This paper proposes an optical system for complex holographic display that enhances the quality of the reconstructed threedimensional image. This work focuses on a new design for an optical system and the evaluation of the complex holographic display, using a single spatial light modulator (SLM) and a circular grating. The optical system is based on a 4
f
system in which the imaginary and real information of the hologram is displayed on concentric rectangular areas of the SLM and circular grating. Thus, this method overcomes the lack of accuracy in the pixel positions between two window holograms in previous studies, and achieves a higher intensity of the real object points of the reconstructed hologram than the original phasereconstructed hologram. The proposed method provides approximately 30% less NMRS (Normal Root Mean Square) error, compared to previous systems, which is verified by both simulation and optical experiment.
I. INTRODUCTION
A digital holographic display is a type of display technology that provides full depth information for real objects. The optical field of an object can be recorded and stored in a digital device such as a CCD camera or optical sensor, with the capability to transfer and display it at any time and place. In a digital hologram, reconstruction devices can only be used for a phase hologram or an amplitude hologram, but not both at the same time. Previous reports have proposed several methods that use a spatial light modulator (SLM) to reconstruct a complex hologram. Such complex reconstructed holography with an SLM may use either of two methods: two SLMs
[1

3]
, or a single SLM
[4

6]
. The method with two SLMs uses a coupled phase and amplitude for each SLM set up on both sides of a beam splitter, for combination into a complex hologram. This solution provides a fulldisplay complex hologram with high spatial resolution, but coupling the two displays with pixel accuracy is very difficult. Using the second method, with a single SLM, some researchers attempted to overcome the pixelmatching problem by using a phasemodulation hologram
[4]
or an amplitudemodulation hologram
[5]
, or both imaginary and real parts of a complex hologram with two small windows and a 4
f
system that integrated the phase or amplitude sinusoidal grating to create a complex hologram where 4
f
is a common optical configuration in holographic recording
[6]
. However, this method only partially avoids the pixelmatching problem and did not yield an entirely complex hologram, with only phase or amplitude information.
In this paper we show how to overcome the constraints of complex holography by using a 4
f
system with an input plane with concentric rectangular data for imaginary and real information, and a circular grating. By mathematically calculating the difference between the two sizes of the concentric rectangular areas in the SLM, we achieve a complex hologram, instead of just adjusting the accuracy of the pixel positions on the SLM, as in previous studies. In our system the optical setup is simple, and the system provides a higher intensity for each reconstructed object point than do previous systems
[2

13]
. Section II describes the proposed method with size variation of concentric rectangular areas for better reconstruction quality. Our results are described in section III by simulation of and experiment with the newly designed optical system.
II. PROPOSED METHOD
The proposed method is based on a 4
f
system using a circular grating to implement the display system of a complex hologram. The principle of this method is illustrated in
Fig. 1
. On the input plane depicted in
Fig. 1(a)
, the SLM is divided into two concentric rectangles such that the outer rectangle contains the real part of the complex hologram, and the inner rectangle contains the imaginary part. The reflected light of the imaginary part from the SLM passes through a device to shift its phase by π/2, before it goes into the 4
f
system (device
D
_{1}
in
Fig. 1(e)
). In the 4
f
system in
Fig. 1(e)
, a circular grating and a pinhole are used to filter the high frequencies.
(a) The data of the input plane, (b) the diffracted light after the circular grating, (c) and (d) the Fourier transform of the circular grating, and (e) system configuration of the proposed method with a circular grating.
The diffracted light after the circular grating is shown in
Fig. 1(b)
. Prior to formulating the mathematical proof of our proposal, we calculated the Fourier transform of a circular cosine function.
Assume that the circular cosine grating function in
Fig. 1(b)
is
or
g
(
r
)=cos(2
πdr
) in the general case,
where
d
is the radial spatial frequency of the grating. From previous research
[7
,
9
, and
10]
, we have a Fourier transform as follows:
where
and
. From Spiegel
et al
.
