Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices
Computer-generated holograms (CGHs) integrated within combined optical-digital security devices (CO/DSDs) are described in this work. They can restore the monochrome and color 3D images in white light. To record them, the Electron Beam Lithography (EBL) is used. Our investigations on optimization of...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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irk-123456789-1211252017-06-14T03:05:02Z Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices Girnyk, V.I. Kostyukevych, S.O. Shepeliavyi, P.Ye. Kononov, A.V. Borisov, I.S. Computer-generated holograms (CGHs) integrated within combined optical-digital security devices (CO/DSDs) are described in this work. They can restore the monochrome and color 3D images in white light. To record them, the Electron Beam Lithography (EBL) is used. Our investigations on optimization of synthesis and recording the CO/DSDs with the integrated in it multilevel CGHs of 3D images possessed horizontal parallax only (HPO) are presented here. The CGH fabrication process is mainly composed of two parts: calculation of the interferogram data (ID) and their recording. Calculation of the ID is done as follows: firstly, the geometrical and optical constants of recording scheme and the object surface represented by the elemental self-radiating areas, are determined, secondly, the basic parameters accounting for discretization of ID in hologram plane is defined. The ID values can be derived by calculation of the necessary elemental object areas bipolar intensities sum. Next, over suitable quantization of ID, recording the rectangle data appropriate for EBL onto glass coated with non-organic photoresist based on As₄₀S₄₀Se₂₀ is performed. We have also investigated reciprocal influence of an optical part of the CO/DSD and a digital one. 2002 Article Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices / V.I. Girnyk, S.O. Kostyukevych, P.Ye. Shepeliavyi, A.V. Kononov, I.S. Borisov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 106-114. — Бібліогр.: 10 назв. — англ. 1560-8034 PACS: 42.40.-i, 42.70.Ln http://dspace.nbuv.gov.ua/handle/123456789/121125 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Computer-generated holograms (CGHs) integrated within combined optical-digital security devices (CO/DSDs) are described in this work. They can restore the monochrome and color 3D images in white light. To record them, the Electron Beam Lithography (EBL) is used. Our investigations on optimization of synthesis and recording the CO/DSDs with the integrated in it multilevel CGHs of 3D images possessed horizontal parallax only (HPO) are presented here. The CGH fabrication process is mainly composed of two parts: calculation of the interferogram data (ID) and their recording. Calculation of the ID is done as follows: firstly, the geometrical and optical constants of recording scheme and the object surface represented by the elemental self-radiating areas, are determined, secondly, the basic parameters accounting for discretization of ID in hologram plane is defined. The ID values can be derived by calculation of the necessary elemental object areas bipolar intensities sum. Next, over suitable quantization of ID, recording the rectangle data appropriate for EBL onto glass coated with non-organic photoresist based on As₄₀S₄₀Se₂₀ is performed. We have also investigated reciprocal influence of an optical part of the CO/DSD and a digital one. |
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Article |
| author |
Girnyk, V.I. Kostyukevych, S.O. Shepeliavyi, P.Ye. Kononov, A.V. Borisov, I.S. |
| spellingShingle |
Girnyk, V.I. Kostyukevych, S.O. Shepeliavyi, P.Ye. Kononov, A.V. Borisov, I.S. Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Girnyk, V.I. Kostyukevych, S.O. Shepeliavyi, P.Ye. Kononov, A.V. Borisov, I.S. |
| author_sort |
Girnyk, V.I. |
| title |
Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices |
| title_short |
Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices |
| title_full |
Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices |
| title_fullStr |
Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices |
| title_full_unstemmed |
Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices |
| title_sort |
multilevel computer-generated holograms for reconstructing 3-d images in combined optical-digital security devices |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121125 |
| citation_txt |
Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices / V.I. Girnyk, S.O. Kostyukevych, P.Ye. Shepeliavyi, A.V. Kononov, I.S. Borisov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 106-114. — Бібліогр.: 10 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Semiconductor Physics, Quantum Electronics & Optoelectronics. 2002. V. 5, N 1. P. 106-114.
© 2002, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine106
1. Introduction
The usage of holographic security devices (HSDs) for
state and commercial needs considerably increases in
recent decade. At the same, time the intensive develop-
ment of holographic technologies and high level of activ-
ity in this area cause high probability of counterfeiting.
