Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

1

Wave optical algorithm for creating digitally synthetic
holograms of three-dimensional surface objects
K. Matsushima and A. Kondoh
Department of Electrical Engineering, Kansai University
Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan
ABSTRACT
Progress in a novel wave-optical algorithm for synthesizing object waves of three-dimensional surface objects isreported. The algorithm has been proposed last year and has a feature of wave-optical calculation in diﬀraction
by object surfaces. Numerical simulation of diﬀraction is implemented by a method based on coordinates rotationin Fourier spectrum. Source ﬁelds on a surface of objects are given as complex property functions deﬁned on eachobject surface. These property functions provide information on each surface, such as the shape and diﬀusiveness.
General formulation of the algorithm has been presented, but its numerical implementation was limited only tothe cases of single axis rotation of plane surfaces.
In this investigation the limitation is removed, i.e. two-axes rotation of plane surfaces is presented and reconstruction of holograms of 3-D objects composed of rotated planes are demonstrated. Furthermore, compensationof surface brightness is discussed for accurate shading objects.
Keywords: computer-generated hologram, digitally synthetic hologram, surface model, shading

1. INTRODUCTION
In digitally synthetic holograms (or computer-generated holograms; CGH), some object models and methods
for synthesizing object waves emitted from the three-dimensional (3-D) objects has been proposed to create realauto-stereoscopic 3-D images. One of the most important models is the layer model, in which the object is sliced
into a plane parallel to the hologram and light ﬁelds emitted from every plane are wave-optically calculated asordinary diﬀraction such as Fresnel diﬀraction. While the method has a deﬁnite advantage of relatively shortcomputation time owing to utilization of FFT especially in full-parallax hologram, there is a major drawback torealistic rendering of 3-D objects such as shading, shadowing, texture-mapping and so on.
An alternative model recently used is the point source model, which is sometimes referred to as the raytracing method in CGH.1, 2 A 3-D object is considered as a collection of numerous point sources of light in themodel, and the object wave is obtained by superimposing all spherical waves from point sources in the hologram
plane. Since computation time of the ray-tracing is proportional to both numbers of point sources and sampling
points in the hologram, slow computation is a major disadvantage of the method, especially in full-parallax
holograms. To overcome the problem of computation time in the ray-tracing some algorithm has been proposed,such as real-valued fringe intensity,2, 3 utilization of geometric symmetry4 and special hardware,5–7 lookup
table,3 recurrence formulas8 and diﬀerence formulas.9
One another method we refer it to as the wave optical algorithm in surface model has been suggested in
the last meeting.10 In the method an object is composed of planar surfaces and the object wave is waveoptically calculated as reﬂection and diﬀraction by the tilted surfaces. The mathematical and physical basics
have been presented, but a hologram of planar objects rotated only upon a single axis has been discussed andits reconstruction was demonstrated. This report follows with creating holograms for true 3-D objects madeup from planes rotated on two-axes, as well as shading the objects by using a numerical model of brightness ofreconstructed planar surfaces.
Further author information: (Send correspondence to Matsushima)
Matsushima: E-mail: matsu@kansai-u.ac.jp, Telephone: +81-6-6368-0933

Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

S u rfa c e -p ro p e rty
fu n c tio nY2

H o lo g ra m

S q(x q, y q)
y qzq

h q ( x q , y q )

y q

x q
x q

X

z q

Z
Figure 1. An object model and coordinate system used to synthesize object waves.

2. OBJECT MODEL AND SURFACE PROPERTY FUNCTION
Coordinates system and geometry used in this investigation is shown in Fig. 1. The global coordinates, referred
to as hologram coordinates, is denoted by (X, Y, Z), and the hologram is placed on the (X, Y, 0) plane. Local
coordinates deﬁned for each surface of the object are (xq , yq , zq ) in a surface q. All local coordinates are deﬁned
so that the planar surface is placed on the (xq , yq , 0) plane. Furthermore, we deﬁne a complex function sq (xq , yq )in the (xq , yq , 0) plane to give the surface q its properties such as the shape, brightness, diﬀusiveness, feel ofmaterial, textures and so on. These complex functions are referred to as the surface property function. In
addition, we deﬁne one another coordinates, the parallel local coordinates (x̂q , ŷq , ẑq ), in which the origin is
identical to the local coordinates but all axes are parallel to that of the hologram coordinates.
All surface property functions are deﬁned in the following form:sq (xq , yq ) = aq (xq , yq )pq (xq , yq )Ψ(xq , yq ),(1)

where aq (xq , yq ) is a real function and mainly keeps the shape of the surface q and its texture. A surface property
function deﬁned only by the factor aq (xq , yq ) do not have any diﬀusiveness, just like as a slide projector, andtherefore it must be multiplied by a given diﬀusive phase function; a diﬀuser given as the complex function:
Ψ(xq , yq ) = exp[ikφd (xq , yq )].