[8]
, we have
Thus the Fourier transform of the circular cosine function is
The principle of the proposed method is described in
Fig. 1(e)
: a 4
f
optical system with an amplitude SLM, lenses
L
_{1}
and
L
_{2}
, the device
D
_{1}
, and a circular grating.
On the SLM, the input hologram is divided into two parts, as shown in
Fig. 1(a)
, with the outside area displaying the pixel values
H_{i}
and the inside area displaying pixel values
H_{r}
. The pixel values on the SLM are described by these functions:
with
a
and
b
being the height and width of
H_{r}
.
The reflected light from the
H_{i}
components of the SLM goes through a device that shifts the light phase by
π
/2:
D
_{1}
, as shown in
Fig. 1(e)
. Thus the function of the input hologram after
D
_{1}
becomes
The function of the light after lens
L
_{1}
is
From this, the light after the circular grating has the following function:
where
.
The function for the light after lens
L
_{2}
is a Fourier transform of Eq. (9), so we have
From Eq. (4), we have the function of light after lens
L
_{2}
as
where
and
.
This can be rewritten as follows:
Then
From this function, we see that
u
_{Plane}
_{2}
(
x
_{2}
,
y
_{2}
) has two components:
G
(
x
_{2}
,
y
_{2}
) and the complex hologram
.
Therefore, if the output of this system is a complex hologram, then the parameter
G
(
x
_{2}
,
y
_{2}
) must be a nonzero constant. From Eq. (13),
G
(
x
_{2}
,
y
_{2}
) has three components:
Among these three components, the function
G
_{1}
(
x
_{2}
,
y
_{2}
) must satisfy the condition
, which means that only the coordinates of a hologram in a circle with radius
f
_{2}
satisfy the condition. The function
G
_{1}
(
x
_{2}
,
y
_{2}
) is described in one and two dimensions in
Fig. 2
.
(a) Schematic plot of a cross section through the spectrum of the function G_{1}(x_{2}, y_{2}). (b) Function G_{1}(x_{2}, y_{2}) in two dimensions.
The position of function (a) G_{2}(x_{2}, y_{2}) and (b) G_{3}(x_{2}, y_{2}), with the size of the SLM.
Because
G
(
x
_{2}
,
y
_{2}
) is a nonzero constant, the parameters
a
and
f
_{2}
must satisfy the following conditions:
• First case:
a
≥
f
_{2}
encircles function
G
_{2}
(
x
_{2}
,
y
_{2}
) inside the rectangle of function
G
_{3}
(
x
_{2}
,
y
_{2}
). The function
u
(
x
_{2}
,
y
_{2}
)) becomes
where A =
G
_{1}
(
x
_{2}
,
y
_{2}
) ×
G
_{1}
(
x
_{2}
,
y
_{2}
), (
x
_{2}
,
y
_{2}
) in the circle with radius
f
_{2}
.
•
Second case:
means the rectangle of function
G
_{3}
(
x
_{2}
,
y
_{2}
) is inside the circle of function
G
_{2}
(
x
_{2}
,
y
_{2}
), so the function
u
(
x
_{2}
,
y
_{2}
) becomes
where A =
G
_{1}
(
x
_{2}
,
y
_{2}
) ×
G
_{3}
(
x
_{2}
,
y
_{2}
), (
x
_{2}
,
y
_{2}
) in the rectangle with
a
×
b
.
III. RESULTS
 3.1. Simulation
In this section we give simulation results from the Matlab program. The simulation parameters are described as follows: The object was created in Matlab as a spherical object or polyhedron, as shown in
Fig. 4
, with 104 object points, 5×104 object points, 105 object points, and 5×105 object points. The wavelength of the laser was 532 nm. The size of the hologram was 1200×1920 pixels. The focal length of lens
L
_{1}
was 10 cm. The size of the SLM screen was 16.39×10.56 mm
^{2}
, and the spectral range of the circular grating was 300 nm to less than 16 μm.
Objects for the experiment: (a) spherical object, (b) polyhedral object with viewing angle 0°, and (c) polyhedral object with viewing angle 90°.
 3.1.1. First Case :a≥f2
The width of the SLM was 16.39×0.56 mm
^{2}
.
