contrast M of the fringes it records.This contrast M is defined as
M = Icp(max) - Icp(min)Icp(max) + Icp(min)
(18)
where the maximum (max) and minimum (min) valuesof Icp [Eq. (6)] are considered.
The complex amplitudes AV and AR incident at CPmay be chosen as
AV = exp(j 1 l) and AR = exp(U02 ), (19)
where the phase terms 01(x',y') and 2(x',y') are relat-ed to the Fresnel transforms of ja(x,y) and ja*(x,y),respectively. Inserting these wave expressions intothe CP intensity of Eq. (6) gives
which reaches a maximum value M = 1 for q = 6.25%.The effect of the fringe contrast M and of the ampli-
tude transmittance To on the diffraction efficiency Dof Kodak 649F photographic plates was investigatedexperimentally by Friesem et al.15 Their results fitthe following empirical relation within 7% for (0.30 •To < 0.70):
D = 0.04M 2 [1 - 6.6 (TO - 0.45)2],
C.)
j8
z
(22)
showing maximum diffraction efficiency at To = 0.45.The amplitude modulus I Acp I for the first-order
wave reconstructed from the hologram at CP is thenapproximately equal to
and the related wave I ARM I reconstructed from thehologram at RM, as observed after transmissionthrough CP, is
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
AMPUTUDE TRMSMITTMCE To
Fig. 3. Fringe visibility [Eq. (25)].
IARMI = A To -
The fringe visibility C for these interfethen found to be
2IARMIIAcpI
|ARM 12 + IAcp 12
1 _OTo 0 M[l - 6.6(To - 0.45)2]1/2
(24)
ring waves is
(09iM 2 [1 - 6.6(To - 0.45)2] + 25TO2
.
where M is given by Eq. (21) if q • 6.25% and M = 1whenever > 6.25%. As expected from Eq. (11) theobserved pattern is a dark-field interferogram as isusually obtained in real-time holographic interferome-try. Figure 3 shows the fringe visibility C computedwith Eq. (25) for X < 6.25 and q = 10%. The amplitudetransmittance corresponding to the highest contrastfor 5 S 6.25% is To 0.65 from Fig. 3 and is seen to beindependent of the RM diffraction efficiency q.
For these small values of 7, the CP reconstructedwave is brighter than I ARM I and the transmittance Tomust be higher (above To = 0.45) to increase IARM l, asseen from Eq. (24), and ensure good fringe contrast.Figure 3 also indicates that the CP transmittance To isnot critical on the final fringe visibility and that a highcontrast (C 2 0.80) is still obtained throughout a largeinterval (0.3 < To • 0.7) for q < 10%.
IV. Experimental Results
Real-time IR holographic interferometry was per-formed with the experimental setup schematized inFig. 1. In this section, typical results are shown forthree experiments in order of increasing degree of com-plexity.
A. Angular Displacement of an IR Object Planar Wave
To show an easily measurable effect, a simple experi-ment was made using a planar IR wave from mirror Ml
in Fig. 1 as the IR object beam (the object and mirrorM2 of Fig. 1 were removed). Keeping mirror Ml in itsinitial position, a photographic plate was exposed atCP to obtain a copy plate of the hologram in RM: theexposure time and intensities were adjusted to obtain atransmittance To = 0.5 for this plate.
Figure 4(a) shows the resulting pattern when thisplate is put back in position CP after being developed.The granular aspect and the barely visible fringes ofthis figure are speckle resulting from the many sur-faces in the path of the He-Ne beam and also noisetypical of phase holograms.'6
An angular displacement of the IR object beam isthen effected by rotating mirror Ml to change the angleOOR [Eq. (13b)] by a value AOR resulting in a differenthologram on RM. Interference with the hologramrecorded on CP for the first position of mirror Ml givesthe pattern shown in Figs. 4(b)-(d). The fringe spac-ing i for this interference is given by
= RAOOR
(26)
For each of these patterns, the value of ic has beencomputed from Eq. (26) and compared in Fig. 4 to thefringe spacing i measured on the pattern: these val-ues are comparable within experimental precision.
To demonstrate the real-time aspect of this tech-nique the experiment was recorded on videotape andthe photographs of Fig. 4 could have been taken fromframes that are 0.5 s apart: a relatively low responsetime of the oil film in RM (0.1 s) is involved whenhigher efficiency iq is needed.
It is also noteworthy to recall that this experimentdoes not involve the double-image problem discussedpreviously since the IR object wave is plane [ro = inEq. (12)].
a. No Rotation of b. Rotation of MirrorMirror M1 M giving
e OR 1.2 ± 0.1 mrad
ic = 9 1 mm
i = 10 1 mm
c. Rotation of Mirror d. Oblique Rotation ofM1 giving Mirror M1 giving
AO O 2.2 + 0.2 mrad
ic = 4.9 ± 0.5 mm
i = 5.2 ± 0.2 mm
a. Vertical b. Horizontal c. AxialDisplacement Displacement Displacement
Fig. 5. Displacement of an IR point object obtained from a Ge lens.
a. Picture of b. Reconstructedthe object Real Image(scale 1:1) (object in
initial position)
LeOR = 3.2 ± 0.3 mrad
ic = 33 ± 0.3 mm
i = 37 ± 0.2 mm
Fig. 4. Angular displacement of an IR plane wave.