(2)

Here, we used a digital diﬀuser proposed for Fourier holograms11 to the phase function φd (xq , yq ).
If a surface property function are given as aq (xq , yq )Ψ(xq , yq ), the carrier frequency of the object wave areplaced at the zero frequency in the local coordinates. This means that the light wave of the surface is emitted
in the direction perpendicular to the surface. Therefore, a phase of the plane wave propagating along the Z axis
(i.e. in the direction perpendicular to the hologram) should be multiplied by the two factors given above. The
plane wave is written by(3)
pq (xq , yq ) = exp[ik(cos αq xq + sin βq yq )],where cos αq and cos βq are the direction cosines in X and Y axis, respectively.
The surface property function given above is transformed into the complex amplitude hq (x̂q , ŷq ) in the local
parallel coordinates by the method described in Ref. 10, 12. This transformation written as
hq (x̂q , ŷq ) = Rθx θy θz {sq (xq , yq )}

(4)

3

Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

(a )

G x y

y

(b )

x

G y

x

h ( x , y )y

G z yxh ( x , y )

G y

G z

x

z

z

s(x , y )s ( x , y )Figure 2. Two types of rotation upon two-axes; rotating (x̂, ŷ, ẑ) coordinates upon the ŷ-axis prior to the x-axis (a),rotating upon the ẑ-axis prior to the y-axis (b)(60 °, 60°, 0 )

λ −1

v

(θ , θ , θ )xy(θ , θ , θ )zx(60 °, 30 °, 0 )uλ(30°, 60 °, 0 )v

z(0, 30°, 30° )(0, 30°, 60° )(30°, 30°, 0 )

−λ −1

y

λ −1

uλ

−λ −1

−1

Sampling areain (uˆ , vˆ )(0, 60°, 30° )(0, 60 °, 60° )

(a)

−1

Sampling areain (uˆ , vˆ )

(b)
−λ −1

−λ −1

Figure 3. Distribution of sampling points in (u, v). The rotation scheme in (a) and (b) agrees with that of Fig. 2.

where θx , θy and θz are rotation angles upon each axis. Finally, ﬁelds emitted from every plane are superimposedin the hologram plane as follows:
H(X, Y ) =hq (x̂q , ŷq ).
(5)
q

3. COORDINATES ROTATION ON TWO-AXES
There are several degrees of freedom to deﬁne local coordinates for each surface. Fig. 2, for example, shows two
types of rotation on two-axes. In Fig. 2(a) the parallel local coordinates (x̂, ŷ, ẑ) is transformed into the local
coordinates by rotating coordinates upon ŷ axis prior to x axis, while the coordinates is rotated upon ẑ axis
prior to y axis in (b). When a position vector r in the local coordinates is transformed into r̂ in the parallel local
coordinates by transformation matrix T , the transformation is written asr̂ = Tr.

(6)

Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

4

The inverse transformation matrix in the case indicated in Fig. 2(a) is given by
T−1= Rx (θx )Ry (θy )
cos θy=  cos θx cos θy− sin θx cos θy0sin θxcos θx− sin θy− cos θx sin θy  .
cos θx sin θy

(7)
(8)

The inverse matrix for Fig. 2(b) is also given by
T−1= Ry (θy )Rz (θz )cos θy cos θz=  − sin θzsin θy cos θz

cos θy sin θz
cos θzsin θy sin θz− sin θy0cos θy(9)(10)

Selection of these rotation schemes are not signiﬁcant in the aspect of the mathematical transformation of
coordinates, nevertheless one should carefully choose the rotation scheme, because an interpolation of Fourier
spectrum is necessary for numerical calculation in the coordinates rotation. Since an object wave is typicallysampled on an equidistant grid in the hologram plane, sampling points of the spectrum in (û, v̂) coordinatesis also distributed on an equidistant grid within a square area, as shown in Fig. 3. The sampling points after
any rotational transformation, however, are not positioned at even intervals, and its envelope does not shapesquare any longer. Fig. 3(a) and (b) shows distribution of sampling points in the local coordinates after rotating
coordinates by using the way indicated in Fig. 2(a) and (b), respectively.
The rotation scheme in Fig. 2(b), described by matrix (10), is adopted in this report, because it seems that
distortion of sampling distribution in Fig. 3(b) is rather less than that in (a). Jacobian necessary for rotationaltransformation by the matrix (10) (See Appendix A of Ref. 10) is given byJ(û, v̂) = sin θy cos θzûv̂
+ sin θy sin θz+ cos θy .
ŵ(û, v̂)ŵ(û, v̂)(11)

4. OBJECT WAVES FROM THREE-DIMENSIONAL OBJECTS
Object waves emitted from 3-D objects can be synthesized by superimposing object waves emitted by planes
constructing the object. A cube, for example, consists of three square planes∗ , as shown in Fig 4(a). An optical∗In this report we do not treat methods for removing hidden surfaces, and thus, such hidden surfaces are originallynot taken into account.

