Optics and Photonics Journal, 2011, 1, 52-58
doi:10.4236/opj.2011.12008 Published Online June 2011 (http://www.SciRP.org/journal/opj/)
Scanning Holography Using a Modulated Linear Pupil:
Simulations
Abdallah Mohamed Hamed
Physics Department, Faculty of Science, Ain Shams University, Cairo, Egypt
Received March 14, 2011; revised April 14, 2011; accepted April 25, 2011
Abstract
In this paper an electron microscopic image of diameter 80 - 120 nm and of dimensions 180 × 220 pixels is
used as a test object to fabricate Fresnel zone plate hologram. The author proposes a different set of pupils:
one pupil still being a delta function and the other being a function of <

> = sqrt(x2 + y2). The obtained re-
constructed images in case of scanning holography are investigated.
Keywords: Scanning Holography, Linear Apertures
1. Introduction
The pioneer work of digital holography or computer-gene-
rated hologram (CGH) was early proposed by Good-
man, et al. [1] and Lohmann, et al. [2], and numerical
hologram reconstruction was initiated by Kronrod et al.
[3] in the early 1970 followed by many other authors.
Improved reconstructed images from CGH are obtained
using an iterative operations [4]. Recently, the possibili-
ties of reconstructing the hologram structure and image
from a digitally recorded specklegram without reference
beam has been considered separately by Hamed [5] and
Gorbatenko, et al. [6]. Also, improved reconstructed im-
age from digital Fourier holograms is attained using su-
perposition of reconstructed images obtained by multiple
wavelengths [7] and separately using a two-step only qua-
dratic phase shifting holography [8] where neither the re-
ference—wave intensity nor an object-wave intensity
measurement is needed in this technique.
The idea of holographic recording accomplished by het-
erodyne scanning was originally proposed by Poon [9-11].
And heterodyne scanning was accomplished using a two-
pupil optical system Lohmann and Rhodes [12]. They real-
ized Fresnel-zone-plate-type impulse response, i.e. its phase
is a quadratic function of x and y, in and out-of focus plane
near the focal plane of lenses L1 and L2. In a precedent pro-
posed work by Poon, one of the pupils is a delta function
and the other has a constant uniform circular aperture.
The original idea, which was later analyzed and called
scanning holography [13], is to scan the 3-D object in a 2
-D raster with a complex Fresnel zone plate—type im-
pulse response created by interference of a point source
and a plane wave emerging from each pupil. A temporal
frequency offset is introduced between the two pupils
and the desired signal from a spatially integrating detec-
tor is obtained using a heterodyne detection.
In the present study, the author investigates scanning
holographic imaging based on two-pupil heterodyne detec-
tion. In the original standard system proposed by Poon, one
of the pupils is a delta function and the other a constant.
In the present case, the author proposes a different set
of pupils: one pupil still being a delta function and the
other being a function of

= sqrt (x2 + y2). The simu-
lated reconstructed images using the above technique of
heterodyne detection are investigated. The proposal of
the linearly modulated aperture [5] was investigated in a
re- cent article of modulated speckle images.
2. Theoretical Analysis
2.1. A Two-Pupil Heterodyne Scanning
Hologram
The optical scanning hologram is based on two—pupil
heterodyne detection as shown in Figure 1. In this study,
the 1st pupil is chosen to be a linear function distributed
within the circular frame of diameter0
2D 
.

10
,;1for linear aperturePxy


(1)
The 2nd pupil remains as before a delta function which
is represented as follows:

A. M. HAMED
53
Figure 1. A two-pupil optical heterodyne scanning system. 1-Laser operating at frequencyω0; 2,11-beam splitters; 3-
acousto-optic frequency shifter; 4-cos Ω t giving a modulated frequency signal at ω0 + Ω; 5,7-reflecting mirrors; 2,5,7,11-form
the Mach-Zehnder interferometer; 6,8-are two-pupils one pupil being a delta function and the other being a linear function of ρ;
9,10-are two converging lenses where the two pupils are located at the front focal planes of lenses L
1 and L2, both with a focal
length of f; 12-two dimensional scanning mirror; 13-object transparency; 14-Collector lens; 15-photo-detector; 16-electronic
band pass filter tuned at the heterodyne frequency Ω; 17-output of scanned and processed current iΩ (x,y).

