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Contours of equal in-plane displacement in holographic interferometry
Volume
7, number
4
OPTICS COMMUNICATIONS
CONTOURS
OF EQUAL
IN-PLANE
IN HOLOGRAPHIC
April 1973
DISPLACEMENT
INTERFEROMETRY
C.H.F. VELZEL Philips
Research
Laboratories,
Received
Eindhovm,
! 2 February
The Netherlands
1973
By using twin apertures in the hologram plane contours of equal in-plane displacement can be obtained from a double exposure hologram. This makes the analysis of three-dimensional deformations of diffusely reflecting objects possible. Some of the possibilities of the method are discussed and a preliminary experiment is described.
A method of measuring in-plane displacements in holographic interferometry has been developed by Boone [I] and others [2,3]. In this method the fringe pattern at infinity caused by the displacement of a small part of a diffusely reflecting object is studied by placing a diaphragm in the image plane. From this interference pattern the in-plane displacement of the’par-t of the object selected by the diaphragm can be determined. The measurement of in-plane displacement by the method outlined above is rather laborious. A further disadvantage is that the in-plane components of the displacement vector become known only for isolated points of the object. Speckle interferometry was used by Leendertz [4] and others [S-7] to obtain contours of equal in-plane displacement. In this paper we present the application of these me mods to holographic interferometry. To obtain the in-plane displacetnent contours from a doubly exposed hologram we operate as follows (see fig. I). The hologram is made with a plane reference wave, so that a real image can be formed by inverting the hologram. This real image is a mirror image of the object with respect to the hologram plane. A screen with two small apertures is placed in front of the hologram. In the image plane moire’ contours are formed, giving the displacement in the direction of the line connecting the centres of the two apertures. The theory of this phenomenon has been given by Duffy [S] The chief results of his theory are that the 302
Fig. 1. Scheme of the experimental set-up used to obtain contours of equal in-plane displacement. R = reference beam; I = illuminating beam; L = laser source; 0 = object; H = hologram; S = screen with two apertures; RI = real image of object 0; F = focal plane where contours are recorded.
lateral displacement per moire fringe is given by h/a, where (Yis the angular distance between the two apertures, and that the maximum displacement that can be measured is about X//3,where 0 is the numerical aperture of each of the apertures. In fig.2 we show the results of a preliminary experiment verifying the principle of this method. A double hologram was made of a diffusely reflecting disc rotated about its axis of symmetry between exposures. Rotating the screen with the two viewing apertures over 90” we made photographs in the image plane of the contours of equal in-plane displacement in two perpendicular directions. We show the contours for two values of the angle of rotation of the disc. As ex-
Volume 7, number 4
(a)
OPTICS COMMUNICATIONS
April 1973
(b)
Fig.2. Recordings are shown of the contours of equal in-plane displacement in two perpendicular directions of a disc rotating about an axis through its centre. In figs.2a and 2b the angle of rotation was about 1.5 min. of arc. In figs. 2c and 2d the rotation angle was 2.5 min. of arc. In figs. 2a and 2c the line connecting the viewing apertures in the hologram plane was horizontal. In figs. 2b and 2d this line was vertical. The contours of equal in-plane displacement are the horizontal and vertical contours in the figures. Besides these, the contours of equal longitudinal displacement are also visible. These contours would also have been obtained in a recording through one of the viewing apertures, as we have verified experimentally. 303
\‘l~lUrnC I.
Illlinhcr
4
OPTICS COMMUNICATIONS
petered, straight line contours are formed parallel to the. line connecting the viewing apertures. 111the photographs of fig.2 we also see the contours (I!-t,clual longitudinal displacement. These contours do 110t change when the screen with the two apertures is i-c;~:~tedabout the point midway between the apertures. Iii a comparable experiment in speckle interferoI:w TI1’ the contours of equal in-plane displacement \*,;~uId have been the only ones visible in the image pl:\nc. The situation in holography is somewhat more Iciiilplicated because two sets of contours are visible in r.;litl ptlotograph. But an advantage of holography o\i‘l speckle interferonietry is that more information I\ ;I\ .lilable: the longitudinal component of the deforIII;IIIOII vector can be obtained and the deforrnations 111‘ IIOI~-plane objects can be analysed more simply be(‘;IUX it is pvssible to obtain contours in different fo<.,il planes from one single hologram. III\IC;I~ of using two viewing directions, moiri con11!II I5 cut also be obtained by using two illuminating \\,i:e:\ [i,]. GI-eater sensitivity can thus be obtained at !!I(, c‘()>I of simphclty of the experimental set-up. The
304
April 1973
method can be made more flexible by using a television tube for recording the images. We should, by analogy with the work of Tiziani [2], also like to point out the application to the analysis of vibrations. The author is grateful to Mr. J. Bruin, of the Philips Medical Systems Deptirtment, who performed the experiment described in this paper.
References
[ 1] P.M. Boone, Opt. Laser Technol. 4 (1972) 162. [2] H. Tiziani, Opt. Acta 18 (1971) 891;Appl. Opt. 11 (1972) 2911. [3] J.W.C. Gates, in: Applications of holography, ed. J. Vi¬. Besaqon (1971). [4] J.A. Leendertz, J. Phys. E 3 (1970) 214. [S] D.E. Duffy, Appl. Opt. 11 (1972) 1778. [6] E. Archbold, J.M. Burch and A.E. Ennos, Opt. Acta 17 (1970) 883. [7] J.N. Butters and J.A. Leendertz. J. Phys. E 4 (1971) 277.