(a )

(b )
S 3

S 1

S 2

Figure 4. A cube built by plane surfaces (a) and an optical reconstruction of a fabricated hologram (b). The dimensionof the cube is 6 × 6 × 6 mm3 .

Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

5

reconstruction of a fabricated hologram of the cube is shown in Fig 4(b). A cube is optically reconstructed as
expected, however the cube is shaded unexpectedly. All amplitudes of the surface property functions were asame constant, i.e. aq (xq , yq ) ≡ a0 , nevertheless the reconstructed cube has a deﬁnite shade. This shading is
unexpected one and should be compensated in order to shade object accurately.

4.1. Numerical model for brightness of reconstructed surfaces
For compensation of the unexpected shading it is necessary to investigate which and how parameters of an object
surface relate to brightness of the surface on reconstruction. Figure 5 is a theoretical model to predict brightnessof a surface, represented by the sampled property function.
Suppose that the amplitude of a surface property function is a constant, i.e. a(x, y) ≡ a0 and |s(x, y)|2 = a20
gives optical intensity on the surface. In that cases radiant ﬂux Φ of a small area δA on the surface is given by|s(x, y)|2 dxdy
(12)
Φ =
δA

= δAσa20 ,(13)

where σ is the surface density of sampling. We assumed that almost the whole light emitted from the small area
δA is bound in a solid angle given byΩπ(r tan ϕd )2
A
=
r2
r2
= π tan2 ϕd ,

=

(14)

where ϕd is a parameter representing degree of the diﬀusiveness of the diﬀuser function Ψ(xq , yq ). Furthermore,assuming that light is diﬀused almost uniformly i.e. dΦ/dA  Φ/A, brightness of the surface, observed at angleof θ with the normal vector, is given by
L =
=

dΦ/dΩ
cos θδA
σa20.
π tan2 ϕd cos θ(15)

As a result, brightness of the surface depends on the surface density of sampling, diﬀusiveness of the diﬀuser
function, and amplitude of the surface property function. In addition, brightness varies dependently on theobservation angle. This means that brightness of the surface changes according to the direction of the surface
and causes unexpected shade.
S u rfa c e o f
o b je c tq

N o rm a lv e c to r

d A

r

j d
d A

d W
A

Figure 5. A theoretical model for predicting brightness of a planar surface represented by the surface property function.

Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

6

A simple theoretical model is discussed so far, therefore Eq. (15) is only partially ﬁt for brightness of opticallyreconstructed surfaces of real holograms. Since brightness given in Eq. (15) diverges in limit θ → π/2 and any
actual hologram cannot produce inﬁnite brightness for its reconstructed surface, Eq. (15) is not suﬃcient for themodel used for compensation. To avoid the divergence of brightness in Eq. (15) we a priori introduced an angle
factor (1 + γ)/(cos θ + γ) instead of 1/ cos θ, i.e. supposed brightness is given as
L=(1 + γ)σa20,
π tan2 ϕd (cos θ + γ)

(16)where γ is a parameter that plays a role to avoid the divergence of brightness and prevent overcompensation
as a result. The γ is dependent on actual methods to fabricate holograms such as encoding of object waves orproperty of recording materials and thus should be determined experimentally. Note that the angle factor in
Eq. (16) is unity in θ = 0 and 1 + 1/γ in θ = π/2.

4.2. Compensation of brightness and shading object
Shading object is based on simple Lambert’s low and the diﬀused reﬂection model. Brightness of a surface q, of
which the normal vector is at an angle of θ̂ with the vector of illumination, is given by
Lq = L0 (cos θ̂q + le ),(17)

where le is the ratio of surrounding light to illuminated one and L0 is brightness in θ̂q = 0 and le = 0. By
substituting Lq in Eq. (17) into L in Eq. (16), amplitude aq of a surface q is given as follows:(cos θ̂q + le )(cos θq + γ),
(18)aq = A01+γL0 π tan2 ϕd
A0 ≡
.
(19)
σ
Here, the observation angle θ is replaced by an angle θq formed between the surface q and the hologram.

5. OPTICAL RECONSTRUCTION OF FABRICATED HOLOGRAMS OF
THREE-DIMENSIONAL OBJECTS
Digitally synthetic holograms were fabricated by a special purpose printer for drawing holograms. Parameters
used to synthesize object waves and fabricate holograms are summarized in Table 1.
Table 1. Parameters used to synthesize object waves and fabricate holograms.