In conditions mentioned, the elaboration of new methods
of synthesis and recording information providing higher
reliability against counterfeiting becomes more and more
actual. The combined technology is one of the powerful
modern protective methods of the HSDs synthesis. We
had reported about creation of combined optical-digital
security devices (CO/DSDs) utilizing both the optical and
e-beam technologies at the SPIE/IS&T conference �Elec-
tronic Imaging �2000� [1]. Based on the results of these
investigations commerce production of HSDs of
CombigramsTM type was started. CombigramsTM were
oriented on protection of the most confidential documents
and goods. They comprises electron-beam holography,
optical holography and computer-generated hologram
of 2D images. It allows them to differ from wide-spreaded
and available dot matrix holograms and considerably
reduce range of potential manufactures of the similar
complexity HSDs. The usage of EBL technology with
submicron resolution and of powerful computing means
enable performing of the synthesis over Fresnel condi-
tions and subsequent recording of computer-generated
holograms that are able to reconstruct 3D images in white
light. As against from CGH of 2D images calculated over
Fourier conditions, they can be reconstructed in non-co-
herent light and can be identified on the first level of
verification. It makes them attractive for further modifi-
cation of CombigramsTM. It is necessary to note that the
synthesis and recording the Fresnel CGHs requires ad-
vanced technical means and considerable efforts, but in
the case of successful embodiment of corresponding ef-
forts it allows to hope for the commercial success.
PACS: 42.40.-i, 42.70.Ln
Multilevel computer-generated holograms
for reconstructing 3-D images in combined
optical-digital security devices
V.I. Girnyk1), S.O. Kostyukevych2), P.Ye. Shepeliavyi2), A.V. Kononov1), I.S. Borisov1)
1) STC �Optronics�, LTD, Kiev Taras Shevchenko University, Radiophysical Dept., 2 build 5 prospekt acad. Glushkov, 03137 Kyiv, Ukraine;
Phone: +380 (44) 266 9235; phone/fax: +380 (44) 252 5597; e-mail: optronics@ukr.net
2) Institute of Semiconductor Physics, NAS of Ukraine, 45 prospekt Nauky, 03028 Kyiv, Ukraine
Abstract. Computer-generated holograms (CGHs) integrated within combined optical-digital security
devices (CO/DSDs) are described in this work. They can restore the monochrome and color 3D images
in white light. To record them, the Electron Beam Lithography (EBL) is used. Our investigations on
optimization of synthesis and recording the CO/DSDs with the integrated in it multilevel CGHs of 3D
images possessed horizontal parallax only (HPO) are presented here. The CGH fabrication process is
mainly composed of two parts: calculation of the interferogram data (ID) and their recording. Calcula-
tion of the ID is done as follows: firstly, the geometrical and optical constants of recording scheme and
the object surface represented by the elemental self-radiating areas, are determined, secondly, the basic
parameters accounting for discretization of ID in hologram plane is defined. The ID values can be
derived by calculation of the necessary elemental object areas bipolar intensities sum. Next, over
suitable quantization of ID, recording the rectangle data appropriate for EBL onto glass coated with
non-organic photoresist based on As40S40Se20 is performed. We have also investigated reciprocal
influence of an optical part of the CO/DSD and a digital one.
Keywords: computer-generated holograms, Electron Beam Lithography, combined optical-digital security
devices, interferogram data, 3D images, horizontal parallax only, multi color, bipolar intensity
Paper received 10.12.01; revised manuscript received 23.01.02; accepted for publication 05.03.02.
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
107SQO, 4(4), 2001
2. Essential principles of holographic synthesis
2.1. 3D imaging on 2D medium
Correct reconstruction of color 3D image is possible only
when a hologram is recorded on 3D registering medium,
because of definite limitations characteristic for 2D reg-
istering medium. Therefore, in the case of the hologram
recorded on the flat registering medium, to make it possi-
ble for the hologram to be reconstructed in white light
according with rainbow holography principles, the an-
gle of view in a definite direction must be considerably
limited. It means that we lose some parallax, as a rule the
vertical one, because of sight physiology stipulates ne-
cessity of keeping the horizontal parallax. So this work
is devoted to synthesis of the Fresnel holograms that have
only horizontal parallax only.
