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On the fundamentals of optical scanning holographyTing-Chung
Poon
Citation: American Journal of Physics 76, 738 (2008); doi:
10.1119/1.2904472 View online: http://dx.doi.org/10.1119/1.2904472
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On the fundamentals of optical scanning holographyTing-Chung
PoonaThe Bradley Department of Electrical and Computer Engineering,
Virginia Tech,Blacksburg, Virginia 24061
Received 13 August 2007; accepted 10 March 2008
Optical scanning holography is a real-time holographic recording
technique in which holographicinformation about a three-dimensional
3D object is acquired using a single two-dimensional activeoptical
scan. Applications of the technique include optical scanning
microscopy, pattern recognition,3D holographic display, and optical
remote sensing. This paper introduces the basics of opticalscanning
holography. 2008 American Association of Physics Teachers.DOI:
10.1119/1.2904472
I. INTRODUCTION
Holography14 is a method of storing three-dimensional3D optical
information and is an important tool for scien-tific and
engineering studies.57 The purpose of this paper isto review the
basic principles of holography using the con-cept of Fresnel zone
plates and to introduce an unconven-tional holographic technique
known as optical scanning ho-lography. The two holographic
techniques will then becompared.
II. BASICS OF OPTICAL HOLOGRAPHY
Consider a planar object located at z=0 and described by
atransparency tx ,y. The object is illuminated by a mono-chromatic
plane wave of wavelength as shown in Fig. 1. Ifwe describe the
amplitude and phase of a light field at z=0by the complex function
x ,y ;z=0, we can obtain thecomplex light field a distance z away,
according to Fresneldiffraction8,9
x,y ;z =x,y ;z = 0 hx,y ;z . 1a
The symbol denotes two-dimensional 2D convolutiondefined by8
fx,y gx,y =
fx,ygx x,y
ydxdy, 1b
and the free-space impulse response function hx ,y ;z isgiven
by8
hx,y ;z = exp ik0zik0
2zexp ik02z x2 + y2 . 2
In Eq. 2 k0=2 / is the wave number of the light field. Ifwe
refer to the situation shown in Fig. 1, we can identifyx ,y ;z=0,
which is given by x ,y ;z=0=1 tx ,y,where the factor of unity in
front of tx ,y assumes planewave illumination with amplitude unity
and with zero initialphase at z=0. Hence, according to Eq. 1, the
complex am-plitude a distance z from the object is given by
x,y ;z = tx,y hx,y ;z = tx,y exp ik0zik0
2z
exp ik02z x2 + y2 . 3If we can record or store this original
complex amplitudeamplitude and phase, and at a later time we are
able torestore the exact complex amplitude, then we do not lose
anyinformation. Because our eyes observe the intensity gener-ated
by the same complex field, it makes no difference toobserve at time
t1, the time the original complex field isrecorded, or t2, the time
when the exact complex field isrestored. The restored complex field
preserves the entire par-allax and depth cue much like the original
complex field andis interpreted by our brain as the same 3D object.
Recordingmedia such as photographic films and CCD cameras
respondonly to light intensity; that is, they record Ix ,y= x ,y
;z2, and hence the phase information of the com-plex field is lost.
However, if x ,y ;z interferes with, say,a normally incident plane
wave of amplitude r at the re-cording medium, the resulting
recorded intensity becomes
Ix,y = r +x,y ;z2 =r2 + 2 +r* +r ,
4
where r has been assumed real for simplicity. Note that appears
in the last term of Eq. 4, even though there areother undesirable
terms introduced due to the nonlinear op-eration, which is the
square of the absolute value of the totalfield r+x ,y ;z. If the
recorded film transmittance is lin-early proportional to the
exposure, which is proportional tothe intensity shown in Eq. 4, the
resulting developed trans-parency is the hologram Hx ,y of the
object, tx ,y, that is,Hx ,y Ix ,y. Interfering the original wave
field withr is the original idea of Gabor10,11 for holographic
record-ing.
