Holographic Multiple-Beam Interferometry
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Vol. 71, No. 6/June 1981/J. Opt. Soc. Am. 635 Restoration of arbitrary finite-energy optical objects from limited spatial and spectral information H. Stark and D. Cahana Department of Electrical and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York 12181 H. Webb Rome Air Development Center, Griffiss Air Force Base, New York 13441 Received November 10, 1980 The mathematical generalization of image restoration by recursive methods furnished by D. C. Youla [IEEE Trans. Circuits Syst. CAS 25, 695-702 (1978)] is used to show that arbitrary L 2 (i.e., square-integrable) images can be re- constructed from two projections without any a priori assumption regarding the mathematical properties of the object, such as space-limitedness or band-limitedness. Recursive algorithms are given to restore images from (1) extended segments and low-pass spectra and (2) short segments and high-pass spectra. Using the alternating pro- jection theorem, we prove monotonic convergence (in the norm) to the original image. 1. INTRODUCTION f = ;+ h, In a recent paper,' Youla generalized the underlying theory of image-restoration algorithms of the type discussed by Papoulis 2 and Gershberg. 3 Youla addressed and furnished the answers to two basic questions: (1) under what conditions is an image f uniquely determined by a single projection, and (2) under what conditions is the restoration of the image from its projection stable or well posed? The opposite of well- posedness is ill-posedness, and a restoration is said to be ill posed or incompletely posed if the restoration depends uniquely but not continuously on the data. In this paper we use some of the mathematical machinery used by Youla to restore a broader class of signals from two projections. We also consider separately the problem of in- terpolation because of its significance in imaging and optics, especially in computer-aided tomography. However, in order to acquaint the reader with the notation and some of Youla's key results, we first consider possible versus impossible res- torations. 2. POSSIBLE VERSUS IMPOSSIBLE RESTORATIONS Following Youla, we consider signals f(x), g(x), h(x), etc. of the Hilbert space X1, consisting of the L 2 (-c-, a) space of square-integrable, Fourier-transformable functions equipped with the inner product (fYg) =f f(x)g*(x)dx. (1) Let P be any closed subspace of X, for example, (1) the space of all space-limited functions that vanish in Ix I > a or (2) the space of all band-limited functions whose Fourier transforms vanish in I wI > b. Each f e X possesses a unique decompo- sition where g is the projection of f onto P and h is the projection of f onto the orthogonal complement of P, written as IP. For g and h, as described above, | g(x)h*(x)dx = 0. We define the projection operators P and Q by the rules g = Pf, h = Qf. (3) (4) (5) * Thus, for example, if P is the subspace of all functions that vanish in Ix I > a, the projection operators P and Q are de- scribed by g(x) = rect(-2) f(x), h(x) = 1 - rect( )I f x. (6) (7) Here the orthogonal complement of P, IP, consists of all functions that vanish in Ix I < a. An important subspace is the space of b-band-limited functions of finite energy, for which the operation g = Pf is described by g W= A) -sin xbI g(x)=f(x) * (8) where * denotes convolution. Here P is an integral operator with the sinc(bx/br) smoothing kernel. The first of Youla's main results can be stated as follows: Suppose that we know that f lies in some closed subspace Pb, but we are given only its projection g = PaJ onto the known closed subspace P, Can we recover f from g? Let Pa,, Qa, Pb, Qb be the projection operators that project onto P,, IP 0 , 'Pb, l'Pb,respectively,withf e Pb. Then(partoneofYou- 0030-3941/81/060635-08$00.50 © 1981 Optical Society of America (2) Stark et al.
Transcript of Holographic Multiple-Beam Interferometry
J O U R N A L O F T H E OPTICAL SOCIETY O F A M E R I C A VOLUME 59, NUMBER 6 JUNE 1969
Letters to the Editor
Holographic Multiple-Beam Interferometry KAZUYA MATSUMOTO
Canon Camera Co., Inc. 30-2, Shimomariiko 3 Chome, Ohtaku* Tokyo, Japan
(Received 5 December 1968) INDEX HEADINGS: Holography; Interferometry; Gratings.
Two-beam interferometry, which enables us to measure the phase variations of an object, is an old and well-established branch of interferometry. However for precise measurement, it is desirable to have sharper and narrower fringes than sinusoidal two-beam fringes, because sharper fringes give more exact fringe location. Such fringes can be obtained with multiple-beam interferometry.
