From OE Reports June 1994
The problems to be solved in order to realize such a camera are essentially: the noninvertibility of the point spread function (PSF) of the basic system (which is the familiar Gabor zone lens (GZL)), the nonlinearity of the shape (i.e., 3D) reconstruction, and the sensitivity of the 3D reconstruction to noise and defects in the impulse response.
The numerical reconstruction of the image of an object (2D reconstruction) is an important step towards the reconstruction of its shape; in principle, this 2D reconstruction is relatively easy since it simulates an optical reconstruction-as in coherent holography-but it faces the same problems as the latter, i.e., the presence of undiffracted light (or DC bias) and of a conjugate image. By linearly combining different PSFs of the system, each of them being obtained by adequately changing the polarization state and the amplitude of the incident light field (by means of a light valve and of a rotating amplitude mask, see figure), we have been able to obtain an invertible PSF-that is, one that has the same mathematical expression as the amplitude transmittance of a lens, and thus yields neither bias nor conjugate image in the reconstruction process.3
The hologram of an object is the incoherent superposition of the GZLs of all the object's points, but is no longer a convolution when the object is 3D. By writing the hologram in a differential form instead of its natural integral form, we have devised an algorithm to recover the shape of an opaque 3D object from its conoscopic hologram. Moreover, this algorithm is not restricted to conoscopic holography and could also be used on other interferometers with similar PSF.
Nevertheless, due to the very physics of the interference phenomenon, the signal-to-noise ratio of the object's shape decreases in the low spatial frequencies, which are the ones that determine the overall shape. This can be understood intuitively in the following way: the object's shape is coded in the variations of the PSF (hologram of a point) when the longitudinal position of the point changes; and it is essentially the fine fringes (and thus the high spatial frequencies) that vary. Consequently, the high frequencies of the shape will be better recorded than the low frequencies, and the reconstruction, although validated on simulations, will be unstable for these latter frequencies. In order to restore them, we have proposed an iterative algorithm4 that takes advantage of the knowledge of the object's support (obtained through the image reconstruction), and makes the shape reconstruction less sensitive to noise and to defects in the experimental PSF. The very encouraging experimental 3D results that have been obtained validate the potential of conoscopic holography as a three-dimensional imaging technique.
D. Charlot, Former Technical Director of Le Conoscope S.A.; and G. Y. Sirat, Le Conoscope S.A. 12 Avenue des Prés, F-78180 Montigny-le-Bretonneux, France.
1. G. Y. Sirat and D. Psaltis, "Conoscopic holography," Opt. Lett. 10, No. 1, January 1985.
2. D. Charlot, "Holographie conoscopique. Principe et reconstructions numériques," Ph.D. dissertation, TELECOM Paris (1987).
3. L. M. Mugnier, G. Y. Sirat and D. Charlot, "Conoscopic holography: two-dimensional reconstructions," Opt. Lett. 18, No. 1, January 1993.
4. L. M. Mugnier, "Vers une inversion des hologrammes conoscopiques," Ph.D. dissertation, TELECOM Paris (1992).