Object and its surface property function
Sampling pitch

2 µm × 2 µm

Dimensions

6 mm × 6 mm × 6 mm (cube)
8 mm in diameter (hexagonal pyramid)

Distance from hologram

100 mmHologram
Number of pixels

8192 × 4096

Pixel size

2 µm × 4 µm

Reconstruction wavelength632.8 nmCodingPoint oriented multi-level amplitude

Reference wave

Plane wave with incidence angle of 4◦

Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

(a )

7

(b )

Figure 6. Optical reconstructions of cubic objects without illumination. Brightness of each surface is compensated in
γ = 0 (a) and 0.5 (b)(-1 ,-1 ,-1 )(a )(+ 1 ,-1 ,-1 )(b )(+ 2 ,-4 ,-1 )(c )Figure 7. Optical reconstructions of 3-D objects shaded with illumination light; cubes illuminated from the upper rightin le = 0 (a) and the upper left in le = 0.7 (b), and a hexagonal prism le = 0.5 (c). Brightness of every object iscompensated in γ = 0.5. An arrow and figures in parenthesis indicates the illumination vector.

Firstly several holograms of cubic objects without illumination light was fabricated to determine the value ofthe parameter γ. Figure 6(a) shows optical reconstruction of a hologram with γ = 0. In contrast to Fig. 4(b),the surface placed at a large angle with the hologram is dark because of overcompensation caused by γ = 0.
Figure 6(b) is also of a cubic object without illumination, but of which brightness is compensated in γ = 0.5.
Diﬀerences of brightness disappear because of proper compensation of brightness, and borders between surfaces
are no longer found out.
Figures 7(a)–(c) shows optical reconstruction of 3-D objects shaded by illumination light, of which brightness
is all compensated in γ = 0.5. As was expected by arrows indicating illumination vectors in the ﬁgures, object
surfaces are shaded in the reconstruction.

6. CONCLUSION
A detailed method for two-axes rotation of plane surfaces has been presented. Object waves emitted fromthree-dimensional objects composed of several rotated planes were digitally synthesized and reconstructions of
the fabricated hologram were demonstrated. Furthermore, the surface brightness dependent on the observation
angle as well as its compensation was discussed.

Publication : SPIE Proc. Practical Holography XVII and Holographic Materials IX, SPIE#5005, Santa Clara, California, 190–197(2003).

8

ACKNOWLEDGMENTS
Authors acknowledge S. Yamanaka and H. Miyauchi for their assistance in fabricating holograms.

REFERENCES
1. J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, pp. 405–407,
1966.
2. A. D. Stein, Z. Wang, and J. J. S. Leigh, “Computer-generated holograms: A simpliﬁed ray-tracing approach,” Computers in Physics 6, pp. 389–392, 1992.
3. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electronic Imaging 2, pp. 28–
34, 1993.
4. J. L. Juárez-Pérez, A. Olivares-Pérez, and R. Berriel-Valdos, “Nonredundant calculation for creating digital
Fresnel holograms,” Appl. Opt. 36, pp. 7437–7443, 1997.
5. A. Ritter, J. Böttger, O. Deussen, M. König, and T. Strothotte, “Hardware-based rendering of full-parallax
synthetic holograms,” Appl. Opt. 38, pp. 1364–1369, 1999.
6. T. Ito, T. Yabe, M. Okazaki, and M. Yanagi, “Special purpose computer holography HORN-1 for reconstruction of virtual image in three dimensions,” Comput. Phys. Commun. 82, pp. 104–110, 1994.
7. T. Ito, H. Eldeib, K. Yoshida, S. Takahashi, T. Yabe, and T. Kunugi, “Special purpose computer forholography HORN-2,” Comput. Phys. Commun. 93, pp. 13–20, 1996.
8. K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, pp. 6587–6594, 2000.
9. H. Yoshikawa, S. Iwase, and T. Oneda, “Fast computation of Fresnel holograms employing diﬀerence,” SPIE
Proc. Practical Holography XIV and Holographic Materials VI #3956, pp. 48–55, 2000.
10. K. Matsushima, H. Schimmel, and F. Wyrowski, “New creation algorithm for digitally synthesized hologramsin surface model by diﬀraction from tilted planes,” SPIE Proc. Practical Holography XVI and Holographic
Materials VIII #4659, pp. 53–60, 2002.
11. R. Bräuer, F. Wyrowski, and O. Bryngdahl, “Diﬀusers in digital holography,” J. Opt. Soc. Am. A8, p. 572,
1991.
12. K. Matsushima, H. Schmmel, and F. Wyrowski, “Fast calculation method for optical diﬀraction on tilted
planes using the angular spectrum of plane waves,” (to be published).