Burch et al.1 attempted to obtain sharper and narrower fringes with holographic interferometry. Their method was restricted to objects that can be bent. I have found that a multiple-interference effect can be introduced by the nonlinear characteristics of photographic materials; this effect can be applied to measure the phase variations of any object.
Suppose that the object is a pure phase object, with the phase ξ{x,y), that the reference wave is a plane wave which is represented by exp[i(2π/λ)x sinθ], and also that the recording is nonlinear; then the following wavefronts are generated from the hologram,2-4
where m is an integer,
and
Qm is the amplitude of the mth-order diffracted wave and δm is the phase retardation of the mth-order wavefront at diffraction. Equations (1) show that many higher-harmonic wavefronts are constructed from the nonlinear hologram, and that each wavefront propagates in a different direction. When the angle θ is small, the directions are given by mθ.
If these wavefronts could be superposed in one direction and have some phase variation εm, then the total irradiance in this direction can be written as
When the Qm and (δm+ εm) are chosen properly, the interference fringes may have a profile sharper than sinusoidal. They can be sharp as multiple-beam interference fringes.
Several methods to obtain the desired superposition can be found. Here a method using a diffraction grating will be described. A grating having the same spacing as that of the hologram diffracts each wavefront that is generated from the hologram; wavefronts of every order travel in the same direction, as shown in Fig. 1
FIG. 2. Holographic multiple-beam interferograms; (a) photograph of holographic multiple-beam interferogram, (b) irradiance curves of aerial fringes in (a).
FIG. 1. Optical system for holographic multiple-beam interferometry. FIG. 3. Holographic two-beam interferograms; (a) photograph of holo
graphic two-beam interferogram, (b) irradiance curves of aerial fringes in (a).
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FIG. 4. Holographic multiple-beam interferogram obtained by rotating a grating around the normal to the hologram.
FIG. 5. Holographic multiple-beam fringe obtained by varying the separation between a hologram and a grating.
(only the wavefronts in the + 1-order direction are described). In this case, the phase variation εm is related to the spacing between the hologram and the grating.
When we observe the hologram from a certain direction with a collimator lens and a small aperture, used as a spatial filter, multiple-beam fringes can be observed. Figure 2(a) shows the fringes observed from the + 1-order direction of diffraction. Figure 3 (a) shows the two-beam interferogram obtained by interference between a reconstructed + 1-order wavefront and a plane wavefront. Figures 2(b) and 3(b) are irradiance curves of the aerial fringes shown in Fig. 2(a) and 3(a), respectively. The difference of fringe sharpness between the holographic multiple-beam fringe and an ordinary holographic two-beam interference fringe is apparent. In this case, the orientations of the fringes produced by the hologram and by the grating were adjusted to be parallel.
I t is possible to vary a fringe pattern, as in ordinary inter-ferometry, by giving a slight tilt to a testing wavefront with respect to a reference wavefront. Tilting effects in x and y directions are obtained by varying the grating constant and by rotation of the grating around the normal to the hologram, respectively. An example of this tilt effect is shown in Fig. 4. This pattern was obtained by a slight rotation of the grating used for Fig. 2. A hologram that was photographed with the same optical setup without an object, can be used as a grating. In these experiments Fuji Process panchromatic plates and Copinal developer were used, and ±6th-order diffracted waves were observed with the unaided eye; therefore Qm was not negligible for w = ± 6 . When the spacing between the hologram and the grating is varied, the phase εm of each diffracted wavefront is changed. As a consequence, the fringe of multiple interference acquires an unusual profile. As an example, the fringes in Fig. 5 are similar to those that can be obtained by "Äquidensitometrie."5
* Mailing address: P.O. Box #50, Tokyo Airport. 1 J. M. Burch, A. E. Ennos and R. J. Wilton. Nature 209, 1015 (1966). 2 O. Bryngdahl and A. W. Lohmann, J. Opt. Soc. Am. 58. 141 (1968). 3 A. A. Friesem and J. S. Zelenka, Appl. Opt. 6, 1755 (1967). 4 J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968). 5 E, Lau and W. Krug, Die Aquidensitometrie (Akademie-Verlag, Berlin,
1957).
