Diffuse illumination in holographic double-aperture interferometry

Diffuse illumination in holographic double-aperture interferometry

Romuald Pawluczyk and Zbigniew Kraska

It is experimentally shown that the diffuse illumination in holographic interferometry influences the visibility function of the fringes and, as a result, the fringe contrast depends on the fringe order. This property can be applied to identification of the local minima of fringe order in double-exposure holographic interferometry. The method has been used in identifying the nodal pattern of a singer’s skin vibrations in double-pulse holographic interferometry.

I. IntroductionThe investigation of a singer’s skin vibrations during singing has been carried out in the Central

Laboratory of Optics in Warsaw for a few years.1-3 Because of the nonstationary (living) character of the investigated object only double-pulse holographic interferometry could be used.

In typical experimental conditions (small illumination and observation apertures) this method gives in-terference patterns with constant contrast independent of the object displacement between exposures, so the nodal pattern of vibrations is sometimes hardly localized. However, for safety reasons diffuse illumination had to be used in these investigations, and in the reconstruction process it was perceived that, in some cases, the fringe contrast had decreased as the object displacement between exposures had increased. This phenomenon has been applied to node localization,’-3 and preliminary interpretation of it on the basis of speckle correlation theory has been attempted.4 An exact theoretical explanation of this effect has been given by Matczak5-7 and some experimental results have been published.4’8’9 This paper includes more complete results of investigations in this field.

II. Preliminary ConsiderationsAt the present time all, even the most general, theoretical work on holographic interferometry of opaque diffusely reflecting objects consider holographic systems with well-defined (plane or spherical) illuminating beams (lists of this work can be found in many books on holography).10—’4 It is clear that the results obtained cannot be used for fringe creation analysis in the case of diffuse illumination. The possibility of using diffuse illumination in zero-order fringe identification was first suggested by Wall.’ 5

Theoretical analysis5-7 shows that the illuminating beam aperture plays the same role as the aperture of the imaging system used in the reconstruction process. This means that, in general, the fringe localization area depends on both illumination and observation apertures, and for an imaging system focused on the object surface the fringe contrast and their visibility also depend on both apertures.As follows from theory, the light intensity distribution I of the interference field in the image plane of an axially symmetric imaging (observing) system OD (Fig. 1) sharply focused on the surface of the object OS illuminated with diffuse light emerging from a circular diffuser (illuminating system) ID is equal to (omitting the speckle structure)5—7

where I~ is the image plane intensity distribution for one exposure, J,(~),J,(i1) are the first-order Bessel functions of the first kind, iii is the object displacement vector between exposures, and K is the sensitivity vector:

where k = 2ir/X, A is the wavelength of light used for hologram exposure, ~ and ~jr are the versors directed from the object point given by the vector i to the aperture centers of the imaging (observing) and the illumi-nating systems, and War and wi,. are the plane angular

The authors are with Central Laboratory of Optics, Department of Physical Optics, Kamionkowska 18, 03-805 Warsaw, Poland.Received 13 December 1984.0003-6935/85/183072-07$02.00/0.

© 1985 Optical Society of America

apertures of the imaging and the illuminating systems, respectively, measured in planes including the dis-placement vector II and the respective versor ~or or ë~r; and Ei~O and U~1 are transversal displacement compo-nents.

As follows from Eqs. (1) and (3) for diffuse illumi-nation used in a wide-aperture imaging system and a wide-aperture illuminating beam, the resulting inter-ference pattern depends on the product of three factors. One of them has the form of a cosine function with the argument dependent on the displacement vector com-ponent in the direction of the sensitivity vector K. This factor describes an interference pattern such as non-diffuse illumination with a spherical wave and a low-aperture imaging system. The other two factors describe the dependence of the fringe visibility on the object displacement and the plane apertures of the illuminating and observing systems.

The visibility of the interference pattern obtained depends on the product of two independent functions which unharmonically and quickly tends to zero as function arguments increase. The visibility of the in-terference pattern reaches zero for the first time when whichever argument of one of the Bessel functions reaches a value equal to 1.22ir. This takes place whenor

where a0,. and air are the angles between the displacement vector II and the corresponding versor ~or or ë~,. and A0,. and At,. denote numerical apertures of observing and illuminating systems, respectively. It follows from the above that fringe visibility drops to zero for the first time when the ~object translation between the compared states in the direction perpendicular to illumination and observation directions is equal to the radius of the speckle16 in a corresponding system. It is clear that fringe visibility cannot depend exclusively on the displacement vector if the apertures of both systems are sufficiently small or if both versors ~or and ~ are parallel to each other and to the displacement vector.

