Phase Multiplication in Holographic Interferometry

OLOF BRYNGDAHL* IBM Research Division, San Jose, California 95114

(Received 14 November 1969) INDEX HEADINGS: Interferometry, Holography.

In interferometric studies of small phase variations, it is often of value to be able to multiply the amount of the phase shift in the wavefront emerging from the object under test. Recently, different possibilities to achieve this have been demonstrated, such as phase amplification by using multipass interferometry1 or by utilizing inherent properties in wavefront-reconstruction techniques.2

Here we want to point out an alternate way and show how a photographic procedure applied on holograms can give the same result.

The idea is the following: The phase information in a hologram (interferogram) is contained in the relative displacement of the fringes. If we could form additional fringes between the fringes in the hologram, the relative fringe displacement would be doubled. The technique called equidensitometry3 gives an equiva­lent result.

The procedure is applicable to image holograms, which will then reconstruct a wavefront with the phase variations amplified. We tested this idea experimentally, using a glass step wedge as object. Figure 1 (a) shows a magnified portion of an image holo­gram of the object. The spatial frequency in the hologram was 25 lines/mm. For phase multiplication, we interrupted develop­ment of the exposed plate, diffusely illuminated it, and then com­pleted the development. The result is shown in Fig. 1(b). The

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JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 60, NUMBER 6 JUNE 1970

FIG. 1. Recording of equidensities in holograms in order to obtain phase multiplication in the reconstructed wavefront. (a) Magnified portion of a conventional hologram; (b) equidensities of the same portion; (c) equi­densities of second degree.

fringes correspond to points of equal density in Fig. 1 (a). He–Ne laser light and Kodak 649 F plates were used to record the holo­grams. The plate in Fig. 1(b) was developed in D-19 for 2.5 min before the darkroom light was turned on for a second exposure, and then further developed for 2.5 min. The relative fringe dis­placement in the conventional hologram [Fig. 1(a)] correspond­ing to the step height in the object is about 1.5 fringes and in the hologram in Fig. 1 (b) about three fringes. Therefore, the hologram in Fig. 1 (b) reconstructs a wavefront with double the phase varia­tion compared to the reconstructed wavefront from the hologram in Fig. 1(a).

We found this method applicable for spatial frequencies up to 200 lines/mm in the original interference pattern. For low spatial frequencies, it is also possible to form equidensities of higher degree. Figure 1(c) shows second-degree equidensities obtained for the same object. Four-times phase multiplication was achieved.

I highly appreciate H. Werlich's skillful assistance with the experiments.

* Present address: IBM Research Center, P.O. Box 218, Yorktown Heights, New York 10598.

1 P. H. Langenbeck, Appl. Opt. 6, 1425 (1967); 8, 543 (1969). 2 K. Matsumoto and T. Ose, Seisan-Kenkyu, J. Inst. Industr. Sci.

(Univ. Tokyo) 19, 18 (1967); M. De and L. Sévigny, Appl. Opt. 6, 1665 (1967); O. Bryngdahl and A. W. Lohmann, J. Opt. Soc. Am. 58, 141 (1968); O. Bryngdahl, J. Opt. Soc. Am. 59, 142 (1969); K. Matsumoto and M. Takashima, J. Opt. Soc. Am. 60, 30 (1970).

3 E. Lau and W. Krug, Die Aquidensitomelrie (Akademie-Verlag, Berlin, 1957).

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