Improving binary computer holograms

Volume 15, number 1

OPTICS COMMUNICATIONS

September 1975

IMPROVING BINARY COMPUTER HOLOGRAMS H. BECKER and W.J. DALLAS

PhysikalischesInstitut der Universitiit,Erlangen, WestGermany Received 12 May 1975

We discuss three algorithms for computer-generating binary holograms. These algorithms yield holograms with differing aperture distributions. Reconstructions from these three hologram types are experimentally compared for image detail and diffraction efficiency. We conclude that the form we term modular-overflow-corrected yields the highest diffraction efficiency and the form we term 3/2-order yields the best image detail of the three hologram types compared. Holograms synthesized by computer offer advantages over conventional holograms. Prominent among these advantages are the ability to create optical wavefronts from objects that do not physically have to exist, just be numerically specified. Since the recording step is not interferometric, the object volume is no longer physically constrained by illumination coherence or air turbulence [1,4,6]. Various techn~ues have been devised for encoding the complex wavefront to be recorded as the hologram. This most often results in a real and non-negative function which is then drawn on a mechanical plotter or displayed on a CRT [1,3,4,7,8]. It is desirable to further restrict the plotted function to being binary, yielding binary transmittance patterns. These binary holograms have advantages in both recording and reconstruction. In recording they exhibit an insensitivity to imperfections in photographic reduction processes. In reconstruction, a binary hologram yields less noise from light scattered by the photographic grain structure [6,10]. The binary hologram consists of an array of apertures on a dark background, whose sizes and positions determine the amplitude and phase of a diffracted wave, and many possible aperture distributions have been presented [3,7,8] which yield acceptable approximations to any one desired wavefront. We offer here an experimental comparison among three methods of positioning the apertures. The first method [1,3] results in the uncorrected Lohmann type III hologram. This type of hologram, however, is afflicted by the "gap and overlap problem" (see below). 50

The other two positioning methods we consider are modifications thereto, aimed at eliminating this drawback. The original Lohmann hologram is presented in ref. [1 ] and justified with rigorous mathematics in ref. [3]. This hologram consists of resolution cells of dimensions v0 by/.t o located on an M × N-element rectangular lattice, each cell containing a rectangular aperture placed so as to encode the amplitude and phase information of the desired wavefront at a corresponding sampling site. The positioning of the aperture within its cell is shown in fig. l a, the numbers Amn, Bmn, Pmn are the parameters used to achieve this encoding at the (m, n)th position within the hologram. Cell transmittivity is then

"CmnQa,v) = rect [(/l-PmntX0)/Amn~0] rect

(v/BmnvO). (1)

The hologram is thus described by H02,v) = E E m n

CmnOX-mlao,v-nuo).

In Fourier holography, with which we work here, the hologram reconstructs an image

U(x,Y) = ~ ~ Cmn(X,y)exp [27ri(mlaOX+nvoY)] , m

n

Cmn(X,y)being the Fourier transform of CmnOl, v):

with

(2)

Volume 15, number 1

OPTICS COMMUNICATIONS

Cmn (X , y) = A mn Bmn #O v0 sinc (A mn #Ox) × sinc (Bmn v0 y) exp (2 rri Pmn #0 x ) .

(3)

Note the similarity of U(x, y) to a Fourier series; we might call the spatially varying Cmn the pseudoFourier coefficients. We have so far considered the reconstruction of the hologram, let's turn now to determining the cell parameters so as to reconstruct a desired image. Let the image to be reconstructed be:

f ( x , y ) = ~ ~ finn exp [27ri(m#ox+nPoY)], m n

(4)

then we can achieve approximate equality of the two series (2) and (4) around'some point by equalizing the Fourier coefficients one by one [3]. Notice also that since the carrier frequency of our hologram is 1/g0, the position of the first diffraction order will be at the coordinate x = Xf/#o, Y = 0, according to Fourier holography ( f b e i n g the focal length of the lens used for transformation, ~. is the wavelength of the monochromatic light). Now to the approximation: if we observe around x = Xf/#o, then the factor exp (2rriPmnt_toX) will become: exp [27riPmn #0 (1/#0 + e)] = exp (27rip mn )

September

19"f5

the (m + 1)th cell. If the phase in this cell is dPm+l,n ~ rr/2, then overlapping of the apertures will occur [6]. One modification which we implemented to avoid overlapping is the "modular overflow-correction" (cf. also ref. [1 I]). It makes use of the fact that the phase is periodic modulo 27r. The overflowing part of the aperture is simply plotted on the other end of the same cell (fig. lb). Fig. 2b shows the reconstruction from a 32 × 32 overflow-corrected hologram. The image is somewhat more differentiated and clearer than in fig. 2a. The diffraction efficiency is also higher than in fig. 2a. Another modification that we implemented to confine the whole aperture within its cell, was to reconstruct in the 3/2-diffraction order. Lohmann and Paris [3] discuss the possibility of reconstructing in higher (integer) diffraction orders. To make the image clear in the Mth diffraction order one assigns: Pmn = (~mn/27rM. They show that the severity of approximation [eq. (5)] is reduced with higher M, but image brightness decreases. Reconstructing in the noninteger 3/2-diffraction order, then, means using the lowest order featuring maximized cell width with no overlap. We observe now at x = 3 )~f/2/a 0, so eq. (5)becomes: exp (2rriPmn#O x) = exp (37riPmn) = exp (i~mn) ,

X exp (2rriPmng, oe ) ~ exp (27riPmn) .

