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fringe visibility and localization in two-beam hologram interperometry


imperial college, london, great britain

in interferometry with optically smooth surfaces and with an extended incoherent light source the classical concepts of fringe localisation and visibility are well understood; ses, e.g./1,2/. in hologram interferometry the fringes are formed between beams scattered from identical rough surfaces (the two images of the object in frozen fringe interferometry) and although the light source is usually a laser and is thus a diffraction-limited source the scattered beams from every region of the surfaces extend over a wide range of solid-angle.

in order to use hologram interferometry quantitatively for stress analysis etc it is important to know how to get the fringe to appear sharp and with good contrast on the surface of the object and it is thus desirable to understand clearly the concepts of visibility and localization. when a two-beam frozen fringe hologram is reconstructed and inspected visually it is common experience that the fringes usually move relative to the surface as the eye is moved; thus there is parallax between the fringes and the surface of the object and it can be said that the fringes do not appear to be localized on the surface. now consider the situation in which the reconstruction is photographed through a camera objective of considerable relative aperture; again it is common experience that if the iris of the lens is closed down to reduce the relative aperture the photographed fringes have good contrast, but as the iris is opened the contrast steadily drops, i.e. the visibility of the fringes decreases. this decrease in visibility with increasing aperture occurs whether or not the camera is focused on the surface of the object; for a given aperture it is possible to find a focal setting which gives the best co-

ntrast and usually this will not be on the surface of the object.

it is important to distinguish between these two effects of visibility and localization; the distinction has been drawn by welford /3,5/ who showed that in general there is no unique region of localization of the fringes. in other words, the parallax of the fringes relative to the surface varies with the direction of movement of the eye. it was also shown that there is not so, namely when the direction of viewing is parallel to the direction of local displacement of the surface; under this condition there is no parallax for small movements of the eye in any direction and the fringes are unambiguosly localized at the surface. in this case it is also possible to obtain a very simple expression for the maximum semi-angular aperture of a lens used to photograph the fringes; it was shown that if this angle Φ is less than


where |a| is the displacement distance and λ is the wavelength, the visibility or contrast of the fringes will not be appreciably lowered.

if the above condition is not fulfilled it is shown in the above references how to find the apparent regions of localization for different directions of movement of the eye; the fringe visibility, however, varies in a very complicated way with position of focus; in general it seems that there is a unique focal setting for given aperture at which the visibility is a maximum. the theory of this is complicated in detail although simple in principle, since the required focal setting depends on all the geometrical parameters involved, namely the distance and direction of viewing and the magnitude, direction and rate of variation of the looal displacement. it is found that -the visibility is a quadratic function of distance from the object surface and the stationary value, in general non-zero, gives a definite position of maximum visibility; however, thin has no simple relation to localization in the parallax sense.

r e f e r e n c e s

1. j.mace'de lepinay, c.fabry. j.de physique, 10, 5, 1891.

2. w.h.steel. "interferometry" (cambridge university prees),1969.

3. w.t.welford. optics communications, 1, 123-125,1969.

4. w.t.welford. optics communications, 1, 311-314, 1970.

5. w.i.welford."applications of holography", paper 4.2 (proceedings edited by j.c.viènot, j.bulabois and j. pasteur, besancon), 1970.

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