2.2. Choice of synthesis methodology
Ignoring the horizontal vertical parallax is equal to di-
viding the object space and holograms plane onto inde-
pendent horizontal sections (Fig. 1). Within each simi-
lar section, points of the object that face into this section
form an elementary hologram at the correspondent part
of the hologram plane. When reconstructed, every elemen-
tary hologram (i.e., each independent section of the holo-
gram plane) forms an image of the respective horizontal
section for the object space. Calculations of an interfer-
ence pattern for interaction of an object wave with the
reference one only within it. Let us proceed to methods of
such calculations.
First of all, we shall consider advantages of the holo-
gram digital synthesis as compared to optical recording.
Consider interaction of iA wave radiated by some point
object with plane reference wave R :
{ }])([exp
||
trrkj
rr
A
A ii
i
i
i ωϕ −+−
−
=
rr
r
rr
. (1)
{ }tjkyRR ωθ −−⋅= )sin(exp , (2)
� for the vertical direction of incidence (when y = 0,
the phase of reference wave is assumed to be zero);
{ }tjkxRR ωθ −−⋅= )sin(exp , (3)
� for the horizontal one (when x = 0, the phase is equal
to zero, too).
In optical recording, there is a possibility to register
spatial distribution only for one characteristic, namely,
for total intensity as a result of interference of all inter-
acting waves:
( )( )
.
),(
****
**2
RARAAARR
RARARAyxI
iiii
iii
+++=
=++=+=
(4)
Digital recording is effectively flexible and enables
to register any term from the right part of the Eq. (4), or
any superposition of them. The first term of the right part
in Eq. (4) is simply some constant shift that reconstructs
plane wave during restoring process. The second one is
an autocorrelation term that describes autointerference.
The third term is responsible for reconstruction of conju-
gated image using a wave that is conjugated with the
reference one. And at last, the latter term describes re-
construction of the original image due to reference wave
action. It would be reasonable to write it, but its complex
character creates some obstacles. To record complex val-
ues, developed are several methods, but they need using
printing devices with much higher resolution as compared
to amplitude recording. By reason of a resolution limit,
the most optimum seems to be registration of non-nega-
tive real value, especially in the case, when the printing
device provides sufficient amount of quantization levels.
There are several methods to choose this value. We shall
use the most spread one, namely, we shall register, as
suggested in [5], the value
bIKpolI += , (5)
where
{ }*** Re2 RARARAbI iii =+= (6)
is a bipolar intensity.
It follows that we shall calculated the next matrix
),(),( nmbIxybI nm = , (7)
where 1,...,1,0 −= Mm is a number of a horizontal sec-
tion, and 1,...,1,0 −= Nn is a number of a vertical band
within this section. For each section, we shall find mini-
mum, maximum, means as well as root-mean-square val-
ues bI for further adequate estimation of an optimum
magnitude for quantization parameters including the
constant shift K.Fig. 1. Recording scheme geometry.
X
Y
Z
q
q
R W (V D )
F H P la ne
R W (H D )
108 SQO, 4(4), 2001
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
2.3. Method for calculations of ID
As mentioned above, the hologram together with an ob-
ject space is divided onto independent sections. Conse-
quently, the calculation of an interference pattern can be
carried out step-by-step for each section.
First, contrary to [2], where an object was represented
as a totality of light point sources, in this work, the object
is defined as a totality of elementary radiating areas.
This representation corresponds better to natural rela-
tions of observing real objects and should improve visual
reception of restored images. Nevertheless, the developed
program algorithm enables to calculate the interference
pattern for point representation, too.
Second, as in [2], to reduce non-uniformities in the
restored image, which are caused by essential differences
of intensity ranges for various sections, the program al-
lows to limit the angle of divergency for the object wave
in the horizontal direction. This limitation is also useful
due to decreasing the upper limit of a spatial frequency
range, so frequencies of the interference pattern down to
values that can be exactly reproduced by a printing de-
vice.
Let us determine limits of the spatial frequencies in-
herent to the hologram. They depend on a direction and
angle of incidence θ for the reference wave as well as on
a maximum angle of divergency HPα for the object wave,
that is a horizontal parallax angle (since the angle α , at
which the object wave spreads, is fixed in the range
HPHP ααα <<− ).