7, number
4
OPTICS COMMUNICATIONS
CONTOURS
OF EQUAL
IN-PLANE
IN HOLOGRAPHIC
April 1973
DISPLACEMENT
INTERFEROMETRY
C.H.F. VELZEL Philips
Research
Laboratories,
Received
Eindhovm,
! 2 February
The Netherlands
1973
By using twin apertures in the hologram plane contours of equal in-plane displacement can be obtained from a double exposure hologram. This makes the analysis of three-dimensional deformations of diffusely reflecting objects possible. Some of the possibilities of the method are discussed and a preliminary experiment is described.
A method of measuring in-plane displacements in holographic interferometry has been developed by Boone [I] and others [2,3]. In this method the fringe pattern at infinity caused by the displacement of a small part of a diffusely reflecting object is studied by placing a diaphragm in the image plane. From this interference pattern the in-plane displacement of the’par-t of the object selected by the diaphragm can be determined. The measurement of in-plane displacement by the method outlined above is rather laborious. A further disadvantage is that the in-plane components of the displacement vector become known only for isolated points of the object. Speckle interferometry was used by Leendertz [4] and others [S-7] to obtain contours of equal in-plane displacement. In this paper we present the application of these me mods to holographic interferometry. To obtain the in-plane displacetnent contours from a doubly exposed hologram we operate as follows (see fig. I). The hologram is made with a plane reference wave, so that a real image can be formed by inverting the hologram. This real image is a mirror image of the object with respect to the hologram plane. A screen with two small apertures is placed in front of the hologram. In the image plane moire’ contours are formed, giving the displacement in the direction of the line connecting the centres of the two apertures. The theory of this phenomenon has been given by Duffy [S] The chief results of his theory are that the 302
Fig. 1. Scheme of the experimental set-up used to obtain contours of equal in-plane displacement. R = reference beam; I = illuminating beam; L = laser source; 0 = object; H = hologram; S = screen with two apertures; RI = real image of object 0; F = focal plane where contours are recorded.
lateral displacement per moire fringe is given by h/a, where (Yis the angular distance between the two apertures, and that the maximum displacement that can be measured is about X//3,where 0 is the numerical aperture of each of the apertures. In fig.2 we show the results of a preliminary experiment verifying the principle of this method. A double hologram was made of a diffusely reflecting disc rotated about its axis of symmetry between exposures. Rotating the screen with the two viewing apertures over 90” we made photographs in the image plane of the contours of equal in-plane displacement in two perpendicular directions. We show the contours for two values of the angle of rotation of the disc. As ex-
Volume 7, number 4
(a)
OPTICS COMMUNICATIONS
April 1973
(b)
Fig.2. Recordings are shown of the contours of equal in-plane displacement in two perpendicular directions of a disc rotating about an axis through its centre. In figs.2a and 2b the angle of rotation was about 1.5 min. of arc. In figs. 2c and 2d the rotation angle was 2.5 min. of arc. In figs. 2a and 2c the line connecting the viewing apertures in the hologram plane was horizontal. In figs. 2b and 2d this line was vertical. The contours of equal in-plane displacement are the horizontal and vertical contours in the figures. Besides these, the contours of equal longitudinal displacement are also visible. These contours would also have been obtained in a recording through one of the viewing apertures, as we have verified experimentally. 303
\‘l~lUrnC I.
Illlinhcr
4
OPTICS COMMUNICATIONS
petered, straight line contours are formed parallel to the. line connecting the viewing apertures. 111the photographs of fig.2 we also see the contours (I!-t,clual longitudinal displacement. These contours do 110t change when the screen with the two apertures is i-c;~:~tedabout the point midway between the apertures. Iii a comparable experiment in speckle interferoI:w TI1’ the contours of equal in-plane displacement \*,;~uId have been the only ones visible in the image pl:\nc. The situation in holography is somewhat more Iciiilplicated because two sets of contours are visible in r.;litl ptlotograph. But an advantage of holography o\i‘l speckle interferonietry is that more information I\ ;I\ .lilable: the longitudinal component of the deforIII;IIIOII vector can be obtained and the deforrnations 111‘ IIOI~-plane objects can be analysed more simply be(‘;IUX it is pvssible to obtain contours in different fo<.,il planes from one single hologram. III\IC;I~ of using two viewing directions, moiri con11!II I5 cut also be obtained by using two illuminating \\,i:e:\ [i,]. GI-eater sensitivity can thus be obtained at !!I(, c‘()>I of simphclty of the experimental set-up. The
304
April 1973
method can be made more flexible by using a television tube for recording the images. We should, by analogy with the work of Tiziani [2], also like to point out the application to the analysis of vibrations. The author is grateful to Mr. J. Bruin, of the Philips Medical Systems Deptirtment, who performed the experiment described in this paper.
References
[ 1] P.M. Boone, Opt. Laser Technol. 4 (1972) 162. [2] H. Tiziani, Opt. Acta 18 (1971) 891;Appl. Opt. 11 (1972) 2911. [3] J.W.C. Gates, in: Applications of holography, ed. J. Vi¬. Besaqon (1971). [4] J.A. Leendertz, J. Phys. E 3 (1970) 214. [S] D.E. Duffy, Appl. Opt. 11 (1972) 1778. [6] E. Archbold, J.M. Burch and A.E. Ennos, Opt. Acta 17 (1970) 883. [7] J.N. Butters and J.A. Leendertz. J. Phys. E 4 (1971) 277.