Weighting method of the Fourier-kinoform synthesis
A new iterative Fourier transform method of synthesis of kinoforms is presented. Two object-depended filters (an amplitude filter and a phase one) are used in the object plane on the iterative calculation of a kinoform instead of a single (phase) filter as usual. The amplitude filter is a system...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2008
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| Цитувати: | Weighting method of the Fourier-kinoform synthesis / A. V. Kuzmenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 3. — С. 303-306. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1190502017-06-04T03:02:31Z Weighting method of the Fourier-kinoform synthesis Kuzmenko, A.V. A new iterative Fourier transform method of synthesis of kinoforms is presented. Two object-depended filters (an amplitude filter and a phase one) are used in the object plane on the iterative calculation of a kinoform instead of a single (phase) filter as usual. The amplitude filter is a system of weight coefficients that vary in the process of iterations and control the amplitude of an input object. The advantages of the proposed method over other ones are confirmed by computer-based experiments. It is found that the method is most efficient for binary objects. 2008 Article Weighting method of the Fourier-kinoform synthesis / A. V. Kuzmenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 3. — С. 303-306. — Бібліогр.: 10 назв. — англ. 1560-8034 PACS 42.40.Eq http://dspace.nbuv.gov.ua/handle/123456789/119050 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
A new iterative Fourier transform method of synthesis of kinoforms is
presented. Two object-depended filters (an amplitude filter and a phase one) are used in
the object plane on the iterative calculation of a kinoform instead of a single (phase) filter
as usual. The amplitude filter is a system of weight coefficients that vary in the process
of iterations and control the amplitude of an input object. The advantages of the proposed
method over other ones are confirmed by computer-based experiments. It is found that
the method is most efficient for binary objects. |
| format |
Article |
| author |
Kuzmenko, A.V. |
| spellingShingle |
Kuzmenko, A.V. Weighting method of the Fourier-kinoform synthesis Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Kuzmenko, A.V. |
| author_sort |
Kuzmenko, A.V. |
| title |
Weighting method of the Fourier-kinoform synthesis |
| title_short |
Weighting method of the Fourier-kinoform synthesis |
| title_full |
Weighting method of the Fourier-kinoform synthesis |
| title_fullStr |
Weighting method of the Fourier-kinoform synthesis |
| title_full_unstemmed |
Weighting method of the Fourier-kinoform synthesis |
| title_sort |
weighting method of the fourier-kinoform synthesis |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/119050 |
| citation_txt |
Weighting method of the Fourier-kinoform synthesis / A. V. Kuzmenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 3. — С. 303-306. — Бібліогр.: 10 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT kuzmenkoav weightingmethodofthefourierkinoformsynthesis |
| first_indexed |
2025-07-08T15:08:56Z |
| last_indexed |
2025-07-08T15:08:56Z |
| _version_ |
1837091874432614400 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 3. P. 303-306.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
303
PACS 42.40.Eq
Weighting method of the Fourier-kinoform synthesis
A.V. Kuzmenko
International Center “Institute of Applied Optics”, NAS of Ukraine,
10G, Kudryavskaya str., 04053 Kyiv, Ukraine, e-mail: avk@iao.kiev.ua
Abstract. A new iterative Fourier transform method of synthesis of kinoforms is
presented. Two object-depended filters (an amplitude filter and a phase one) are used in
the object plane on the iterative calculation of a kinoform instead of a single (phase) filter
as usual. The amplitude filter is a system of weight coefficients that vary in the process
of iterations and control the amplitude of an input object. The advantages of the proposed
method over other ones are confirmed by computer-based experiments. It is found that
the method is most efficient for binary objects.
Keywords: digital holography, holographic optical elements, phase retrieval, phase-only
elements.
Manuscript received 18.06.08; accepted for publication 20.06.08; published online 30.09.08.
1. Introduction
There exists a high number of iterative algorithms aimed
at the solution of phase problems, including the problem
of a kinoform. The majority of algorithms was
developed on the basis of the so-called error-reduction
(ER) algorithm [1]. It was refined in works by Fienup
[2, 3] on solving the problems of star interferometry (a
group of input-output algorithms). I mention also the
subsequent works, e.g., by Wyrowski and Bryngdahl [4],
Wyrowski [5], and Yang, Gu [6] in which the iterative
algorithms were further improved. At the present time,
all they form a family of the so-called iterative Fourier
transform (IFT) algorithms. The comparative analysis of
a number of algorithms from this family is available in
some surveys (see, e.g., [7, 8]).