For the horizontal incidence direction of the refer-
ence wave, the minimum spatial period in an inteferogram
(which corresponds to the maximum spatial frequency)
can be estimated using the formula:
HP
d
αθ
λ
sinsin
min +
= (8)
When λ = 0.6 µm, θ = 30°, αHP = 30°, we obtain
dmin = 0.600 µm, and assuming θ = 45°, αHP = 45°
dmin = 0.424 µm for the same l value.
In the case of vertical incidence of the reference wave,
the spatial period of the interference pattern can be de-
fined as:
� in a vertical direction:
θ
λ
sin
=Vd ; (9)
� as the minimum one in the horizontal direction:
HP
Hd
α
λ
sinmin = ; (10)
For the same values of parameters (λ = 0.6 µm,
θ = 45°, αHP = 45°) the formulae (9) and (10) yield in the
next values of spatial periods: == HV dd min 0.848 µm.
In accordance with the Kotelnicov�Shannon theorem
about coding the analog signal, the spatial frequency of
coding and recording an image should be at least twice
as much the maximum spatial frequency presented in the
image. It means that elementary horizontal and vertical
intervals DX and DDY should be smaller than the half of
minimum spatial periods. For the horizontal incidence,
direction of the reference wave DDY is not limited, DX
should not exceed 0.3 µm for λ = 0.6 µm, and should not
exceed 0.2 µm, when λ = 0.6 µm, θ = 45°, αHP = 45°. In
the case of vertical incidence and quantization param-
eters λ = 0.6 µm, θ = 45°, αHP = 45°, DX and DDY should
not exceed the value 0.4 µm. (Accuracy of positioning
and stump sizes of the accessible EBL is about 0.1 µm). It
is worth to note that for normal operation of a diffraction
grating, the latter must content at least 10�15 fringes.
This limits the minimum value of the thickness for the
horizontal section DY in the case of vertical incidence of
the reference wave. Therefore, we chose DYmin = 20 µm,
in order to set 23�24 diffraction fringes within the sec-
tion, when θ = 45°, and 16�17 fringes at θ = 30°. As to the
horizontal incidence of the reference wave, there are no
limitations on the lower limit of the value DY = DDY for
such geometry. At the same time, the upper limitation on
the horizontal section thickness DY is determined by the
desired higher resolution of a reconstructed image. The
matter is that the spatial resolution in the object space in
chosen to be equal to DY.
Third, to eliminate a vertical-fringe structure, we shall
prescribe a random initial phase to every elemental radi-
ating area, as it have been proposed in previous investi-
gations [2]. This way of randomizing initial phases trans-
forms the interference pattern into a non-regular struc-
ture. In optical interpretation, this procedure is similar
to placing a diffuser between the object and hologram,
which eliminate clearly seen streak noise and replace it
by less seen speckle noise.
Taking into account all above considerations, for
each n-th vertical band ( 1,...,1,0 −= Nn ) within the fixed
m-th section ( 1,...,1,0 −= Mm ), when the reference beam
falls in horizontal direction, we summed up all contribu-
tions to the bipolar intensity (4) from each elementary
radiating area belonging to the section. The contribu-
tion of one such radiating area can be represented by the
next expression:
)cos(
)(
3 FHxrk
r
rnsqA
visCtbI ⋅++⋅±⋅⋅⋅⋅= ηϕ
rr
, (11)
where (as seen in Fig. 2) DXnxFH ⋅= is a horizontal
coordinate of this vertical band; θη sin⋅= k is a spatial
frequency for the reference wave phase distribution in the
hologram plane; ϕ � is an initial phase of the wave emit-
ted by the area;
λ
π2=k is a wavenumber; 22 zrr x +=
is a modulus of vector that is drawn from this radiating
area to the fixed vertical band in the hologram plane;
xxr FHx −= is a projection on this vector on the holo-
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
109SQO, 4(4), 2001
gram plane; znrnrn zxx ⋅−⋅=⋅ )(
rr
is a projection of this
vector on the normal to a given radiating area;
),,( zyx nnnn =
r
is a unity normal to an elementary area
surface; x,z are coordinates of the area; sqA is a square of
the wave amplitude emitted by this area, it characterizes
its brightness; vis is a coefficient taking into account vis-
ibility of the area from the position of the n-th vertical
band (i.e., reflect the degree of overlapping it by other
areas); positive (+) sign before rk ⋅ is chosen for elemen-
tary radiating areas placed below the hologram plane
(z < 0), and negative (�) sign is used for those placed
above (z > 0). It is worth to note that the term
),(cos
),(cos)(
rn
r
rnnr
r
rn rr
rrrr
∠=∠=⋅
, (12)
which normalizing brightness sqA, is responsible for the
natural behavior of this radiating area; for majority of
natural light sources, flux emitted at an arbitrary direc-
tion r
r
can be represented by the expression
),(cos0 rnSS
rr
∠= , (13)
where S0 is a light flux emitted along the normal n
r
to the
source surface.