Figure 5(a)
describes a normal root mean square (NRMS) error for reconstructing the object, depending on the parameters
a
and
b
, fixing the value of
f_{2}
at 5 mm. The experiment revealed that the NRMS error changed very little when
a
and
b
changed from 5 to 10 mm, or from 5 to 16 mm.
Figure 5(b)
plots the changes in the NRMS error when
a
= 9 mm and
b
= 14 mm as
f_{2}
changed from 9 to 2 mm. At
f_{2}
= 5.4 mm, the NRMS error was at its lowest value; at this position, the NRMS error was decreased by 30%, compared to a reconstructed original phase hologram.
(a) When fixing the value of f_{2} and changing a and b; (b) when a × b = 9 mm ×14 mm and changing the focal length of lens L_{2} (parameter f_{2}).
 3.1.2. Second Case:a2+b2≤f22
If the ratio of the rectangle depends on the width and length of the SLM to satisfy this condition, the maximum values are
b
= 8.8 mm and
a
= 5.4 mm. Therefore, values of
a
from 5.4 to 0.44 mm and
b
from 8.8 to 3.8 mm are chosen, with steps of 200 µm.
Figure 6(a)
describes the NRMS error when
f_{2}
= 10 mm, changing the size of the rectangle from 5.4×8.8 mm
^{2}
to 0.4×3.8 mm
^{2}
. When
a
= 4.8 mm and
b
= 8.2 mm the NRMS error reaches its lowest value, decreasing by about 40% compared to a reconstructed phase hologram.
Figure 6(b)
plots the NRMS error when
a
×
b
= 3.2×6.6 mm
^{2}
, changing
f_{2}
from 8 to 10 mm. This figure shows that the NRMS error changes very little due to the rectangle inside the circle, as expressed by Eq. (18).
(a) When fixing f_{2} at 10 mm and changing the size of the rectangle, and (b) when fixing the rectangle at a × b = 3.2 × 6.6 mm^{2} and changing the focal length of lens L_{2} (parameter f_{2}).
The simulation of the reconstructed hologram is shown in
Fig. 7
, for a spherical object of 104 points.
Reconstruction of the hologram: (a) first case, a = 9 mm and b = 14 mm; (b) second case, f_{2} = 10 mm, a = 4.2 mm, and b = 7.6 mm.
 3.2. Experiment
 3.2.1. Using a Quarterwave Plate forD1
In the experiment, a laser of wavelength 532 nm was used; the focal lengths of lenses
L
_{1}
and
L
_{2}
were set to 10 cm and 10 mm respectively; and the size of the rectangle on the SLM was 4.8×8.2 mm
^{2}
. The SLM had the following characteristics: pixel size 8.1 μm, total size 1200×1920 pixels, and pinhole size 2 mm. The radius of the quarterwave plate used before
L
_{1}
was 8.5 mm (the device
D
_{1}
in
Fig. 1(e)
). The reconstructed hologram is shown in
Fig. 8
.
Reconstructed spherical hologram: (a) proposed method in the first case with f_{2} = 6 mm, (b) proposed method in the second case with a × b = 3.2 × 6.6 mm^{2}, (c) previous method using a sinusoidal grating, and (d) original phase hologram.
Figure 8
shows the reconstructed holograms of spherical objects with 104 object points (
Fig. 8(a)
), 5×104 object points (
Fig. 8(b)
), 105 object points (
Fig. 8(c)
), and 5×105 object points (
Fig. 8(d)
). Moreover, these reconstructed holograms shown in
Figs. 8 (a)
,
(b)
,
(c)
, and
(d)
were obtained by applying respectively the techniques of the first case, second case, previous 4
f
method with a sinusoidal grating, and original phase hologram.
Figure 8(d)
shows that the reconstructed original phase hologram has better overlap with the object points than does either the proposed method or the previous 4
f
method with a sinusoidal grating. On the other hand, in
Fig. 8(a)
with parameter
f_{2}
= 6 mm for the first case, and in
Fig. 8(b)
with parameter
a
×
b
= 3.2×6.6 mm
^{2}
for the second case, the quality and intensity of the object points of the reconstructed hologram are also better than both the previous method as shown in
Fig. 8(c)
, and the original phase hologram as shown in
Fig. 8(d)
.