In the terminology of holography the complex fieldx ,y ;z
generated from the object is called the object waveand r is called
the reference wave. Once we have the ho-logram, we can retrieve x
,y ;z, the original light field byilluminating the hologram, say,
by an incident plane wave ofamplitude rec. We then have
recHx,y =recr2 + 2 +r* +r , 5
which is the light field immediately after the hologram. For
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simplicity, if rec is a constant such as is the case for
normalillumination by a plane wave, we have retrieved the
originallight field the last term in the bracket, but with some
un-wanted contributions the first three terms inside
thebracket.
III. HOLOGRAM AS A COLLECTION OF FRESNELZONE PLATES
We first give an illustrative example of holographic re-cording
of a single point object. In Fig. 2 we show a setupfor holographic
recording. The point source is the object,which is given by the
illumination of a pinhole aperture. Thereference wave is provided
by the collimated beam directedby beam splitters BS1, BS2, and
mirror M1 in Fig. 2.
The object wave 0 reaching the recording medium z0away from the
pinhole aperture is a diverging spherical wavegiven by see Eq.
3
0x,y ;z0 = x,y hx,y ;z0 = hx,y ;z0
= exp ik0z0ik0
2z0exp ik02z0 x2 + y2 ,
6
where the delta function x ,y has been used to model
thetransparency of the point source and the sampling propertyof the
delta function; that is,8
fx,yx x0,y y0dxdy = fx0,y0 7
has been used to evaluate the convolution in Eq. 6. For
thereference plane wave we assume that the plane wave has thesame
initial phase as the point object at a distance z0 from thefilm.
Therefore, its field distribution on the film is r=a expik0z0,
where a is the amplitude of the plane waveon the film. The recorded
intensity distribution on the filmand hence the hologram is given
by
Hx,y Ix,y = r +0x,y ;z02 = a exp ik0z0+ exp ik0z0
ik02z0
exp ik02z0 x2 + y22
8a
or
Hx,y = a2 + k02z02
+ a ik02z0
exp ik02z0 x2 + y2+ a
ik02z0
exp ik02z0 x2 + y2 , 8bwhere for brevity we have replaced the
proportional sign bythe equality sign. Note that the last term,
which is the desiredterm in Eq. 8b, is the total complex field of
the originalobject wave see Eq. 6 aside from the constant,exp
ik0z0. Equation 8b can be simplified to
Hx,y = A + B sin k02z0 x2 + y2 , 9where A=a2+ k0 /2z02 and B=ak0
/z0. The expression inEq. 9 is called the sinusoidal Fresnel zone
plate, which isthe hologram of the point source object.
A plot of the Fresnel zone plate is shown in Fig. 3a,
Fig. 1. Fresnel diffraction of x ,y ;z=0 at a distance z. The
complex field is obtained by the illumination of a transparency tx
,y by a plane wave.
Fig. 2. Holographic recording of a point source object M and M1
aremirrors. The point source is created by the illumination of a
pinhole aper-ture by a plane wave. The film records the reference
wave and the objectwave simultaneously.
Fig. 3. Plots for a Fresnel zone plate based on Eq. 9 for a z0=4
and bz0=8.
739 739Am. J. Phys., Vol. 76, No. 8, August 2008 Ting-Chung Poon
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where we have set A=B=1, k0 /2=1 and z0=4 for plottingpurposes.
Note that the spatial rate of change of the phase ofthe Fresnel
zone plate, say, along the x-direction, is given by
f local =1
2ddx k02z0x2 = k0x2z0 . 10
f local has units of inverse length and is called the local
fringespatial frequency, which increases linearly with the
spatialcoordinate x. In other words, the farther from the
zonescenter, the higher the local spatial frequency.
In Fig. 3b we have doubled the value of z0. The localfringe
frequencies have become smaller than those of Fig.3a, as is evident
from Eq. 10. Hence, the local frequencycarries information about
z0; that is, from the local frequencywe can deduce how far the
object point source is away fromthe recording filman important
aspect of holography.