In the case of the vertically incident reference wave,
for each n-th vertical band ( 1...,1,0 −= Nn ), belonging
to the m-th fixed section ( 1...,1,0 −= Mm ), instead of one
value of bipolar intensity one should calculate a respec-
tive massif with amount of elements
DDY
DY
Counts≤ , (14)
where DY is a thickness of a horizontal section, and DDY
is an elemental vertical interval.
Thus, the elemental radiating area makes contribu-
tion as a massif with Counts elements, each of them being
calculated using the formula
)cos(
)(
3 ii yrk
r
rnsqA
visCtbI ⋅++⋅±⋅
⋅⋅
= ⋅ ηϕ
rr
,
DDYiyi ⋅= , 1,...,1,0 −= Countsi (15)
The sign �+� before iy⋅η corresponds to the refer-
ence wave incident from above.
Let us consider expressions (11) and (15) from (6) in
detail. Eq. (6) foresees summing the next values in the
hologram plane
{ }*Re2 RACtbI = , (16)
where
{ }ϕ+⋅
⋅
= rk
r
r
rn
sqA
A
r
r
rr
exp
)(
2
(17)
is the wave, that is radiated by the elemental area (nota-
tions here are the same as in Eq. (11)). Wave
{ } HDRWxjxpR FH −⋅= ηRe*
( { } VDRWyjxpR i −⋅= ηRe* ) (18)
is conjugated to the reference wave (HDRW means the
horizontal incidence of the reference wave , VDRW � the
vertical one). Assuming R = 1/2, substituting (17), (18)
into (16), and taking into consideration mutual overlap-
ping elements of the object, one can deduce (11) and (15),
respectively.
2.4. Definition of optimal quantization
parameters.
During the synthesis CGH need discretization in space
and quantization of amplitude for every count with using
limited number of levels. In quantization process the count
value is compared with the threshold levels di standing
over the quantization step q. If amplitude of counts get
between two neighbour threshold levels (di ≤ T[r,s] < di+1),
it get value of the fixed quantization level ri correspond-
ing for the interval. It is known that the center of interval
between two neighbour threshold levels is the optimal
position of quantization level for quantization function
( )
2
11 −+= +ii
i
dd
r , (19)Fig. 2. Horizontal slice of object space and hologram plane: for
calculation of definite bipolar intensity in the vertical band.
X
Z
O P ( , )x z
O P ( , )x z
n n n ( , )
n n n ( , )
2
1
2
1
2
1
2 x
1 x
2
1
2 z
1 z
X F H
F H
F H
v is = 1.0
v is = 0.0
1
2
FH P lane
r x x z ( , 0 )- -
r x x z ( , 0 )- -
2
1
2
1
2
1
110 SQO, 4(4), 2001
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
in this case dispersion of quantization noise can be de-
fined as
12
2
2
0
q=σ . (20)
This correlation corresponds to the so-called linear
model, because it is obtained under presumption that the
quantization noise source is additive and independent.
Unfortunately, photoresist (phase�registering medium)
that was used for CGH registration has essentially non-
linear characteristics, and it cannot be described by lin-
ear model for substantial phase shifts over 0.6 rad. Abun-
dance of factors which influence on diffraction efficiency
and SNR make it expedient to estimate these parameters
applying numerical methods during modeling process and
experimental methods during recording process. Fig. 3
presents the results of investigations of quantization lev-
els number influence on reconstructed image quality. We
can see that it is expedient to select 16 ≤ qln ≤ 128 if it
allows the dynamic range of EBL-photoresist system. It
provides high quality of CGH under the low cost.