(5)

(e = small number). The factor sinc (Amn gOx) becomes sinc (Amn) and sinc (Bran v0 y) ~ 1. Equating the Fourier coefficients thus yields [with finn = amn exp (iq~mn)l: exp (iq~mn) = exp (2 7riPmn) , so

Pmn = 49mn/2rr ,

Bmn = amn •

(6)

The factor sin (Amn 7r)/Trwill influence image brightness; Amn = 1/2 resulting in brightest images, #0 and v0 remaining as scaling factors. All there is to do, then, is to calculate fmn = amn exp O(~mn) by means of the Fast Fourier Transformation [9], then t 9 set Pmn and Bran as indicated above. Fig. 2a shows the reconstruction of a 32 × 32cell hologram produced this way. As is evident from fig. la, if (~mn > 7r/2, the aperture will cross over into

and thus

Pmn = ~mn/3 7r .

(5a)

Again: Bmn = amn, and sin (Amn 3 n/2)/n has its maxim u m at Amn = 1/3. The resulting cell structure is shown in fig. 3a. A minor detail left to consider is to somehow position the reconstructed image, which is still at x = Xf/#O, at the x = 3 kf/la 0 location, where the reconstruction is now clearest. By simply defining as o u r n e w object:

g(x, y) = f ( x - 1/2UO, Y) (that is: shifting the object periodically by half its lateral dimension) the reconstruction will appear to be located at the 3/2-order. Fig. 3b shows a reconstruction thus obtained. Notice how the image is now further out of the noisy center and image detail is noticeably sharper than in fig. 2b.

Conclusion: Two modifications to the original 51

((n+1)~o,

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l

It

~t

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i~

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p'~ mn

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n

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I

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.

(m+1) %Po) - I,../

((n+1) ~to ,re%Do)

i

t I

Amn

'm

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Fig. 1a. The structure o f the cell in the (n, m) th position 1" within a Lohmann-type III computer hologram. The darker area denotes the aperture within this cell. Fig. lb. Reconstruction from a 32 by 32-cell binary Lohmann ~ type lII computer-generated hologram.

f

/ / / / / / / / /

/

/

/

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mn

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mn

mn

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Amn=Amn+Amn

Fig. 2a. Structure o f the same cell as in fig. la, however including overflow correction modulo 2 ~r for - 1 / 4 ~< Pmn < 1/4. Fig. 2b. Reconstruction from a 32 by 32-cell binary hologram featuring modular overflow-correction. 52

Volume 15, number 1

OPTICS COMMUNICATIONS

September 1975

.

i !n !1

I I

I I

I I

I I

Amn

"]

Fig. 3a. Structure of a cell to reconstruct in the 3/2-diffraction order. Amn = 1/3, -1/3 <~Pmn < 1/3. Lohmann type III computer-generated hologram have been presented. Experimental comparison shows that both modifications yield improved images as compared with the original version. The modular overflowcorrected type delivering higher light efficiency than either the original Lohmann version or the 3/2-order version. The 3/2-order version, in turn, leads to sharper and clearer images than the other two. We have found that reconstructing computer-generated holograms o f the Lohmann type in the 3/2-order as presented is a simple and rather straightforward modification that can easily be implemented and requires no additional computer time. Holograms o f 3-dimensional objects have also been successfully reconstructed in the 3/2order. The authors wish to thank Professor A. Lohmann for his encouragemeiit in presenting this work as well as for many helpful discussions.

Fig. 3b. Reconstruction in the 3/2-diffraction order from a 32 by 32-cell binary computer generated hologram.

References [1] B.R. Brown and A.W. Lohmann, Appl. Opt. 6 (1966) 967. [2] A.W. Lohmann and D.P. Paris, Appl. Opt. 6 (1967) 1139. [3] A.W. Lohmann and D.P. Paris, Appl. Opt. 6 (1967) 1739. [4] L.B. Lesem, P.M. Hirsch and J.A. Jordan, Comm. Assoc. Comp. Mach. 12 (1968) 661. [5] A.W. Lohmann and D.P. Paris, Appl. Opt. 7 (1968) 651. [6] B.R. Brown and A.W. Lohmann, IBM J. Res. Dev. 13(2) (1969) 160. [7] W.H. Lee, Appl. Opt. 9 (1970) 639; C.B. Burckhardt, Appl. Opt. 9 (1970) 1949. [8] P.R. Ransom and R.M. Singleton, Appl. Opt. 13 (1974) 717. [9] W.T. Cochran, J.W. Cooley et al., Proc. IEEE 55 (1967) 1664. [10] T.C. Strand, Opt. Eng. 3 (1974) 219. [11] W.J. Dallas, Appl. Opt. 13 (1974) 2274.

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