To reconstruct the original light field from Hx ,y, we
canilluminate the hologram with a plane wave called the
recon-struction wave in holography, which gives, according
toFresnel diffraction or Eq. 1 and Eq. 8b,
recHx,y hx,y ;z
=recA + a ik02z0 exp ik02z0 x2 + y2+ a
ik02z0
exp ik02z0 x2 + y2 hx,y ;z . 11The evaluation of Eq. 11 gives
three light fields emergingfrom the hologram. The light field due
to the first term in Eq.11 is a plane wave because recA hx ,y
;zrecA,which is reasonable because a plane wave propagates with-out
diffraction. This plane wave is called the zeroth-orderbeam. The
field due to the second term is
rec ik02z0
exp ik02z0 x2 + y2 hx,y ;z
ik02z0
ik02z
exp ik02z0 x2 + y2 exp ik02z x2 + y2=
ik02z0
ik02z
exp ik02z0 z x2 + y2 . 12Equation 12 represents a converging
spherical wave if zz0. If zz0, the wave is diverging. For z=z0 the
wavefocuses to a real point source z0 away from the hologram andis
given by a delta function, x ,y. For the last term in Eq.11 we
have
recik0
2z0exp ik02z0 x2 + y2 hx,y ;z
ik02z0
ik02z
exp ik02z0 x2 + y2 exp ik02z x2 + y2=
ik02z0
ik02z
exp ik02z0 + z x2 + y2 , 13which is a diverging wave with its
virtual point source ap-pearing to come from a distance z0 behind
the hologram.This reconstructed point source is at the location of
the origi-nal point source object. The situation is illustrated in
Fig. 4.The real point source is called the twin image of the
virtualpoint source.
Let us see what happens if we have an object consisting oftwo
point sources given by b0x ,y+b1xx0 ,yy0,where b0 and b1 denote the
amplitude of the two pointsources. We assume that the two point
sources are located z0away from the recording film. The object wave
on the filmthen becomes
0x,y = b0x,y + b1x x0,y y0 hx,y ;z0 .14
Using Eq. 14 the hologram becomes
Hx,y = r +0x,y ;z02 = a exp ik0z0+ b0 exp ik0z0
ik02z0
exp ik02z0 x2 + y2+ b1 exp ik0z0
ik02z0
exp ik02z0 x x02+ y y022. 15
Equation 15 can be written in real form as
Hx,y = C +ab0k0z0
sin k02z0 x2 + y2+
ab1k0z0
sin k02z0 x x02 + y y02+ 2b0b1 k02z0
2cos k02z0 x2 + y2 x x02
y y02 , 16where C is a constant obtained as in Eq. 9. The second
andthird terms are the familiar Fresnel zone plate associatedwith
each point source, and the last term is a cosinusoidalfringe
grating whose origin is due to the nonlinear operationof
photographic recording. Hence, there are a total of twoFresnel zone
plates for two point sources. Only one termfrom each of the
sinusoidal Fresnel zone plates contains thedesired information
because each contains the original lightfield for the two points.
In general, the cosinusoidal fringegrating introduces noise on the
reconstruction plane. Givenshas given a general form of such a
hologram due to n pointsources.
12
Fig. 4. Reconstruction of a Fresnel zone plate hologram with the
existenceof the twin-image which is the real image reconstructed in
the figure.
740 740Am. J. Phys., Vol. 76, No. 8, August 2008 Ting-Chung Poon
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IV. OPTICAL SCANNING HOLOGRAPHY
Optical scanning holography is a form of electronic ordigital
holography.13,14 The latter is a general name that re-fers to the
fact that holographic recording is done electroni-cally, thereby
avoiding the nonreal-time darkroom process-ing of the film. Digital
holography traditionally employs aCCD camera for recording. Optical
scanning holography is areal-time technique in which holographic
information of a3D object can be acquired by using a single 2D
optical scanwhere scattered light from the object is detected by a
photo-detector. Hence, optical scanning holography is a form
ofdigital holography. Optical scanning holography was firstproposed
by Poon and Korpel,15 and the original idea waslater formulated in
Ref. 16. The technique was eventuallycalled optical scanning
holography to emphasize that holo-graphic recording can be achieved
by active opticalscanning.17 Applications of optical scanning
holography in-clude scanning holographic microscopy, 3D image
recogni-tion, 3D optical remote sensing, 3D TV and display, and
3Dcryptography.18 Optical scanning holography involves theprinciple
of optical heterodyne scanning. We shall thereforediscuss optical
scanning first.