B. Displacement of an IR Point Object
The IR point source is obtained by inserting a goodquality Ge lens in the path between Ml and RM in Fig.1: the image focal point of the lens is the point object.A photographic plate is first exposed at CP and subse-quently developed (To = 0.36) and put back in positionat CP. When the Ge lens is displaced in a directionperpendicular to the path between Ml and RM, thehologram changes at RM give rise to interferences onthe observation screen. The nonlocalized fringes ob-served, as shown in Figs. 5(a) and (b), are dependent onthe amplitude and direction of the lens displacementand may be considered as the interference pattern oftwo sheared spherical wave fronts. Calculation of theexpected interfringes due to the Ge lens displacementshowed good correlation with the measured values ofFigs. 5(a) and (b). This potential application to shear-ing interferometry is an important feature of the meth-od and may lead eventually to interesting results usingdiffusing objects. For an axial displacement of the Gelens circular rings appear on the interferogram, as seenin Fig. 5(c). In these experiments the double-imageproblem was prevented by keeping the angle OOR small-er than 5.4° as specified in Eq. (17).
C. Inspection of an Object Having Diffusing Surfaces
The diffusing surfaces were made by rough machin-ing the letters CMR on the surface of an aluminumplate [Fig. 6(a)]. This plate was used as the object inFig. 1 where mirror M2 illuminates its surface. Aphotographic plate is first exposed at CP and thendeveloped giving a transmittance To = 0.64.
With the copy plate at CP and the object still in itsoriginal position, a real image of the object is recon-
c. Horizontal rotationby a = 0 .9 + 0.1 mrad
i = 59 + 0.6 mm
i = 65 0 . mm
d. Vertical rotationby a = 1.5 ± 0.1 mrad
F i = 3.6 ± 0.2 mmI 0
Ii = 3.3 .3 mmI l m
Fig. 6. Displacement of an IR diffusing object by IR holographicinterferometry.
structed as shown in Fig. 6(b). The inverted image isof the same size as the object but its poor quality is dueto intrinsic nonlinearities of the phase hologram16 atRM and to degradation in the development process ofthe copy plate; no interference pattern is observed asexpected.
Any displacement of the object will cause an inter-ference pattern to appear over this image as in Figs.6(c) and (d) where the object was rotated by an angle a
around an axis parallel to its plane as shown in Fig. 7.The fringes can be localized very near the real image
so that they can be clearly recorded on the same photo-graph presenting an advantage over the usual holo-graphic interferometry where the fringes are visibleonly near the virtual image and cannot be photo-graphed superimposed on the real image.
The plane in which the fringes are located in thisexperiment is at a distance hR from the object given byCollier et al.8
cos(GP) sin2 (0 )1hR = X I,Lsin(oi) + sin(o,)J
(27)
where x, q5i, and q5s are defined in Fig. 7. Using ourexperimental values we compute hR = 0.0794x. Sincethe holographic reconstruction is made with a wave-length Xc, different from the IR wavelength XR, the realimage is localized at a distance of -14 m, which is pu
Fig. 7. Definition of the coordinates: rotation of the dobject. Experimental values are pi = 8503 and qs = 99
VI. Conclusion
M2 Holographic interferometry on diffusing objects wasshown to be possible in real time adding to the usualmethods the advantage of little vibration isolation andthe interesting feature of direct observation of thefringe pattern superimposed on the image, thus com-
M plementing the technique already published" for thecase of nondiffusing objects. Also noteworthy is thepossible application of the method to IR shearing in-terferometry thus permitting the testing of IR materi-als and optical systems. Use of the long wavelength(p10 ,gm) decreases the scattering properties of roughsurfaces and allows the interferometric study of large
n<J wave front deformations.03. U01115
b03
times larger than the distance between the object andRM [ = (R/XC) = 16.75]; similarly the localization ofthe fringes is at a distance h from this real image:
h =uhR = 1.33x. (28)
Because in our experiments x < 3 cm, the value of his smaller than 4.0 cm and it is very close to the realimage so that the fringes exhibit good contrast in theplane of the real image.
The fringe spacing i in the observed pattern can beshown to be
a[sin(oi) + .in() (29)
The experimental values of 0 and s as well as therotation angle a measured by other means provide acomputation of the expected interfringe i. The mea-sured fringe spacing im in Figs. 6(c) and (d) is seen to becomparable, within experimental precision, with thecomputed values i obtained by Eq. (29).
In this case also the angle OR is maintained to a valuesmaller than 5.4° so as to elude the double-image prob-lem.
V. Discussion
Because of the long wavelength in IR, the fringespacing of the hologram at RM is large and its copy on aphotographic plate can be easily put back in its initialposition with less stringent constraints than in the caseof visible holographic interferometry.
No particular difficulty is implied in meeting thespecified conditions to elude the double-image prob-lem and to obtain good fringe visibility.
Moreover the technique proposed in this papermakes it possible to observe in real time directly on ascreen the real image of the object inspected togetherwith the fringe pattern characteristic of the displace-ment or deformation of the object. The image qualitydegradation through the various processes is consid-ered of least importance since the experiment is mainlyconcerned with the interpretation of the interferogramin terms of object displacement or deformation.While no particular care is taken to isolate the systemfrom vibrations the method is still limited to laborato-ry applications.
The authors would like to thank M. A. Malo for hiscontribution to the experimental work. This projectwas partly supported by a grant from the CanadianDND ARP (3610-371).
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