Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

1

New Creation Algorithm for Digitally Synthesized Hologramsin Surface Model by Diﬀraction from Tilted Planes
Kyoji Matsushima* , Hagen Schimmel† and Frank Wyrowski‡* Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan† LightTrans GmbH, Jena, Germany‡ Friedrich-Schiller-University of Jena, Jena, Germany

ABSTRACT
Novel method for synthesizing light waves from objects expressed in surface model is presented for fast creationof digitally synthetic holograms. This method has a feature of coordinates rotation in Fourier domain, in which
only twice FFTs and an interpolation of spectrum are necessary for calculation of light waves from each object’s
plane. Therefore, presented method is rather faster than conventional ray-tracing when large scale full-parallax
holograms are synthesized in a surface model. Phases of a surface object are also discussed in relation tocontrolling the direction and diﬀusiveness of object waves. Furthermore, fabrication of a hologram based on themethod and its reconstruction are presented for a demonstration.
Keywords: Computer-generated Holograms, Digital hologram, Angular Spectrum of Plane Waves, Surface
Model

1. INTRODUCTION
Computer-Generated Holograms or digitally synthetic holograms are desired media to create three-dimensional(3-D) autostereoscopic images of virtual objects. However, the technology has several major problems should
be solved. One of them is to need very high spatial resolution to fabricate the holograms. Recent electron-beamlithography techniques enable us to fabricate synthetic holograms with sub micrometer structures.1 In suchlarge-scale holograms, a computational explosion in the numerical calculation of complex amplitude distributionsfor 3-D objects causes another problem.
Recently, a ray-tracing method2, 3 is commonly used to generate the complex amplitudes of light waves
emitted from 3-D objects, in which objects are composed of a large number of point sources of light, andcomplex amplitudes of spherical waves from each point source is superimposed on the hologram plane. The raytracing method is simple in principle and is potentially the most ﬂexible in synthesizing holograms for true 3-D
objects. The method, however, requires an enormous computation time for generating the complex amplitudes
or the fringe patterns, especially in full-parallax holograms. A typical method for reducing computation time in
ray tracing is to calculate the real-valued fringe intensity instead of the complex-valued amplitude of the lightwave from objects.3, 4 There are also other methods using geometric symmetry to avoid redundant calculation5
and lookup table4 that remarkably improves the computation time. Some arithmetic acceleration algorithms
such as diﬀerence formulas6 and recurrence formulas7 has been presented for accelerating the computation.
A computer-graphics hardware is also used to assist in calculating the diﬀraction pattern of point sources.8
Moreover, a special purpose computer for fast synthesis of 3-D holograms is another attractive approach.9, 10
In general, computation time in ray-tracing is proportional to both number of sampling on the hologramand point sources on the objects. Therefore, the computation time can be written as7
Ttotal = τa M N,

(1)

where M and N is number of pixels of a hologram and point sources of an object, respectively. τa is, here, anelemental time coeﬃcient that depends on the processors, methods, algorithms and so on. The coeﬃcient isCorrespondence : *E-mail: matsu@kansai-u.ac.jp

Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

H o lo g ra mYs q(x q, y q)
y q
O b je c t2zqh (X , Y )

x q

X
Z

Figure 1: An object model and coordinate system used for synthesizing object waves.

deﬁned as the time necessary for calculating the fringe intensity of a single point source in a single pixel. τa is,for example, 47 ns with the recurrence formula on an Alpha 21164A (600 MHz) processor,7 or 0.6 ns with thediﬀerence formula on Pentium II (450 MHz)∗ . Suppose that we synthesize a hologram of which dimensions are
5cm × 5cm. If the hologram would be sampled with interval of 1 µm and the objects have 105 point sources,
its computation time by the diﬀerence formula is approximately 40 hours in a rough estimation. Processor
performances are continuously increasing year by year, and new technology such as parallel processing will be
introduced into synthesis of digital holograms in the future. However, it seems to be hard to get over the
essential drawbacks of ray tracing in the point source model.
Another object model to synthesize object waves has been suggested in a Fresnel hologram.11 In themodel, an object is composed of several small planar segments, and waves diﬀracted by the planar segments
are superimposed on a hologram plane. To implement this model, numerical algorithm to calculate diﬀraction
by tilted apertures is necessary for synthesizing object waves. In near ﬁeld diﬀraction, a similar scheme has
been suggested by a method based on the angular spectrum of plane waves,12 in which an idea on coordinatesrotation in the Fourier domain has been mentioned brieﬂy. However, any numerical and experimental results
as well as complete formulation are not provided in the literature. We have independently developed algorithm
based on the angular spectrum of plane waves to simulate optical diﬀraction by tilted apertures as well as
diﬀraction onto tilted screens.13 By use of this method, we can calculate complex amplitudes of light waves
emitted from objects structured upon the plane surface model.

