Optimization of phase-only computer-generated holograms using

Optimization of phase-only computer-generated
holograms using an ion-exchange process
Hans Christian Boistad
Toyohiko Yatagai, MEMBER SPIE
University of Tsukuba
Institute of Applied Physics
Tennoudai, Tsukuba
lbaraki 305, Japan
Masafumi Seki
Nippon Sheet Glass Company, Ltd.
Tsukuba Research Laboratory
5-4, Tokodai, Tsukuba
lbaraki 300-26, Japan
1 Introduction
In recent years there has been a considerable resurgence of
interest in computer-generated holograms (CGHs) and kinoforms.
1-6 The primary motivation is the possible use in
flexible interconnections between operation logic gates in
optical computing. The increase of diffraction efficiency is
one of the key issues in recent studies of CGHs and the
kinoform. Dammann described blazed phase-only holograms
theoretically.7 Takeda and Yatagai described a bleached
binary hologram to obtain an efficient binary CGH with
high efficiency in reconstruction.8
In a previous paper,9 we proposed an approach to increase
the diffraction efficiency of CGHs by an ion-exchange
1013 A phase-only hologram was produced by
changing a transmittance distribution of a conventional binary
CGH to a refractive index distribution to obtain a binary
phase hologram. A simplified theory of the binary phase
hologram and experimental verification were presented.
In this paper, we discuss an optimum condition of the
phase profile of binary phase-only CGHs made by ion exchange.
First, we describe briefly diffraction efficiencies of
ideal phase-only holograms, including a binary grating, a
binary phase-only grating, and a bleached gray grating.
Then computer simulations of the ion-exchange process and
phase distribution calculation are presented. The diffraction
efficiency is calculated by using calculated phase distribution.
2 Ion-Exchange Method for Phase-Only CGH
Figure 1 shows an illustration of ion concentration profile
formed by ion exchange, resulting in a phase-only hologram.
The substrate is an optically homogeneous glass plate
of borosilicate glass containing sodium and potassium ions
Paper 02091 received Sep. 4, 1991; revised manuscript received Nov. 20, 1991;
accepted for publication Nov. 20, 1991. This paper is a revision of a paper
presented at the SPIE conference on Computer and Optically Formed Holographic
Optics, January 15—16, 1990, Los Angeles, Calif. The paper presented there
appears (unrefereed) in SPIE Proceedings Vol. 1211.
1992 Society of Photo-Optical Instrumentation Engineers. 0091-3286/92/$2.00.
Abstract. A highly efficient phase-only hologram has been produced using
an ion-exchange method. An optimum condition for the phase profile
of the hologram is described. The ion density and the phase profile generated
by the ion exchange were calculated by computer simulations.
The diffraction patterns of different gratings are calculated based on these
data, and an optimum phase distribution is estimated. The optimum condition
has been verified experimentally.
Subject terms: computer-generated holograms; ion-exchange processes.
Optical Engineering 3 1(6), 1259-1263 (June 1992).
___ [1 ri I 1
___
4' ___ ___
1
4, ___ ___
4,
Metal mask
depos I t Ion
Photo 1 ithography
and patterning
Thermal
Ion exchange
Mask removal
Fig. 1 Schematic diagram of fabrication of a phase-only hologram
by means of an ion-exchange method.
for ion exchange. The mask of the binary hologram pattern
was made by 25 time reductions of the original mask pattern
drawn by a VAX 8200 computer and a DEC LNO3 PLUS
laser beam plotter on A4 paper. The mask was delineated
onto a titanium mask by standard photolithography and
chemical etching by a constant exposure. Then the substrate
was immersed in molten salt composed of 90% thallium
ions and 10% potassium ions to perform one-step purely
thermal ion exchange, which is the first part of the two-step
purely thermal ion-exchange process13 proposed for singlemode
waveguides . Ion exchange was carried out at 450°F
for 2 h. The ions were diffused only through the window
area of the mask in the glass substrate, increasing the refractive
index of the substrate. Because the ion radius of
thallium ions is larger than that of alkaline ions to be exchanged,
the surface inside the apertures swelled slightly.