Practically, the observing direction is almost never parallel to that of the illumination so, generally, fringe visibility is always dependent on the displacement value if wide-aperture illumination and observation systems are used. As the displacement increases the fringe visibility decreases. This dependence constitutes the foundation of the proposed method for lower-order fringe localization in double-exposure holographic interferometry.

IlI. Experimental VerificationFor experimental verification of the proposed method

some holographic interferograms of simple objects were

made on the holographic setup shown in Fig. 2. It is atypical two-beam holographic setup with diffuser D

Fig. 1. Coordinate system and geometry of the holographic setup: OS, object surface; ID, illumination system aperture (diffuser); OD, observation system aperture.

Fig. 2. Optical scheme of the holographic setup used: He—Ne, helium—neon laser; M, mirrors; X/2, halfwave plates; BS, interference-polarizing beam splitter; L, positive lens; D, iris diaphragm; GP, ground glass plate; 0, investigated object; LC, laser collimator; and H. the hologram.

placed in the illuminating beam ‘~-~30 cm in front of object 0. The illuminated spot diameter at the diffuser is changeable as demanded over a range of 5—100 mm with an iris diaphragm placed close to the diffuser. For comparison sake, a slight modification of the setup makes it possible to interchange the diffuse illumination with an ordinary spherical wave emerging from the focus located exactly at the center of the diffuser. Reconstructed images were photographed with an ordinary short-focus photographic camera placed ‘--‘50 cm from the reconstructed virtual images. It is clear that the effective aperture of the imaging system was very small and that it had insufficient influence on the visibility of the fringes. For this reason the influence of the imaging system on the interference-fringe visibility is omitted from the following considerations.



M

The influence of wide-aperture diffuse illumination on holographic interference-fringe visibility is well il-lustrated in Figs. 3 and 4. Figure 3 depicts four inter-ferograms of a plane rectangular plate rotated around a vertical axis by an angle ‘—‘20 sec at different angles of illumination (aj = —20°, 0°, 200, and 40°). For comparison, Fig. 4 shows interferograms of the same plate



Fig. 7. Same plate as in Fig. 6 at normal illumination and a 5.5-cmdiffuser diam; rotation angle ~t’ = 2 mm.

the 11-cm light spot at the diffuser is not two times greater than that for the 5.5-cm light spot. This means that the ground plate used diffuses light in a limited angle; and for the 11-cm light spot diam the effective aperture of illumination is smaller than follows from geometrical relations.

A good illustration of the diffuse illumination appli-cation in holographic interferometry is the series of photos in Fig. 8 presenting holographic interferograms of the closed empty cylinder wall deformation caused by inside pressure changes with different values be-tween the first and second expositions. For compari-son, every interferogram obtained in the previously described holographic setup with wide-aperture diffuse illumination is accompanied by an interferogram obtained in the same conditions but with nondiffuse illumination.

For nondiffuse illumination (the upper set of photos) the displacement of the cylinder walls associated with object deformation is depicted and practically no conclusion as to the object elongation can be derived. On

Fig. 9. Interferograms of a vibrating rectangular plate excited with the same frequency obtained by (a) time-averaging and (b) double-pulse with diffuse illumination holographic interference methods.

the other hand, the object elongation causes a transla-tion of particular cylinder points along its walls. It is clear that the exact value of this translation in a given point is proportional to the distance of this point from the cylinder base and depends on the inside pressure

Fig. 8. Interferograms of an empty cylinder at increasing inside pressure differences obtained with (a)—(c) nondilfuse illumination and (d)—(f) diffuse illumination.



Fig. 10. Sample holograms of singers during singing obtained by the double-pulse holographic method with diffuse illumination.

change between hologram exposures. In an applied holographic system this translation is perpendicualr to illumination and observation directions so the fringe visibility decline caused by this transversal object dis-placement depends on the observed point distance from the base; the range of this decline along the cylinder walls must depend on the inside pressure change between exposures. This dependence is well illustrated by the lower set of photos in Fig. 8, where the fringe visibility decline in dependence on the mentioned parameters is clearly visible.

IV. Exemplary ApplicationsAs shown above for wide-aperture and for noncollinear

illumination and observation systems, the interference-fringe visibility depends directly on the value of object displacement between exposures in directions perpendicular to both illumination and observation directions independently. This means that if both these directions are not coaxial the fringe visibility decrease as the object displacement increases must always exist. Therefore the possibility arises of differentiating the fringe visibility depending on the fringe order. As a result diffuse illumination can be used for identification of the local minima of fringe order in holographic double-exposure interferometry.