2. OBJECT MODEL
Coordinates system and geometry used in this paper is shown in Fig. 1. The global coordinates is denoted by(X, Y, Z), and the hologram is placed on the (X, Y, 0) plane. Objects composed of plane surfaces are usually
positioned in Z < 0. Each surface of the object, speciﬁed by q, posses its own local coordinates (xq , yq , zq ),in which the surface is placed on the (xq , yq , 0) plane. Distribution of complex amplitudes sq (xq , yq ), referred
to as the surface property function in this paper, is deﬁned on the (xq , yq , 0) plane for each surface segment.
Each surface property function contains information on the surface segment such as the shape, brightness,
diﬀusiveness, feel of material, textures and so on. To obtain distribution of complex amplitudes H(X, Y ) of
object waves on the hologram, light waves hq (X, Y ) diﬀracted by each surface property function are calculated∗

We estimate the coeﬃcient τa for the diﬀerence formula from computation time reported in Ref. 6

Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

y

y

Y

3

P ic tu re

x

j

x
X

d

z
Z

Figure 2: Geometry in single-axis rotation.

by the method described in the following section, and then they are superimposed on the hologram plane asfollows:hq (X, Y ).
(2)
H(X, Y ) =q

Two steps are necessary to calculate individual light waves emitted from each surface segment; local coordinates (xq , yq , zq ) is made to rotate so as to be parallel to the hologram, and then object waves on the hologram
are calculated from the rotated complex amplitudes. In the Appendix A and the following section we summarize
the step for rotation of local coordinates.

3. SINGLE AXIS ROTATION
In conventional simulation methods for diﬀraction such as the Fresnel-Kirchhoﬀ formula, its Fresnel and Fraunhoﬀer approximation, and ordinary angular spectrum of plane waves,14 a common constraint is that a screen
must be parallel to the aperture. Therefore, object waves hq (X, Y ) on the hologram can not be given by meansof these methods. We derived general formulation for numerical simulation in diﬀraction by tilted apertures13
and summarize it in the appendix A.
In this Section and the Appendix A, (x, y, z) denotes local coordinates not parallel to global coordinates,and (x̂, ŷ, ẑ) denotes another coordinates locally deﬁned, which shares its origin with (x, y, z) coordinates but
is parallel to global coordinates. This local coordinates is referred to as rotated local coordinates.

3.1. Formulation and Performance
Object waves emitted from plane surface rotated upon single-axis was synthesized and its hologram reconstructing the 3-D images was fabricated to verify and demonstrate the method. In this experiment, planes composing
an object are rotated on ŷ axis with angle of ϕ, and origin of the rotated local coordinates is positioned on(0, 0, −d) in the global coordinates as shown in Fig. 2.
Transformation matrix for the single-axis rotation is given ascos ϕ 0 sin ϕ010 .
T−1 = − sin ϕ 0 cos ϕ(3)

Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

4 0

3 5 .6In v e r se F F T

T im e [s e c ]1 2 .1

P r o p a g a tio n

3 0

4

R o ta tio n (I n te r p o la tio n )
3 .1

F F T

2 0

8 .3
8 .5

1 0
0

0 .5
2 5 6 2

1 2 .12 .0
5 1 2 21 0 2 4 2
N u m b e r o f s a m p lin g

2 0 4 8 2

Figure 3: Performance of the single axis rotation accompanied with translational propagation.