OPTICAL ENGINEERING / June 1 992 / Vol. 31 No. 6 / 1259

1.0 -
0.5
0
0
Amp ii tude
(a) (b)
Fig. 2 Transmittance functions of a binary amplitude
gray grating (c).
1260 / OPTICAL ENGINEERING / June 1992 / Vol. 31 No. 6
x
BOLSTAD, YATAGAI, and SEKI
0.5
Phase Phase
3 Diffraction Efficiency of Ideal Phase-Only
Holograms
The diffraction efficiencies of gray and binary gratings were
14
originally described by Brown and
First we
will describe these and then the diffraction efficiencies of
their phase-only counterparts. For simplicity, consider the
diffraction efficiency difference between a binary grating
and its phase version, as shown in Fig. 2. The transmittance
function of a binary grating with a period of L\v is given
0
Amp] itude Amp] itude
grating (a), its phase version (b), and a bleached
0
4.)
>.,
u
C
a)
a
4-
4w
1.e
0.5
0
x x
Phase
x x x
by 0
211'
Phase Difference
_1 m1 . Tb (2irnx
2 7T'fl Lv ) ' (1) Fig. 3 Theoretical plot of diffraction efficiency ratio R and R' versus
phase difference 0.
where m denotes the modulation level of the grating and
O

OPTIMIZATION OF PHASE-ONLY COMPUTER-GENERATED HOLOGRAMS
Fig. 4 Flow diagram of computer simulation for phase profile estimation.
This is also plotted in Fig. 3 for the case m =1 . The maximum
diffraction efficiency is not obtained at 0 =ir, but
0=3.6 rad or 220 deg.
4 Phase Profile Simulation
It is possible to simulate some of the processes described
so far, diffusion, phase-shift, and reconstruction, by using
a digital computer. The flow diagram of the simulation is
shown in Fig. 4. First we consider the calculation of the
ion-diffusion profile (ion-concentration profile). The resulting
ion-concentration after ion exchange can be calcu-
lated by solving the 2-D diffusion equation numerically
12
using the finite difference An actual profile can
be calculated/simulated from the given data for the experiment.
This was done using a VAX 8200 computer. In order
to simplify programming, we assumed the diffusion constant
to be concentration independent. This is a rough approximation,
if the ion concentration is very high. Necessary
inputs are time of diffusion, aperture size, and diffusion
constant. A relation between ion concentration in a diffused
area and the change in refractive index has been found
empirically, and can be expressed as follows:
n=aC+b,
where n is the index change in the local area, C is the local
concentration of ions of thallium exchanged with sodium,
and a and b are constants.
Using an experimental relation,9 the final index increase
in a local area can be found using a linear relation. The
concentration of ions (thallium) exchanged with sodium!
potassium ions is 90% near the surface of the aperture opening,
and gradually decreasing to 0% in substrate material.
To find the actual index change over the full depth of
the diffusion area, we calculate as follows. Consider the
coordinate in the depth direction as x. A small element along
(9)
V
4.)
a)
0
E
'1e
C
o8
4.,
.; 6
0
0_4
2
0
—2
—4
—6
—6
—10 0 2 4 6 0 10
Diffusion Depth (urn)
Fig. 5 Simulation result of index change.
this direction is called zx. The ion concentration of thallium
is found in each small element according to Eq. (9). This
is actually done in 20 elements. We can now find the total
phase change by the following equation:
(pi= , (10)
where and L1x1 denote the local change of the refractive
index in the position of the (i,j)'th element and a small
distance element, respectively. The result of inserting the
experimental data for 120 mm of diffusion time and a mole
percentage of 90% thallium is shown in Fig. 5. For this
case, the maximum index change along the depth direction
is n =0.01, and this corresponds to a total phase change of
240 deg when using 633-nm laser light. The diffusion length
of thallium is measured to be 6 xm.
5 Phase-Only Grating and Their Diffraction
Efficiency
There has been no earlier report on holograms made by this
technique, and making such an element was also considered
as a test of the limitations of ion-diffusion-made elements.
In our experiment, the smallest aperture was about 3 rim.