2(, ),Pxy xy (2)
The Fourier transform of Equation (1) is previously
computed as follows [14]:
   
10
122
2
.i
i
Jk Jk
PkconstJ k
kkk




 (3)
J0, J1 are Bessel functions of zero and first orders.
The optical transfer function is obtained as [10]:

,;
xy
OTF kkz


22 *
12
00
0
exp (),
2
exd '
,
pd
xy x
yxy
jz f
kkPPx ky
kk
fz
kjxkykxy
kf
xy
 

 
 
 

 






(4)
In the present work, we have assumed linear function
for the 1st pupil and the same delta function for the 2nd
pupil is used, hence substitute from (1) and (2) in Equa-
tion (4), we can write the OTF as follows:

,; xy
OTF kkz

22 22
00
0
exp (),
2
expdd
xy x
yxy
jz f
kkx ky
xy
kk
fz
kjxkykxy
kf

 

 
 

 

  



(5)
This equation can be rewritten symbolically as follows:

22
0
22
00
,;exp() ..
2
,
xyx y
xy
jz
OTFkkzkkF T
k
ff
xy xkyk
kk







 
 
 



(6)
Since the Fourier transform of multiplication product is
transformed into a convolution product of the Fourier spec-
trum of each function [15], then Equation (6) becomes


222 2
0
00
,;exp()..
2
*.. ,
xyx y
xy
jz
OTFkkzkkFTxy
k
ff
FTxk yk
kk


 








 



(7)
The Fourier transform of a shifted delta function is
calculated to give this result:

22
00 0
.,exp
xy xy
ff jz
FTxkykkk
kk k




 
 




(8)
Substitute from Equation (8) to Equation (7), we obtain:



22
0
22 22
0
,; exp()
2
.. *exp
xyx y
xy
jz
OTF kkzkk
k
jz
FTxyk k
k





















(9)
It is shown that the F.T. of the linear function p1(x,y) =
ρ = (x2 + y2)1/2 appeared in Equation (9) is obtained in
Equation (3).

A. M. HAMED
54
2.2. Special case (Poon results):
In case of uniform circular aperture to represent the 1st
pupil instead of the linear aperture, then the F.T. becomes:

 
22
....1 ,
x
y
F
Tx yFTkk 

  (10)
In this case, Equation (9) is reduced to
 
 
22
0
22
0
,; exp
2
,*exp
xyxy
xyx y
jz
OTFk kzkk
k
jz
kkk k
k















(11)
From the properties of convolution product of a func-
tion with a delta function leave it unchanged, then Equa-
tion (11) is reduced to the OTF of Poon [13] to give:
 
22
0
,;exp 2
xyx y
jz
OTFk kzkk
k


 


; Poon’s result (12)
The intensity distribution of the complex optical scan-
ning hologram, obtained in case of uniform circular ap-
erture for the 1st pupil and delta function for the 2nd pupil,
is represented as :
 


cos sin
,, ,
c
H
xy HxyjHxy



22
0
20
0
,;*dexp
22
D
jk xy
jk
x
yz z
zz











 (13)
Where
 

22
20
0
cos 0
,,;*cos d
22
D
kxy
k
H
xy xyzz
zz











(14)
 

22
20
0
sin 0
,,;*sin d
22
D
kx y
k
H
xy xyzz
zz











(15)
While in case of the linear pupil combined with the delta
function for the 2nd pupil, the intensity distribution of the
complex optical scanning hologram is written as follows:
 

22
20
0
0
22
00
,,;*exp *
22
,d
cD
xy
jk xy
jk
Hxyxyz zz
ff
xy xkykz
kk












 
 


 



(16)
3. Results and Discussion
The original image of dimensions 180 × 220 pixels is
plotted as shown in Figure 2. The actual dimension of
the image range is 80 - 120 nm. The auto-correlation in-
tensity of the image is shown in Figure 3.
A cosine FZP hologram obtained using a linear aperture
and a pinhole aperture is shown in Figure 4. The recon-
structed images are obtained by operating the Fourier trans-
form upon the holographic images which are plotted as in
Figure 5(a)-(f). The reconstruction from the complex ho-
lographic FZP images are shown as in Figure 5(e) for
constant pupil and in Figure 5(f) for linear pupil. It is
shown that the reconstructed images from the complex
holograms are much better in resolution than the recon-
structed images obtained from cosine and sine holograms.
Also, it is shown that the reconstructed FZP image ob-
tained in case of linear modulation for the 1st pupil as in
Figure 5(b) is better in resolution than the reconstructed
image obtained from the sine FZP hologram using circular
uniform pupil Figure 5(a). This improvement in image
resolution is attributed due to the resolution improvement
occurred for apodized linear pupils [5,14] as compared
with the constant uniform circular pupils. The image pro-
file of the original image of H1N1 virus is shown as in
Figure 6(a) while the image profile of the sine FZP re-
constructed image, is shown in Figure 6(b) and the image
profile of the sine FZP reconstructed image using linear
amplitude aperture is shown in Figure 6(c). Also, image
profile of the cosine FZP reconstructed image is repre-
sented in Figure 6(d), image profile of the cosine FZP
reconstructed image using linear amplitude modulation is
shown in Figure 6(e), image profile of the complex FZP
reconstructed image is shown in Figure 6(f), and the im-
age profile of the complex FZP reconstructed image using
linear amplitude modulation is shown in Figure 6(g). All
image profiles represented in Figures 6(a)-(g) are taken at
slice x = [12,127,575] and slice y = [1,180,100,100].
20 40 60 80 100 120 140 160 180
20
40
60
80
100
120
140
160
180
200
Figure 2. Electron microscope image of the reasserted H1N1
influenza virus photographed at the CDC influenza Labo-
ratory. The viruses are 80 - 120 nm in diameter. The image
has dimensions of 180 × 220 pixels.