In Fig. 5 we show a typical optical scanning imaging sys-tem
such as a laser scanning microscope. A collimated laserbeam is
projected through the x-y optical scanner to scan outthe input
object specified by transparency 0x ,y. The pho-todetector converts
the light to an electrical signal that con-tains the processed
information for the scanned object. If thescanned electrical signal
is digitally stored in a computer insynchronization with the 2D
scan signals of the scanningmechanism such as the x-y scanning
mirrors, the storedrecord is a processed 2D image of the scanned
object.
Assume that the scanning optical beam is specified by acomplex
field bx ,y on the object transparency, and thecomplex field
reaching the photodetector is 0x ,ybxx ,yy. The coordinate shifts
in the argument of bxx ,yy signify the motion relative to the
transparency.The photodetector then delivers a current ix ,y as the
output
by spatially integrating the intensity, bxx ,yy0x ,y2, over the
active area S of the detector. Thecurrent is displayed on the
monitor as a 2D record:
ix,y S
0x,ybx x,y y2dxdy, 17
where xt and yt represent the instantaneous position ofthe
scanning beam. For uniform scanning speed, v, of thebeam, we have
xt=vt and yt=vt. In terms of correlationsin two dimensions for real
signals f and g, we have18
fx,y gx,y =
fx,ygx + x,y
+ ydxdy, 18
and Eq. 17 can be written as
ix,y = bx,y2 0x,y2. 19
Note that the beam field distribution bx ,y and the
pupilfunction px ,y located in the front focal plane of Lens Lsee
Fig. 5 are related by a Fourier transform. For instance,assume that
the light field that illuminates the pupil functionis uniform and
px ,y=1. Then bx ,y becomes a deltafunction. Equation 17
consequently gives an exact replicaof 0x ,y2.
In general, 0x ,y2 is processed by bx ,y2, eventhough the object
0x ,y originally may be complex. Thefact that the objects
intensity, 0x ,y2, is manipulated by areal and non-negative
quantity, bx ,y2, is known as in-coherent optical image
processing.18 Such an optical scan-ning system cannot manipulate
any phase information. Be-cause holography requires the
preservation of the phase, weneed to find a way to preserve the
phase information duringphotodetection if we expect to use optical
scanning to recordholographic information. The solution to this
problem is op-tical scanning heterodyning. In the latter the object
isscanned by a time-dependent Fresnel zone plate, which is the
Fig. 5. Standard optical scanning system for imaging. The object
transparency 0x ,y is scanned two-dimensionally by an optical beam.
The photodetectorPD converts the light into electrical information,
which is displayed or stored on a computer as a 2D record. Note
that the pupil function px ,y located in thefront focal plane of
lens L with focal length f can modify the shape of the scanning
beam.
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superposition of a spherical wave and a plane wave of dif-ferent
temporal frequencies. The top part of Fig. 6 shows
theconfiguration. The lens is used as a light collector that
col-lects all of the transmitted light to the photodetector. In
theactual experimental setup, the plane wave and the sphericalwave
are combined, though a beam splitter and the sphericalwave can be
derived from a focusing laser beam. The tem-poral frequency
difference between the two waves can beprovided by using an
acousto-optic modulator8,19 or anelectro-optic modulator8 in the
path of the plane wave.
Assume that the spot of the focusing laser beam that gen-erates
the spherical wave is a distance z from the objecttransparency 0x
,y ;z. We can express the scanning beampattern at the transparency
as
bx,y ;z = a expi0 +t
+ik0b2z
exp ik02z x2 + y2expi0t , 20where 0 and 0+ are the frequencies
of the sphericalwave and the plane wave, respectively. As in Eq.
17, thephotodetector generates a current given by
ix,y = S
0x,y;zbx x,y y ;z2dxdy.