2.5. Modified approach to object wave
amplitude calculation
In some cases, there is a necessity to record a hologram
of an object or its part with high resolution. As an exam-
ple, we can indicate the need to include microimages into
3D synthesized holographic protective elements, these
microimages being not observed with the naked eye but
could be simply controlled under sufficient optical mag-
nification. To use the approach based on Eq. (15) for
synthesizing such holograms, it is necessary to reduce
the object space discretization step h by 10 times or more.
It results in respective increase of the number of point
sources that represent the object surface. Accordingly,
time expenses for calculations increase by (h0/hm) to
(h0/hm)2 times as dependent on microobject complexity
(here h0/hm is the ratio of discretization step for the main
part and microobject, respectively). To overcome this
problem and speed digital synthesis, we suggest using
the next approach for calculation of microobject surface
element contribution.
As known, Eq. (15) arises as a result of digital calcu-
lation of the Fresnel integral
{ }∫ += xdjjkr
rj
zxA
xO ~exp
)~,~(
)( ϕ
λ (21)
by the trapezium method, and further extracting the in-
formational part of interference of the obtained object
wave amplitude with reference one. )~,~( zx is a point of
the object space in the parallax plane, r is a distance
from this point to the point where the object wave ampli-
tude O(x) is calculated, )~,~( zxA is a wave amplitude on
the object surface.
To calculate the object wave amplitude, we divide the
object surface section by finite segments and integrate
Eq. (21) analytically within each of these segments. To
make the approximate estimation of the integral, we can
use the Taylor expansion of the integral in the vicinity of
the middle point M of the each segment [A,B] in the ob-
ject section using smallness parameters
θα cos
2
r
L= , 2
2
r
L=β (22)
For simplicity, we use an approximation of the sec-
tion segment by a straight line, however, analogous ex-
pressions can be also deduced for any analytical curves
that have a curvature radius sufficiently large compared
to the segment length L⋅r is a length of a vector drawn
from the point x of the elemental hologram into the seg-
ment middle point M, and θ is an angle between vectors
r
r
and AB .
Fig. 3. Influence of quantization levels number on quality of
reconstructed image.
(23)
( ) ( ){ }
( )( ) { }∫
∫
−
−
++++++−≈
≈++++++++−=
2/1
2/1
3
3
2
2
4
4
3
3
2
20
2/1
2/1
4
4
3
3
2
2
4
4
3
3
2
20
2
1
2
1
2
1
2
1
exp...11
...exp...1)(
dsskrjsjkrbsjkrbsasasasO
dssbsbsbsjkrsasasasOxO
αα
αα
h, %
h
SN R
SN R
4.0
3.6
3.2
2.8
2.4
2.0
1.6
1.2
0.8
0.4
4 8 16 32 64 128 256
8
16
32
64
24
40
48
56
72
80Y = 0.8 ra d
Y = 0.8 ra d
Y = 2.2 ra d
Y = 2.2 ra d
Y = 1.4 ra d
Y = 1.4 ra d
m
m
m
m
m
m
Q ua n tiza tio n lev e ls nu m ber
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
111SQO, 4(4), 2001
where { }ϕ
λ
ijkr
rj
AL
O += exp0 ,
λ
π2=k is a wavenumber,
λ is a wavelength of radiation, s is a curve parameter
taking values �1/2 and +1/2 at the segment ends and zero
in the middle; βα
2
1
8
3 2
2 −=a , αβα
4
3
16
5 3
3 +−=a ,
224
4 8
3
16
15
128
35 ββαα +−=a , βα
2
1
8
1 2
2 +−=b , =3b
αβα 4
1
16
1 3 −= , 224
4 8
1
16
3
128
5 ββαα −+−=b . The pa-
rameter θ
λ
παγ cos
4 L
kr == is large (>>1) and the lat-
ter exponent in the integral (23) is fast oscillating. The
existence conditions of the representation (23) are as fol-
lows:
1<<
r
L
, (24)
1sin
4
2
2
24
1 <<= θ
λ
π
r
L
krb , (25)
1sincos
8
2
2
3
38
1 <<≈ θθ
λ
π
r
L
krb (26)
The first condition is easily achievable, and the sec-
ond and third ones give some limitations for the permissi-
ble segment length of integration. The term generating
the condition (24) is taken into account during the fol-
lowing integration and included into a correction multi-
plier, therefore, its value does not contribute any supple-
mentary error into the final result. It is sufficient to choose
it to be approximately 0.4 for the expansion into series to
take place. At given accuracy 1% and the wavelength
0.4 mm, using the conditions (25) and (26) we obtain
the next limitations: Lmax < 0.6 µm at r = 10 mm,
Lmax < 3 µm at r = 0.1 mm, Lmax < 14 µm at r = 1 mm,
Lmax < 45 µm at r = 10 mm.