Because of sideways diffusion, a good result was achieved
mainly for apertures larger than 5 jim and more than 20 pm
apart. With a smaller spacing between the apertures, they
tended to ''fuse' ' together.
By knowing the index increase in a local area, we can
find the phase profile of a plane wave passing through the
aperture in a vertical direction. The optical path length is
an integral over the index distribution multiplied by the
length. Then, multiplying the difference in optical path length
with the wave number k gives the phase profile after a single
pass. We have here neglected refraction due to the small
depth (5 to 10 iim) of an aperture. The integral mentioned
above can be found by summing the local index values
multiplied a small distance x. Doing this for all columns
gives the final profile.
OPTICAL ENGINEERING / June 1992 / Vol. 31 No. 6 / 1261

0
4-
-C
U)
6)
0
-C
0
63
>
6)
Position x
Fig. 6 Phase profile of the wavefront through an aperture.
Figure 6 shows an example of a phase profile. A high
index difference (aperture center area) gives a large phase
shift. Because of the diffusion that takes place in all directions,
there is also a phase shift outside the aperture region.
This causes only small reductions in diffraction efficiency,
but is a major problem when making aperture holograms.
Fourier transforming the phase profile of Fig. 6 gives the
diffraction pattern, and therefore also the diffraction effi-
0.40
0.35
> 0.30
6)
0.25
4-
4- 0.20
Li
0.15
L
4-
4- 0.10
6)
0.05
0.00
0.40
0.35
8.30
0.25
4-
4. 0.20
Li
0.15
0 8 16 24 32 40 48 56 64 72 80 88 96 104112120128136
Pos I t I on
1262 / OPTICAL ENGINEERING / June 1992 / Vol. 31 No. 6
(a)
C4
4- 0.10
8.05
I I I I I Ji L I I I I
a a 18 24 32 40 48 56 84 72 80 88 96 104112120128136
Pos I t I on
(c)
BOLSTAD, YATAGAI, and SEKI
Fig. 7 Calculated diffraction patterns: (a) 0=182.1
(d) 0=240.3 deg.
ciency, as shown in Fig. 7. In this figure we can see the
result for four different phase profiles. It is drawn with the
maximum phase shift as a variable. We can see that at near
180 deg, there is still a considerable energy left in the zeroorder
(30%), whereas there is about 35% in the first orders.
Increasing the phase shift to about 220 deg gave the best
result for this particular profile with 13% left in the zero
order and about 37% in the first orders, compared to the
theoretical limit of 40% for a binary phase-only grating.
Other profiles had even higher efficiencies. Figure 8 shows
the best result with maximum of 38.5% in the first orders
at 210-deg phase shift, which is very close to theoretical
prediction of 0 — 220 deg. A corresponding low diffraction
efficiency of 5% is obtained at 250 deg for the zero order.
Because of the broadening of the phase profile in the ion
exchange there is a shift in the efficiency curve to a somewhat
higher phase shift compared to the theoretical result
for a rectangular grating.
6 Conclusion
We have proposed the use of an ion-exchange technique to
produce binary phase-only CGHs. The optimum ion-exchange
condition is discussed. For binary phase-only holograms,
the optimum amount of phase change is IT, whereas this
condition is shifted to 220 deg in the ion-exchange case
0.40
0.35
> 0.30 -
a)
cj 0.25 -
4-
0.20
LI
0.15 -
L
IJI
4-
0.10
0.05 -
0.00 — I I I I I I I I I
0 8 16 24 32 40 48 56 64 72 80 88 96 104112120128136
>,.
U
C
SI
U
4.
1.
Li
4-
4.
0.35
0.30
Pos I t I on
(b)
I I IiI
0 8 18 24 32 40 48 58 84 72 80 88 88 104112120128138
Pos I t ion
(d)
deg, (b) 0 = 204.5 deg, (c) 0 = 226.9 deg, and

C)
C
a)
ci
Li
C
0
4-)
C)
a)
L
'4-
'4-
50
40
30
20
10
0
-
0 0.51T TI 1.511
Phase Shift (rad)
OPTIMIZATION OF PHASE-ONLY COMPUTER-GENERATED HOLOGRAMS
2ff
Fig. 8 Plot of calculated diffraction efficiency versus phase shift.
when ion diffusion is considered. The technique presented
here is expected to provide an efficient CGH that can be
used in optical interconnection, three-dimensional image
display, and other devices.