By substituting elements of the matrix (3) into Eqs. (10) and (13), complex amplitudes on the rotated plane
are given by the inverse Fourier transformation as follows:h(x̂, ŷ) = F −1 {T̂ (û cos ϕ + ŵ(û, v̂) sin ϕ, v̂)| cos ϕ +û
sin ϕ|}.
ŵ(û, v̂)(4)

Performances of the single-axis rotation, measured on a Pentium III (1 GHz) processor, are shown in Fig. 3.
The FFT and inverse FFT consume approximately 68 % of processor time, and therefore total computation
time as a function of number of sampling points approximately agrees with that of FFT.

3.2. Phases of Surface Property Functions
In this section, we discuss single-axis rotation of a two-dimensional picture, i.e. amplitudes of a picture is set
to that of the surface property function. Question in the case is how to determine its phase. When a constantphase is given to the picture, light waves emitted from the picture propagate in a direction perpendicular to the
picture surface, and therefore, in cases of large ϕ or d, the object waves may not reach to the hologram plane.
This is shown in the row (1) of Fig. 4. Amplitudes and phases on the hologram plane in d = 0 are shown incolumn (a) and (b), respectively. When d = 0, the amplitude image of the picture is approximately placed at
the center of the hologram. The amplitude image is a little unclear only around the right edge of the picture
because of optical diﬀraction. However, the spectrum (c) has a sharp spot and the spot shifts from the center.
As a result, when d > 0, the amplitude image in (d) is not only unclearer than in (a) but also shift right.
In the row (2) of Fig. 4, phase of a plane wave, which propagates along Z axis, is set to the phase of thepicture object. By this linear phase, the light emitted from the picture irradiates the hologram perpendicularly.
Accordingly, the amplitude image keeps the center of the hologram, as shown in column (d).
Direction of object waves can be controlled by the linear phase, but the spectrum still keeps a sharp spot,
which means that the object wave is hardly diﬀracted by the surface object and therefore does not reach to
the whole of the hologram. This is attributed to deﬁciency of diﬀusiveness of the surface. Randomizing theobject phase can achieve increase in diﬀusiveness, but it also causes speckle noise. For numerical diﬀusion oflight without speckle noise a special phase called the digital diﬀuser has been suggested in Fourier hologram.15
The row (3) of Fig. 4 shows results of superimposing the digital diﬀuser on the linear phase. The object ismuch more diﬀusive than that only with the linear phase. As a result, the original picture image disappears in
amplitude images on the hologram because of strong diﬀraction.

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Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

(1 )

(2 )
X

P h o toZd(3 )P h o to(d ) A m p litu d e(d > 0 )Zd

P la n e
W a v e(c ) A m p litu d e
o f S p e c tru mX

P h o toP la n e
W a v e(b ) P h a se
(d = 0 )(a ) A m p litu d e(d = 0 )Y
X

ZX

Figure 4. Phases for a picture object; (1) a constant phase, (2) a linear phase, (3) superimposition of a digital diﬀuser
on the linear phase.

Y

°
°

d

X
Z

H o lo g ra m

Figure 5: An object numerically constructed for fabricating a hologram.

Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

6

4. FABRICATION AND RECONSTRUCTION OF A HOLOGRAM FOR A 3-D
SURFACE OBJECT
For verifying the method described in above sections, an object wave is synthesized from surface planes with alinear and diﬀused phase in a setup for the single-axis rotation. The object numerically constructed is shownin Fig. 5. The object is composed of two tilted plane surfaces, which have binary images including the letter‘FSU’ and ‘KU’ as their amplitude distribution and the linear and diﬀused phase as their phase distribution.
Parameters used for synthesizing the object wave and for fabricating its hologram are summarized in Table 1.
Table 1: Parameters used for synthesizing an object wave and fabricating a hologram.

Surface property function
Number of sampling

8192×4096

Sampling pitch

2µm×4µm

Distance (d)

60 mm

Phase

linear & Bräuer digital diﬀuser

Hologram
Pixels16384×4096

Pixel size

2µm×4µm

Type

Binary amplitude hologram

CodingPoint oriented coding by simple threshold

Reference wave

Plane wave with incidence angle of 4◦

The hologram was fabricated by a special laser printer originally developed for printing holograms. The
dimensions of the hologram is 32.8 mm × 16.4 mm and its viewing zone is 18◦ × 9◦ according to the pixel size.
An optically reconstructed image of the fabricated hologram is shown in Fig. 6.