Then, multiplying expressions in parentheses in the
latter integral in (23), and integrating it term-by-term,
we obtain the following equation (keeping only the lead-
ing terms of the expansion):
( ) { }
+
+=
γ
γθ
λ
πϕ
λ
sin
sin
4
1exp 2
2
r
L
jijkr
rj
AL
xO (27)
Here, the term in brackets coincides with the ampli-
tude of the object wave from a point source localized in
the middle of segment [A,B]. The second multiplier gives
a correction to the amplitude, which does not exceed 40%.
And the latter multiplier is an interference term taking
into account coherent summation of partial waves from
the points of the segment [A,B]; in the limit, 0→γ , which
means the case of a point source, this multiplier trans-
forms into unity.
Using the formula (27) enables to analytically sum
contributions from the whole group of closely located
point sources, distances between which are less than the
value Lmax deduced above. This allows us to shorten the
calculation time by orders when synthesizing 3D holo-
gram with high resolution.
3. Basis and peculiarities of algorithm
3.1. Object representation
With the aim of simplifying program means, we used
model representations of objects in the form of a set of
plane images having arbitrary orientations in the object
space. However, this model representations enables to
adequately investigate workability of the above approach
in general, and even to emphasize realization peculiari-
ties of characteristic variants for locating image elements
within the object space. Investigations for optimization
of all managing parameters can be completely carried
out in the case of characteristic test elements of the object
space within this model representation, and their results
can be used in the following study with improved repre-
sentations of the object, namely, not quasi-3D, but by the
real bulk one. As to such improvement, to some extent, it
can be made in the framework of using plane images,
stretching them on simple surfaces that differ from the
plane ones: cylindrical, spherical, conical, parabolical
etc., but the valid bulk representation all the same is more
desirable.
3.2. Basic parts of the algorithm
Algorithm is mainly composed from two parts: synthesis
and recording. Synthesis stage implies calculation of the
interferogram data, and the recording stage � its suitable
transformation and registration. Fig. 4 shows the ex-
tended scheme of the main production algorithm. It is
necessary to mention the influence of quantization pa-
rameters on obtained results. So, analysis of statistic char-
acteristics of ID is sufficiently necessary.
3.3. Looking for shielded points
Separate parts of the object can be overlapped and shielded
by one another. Shielded elements are not seen from the
considered point of the elementary hologram (PEH) and
do not contribute into an amplitude of the object wave.
Various parts of the object are shielded for different PEHs.
To find shielded elements of object, represented as a set
of N point sources, using the method of direct ray tracing
needs ~N2 operations for each PEH. It increases the time
of a hologram synthesis up to unacceptable values (ap-
proximately 1 year). To overcome this problem, we have
developed the algorithm for fast searching these shielded
object parts. It requires about 20N operations with a float-
ing point. To this aim, the object section is approximated
by a broken line, each segment of which comprises one or
several radiating sources. Using the known method of
chain cells all segments are ordered along the z-coordi-
nate and are stored in a z�buffer.
112 SQO, 4(4), 2001
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
C h o ice of sy nth et ized ty pe :
1 ) co lo r o r m o n o ch ro m e?
2 ) d irec tio n o f re feren ce w ave – h o rizo n ta l o r ve r tica l?