For a stationary vibrating object the order of interference fringes obtained by the double-pulse method attains a minimum in nodal areas. So, by using oblique diffuse illumination it is possible to obtain a differentiation of the fringe contrast (and the visibility) and to identify nodal patterns as well. This possibility is illustrated in Fig. 9 where two interferograms of a vibrating plate are presented. The first was obtained by the time-averaging method and the second by the double-pulse method with a ruby laser and diffuse illumination. It can be seen that the fringe contrast decreases in both interferograms. (There must be some differences in fringe numbers as the fringe functions are different and different wavelengths of laser light were used in the experiments.)

The method under consideration has been used in the Central Laboratory of Optics for nodal pattern identification in the experiments on holographic vibration analysis of the front parts of the human neck during singing.~ Some sample interferograms are presented in Fig. 10. The nodal patterns are visible as the areas with the best fringe contrast.

V. ConclusionsPrevious considerations allowed us to conclude that, for

double-exposure holographic interferometry of diffuse opaque objects using wide-aperture diffuse illumination and wide-aperture observation systems, the resulting interference-pattern visibility depends equally on the apertures of both systems. For nonparallel axes of observation and illumination systems the visibility of the interference pattern always depends on the absolute object displacement; this enables us to localize the interference fringes corresponding to the minimal displacement. Practically, it brings about the possibility of localizing interference fringes of the lowest order which correspond to the nodal pattern of the vibrating object when studied by double-pulse holographic interferometry.

The Authors wish to express their sincere thanks toM. Sochacka for her helpful assistance in preparation of this paper.

References1. R. Pawluczyk, Z. Kraska. and Z. Pawjowski, “Holographic Vibration

Analysis of the Frontal Part of the Human Neck During Singing,” in Optics in Biomedical Sciences, G. von Bally and P. Greguss, Eds. (Springer, Berlin, 1982), pp. 13 1—133.

2. R. Pawluczyk, Z. Kraska, and Z. Pawjowski, “Holographic In-vestigation of Skin Vibrations,” Appl. Opt. 211, 759 (1982).

3. Z. Pawjowski, R. Pawluczyk, and Z. Kraska, “Epiphysis Vibrations of Singers Studied by Holographic Interferometry,” paper presented at SMAC-83 Stockholm Music and Acoustic Conference, Sweden, 28 July—1 Aug. 1983.

4. R. Pawluczyk and Z. Kraska, Diffuse Illumination in Holographic Interferometry of Living Objects,’’ in [let. 1.



5. M. J. Matczak: “Interpretation of Holographic Interferograms Using the Scalar Diffraction Theory,” paper presented at Seventh SESA International Conference on Experimental Stress Analysis, Haifa, Israel, (Aug. 1982).

6. M. J. Matczak: “Fringe Visibility Method as a New Method of Holographic Interferograms Interpretation,” Proc. Soc. PhotoOpt. Instrum. Eng. 370, 163 (1982).

7. M. J. Matczak, “Diffractional Description of Interference ImageFormation and Its Consequences in Holographic Interferometry,”paper presented at EOC-83 European Optical Conference onOptics in Science and Technology, Rydzyna, Poland, 30 May—4June 1983.

8. Z. Kraska, R. Pawluczyk, and M. J. Matczak: “Identification of the Local Minima of the Fringes Order in Holographic Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 370, 206 (1982).

9. M. J. Matczak, R. Pawluczyk, and Z. Kraska, “Diffuse Illumination in Holographic Interferometry,” Proc. Soc. Photo.-Opt.Instrum. Eng. 370, 216 (1982).

10. C. M. Vest, Holographic Interferometry (Wiley, New York,1978).

11. W. Schumann and M. Dubas: Holographic Interferometry (Springer, Berlin, 1979).

12. Yu. I. Ostrovskii, M. M. Butusov, and G. V. Ostrovskaja, Holo-graphic Interferometry (Nauka, Moscow, 1977), in Russian.

13. H. J. Caulfield, Ed., Handbook of Optical Holography (Academic, New York, 1979).

14. M. Pluta, Ed., Optical Holography (PWN, Warsaw, 1980), in Polish.15. M. R. Wall, “Zero Motion Fringe Identification,” paper presented at

International Symposium on Applications of Holography,Besancon, France, 6—11 July 1970.

16. J. C. Dainty, Ed., Laser Speckle and Related Phenomena (Springer, Berlin, 1975).



Diffuse illumination in holographic double-aperture interferometry

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