It is known that processing the amplitude of a field
(A-field) in the spectral plane is the same in all IFT
algorithms of calculation of a kinoform. This operation
is nonlinear and consists in the reduction of the A-field
to unity. The algorithms differ from one another by a
mean of processing the field in the object plane which
depends on the final purpose, namely, on a function
which should be realized by a kinoform (the beam
splitting, beam shaping, image generation, and so on). It
is worth to note that, despite the diversity of operations
used in the object plane, all they, are linear (except the
ER-algorithm).
2. Algorithm
In the present work, I give a modified ER-algorithm of
calculation of a kinoform, whose main unique distinction
from the classical ER-algorithm [1] consists in the use of
a new nonlinear operation [9] of processing the A-field
in the object plane. This operation in its complex-valued
version was used earlier by us in the synthesis of double-
phase holograms [10]. To clarify its application, the
work of the algorithm is illustrated in Fig. 1. First, one or
several iterations (Ker) are realized by the classical ER-
scheme which requires no explanations. Then, in all
iteration with k > Ker on the formation of an input, f will
be replaced by a new function defined as
kf = ,fkα (1)
where the weight coefficients kα are determined by the
recurrence relation
11 −− βα=α kkk ( 1>k ), (2)
)./( 11 ε+=β −− kk gf (3)
Fig. 1. Weighting IFT algorithm.
Here, |gk–1| is the reconstructed amplitude on the
(k–1)th iteration, and ε is a small number ~10-10,
excluding the division by zero. It should be mentioned
that f is real. Operations (1)-(3) are heuristic and have no
strict mathematical justification. The physical sense of
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 3. P. 303-306.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
304
the coefficients kα becomes clear if the block of the
algorithm separated by a dashed line in Fig. 1 is
considered, according to Fienup [3] as a nonlinear unit
with the input αfexp(iφ) output g, and action operator
ℱ +1CF ℱ
-1. From the viewpoint of the theory of image-
processing system, the coefficients α is nothing but the
collection of coefficients of a negative feedback “output-
input'”: if the amplitude |gk–1| on the (k–1)th iteration at
some point (x′, y′) of the plane of images is more than a
given value f, then, on the next kth iteration, the input f
at the corresponding point (x, y) will be corrected.
Namely, it will be decreased by αk times, and vice versa.
At the same time, from the viewpoint of optics, the
system of coefficients αk normalized to unity can be
interpreted as some object-depended amplitude filter
which acts on the initial object f and varies in the process
of iterations. It is clear that, for all ER-iterations, α(x, y)
= α0(x, y) = 1(x, y).
3. Experiment
A number of model experiments with various objects
was realized with the purpose to study the potentialities
of the method. For the sake of comparison, analogous
experiments were also performed with the use of a
kinoform version of the input-output (IO) algorithm by
Fienup [3]. In all the cases, the same phase starting
diffuser with a uniform distribution of phases in the
interval 0-2π is used. The variance of the amplitudes of
images reconstructed in the process of iterations was
evaluated using the formula
gσ )(k = ,
)(
))()((
,
2
,
,
2
.,
∑
∑ µ−
ml
ml
ml
mlml
f
kgkf
(4)
where
2/1
,
2
,
,
2
,
)(
)(
)(
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
=µ
∑
∑
ji
ji
ji
ji
kg
f
k (5)
is the scale factor, the indices l, m and i, j run over the
points, where the amplitude of an initial object f is
nonzero, and k is the iteration number.
In the experiments involving the IO algorithm, the
optimum value of the object-depended coefficient
β = βopt in the equations for the input function (see Eqs
(8) and (9) in Ref. [3]) is used by attaining the best
behavior of the function σg(k) during iterations in all the
cases. The value of βopt was determined by means of the
cyclic repetition of the procedure of synthesis for various
values of β (from the interval 0.1-5.0 with a step of 0.1).