D e te rm in at io n o f bas ic p aram ete rs o f sy n th esis:
1 )
1 )
w ave len g th s λR, λG , λB, o r w av e len gth, λ;
2 )
2 )
re feren t w ave in cid en ce an g le , θ;
3 ) ob jec t w av e d iv e rg en cy an g le , αH P;
4 ) w id th o r h o rizo n ta l sect io n , D Y;
5 ) sec tio n am o u nt, M ;
6 ) w id th o f v er tica l strip , D X ;
7 ) am o u nt o f v er tica l strips p e r, N ;
8 ) e lem entary v er tica l in te rva l, D d y, an d am ou n ts o f cou n ts , C ou n ts , w ith in
e lem entary v er tica l strip (o n ly for v e rtical d irec tio n of the re fe ren ce w av e) ;
9 ) p a ram ete r S , u sed fo r ca lcu la tion o f re -cov e red im ag e e lem ents p recisio n.
D e te rm in at io n o f o b ject space geo m etr ical an d o p tica l p rop e rt ies (b righ tness,
transp aren cy)
In ter fe ro g ram da ta ca lcu la tion
D e te rm in at io n o f th e area in w h ich the F resn el h o lo g ram to p olog y b o rd e rs a re
lo cated ( fo r p ro v id in g p o ss ib i lity o f co m b in a tion w ith th e o th er im ag e tech no log ies )
R eco rd ing to p olog y u s in g E B L eq uip m en t
S ta tist ic ch arac te r is tics o f in te rfe ro g ram d a ta an alys is . S elec tio n of o p tim al
p aram eters o f qu ant iza tio n
S
Y
N
T
H
E
S
IS
R
E
C
O
R
D
IN
G
S tep 1
S tep 2
S tep 3
S tep 4
S tep 5
S tep 7
S tep 6
Fig. 4. Flowchart of macro-algorithm.
Each segment forms a shadow sector limited by rays
passing through its extreme points (A,B) and the point x
of elementary hologram. Object elements falling into the
shadow sector are invisible from the point x.
The algorithm operates as follows. Object elements
(i.e. section segment) taken from the z-buffer are sorted
out in the order of increasing distances from the x-point.
There are calculated angle equivalents for rays limiting
the shadow sector of that segment. The angle equivalents
are integers, they are calculated using a special algo-
rithm without calling trigonometrical functions. The value
of a visibility coefficient for a given section element is
estimated throw the degree of overlapping the sector,
confined by its limiting angles, with a total shadow zone.
After this, the total shadow zone is enlarged by this sec-
tor. At a computer with the clock frequency 1000 MHz
the algorithm work time can be reduced down to 25�35
ms (for N = 105).
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
113SQO, 4(4), 2001
4. Experimental results
Results of investigations of test hologram recording are
presented in the next figures. Synthesized hologram of
Fig. 5. Analysis of the hologram recorded using non-organic resist on different scale levels. a) Examples of holograms recorded with
different exposures which corresponds to different phase shifts. b) Microstructure of the hologram recorded by EBL.
Fig. 6. Influence of quantization method. a) linear quantization. b) nonlinear quantization with cutting amplitudes deviation of which
from the middle value exceeds some limit. Both relief are made using non-organic resist As40S40Se20.
test object (logotype �OES� splitted at 3 planes) has been
printed on electron resist and on As40S40Se20 photoresist
[8, 9, 10] by EBL device. Structure of resist relief has
been investigated by using electron beam microscope.
a b
a b
114 SQO, 4(4), 2001
V.I. Girnyk et al.: Multilevel computer-generated holograms ...
5. Conclusions
We had carried out investigations on the advantages of
the original approach which imply representation of an
object surface as the aggregation of the elemental self-
radiating areas in comparison with the traditional ap-
proach of representation of an object as the point light
sources aggregation. Investigations was carried out on
quality level and gave us positive results. Quantitative
evaluation of the advantages is planned in the nearest
future. Investigations on the optimization of quantization
parameters was also carried out. It was ascertained that
the number of quantization levels of 16 value is optimal
for the AsGa registering medium in the application of the
recording of CGH of 3D-images. We have to mention
that there still remains a need in the determination of the
appropriate parameters of count discretization in CGH
plane for characteristic parameters of recording scheme,
and also in the optimization of object parameters for
CGH of different types (monochrome and color) and of
any reference wave direction.
Fig. 7. The example of a non-optimal fragment of the CGH recording using non-organic resist As40S40Se20.
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