A. M. HAMED
55
Figure 3. The auto-correlation intensity of the H1N1 image
shown in Figure 2. The correlation image has dimensions of
180 × 220 pixels.
Figure 4. A cosine FZP hologram using two pupil model
with one of linear distribution while the other remains the
same pinhole aperture.
Rec onstruc t i on of s i ne-F ZP hol ogram
Rec onst ruction of s i ne -FZP h ol ogram usi ng l i near aperture
(a) (b)
Rec onstruc t ion of cosi ne-F ZP hol ogram Rec ons truction of cosine-FZP hologram usi ng l i near apert ure
(c) (d)

A. M. HAMED
56
Real im age rec onstructi on of com pl ex F ZP hologram,Hc+
Real i m age reconstructi on of com pl ex F ZP hol ogram , Hc+ us i ng l inear apert ure
(e) (f)
Figure 5. (a) Reconstruction of the sine-FZP hologram computed using two-pupils heterodyne detection, where the 1st pupil is
uniform circular and the 2nd is a delta function; (b) Reconstruction of the sine-FZP hologram computed using two-pupils
heterodyne detection, where the 1st pupil is linearly distributed while the 2nd remains a unchanged (delta function); (c) Re-
construction of the cosine-FZP hologram computed using two- pupils heterodyne detection, where the 1st pupil is uniform
circular and the 2nd is a delta function; (d) Reconstruction of the cosine-FZP hologram computed using two-pupils hetero-
dyne detection , where the 1st pupil is linearly distributed while the 2nd remains a unchanged (delta function); (e) Reconstruc-
tion of the complex-FZP hologram computed using two-pupils heterodyne detection, where the 1st pupil is uniform circular
and the 2nd is a delta function; (f) Reconstruction of the complex-FZP hologram computed using two-pupils heterodyne de-
tection, where the 1st pupil is linearly distributed while the 2nd remains a unchanged (delta function).
(a) (b)
(c) (d)

A. M. HAMED
57
(e) (f)
(g)
Figure 6. (a) Image profile of the original image of H1N1 virus at slice x = [12,127,575] and slice y = [1,180,100,100]; (b) Image
profile of the sine FZP reconstructed image at slice x = [12,127,575] and slice y = [1,180,100,100]; (c) Image profile of the sine
FZP reconstructed image at slice x = [12,127,575] and slice y = [1,180,100,100] using linear amplitude modulation; (d) Image
profile of the cosine FZP reconstructed image at slice x = [12,127,575] and slice y = [1,180,100,100]; (e) Image profile of the
cosine FZP reconstructed image at slice x = [12,127,575] and slice y = [1,180,100,100] using linear amplitude modulation; (f)
Image profile of the complex FZP reconstructed image at slice x = [12,127,575] and slice y = [1,180,100,100]; (g) Image profile
of the complex FZP reconstructed image at slice x = [12,127,575] and slice y = [1,180,100,100] using linear amplitude modula-
tion.
4. Conclusions
Firstly, we conclude that the complex FZP hologram
gives better resolution for the reconstructed images as
compared with the reconstructed images obtained from
the sine and cosine FZP holograms.
Secondly, the reconstructed images in case of the sine-
FZP hologram provided with linearly modulated aperture
is better in resolution than the reconstructed images ob-
tained in case of uniform circular pupil. This resolution
improvement of the reconstructed holographic images in
case of linear pupils is due to the sharp distribution of the
PSF obtained in case of linear pupils as compared with the
corresponding PSF obtained for circular uniform aperture.
5. References
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A. M. HAMED
58
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