In Figs 2 to 4, the results of model experiments on
the synthesis of the kinoforms of binary and halftone
objects with a dimension of 64×64 counts are presented.
20 40 60
20
40
60
20 40 60
20
40
60
0
0.5
1
20 40 60
20
40
60
0
0.5
1
(a) (b)
(c)
Fig. 2. Objects 64×64: (a) binary; (b), (c) halftone without and
with a base (equal 0.17).
3.1. Binary object
In Fig. 3, the plots characterizing the quality of the
image of a binary object (Fig. 2a) reconstructed by a
kinoform are presented. As seen from Fig. 3a, the
weighting algorithm allows one to decrease the dispersal
of the one-bit-intensity ∆Ione-bits given by the ER-
algorithm practically to zero, i.e., the algorithm does not
reveal the effect of stagnation relative to binary objects.
In our example, 180 weighting-iterations reduce ∆Ione-bits
from 0.008 to 7.6×10-7 (Fig. 3b), whereas 1500 such
iterations result in ∆Ione-bits = 2×10-12.
20 40 60 80 100 120 140 160 180 200
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
K, iteration
ou
tp
ut
in
te
ns
ity
k=K
er
zero−bits
∆I
one−bits
Weighting
IO, eq.(8) [3], β
opt
=3.5
(a)
20 40 60 80 100 120 140 160 180 200
10
−6
10
−5
10
−4
10
−3
10
−2
k, iteration
σ g(k
)
(b)
k=K
er
Weighting
IO, eq.(8) [3], β
opt
=3.5
Fig. 3. The kinoform of the binary object (Fig. 2a): (a) range of
output intensities, (b) variance of amplitude of the
reconstructed images vs iteration number.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 3. P. 303-306.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
305
At the same time, the IO algorithm (with optimized
β) “stops” at the value ∆Ione-bits = 2.5×10-5. That is, it
falls in a minimum of σg(k) which is quite deep, but,
nevertheless, is local. As for the ratios of the minimum
one-bits-intensity to the zero-bits intensity for three
algorithms, they are equal, respectively, to 4 (ER), 4.53
(IO), and 7 (weighting). Figure 3b demonstrates the
effect of diminution of the variance σg(k) on the
transition from one algorithm to another one. Analogous
results were obtained also for other binary objects with
dimensions of 64×64 and 128×128.
3.2. Halftone object
A somewhat more complicated situation is observed for
halftone objects, one of which is presented in Fig. 2b. As
was shown by model experiments, the kinoforms of such
objects calculated with the help of the weighting
algorithm reconstruct a high-quality image only in the
range of amplitudes from ~0.15 to unity (on the
normalization of the image to unity). The rest amplitudes
are distorted to a variable degree. The proper
reconstruction of all the amplitudes, including those
close to zero, can be reached if the initial object is
positioned on a pedestal (Fig. 2c) whose height is ~15-
20 % of its maximum amplitude and if the reconstructed
image amplitude (the intensity in an optical experiment)
is cut off by the pedestal level. It is obvious that, in this
case, the useful diffraction efficiency of a kinoform
decreases. The dependences of σg on the iterations for
both compared algorithms given in Fig. 4 indicate that,
in the case where a base is supplemented to an object,
the weighting algorithm begins to surpass the IO
algorithm after a certain number of iterations. In our
example with the object in Fig. 2c, the advantage of the
weighting algorithm begins to manifest itself after 90
iterations (see Fig. 4), and is characterized almost by the
six-order difference by the 1500th iteration (1×10-12
against 9.5×10-6). But if the base is absent, the IO
algorithm has some advantage.
20 40 60 80 100 120 140 160 180 200
10
−4
10
−3
10
−2
k, iteration
σ g(k
)
4
3
2
1
Object, Weighting
Object, IO, eq.(9) [3], β
opt
=1.1
Object+base, IO, eq.(9) [3], β
opt
=1.1
Object+base, Weighting
1 −
2 −
3 −
4 −
Fig. 4. Variance of amplitude of the reconstructed images for
halftone object without and with a base (Fig. 2b, c).
The calculated efficiencies of kinoforms (in
parentheses, the values obtained within the IO algorithm
are given) are as follows: 91.39 (91.25) % for the object
in Fig. 2a, 94.82 (92.48) % for the object in Fig. 2b, and
96.91 (95.02) % for the object in Fig. 2c.
