Multiple-Exposure Holography P. C. Mehta

Instruments Research and Development Establishment, Dehra Dun, India Received 19 November 1973.

Holograms are usually recorded by using a high refer­ence-to-object beam intensity ratio. But the intense ref­erence beam produces ghost images and flare light in the reconstructed images and thus degrades the image quali­ty. In the case when the power of the laser available is low and the object is large, it becomes difficult to record good holograms. To eliminate the use of an intense refer­ence beam and high reference-to-object beam intensity ratio for recording the hologram and at the same time to retain the quality of the reconstructed images, a tech­nique has been suggested1-4 in which a large number of holograms are recorded on the same area of the emulsion by dividing the object in a number of parts and scanning the whole object by illuminating only one part during each exposure. However, the multiple exposures decrease the reconstruction efficiency and the recording material becomes overexposed if the number of exposures and each exposure time are not adjusted properly. Lang et al1

have shown that it is possible to produce a multiple-expo­sure hologram with the same diffraction efficiency as that of a single-exposure hologram, if appropriate exposure conditions are used. They compared the multiple-expo­sure, multiple-object hologram with a single-exposure, multiple-object hologram. In the present communication the multiple-exposure, multiple-object hologram has been compared with a single-exposure, single-object hologram and useful conditions have been derived in order to record good multiple-exposure holograms.

In the conventional single-exposure, single-object tech­nique the exposure E to the emulsion is given by

where T is the recording time, 10 the object beam intensi­ty, ø the phase difference between the object and the ref­erence beam, and ρ the reference-to-object beam intensity ratio.

In the multiple-exposure technique, the object is divid­ed into N parts and a hologram of each part is recorded by N exposures on the same recording emulsion. Let the object beam amplitude of the nth exposure at the record­ing plane be On exp (i øn) and that due to the reference beam be Rexp (i ør). If T is the exposure time for each exposure, the total exposure En to the emulsion for all the N exposures is given by

same. Or more specifically, the values of the bias and the fringe visibility in both the cases must be equal. The bias E and the fringe visibility Vare given by

and

Equations (1) and (2) with the help of Eqs. (4) and (5) give

and

with the assumption1 t h a t g m i n (N) = -g m a x From Eq. (3) it is evident that [gmax(N)/N] <1 . In the

extreme case this may be equal to 1. This with Eq. (7) yields a condition

The required condition is, therefore,

and

Equations (6)-(8) are useful for obtaining the values of each exposure time and the number of times the recording medium may be exposed in order to avoid the overexpo­sure of the emulsion. Figure 1 shows the variation of (Nt) with ρ for different values of ρ′ with Io = Io′. The re­quired exposure time for any value of N can be found. For Io/Io′ = x, (Nt) may be obtained by multiplying the corresponding values of t from the graph by x. In Fig. 2, (gmaχ(N)/N) has been plotted against ρ for different values of ρ′. For a given value of gmax(N), N may be ob­tained with this graph.

If we assume 0 = On, T = T′ and R2 ≫ 02 Eq. (6)

where lo' and ρ' are the object beam intensity and the ref-erence-to-objéct beam intensity ratio, respectively; and g(N) is known as the interference function given by

In deriving Eq. (2) it has been assumed for simplicity that the recording time for each exposure is the same (T' = tT) and the absolute amplitude of the object beam at the recording plane is the same in each case.

In order to retain the characteristics of conventional single-exposure, single-object recording in the multiple-exposure case, the exposures in the two cases must be the Fig. 1. Variation of Nt with ρ for different values of ρ′.

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Fig. 2. Variation of [gmax(N)/N] with ρ for different values of ρ′.

would lead to a relation which connects the two hologram memories under consideration by the relation:

If we compare Eq. (11) with Eq. (2.16) of Ref. 1, we get the following relation connecting different hologram memories:

It may be mentioned that the technique of multiple ex­posure can be thought of in terms of pre- or post-exposure. In the multiple-exposure method each exposure serves as a bias for the previous or next recording. If the multiple-exposure technique is required for holographic multiplex­ing, i.e., to store different signals on the same recording emulsion, then On can no longer be regarded as the same. In this case the treatment will be slightly different.

Multiple-exposure holography has several useful appli­cations, viz., holographic multiplexing,3,5,6 optical data processing,3,7 pre- or post-exposing the hologram for dif­fraction efficiency optimization,8,9 holographic interfero-metery,3 , 1 0etc.

The author is grateful to P . K. Katti for helpful discus­sions.

References 1. M. Lang, G. Goldmann, and P. Graf, Appl. Opt. 10, 168

(1971). 2. M. Lang, Opt. Commun. 3, 141, 229 (1971). 3. R. J. Collier, C. B. Burchardt, and L. H. Lin, Optical Holog­

raphy (Academic Press, New York, 1971). 4. C. S. Vikram and H. Vardhan, Nouv. Rev. Opt. Appl. 3, 185

(1972). 5. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).

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6. P. C. Mehta, Information Storage by Holographic Multiplex­ing, All India Symposium on Digital Image Processing, Indian Institute of Science, Bangalore, November 1973, Symposium reprints, p. I.l.

7. H. Akahori and K. Sakurai, Appl. Opt. 10, 665 (1971). 8. H. J. Caulfield, S. Lu, and J. L. Harris, J. Opt. Soc. Am. 58,

1003 (1968). 9. J. T. LaMacchia and C. J. Wincelette, Appl. Opt. 7, 1857

(1968). 10. 0. Bryngdahl, Appl. Opt. 7, 2322 (1968).