Acknowledgment
The authors are grateful to Hideki Hashizume and Shigeru
Kobayashi of Nippon Sheet Glass Co. for their experimental
assistance.
References
"
— First order
———— Zero order
1 . W H. Lee, ''Computer-generated holograms: Techniques and applications,"
Prog. Opt. 16, 121 (1987).
2. W. J Dallas, The Computer in OpticalResearch, B. R. Frieden, Ed.,
p. 291, Springer-Verlag, Berlin (1980).
3. A. W. Lohmann and D. P. Paris, "Binary Fraunhofer holograms generated
by computer," Appi. Opt. 6, 1739 (1967).
4. W. H. Lee, ' 'Sampled Fourier transform hologram generated by computer,"
Appi. Opt. 9, 639 (1970).
5. W. H. Lee, "Binary computer-generated holograms," Appi. Opt. 18,
3661 (1979).
6. L. B. Leesem, P. M. Hirsch, and J. Jordan, Jr. , ' 'The kinoform: A
new wavefront reconstruction device,' ' IBM J. Res. Dev. 13, 150
(1969).
7. H. Dammann, "Blazed Synthetic Phase-Only Holograms," Optik 31,
95 (1970).
8. M. Takeda and T. Yatagai, "Computer-generated hologram and kinoform,"
Oyobutsuri 41, 1039 (1972) (in Japanese).
9. T. Yatagai, R. Sugawara, H. Hashizume, and M. Seki, "Phase-only
computer-generated hologram produced by an ion-exchange technique,"
Opt. Lett. 13, 952 (1988).
10. T. Yamagishi, K. Fujii, and I. Kitano, "Gradient-index rod lens with
high NA. ," App!. Opt. 22, 400 (1983).
11. M. Oikawa and K. Iga, "Distributed-index planar microlens," App!.
Opt. 21, 1052 (1982).
12. E. Okuda, I. Tanaka, and T. Yamagishi, "Planar gradient-index glass
waveguide and its applications to a 4-port branched circuit and star
couplar," App!. Opt. 23, 1745 (1984).
13. M. Seki, H. Hashizume, and R. Sugawara, "Two-step purely thermal
ion-exchange technique for single-mode waveguide devices in glass,"
E!ectron. Lett. 24, 1258 (1988).
14. B. R. Brown and A. W. Lohmann, ''Computer-generated binary holograms,'
' IBM J. Res. Dev. 13, 160 (1969).
Hans Christian Boistad received the MS
degree in physics from the Norwegian Institute
of Technology (NTH), Trondheim, in
1987. Following this he was a research fellow
for two years, working at the University
of Tsukuba and later with Nippon Sheet
Glass, Inc. , where he was engaged in modeling
and fabrication of computer generated
holograms. He is currently working toward
a PhD degree in fiber optics and MBE-
____________ — grown GaAs/AIGaAs optical waveguide
technology at . . I. He is a student member of IEEE.
Toyohiko Yatagai received his DE degree
in applied physics from the University of
Tokyo in 1 980. He joined Institute of Physical
and Chemical Research in 1 970. Since
1983, he has been an associate professor
in the Institute of Applied Physics, University
ofTsukuba. He received an Optical Research
Award from the Japan Society of
Applied Physics in 1978. He is active in
optical computing and optical measurement
research.
Masafumi Seki received BS and MS degrees
from the University of Tokyo in 1972
and 1 974, respectively. From 1 974 to 1983,
he was with the Central Research Laboratories
of NEC Corporation and worked in
the fields of optical fibers, microoptic devices,
laser diodes, and related integrated
optics. He pioneered WDM multiplexers,
optical isolators for fiber optic communication
and LD-isolator-SMF modules. In
1985 he joined Tsukuba Research Laboratory
of r! ,. I Glass Co., Ltd., where he has been working
on guided-wave devices in glass and in LiNbO3. He is a member
of OSA.
OPTICAL ENGINEERING / June 1 992 / Vol. 31 No. 6 / 1263