Coding and binarization in digital Fresnel holography

Volume 77, number 1

OPTICS COMMUNICATIONS

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C O D I N G A N D B I N A R I Z A T I O N IN D I G I T A L F R E S N E L H O L O G R A P H Y M. B E R N H A R D T , F. W Y R O W S K I and O. B R Y N G D A H L

Department of Physics, Universityof Essen, 4300Essen 1, bed. Rep. German)' Received 6 February 1990

Aspects of coding and binarization in digital Fresnel holography are discussed. A binarization scheme, based on the iterative Fresnel transform algorithm, is described. Optical and digital reconstructions are presented.

I. Introduction A Fresnel hologram reconstructs a signal in a finite distance behind the hologram plane. In comparison to digital Fourier and image plane holography [ 1 ] the effort to calculate a Fresnel hologram is comprehensive. To fabricate a digital Fresnel hologram as a transmission mask, i.e. an amplitude hologram, it is necessary to introduce a coding scheme. In the following a procedure to generate a Fresnel hologram is described which utilizes a coding scheme known from digital Fourier holography [2]. To meet requirements set by recording media and to avoid disturbing influences due to nonlinearities the holograms are made binary. A digital binarization scheme based on an iterative Fresnel transform algorithm is presented.

2. Relation between signal and hologram To produce a digital Fresnel hologram we need to perform the inverse diffraction from the signal to be reconstructed to the hologram plane. The complex amplitude to be stored in the Fresnel hologram is FRT - ' [f(x ) ] = f e x p [ - i z c ( 2 z ) -1 ( u - x ) 2 ] f ( x ) d Z x ,

signal . f ( x ) = f ( x , y). Hereby it is assumed that a plane wave illuminates the hologram and that the Fresnel conditions are fulfilled [ 3 ]. The coordinates in the signal and hologram planes are x = (x, y) and u = (u, v), the distance between these planes z and the wavelength 2. The calculation of the complex amplitude in the hologram plane can be performed as a Fourier transform ( F T ) , if the signal and its calculated inverse Fourier transform are multiplied with spherical phase factors, i.e., FRT -1 If(x) ] = e x p [ -iTr(2z) ×FT

l{f(x)exp[-i~r(2z)

lu2] 1x2]}.

(2)

In the case the phase factor e x p [ - i n ( 2 z ) - ~ x 2] is omitted a phase error will occur in the reconstruction. In display situations, where the intensity i(x) is the observed quality, this is of no concern and it is enough to compute the inverse Fourier transform o f f ( x ) = i(x)1/2 exp [i~0(x) ]. We have then reduced the calculation of the complex amplitude to be stored in the hologram to the synthesis of the FRT pair

F(z, u) = F R T - i [~(x) ] = e x p [ - i z t ( 2 z ) - 1 u 2 ] FT -~ [J(x) ]

(3a)

with (1)

~(x)=i(x)~/2exp[i(q~(x)+rc(2z)-~x2)] .

(3b)

i.e., the inverse Fresnel transform ( F R T - I )

of the

4

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3. Coding procedure To store the complex amplitude of eq. (3a) in an amplitude hologram a coding step has to be performed. In digital holography this can be achieved by making the real distribution R e [ F ( z , u)] positive by adding a bias function [ 2 ]. A carrier is introduced by positioning the s i g n a l f ( x ) with the extent AX= (AX, AY) and x ~ = [ - Ax/2, Ax/2] in Xo= (Xo, Yo) and an autocorrelation distribution is avoided in the reconstruction by the constant bias B(z, u)=min{Re[F(z, u)]}. A continuous transmission variation

G(z,u)=½ [F(z,u)+F*(z,u)]+B(z,u)

(4)

can be formed on a holographic material. Illuminating this hologram with a plane wave, the diffraction pattern g (x) = FRT [ G ( z, u ) ] is generated. Thus, in addition to the signal f(x-xo) reconstructed in the signal window xe ~ = [Xo-AX/2, Xo+ zkx/2], a defocussed twin signal and an extended zeroth order occur (fig. la). The extent of the zeroth order does not allow the signal to be positioned as close to the optical axis as in digital Fourier holography. In order to account for the extended zeroth order the discrete signalf(m-mo) =f(x-xo) comb(x, 8x) is displaced off-axis. This is achieved if f (m-too) with the extent My= (My, Nf) = (AX/Sx, AY/Sy) and me[mo-Mf/2+l, mo+Mf/2] is displaced by mo= (too, no) in a data array of M = (M, N) zeros.