 3.2.2. Using Thick Glass forD1
A thick glass device changes the phase of light that passes through it. We used this property to calculate the phase change between light passing through thick glass and light not passing through it. We assume that the thick glass has a thickness
d
as in
Fig. 9
.
(a) The light goes through thick glass, (b) the first and second beams differ by λ/2 wavelengths, (c) the angle of illuminated light for a thick lens with K = 3×10^{5} +1, and (d) with K = 1.
In this figure we see that the light passing through the thick glass travels a longer distance than the light that does not pass through it. This distance is described as
For incident light passing through the thick glass and light not passing through to have a phase difference of π/2, the distances traveled must differ by an odd number of halfwavelengths:
where
K
is odd. From this we have
From Eq. (21), when
K
is given, the angle of incident light is calculated to yield a phase difference of π/2 between the light passing through the thick glass and the light that does not pass through it.
In
Fig. 10
an experiment was conducted with the wavelength of the laser at 532 nm, where the refractive index of the glass was
n
= 1.524. Thick glass was placed before lens
L
_{1}
with a size of 4.8×8.2 mm
^{2}
, thickness of 2 mm, and angle of inclination 1.9256 degrees with respect to the reference beam. The size of the rectangle on the SLM was 4.8×8.2 mm
^{2}
. The relative measurement error due to angle adjustment was 5×10
^{6}
.
Reconstructed polyhedral hologram: (a) proposed method in the first case with f_{2} = 6 mm, (b) proposed method in the second case with a × b = 3.2 × 6.6 mm^{2}, (c) previous method using a sinusoidal grating, and (d) original phase hologram.
Figure 10
shows the reconstructed hologram of a polyhedral object with two different viewing angles of 0˚ and 90˚. The parameters in the first and second cases were similar to those for a spherical object. This shows that the quality of the reconstructed hologram was better than with either the previous method or the original phase hologram.
Figure 11
shows the average of the intensity of ten objectpoint layers from 121 to 132 of the reconstructed hologram from three methods: the reconstructed original phase hologram, a complex hologram using a sinusoidal grating method, and the proposed method.
Figure 11(a)
shows the average intensity of the object points from layers 121 to 132 of the reconstructed hologram for a spherical object using quarter wave plate. It can be seen that the average intensity of object points with the phase hologram was mainly in the range from 30 to 40, while with the sinusoidal grating method it was mainly in the range from 50 to 60, and from 80 to 100 with our proposed method in the second case.
(a) The average intensity of reconstructed objectpoint layers from 121 to 132 with the second case, spherical object, and using a quarterwave plate; (b) reconstructed objectpoint layers from 121 to 132 with the second case, polyhedral object, and using thick glass with K = 1.
Figure 11(b)
shows the average intensity of objectpoint layers from 121 to 132 of the reconstructed hologram of a polyhedral object. In this figure the intensity of the object points in the original phase hologram was mainly in the range from 10 to 20, whereas with the sinusoidal grating method it was in the range from 20 to 40, and in our proposal method using thick glass with
K
= 1, the range was from 50 to 70.
Figure 8
,
10
, and
11
confirm that the quality of the reconstructed hologram for a spherical object using the proposed method is better than the quality of either the original phase hologram or the hologram obtained by the previous method using the 4
f
system with a sinusoidal grating.
IV. CONCLUSION
In this study we proposed a method to display a fulldisplay complex hologram with a single SLM and a circular grating, and demonstrated it experimentally. By calculating the difference between the size of the area of the changing phase of the hologram and the SLM, we were able to generate a complex hologram through a 4
f
system with a circular grating. This method does not require an adjustment of the accuracy of pixel positions on the SLM, as in previous methods, and provides approximately 30% less NMRS error compared to previous systems. Also, the proposed method features ease of implementation and stability of the optical system. In this experiment, the reconstructed hologram with good quality has been achieved from the two optimized cases for focal length of the lens
L
_{2}
, where 9 to 16 mm in the first case, and from 8 to 4 mm in the second case.