21
If we substitute Eq. 20 into Eq. 21 and keep only theheterodyne
current by using a bandpass filter tuned at theheterodyne frequency
, we have
ix,y = abk0z
sint + k02z x2 + y2 0x,y ;z2.22
This heterodyne current shows that the information of theobject
is carried by the heterodyne frequency. To extract theinformation
associated with the object, we can electronically
mix it with a sine and a cosine function at the
heterodynefrequency to obtain the in-phase and the quadrature
compo-nents of the heterodyne current, respectively. We have
ix,y sint =abk02z cos k02z x2 + y2 0x,y ;z2
cos2t + k02z x2 + y2 0x,y ;z2 , 23
where we have used the identity sin sin = 12 cos
12 cos+. By using an electronic lowpass filter, we can
extract the first term of Eq. 23 and finally obtain the
cosine-Fresnel zone plate hologram:
isx,y Hcosx,y =k0
2zcos k02z x2 + y2
0x,y ;z2. 24
Similarly, if a cosine function in Eq. 23 is used, that is,cost,
we obtain the product of sine and cosine functions.After lowpass
filtering, we have the sine-Fresnel zone platehologram:
icx,y Hsinx,y =k0
2zsin k02z x2 + y2
0x,y ;z2. 25
Electronic processing is shown in the lower part of Fig.
6.Equations 24 and 25 are for a planar object
0x ,y ;z2 located a distance z from the scanning pointsource. If
we consider a 3D object as a collection of planesalong the
z-direction, the holograms of the 3D object become
Hcosx,y = k02z cos k02z x2 + y2 0x,y ;z2dz;26a
Hsinx,y = k02z sin k02z x2 + y2 0x,y ;z2dz .26b
That is, we integrate the results of Eqs. 24 and 25 along
zassuming that the 3D object is weakly scattering. Equation26 is
the major result of optical scanning holography. Foreach 2D
scanning of the 3D object, we have two holograms.
V. EXAMPLE OF THE OPTICAL SCANNINGHOLOGRAPHIC RECORDING OF POINT
SOURCES
As an example, we let the object be a pinhole a distance z0from
the scanning point source, that is, 0x ,y ;z2=x ,yzz0. From Eq. 26b
we have
Fig. 6. Optical scanning holography setup. The thick specimen is
scannedtwo-dimensionally by the combination of a spherical wave and
a plane waveof different frequencies. The photodetector PD collects
all the light anddelivers a heterodyne signal at frequency . The
heterodyne signal is thenelectronically processed to give two
processed outputs. BPF is a bandpassfilter and LPF is a lowpass
filter.
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Hsinx,y = k02z sin k02z x2 + y2 x,yz z0dz=
k02z0
sin k02z0 x2 + y2 x,y=
k02z0
sin k02z0 x2 + y2 , 27where we have used the sampling property
of the delta func-tion to evaluate the integration along z. The
expression in Eq.27 is almost Eq. 9 with the exception of the DC
term, andhence is the holographic recording of a single point
withoutany DC.
What happens when we have a two-point object given by0x ,y ;z2=x
,yzz0+xx0 ,yy0zz0. Weuse this form in Eq. 26b and obtain
Hsinx,y =k0
2z0sin k02z0 x2 + y2
+k0
2z0sin k02z0 x x02 + y y02 . 28
Equation 28 represents two Fresnel zone plates without
theundesired intermodulation term shown in Eq. 16. This re-moval of
the intermodulaton term is possible because holo-graphic
information is generated by heterodyne scanning. Inaddition,
because of heterodyning, we have two hologramssee Eq. 26 for each
2D scan. The cosine-Fresnel zoneplate hologram is not redundant and
is useful for the elimi-nation of the twin-image.