The sampling distance is 8x= (Sx, By) and comb(x, 8x) =E,,,,,8(x-mSx) 8(y-nSy). The complex amplitude (eq. ( 3 a ) ) is then calculated utilizing the discrete Fourier transform. The coding procedure of eq. (4) results in a discrete hologram distribution

G(z, k)=G(z, u) rect (u, AU)

comb(u, 8u) ,

(5)

with the extent A U = ( M S u , NSv) on the raster 8u= (Su, By). The constant bias B(z, u) implies that the extent of the zeroth order is approximately equal to the extent of the hologram AU. To spatially separate the signal from the zeroth order, the vector mo has to be

(mo, no)> ( M28u2 \

Mf. N28v2 ~)

2)~z + 2 - '

22~-7-+

"

(6)

Assuming that M=mo+MJ2, a proper mo according to eq. (6) can then be found, if '

-

-~z/O

(7)

is fulfilled. The condition (7) is not sufficient to avoid signal disturbances caused by superposed higher diffraction orders, generated due to the rastering of the hologram distribution (fig. 1a). This aliasing effect can be avoided (fig. lb) by choosing mo according to condition (6), if

Fig. 1. (a) Computer simulated reconstruction in z = 0.05 m with mo= - ( - 184, 184). (b) Optical reconstruction in z = 0.15 m with m= ( - 136, 136). Parameters: Mf=Nf=145, M=N=512.

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(8) is valid [4]. It is interesting to note that to reconstruct an undisturbed signal in digital Fourier holography, Mr
4. Binarization procedure

A simple method of binarization is to apply a hardclip operator (32 on the distribution G(z, u). The binary distribution is then

G ( z , u ) = ( 3 2 [ G ( z , u ) ] = l , G(z,u)>~0.5, =0,

G(z,u)<0.5,

crate a binary hologram of optimized diffraction etficiency. This hologram will reconstruct an almost undisturbed intensity signal. The iterative binarization scheme is applicable if the reconstructed intensity ]g(x)[2=i(x) is of concern and the phase ~ ( x - x o ) of the signal f ( x - x o ) = i ( x - x o ) ~/~ exp [i~0(x-xo) ] can be used as a free parameter. This phase freedom allows us to find a phase ~o(x-xo) so that the intensity of the signal is reconstructed correctly by the binary hologram. Then, we have Ig(x) j x = I f ( x - x o ) +q(x)l 2 ~ i ( x - x o ) within the signal window xe.Y. This means that the noise superposcd on the signal in the reconstruction is adapted to the signal in xe.Y. Therefore, in the iteration (fig. 2) a special operator × is applied in each iteration cycle j to obtain the phase ~0j(X-Xo). The resulting complex amplitude is g,+~ (x) = × [g,(x) ]

=cti(x-xo)J/2exp[i~oAX-Xo) ] , x~ ~ , =g,(x),

(9)

where G(z, u) is normalized to 1. The influence of this operator can be described by an additional term Q(z, u), i.e.

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otherwise.

12)

with the scale factor

C,=I i(X--Xo)l/2 ' g s ( x ) d 2 X ( f i(x)d2A- )

' 13)

G(z, u) =G(z, u) +Q(z, u ) .

(10)

The corresponding reconstructed pattern is g(x)=FRT[(~(z,u)]

=g(x)+q(x)

.

Input :

initial signal go (x)

(11 )

The binarization noise q(x) caused by (32, is coherently superposed onto the signal in the reconstruction. As a consequence the diffraction efficiency of the binary hologram is increased compared to the analog hologram. Unfortunately, the signal is distributed by the noise q(x). Utilizing the error diffusion algorithm [5,6] a binary hologram can be generated which reconstructs the signal spatially separated from the noise. As a consequence, the diffraction efficiency o f this hologram is about equal to that of the corresponding analog hologram [ 1 ]. Furthermore, the spatial separation to obtain an undisturbed signal reconstruction is possible to achieve if the SBWP is sufficiently increased compared to the analog hologram. We present a binarization method based on an iterative Fresnel transform algorithm [7,8] to gen-

1 Phase adjustment g] +1 (X)=_X[g] (X)]

FRT" I

Coding

FRT

and

binarization ~j (z, u) =Q2 p [Gj (z, u)

]l

1 Output : binarized hologram

~j (z,u) Fig. 2. Binarization scheme based on an iterative Fresnel transform algorithm applying the operator O2p in lhe hologram plane and 1he operator X in the signal plane.

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step 10 1 cycle

step p 8 cycles

step 1 8 cycles

Q21o

G

?,

G

Fig. 3. Illustration of the stepwise introduction of the operator Q 2 p.