Acknowledgements
This research was supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the ITRC (Information Technology Research Center) support program (IITP2015R0992151008) supervised by the IITP (Institute for Information & communications Technology Promotion) and ‘The CrossMinistry Giga KOREA Project’ grant from the Ministry of Science, ICT and Future Planning, Korea.
Stolz C.
,
Bigue L.
,
Ambs P.
(2001)
“Implementation of highresolution diffractive optical elements on coupled phase and amplitude spatial light modulators,”
Appl. Opt.
40
6415 
6424
DOI : 10.1364/AO.40.006415
Tudela R.
,
Badosa E.
,
Labastida I.
,
Carnicer A.
(2003)
“Full complex Fresnel holograms displayed on liquid crystal devices,”
J. Opt. A: Pure Appl. Opt.
5
S1 
S6
DOI : 10.1088/14644258/5/4/001
Neto L.
,
Roberge D.
,
Sheng Y.
(1996)
“Fullrange, continuous, complex modulation by the use of two coupledmode liquidcrystal televisions,”
Appl. Opt.
35
4567 
4576
DOI : 10.1364/AO.35.004567
Arellano N.
,
Zurita G.
,
Fabian C.
,
Castillo J.
(2008)
“Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for oneshot phaseshifting interferometry,”
Opt. Express
16
19330 
19341
DOI : 10.1364/OE.16.019330
Ulusoy E.
,
Onural L.
,
Ozaktas H.
(2011)
“Fullcomplex amplitude modulation with binary spatial light modulators,”
J. Opt. Soc. Am. A
28
2310 
2321
Liu J.
,
Hsieh W.
,
Poon T.
,
Tsang P.
(2011)
“Complex Fresnel hologram display using a single SLM,”
Appl. Opt.
50
H128 
H135
DOI : 10.1364/AO.50.00H128
Reichelt S.
,
Hausler R.
,
Futterer G.
,
Leister N.
,
Kato H.
,
Usukura N.
,
Kanbayashi Y.
(2012)
“Fullrange, complex spatial light modulator for realtime holography,”
Opt. Lett.
37
1955 
1957
DOI : 10.1364/OL.37.001955
Spiegel M.
,
Lipschutz S.
,
Liu J.
2009
Mathematical Handbook of Formulas and Tables
McGraw Hill
NY, USA
155 
Goodman J. W.
1996
Introduction to Fourier Optics
McGrawHill
Singapore
32 
55
Stein M.
,
Weiss G.
1971
Introduction to Fourier Analysis on Euclindean Spaces
Princeton University Press
NJ, USA
133 
172
Lee K.
,
Jeung S.
,
Kim N.
(2007)
“Holographic demultiplexer with low polarization dependence loss using photopolymer diffraction gratings,”
J. Opt. Soc. Korea
11
51 
54
DOI : 10.3807/JOSK.2007.11.2.051
Park J.
,
Kim M.
,
Ganbat B.
,
Kim N.
(2009)
“Fresnel and Fourier hologram generation using orthographic projection images,”
Opt. Express
17
6320 
6334
DOI : 10.1364/OE.17.006320
Mills G.
,
Yamaguchi I.
(2005)
“Effects of quantization in phaseshifting digital holography,”
Appl. Opt.
44
1216 
1225
DOI : 10.1364/AO.44.001216
Javidi B.
,
Tajahuerce E.
(2000)
“Threedimensional object recognition by use of digital holography,”
Opt. Lett.
25
610 
612
DOI : 10.1364/OL.25.000610
Li J.
,
Li Y.
,
Wang Y.
,
Li K.
,
Li R.
,
Li J.
,
Pan Y.
(2014)
“Twostep holographic imaging method based on singlepixel compressive imaging,”
J. Opt. Soc. Korea
18
146 
150
DOI : 10.3807/JOSK.2014.18.2.146