Figure 6. Photograph of optical reconstruction of a fabricated hologram. The object is composed of two plane surfaces
not parallel to the hologram.

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Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

5. CONCLUSION
A novel method for calculating complex amplitudes of light wave from virtual 3-D objects was presented for fastcreation of digitally synthetic full-parallax holograms. In most of synthetic 3-D holograms, objects are composedof point sources, and the spherical waves emitted from the point sources are superimposed on a hologram plane,
whereas our concept has a feature that 3-D objects are composed of small planar surfaces and diﬀraction by
the surfaces is simulated. For realization of the concept i.e. for eﬀective calculation of optical diﬀraction by atilted surface, we developed algorithm based on the angular spectrum of plane waves.
In the method, a complex function representing optical properties of a surface in the object is deﬁned inlocal coordinate system. This surface property function includes information on the shape, diﬀusiveness andadditional properties such as texture of the surface. The local coordinates is rotated by using the algorithm,
so that the plane including the surface property function becomes parallel to the hologram. Finally the surface
property function on the rotated plane is numerically diﬀracted onto the hologram by conventional method.
Just twice FFTs and an interpolation of spectrum are necessary for the rotation step, and therefore computation
can be faster than ray tracing of point sources.

APPENDIX A. SUMMARY OF GENRAL FORMULATION FOR COORDINATESROTATION IN THE FOURIER DOMAIN
This section summarizes derivation of general formulation for coordinate rotation in the Fourier domain,13
based on the angular spectrum of plane waves.
Fourier spectrum of a surface property function is given as ∞T (u, v) = F {s(x, y)} =s(x, y) exp[−i2π(ux + vy)]dxdy.

(5)

−∞

where F {s(x, y)} denotes the Fourier transformation of s(x, y). By physical interpretation of integrand in
the inverse Fourier transformation of T (u, v), the wave vector of plane waves can be associated with Fourier
frequency on the (x, y, 0) plane as follows:
k =
=

2π
[kx ky kz ] ,λ2π [u v w(u, v)] ,

(6)
(7)

√
where the wave vector is deﬁned as |k| = 2π/λ and kx2 + ky2 + kz2 = 1, and w(u, v) = λ−2 − u2 − v 2 . There is
the same relation in the rotated local coordinates. Therefore, when rotation matrix from (x, y, z) to (x̂, ŷ, ẑ) is
deﬁned as
k̂
T−1

= Tk,

a1
=  a4a7

k = T−1 k̂,
a2 a3
a5 a6  ,
a8 a9

(8)
(9)

the Fourier frequencies (u, v) in local coordinates are given as functions of frequencies (û, v̂) in rotated local
coordinates as follows:u =α(û, v̂) = a1 û + a2 v̂ + a3 ŵ(û, v̂),v

β(û, v̂) = a4 û + a5 v̂ + a6 ŵ(û, v̂).

=

(10)

Thus, the Fourier spectrum T̂ (û, v̂) in rotated coordinates is given by using the Fourier spectrum in local
coordinates.
T̂ (û, v̂) = T (α(û, v̂), β(û, v̂)).
(11)

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Publication : SPIE Proc. Practical Holography XVI and Holographic Materials VIII, SPIE#4659, 53–60(2002).

The complex amplitudes h(x̂, ŷ) in rotated local coordinates is given by integration of the spectrum T̂ (û, v̂).
h(x̂, ŷ) = F −1 {T̂ (û, v̂)|J(û, v̂)|},(12)

where Jacobian J(û, v̂) deﬁned asJ(û, v̂) =
=

∂α ∂β
∂α ∂β
−
∂ û ∂v̂
∂v̂ ∂ û(a1 a5 − a2 a4 ) + (a2 a6 − a3 a5 )ûû+ (a3 a4 − a1 a6 ),ŵ(û, v̂)ŵ(û, v̂)(13)is necessary for energy conservation of the ﬁelds because of non-linearity of the rotational transformation of
Eq. (10).

ACKNOWLEDGMENTS
The authors thank A. Joko for his assistance in fabrication of holograms.

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