In this procedure the coded and binarized hologram distribution Gj(z, u) = 0 2 [Gj(z, u) ] is calculated in each iteration cycle. This iterative algorithm stagnates. The phase ~j(X-Xo) cannot be optimized to adapt the noise to the signal. To circumvent this stagnation we introduced the operator Q2 stepwise. The binarization operator O2 p was introduced as illustrated in fig. 3, i.e., so that the range of the analog hologram values is more and more restricted in p steps [ 9 ]. This iterative binarization method can be applied successfully to generate a binary Fresnel amplitude hologram of a discrete Signal.

5. Experimental results To show the influence of the presented coding scheme two reconstructions of the same signal are shown reconstructed at different distances behind the hologram (fig. 1). As a discrete s i g n a l f ( m ) 10× 10 points were used. Each signal point was spatially separated from its neighbor by 16 pixels. Then, the signal has an extent of My= ( 145, 145 ) pixels. To form a hologram whicl~ reconstructs the signal in z = 0 . 0 5 m spatially separated from the zeroth order, the signalf(m-mo) according to conditions (6) and (7) was located in a data array of M = (512, 512) zeros centered at the position too= ( - 184, 184). The hologram was written by a laser plotter on a raster of ~ u = 6v= 5 lam. Illuminating this hologram with a plane wave o f 2 = 6 3 3 nm the signal was reconstructed spatially separated from the zeroth order as shown in fig. l a. The light distribution between the discrete signal points is caused by

superposed higher diffraction orders. This signal disturbance cannot be eliminated by an offset mo o f the signal condition (6) because condition (8) is not fulfilled. Therefore, to generate a hologram which reconstructs the signal undisturbed in z = 0 . 1 5 m the signal f ( m - m o ) was located in the data array of M = (512, 512 ) zeros in mo= ( - 136, 136 ). This signal position mo fulfills conditions (6) and (8). The materialized hologram reconstructs the signal spatially separated from the zeroth and higher diffraction orders (fig. lb). The presented iterative algorithm was used to calculate a binary Fresnel amplitude hologram. During the iteration the number o f grey levels was restricted by Q2 p in p = 10 steps (fig. 3). In each step p the operator Q2 p was applied during 8 cycles except in the 10th step, in which it was only applied once before

Fig. 4. Computer simulation of the binarization noise. This noise is superposed onto the signal of similar shape.

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Fig. 5. Optical reconstruction in z = 0.15 m of a binarized hologram. Parameters: Mr=Nr= 145, M = N-- 512, and mo = ( - 136, 136).

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tensity signals have been generated using the presented iterative Fresnel transform algorithm. With this binarization scheme it was possible to adapt the noise to the signal by the use of phase freedom, i.e.. a complete spatial separation of signal and the binarization noise is not necessary. The reconstruction of a signal with a high SNR is possible from a binary Fresnel hologram with a SBWP compared to that of the analog hologram. Furthermore, adaption of the noise to the signal leads to an increase of the diffraction efficiency. In conclusion we like to stress that the use of the presented iterative binarization scheme is attractive for the generation of binary Fresnel amplitude holograms of intensity signals with an optimized SBWP and an improved diffraction efficiency.

Acknowledgement the iteration was terminated. Compared to the reconstruction of the hologram binarized by (32 according to eq, (9), the signal to noise ratio is increased from 31 to 152. Adapting the binarization noise (fig. 4) to the signal (fig. 5) results in an increase of the diffraction efficiency from 1% of the analog to 7% of the binary hologram (these values are due to computer simulations).

6. Conclusion In this paper we describe the influence of the Burch coding scheme [2 ] on digital point oriented Fresnel amplitude holograms. To spatially separate the signal from disturbing distributions the number of signal to hologram samples has to fulfil condition (8). Binary Fresnel amplitude holograms of discrete in-

We like to thank Ingo Kummutat for his help with the illustrations. M. Bernhardt was supported by the Graduiertenf6rderung of Nordrhein-Westfalen, Fed. Rep. Germany.

References [ 1 ] O. Bryngdahl and F. Wyrowski, in: Progress in optics, Vol. XXVIII, ed. E. Wolf (North-Holland. Amsterdam, 1990 ) Ch. I. [2] J.J. Burch, Proc. IEEE 55 (1967) 599. [ 3 ] W.H. Southwell, J. Opt. Soc. Am. 71 ( 1981 ) 7. [4] A. Macovski, J. Opt. Soc. Am. 60 (1970) 21. [ 5 ] R.W. Floyd and L. Steinberg, Proc. SID 17 ( 1976 ) 75. [ 6 ] S. Weissbach, F. Wyrowski and O. Bryngdahl, Optics Comm. 67 (1988) 167. [ 7 ] R. Rollestone and N. George, Appl. Optics 25 ( 1986 ) 178. [8] G. Liu and P.D. Scott, J. Opt. Soc. Am. A4 (1987) 159. [9] F. Wyrowski, Appl. Optics 28 (1989) 3864.