In the course of calculations, the criticality of the
weighting algorithm with respect to a value of the
parameter ε in the formula (3) is verified. By varying ε
from 1×10-22 to 1×10-6, the deviation of σg(ε) from
σg(ε10) = 10-10 is determined by the formula
∆σ = 100 % [σg(ε) – σg(ε10)]/σg(ε10)
for various objects with the number of iterations equal to
50. On the average, ∆σ was (0.002-0.05) %. Thus, the
variation of ε in the indicated limits did not influence
practically the exactness of the calculation of a kinoform
and, at the same time, excluded the situation where one
should divide the numerator in formula (3) by zero.
3.3. Super-Gaussian (SG) beam shaping
Within the weighting and IO algorithms, the calculations
of kinoforms that are the transducers of the intensity of a
Gauss beam of the form exp[–(x2 + y2)/2r0
2] in a SG
beam of the form exp[–(x'2/2r0
'2)M – (y'2/2r0'2)M] are
performed, where r0 and r0' are the inflection radii of the
Gauss curves, and M is the SG order (as known, the
calculation of a kinoform involves the square root of the
both indicated intensities). In Fig. 5, the input (r0 = 70)
and the output (r0' = 25) intensity profiles for M = 4 and
M = 100 with a dimension of the object (kinoform) of
256×256 counts are presented. The iteration process with
Ker = 10 was truncated at the 100th iteration. With regard
for the separation of the working part of a SG beam so as
shown in Fig. 5, the intensity variance σI and the output
efficiency η are as follows: σI = 2.9×10-4 (6.59×10-4), η =
96.46 (89.72) % for M = 4; σI = 3.7×10-5 (1.98×10-4), η =
93.63 (91.2) % for M = 100. The calculation of σI was
performed by a formula analogous to (4), but for
intensities.
20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
in
te
ns
ity
n
M=4
M=100
Weighting
IO, eq. (9)[3], β
opt
=1.2
ideal Super−Gaussian
output
output screening
Gauss, r
0
=70
Fig. 5. The profiles of intensities of super-Gaussian beams
with r0' = 25 of the 4th and 100th orders obtained by
calculations of the 256×256 kinoform within the weighting and
IO algorithms. Curves for M = 4 and M = 100 are vertically
shifted up for clarity.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 3. P. 303-306.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
306
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250
50
100
150
200
250
Gauss, r
o
=70 SG, r’
o
=25
Gauss, r
o
=35 SG, r’
o
=25
Fig. 6. Proper (r0 = 70) and erroneous (r0 = 35) choices of the
effective width of an illuminating beam on the calculation of
the kinoform-former of a super-Gaussian (r0' = 25). In the
second case, the cross-section of a super-Gaussian is covered
by a speckle.
I note that, in the calculation of a kinoform-former
of a SG, a special attention should be paid to a choice of
r0 defining the effective width of a beam illuminating the
kinoform. For small r0, the kinoform is illuminated by a
narrow Gauss beam, which means the actual
nullification of light amplitudes on the edges of the
kinoform. This is equivalent to a reduction of the band
of space frequencies forming a SG. As a result, the
pattern of a SG will be covered by a speckle irrespective
of the value of r0' (see Fig. 6). In more details, the
problem of restriction of the frequency band and its
relation to the quality of images are considered in [4].
4. Conclusions
Thus, the weighting algorithm has high efficiency in the
synthesis of the kinoforms of binary objects. It is
basically important that the effect of stagnation of the
algorithm is absent in this case, i.e., the one-bits variance
σg(k) in a reconstructed image tends to zero with
increase in the number of iterations and the noise level
(zero-bits) is the same as that of other algorithms or
lower. The weighting algorithm is also efficient in
calculations of kinoforms as the formers of SG laser
beams. It must be emphasized that the weighting
algorithm contains no parameters requiring the
optimization (like β in the IO algorithm), which
essentially accelerates the counting rate. However, it is
expedient to apply this algorithm to halftone objects only
in the case where the minimum amplitude of the
normalized distribution of an object is ~0.15-0.20.
References
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algorithm for the determination of phase from
image and diffraction plane pictures // Optik 35,
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