We can form the complex hologram given by
Hc x,y = Hcosx,y iHsinx,y , 29
where is a constant. Reconstruction of such a complexhologram
will not give rise to a twin-image. As an example,the complex
hologram for a single point object is given bysubstituting Eqs. 24
and 25 into Eq. 29 with =1,
H
cx,y =k0
2z0exp ik02z0 x2 + y2 , 30
where we have taken the negative sign in Eq. 29.Equation 30 is
for the complex field see Eq. 6, aside
from a constant, due to a single point source. This
hologramcontains only the desired information of the point
sourcebecause there is no term leading to the twin-image and
theannoying background noise due to intermodulation,
whichunavoidably comes from conventional holographic recordingsee
Eq. 8. To see how only the desirable point source isrestored, we
reconstruct the complex hologram by the illu-mination of a plane
wave. Using the notation of Eq. 11, wehave
recHcx,y hx,y ;z
k02z0
exp ik02z0 x2 + y2
ik02z
exp ik02z x2 + y2 x,y 31
for z=z0. We see that a virtual point object has been
recon-structed at a distance z0 behind the hologram, and there is
noreal point object the twin image formed in front of the
hologram. In addition, there is no zeroth-order beam. If weuse
the + sign from Eq. 29, we obtain H+cx ,y and uponthe illumination
of a plane wave of such a hologram, we willreconstruct only a real
point object in front of the hologram.
VI. SUMMARY AND REMARKS
The hologram of a point source object has been discussed,and we
showed that a Fresnel zone plate is the hologram of apoint object.
We then demonstrated the reconstruction of theFresnel zone plate
and showed that in addition to the recon-struction of the original
point object, a twin-image has beenformed. For a two-point object,
the recorded hologram is acollection of two zone plates and a
grating, with the twozone plates corresponding to each of the two
point sourcesand the grating resulting from the interference or
intermodu-lation of the two point sources. The two zone plates
corre-sponding to the two point sources generate the original
twopoint sources, but also create twin-images, and the
gratingarising from the intermodulation in general introduces
noiseon the reconstruction plane.
We then introduced optical scanning holography, which
isunconventional in the sense that its holographic recording ofa
point source object gives a sine-Fresnel zone plate holo-gram and a
cosine-Fresnel zone plate hologram. From thetwo Fresnel zone plate
holograms, we can construct a com-plex hologram that does not
create a twin-image upon recon-struction. For multiple point
sources, no intermodulations arerecorded. This explanation of
holographic recording is calledthe zone plate approach and was
pointed out by Givens as analternative approach to the
understanding of holography.12 Atthat time, it was pointed out the
approach was slightly lessthan perfect because the zone plate
approach could not re-produce the hologram generated by Gabors
approach.10 Wefound, however, that the zone plate approach can be
imple-mented and is a better way to generate holograms because
nointermodulation is recorded and we can eliminate the twin-image
when a complex hologram is used.
For a complete discussion of optical scanning holographyand its
current applications, readers are referred to Ref. 18.In the
Appendix, we show some examples of optical scan-ning
holography-based holograms and their numerical recon-struction.
APPENDIX: EXAMPLES OF OPTICAL SCANNINGHOLOGRAPHY-BASED HOLOGRAMS
ANDTHEIR NUMERICAL RECONSTRUCTION
In this Appendix we first formulate optical scanning ho-lography
in the frequency domain. We then show some nu-merical results. An
example of the MATLAB code can befound in Ref. 18. For some basic
use of MATLAB in opticalimaging, readers are encouraged to consult
Ref. 20.
For a planar intensity object located a distance z0 from
thepoint source of the scanning beam, the holograms
obtainedaccording to Eqs. 24 and 25 are
Hcosx,y =k0
2z0cos k02z0 x2 + y2 0x,y ;z02,
A1
and
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Hsinx,y =k0
2z0sin k02z0 x2 + y2 0x,y ;z02.
A2
We have replaced the correlation operation by the convolu-tion
operation, which is possible because the cosine- andsine-Fresnel
zone plates are symmetrical. We then constructa complex hologram
according to the general guideline givenby Eq. 29. We construct
H+cx,y = iHcosx,y + iHsinx,y
=
ik02z0
exp ik02z0 x2 + y2 0x,y ;z02,A3
where we have chosen =i for convenience.We define the 2D Fourier
transform as
Fgx,y =
gx,yexpikxx + ikyydxdy
= Gkx,ky Ref. 8 , A4
where kx and ky are spatial frequencies9,18 corresponding tothe
spatial coordinates x and y, respectively. Spatial frequen-cies are
commonly referred to as wave numbers in physics.By taking the 2D
transform of Eq. A3, we have
FH+cx,y = F ik02z0 exp ik02z0 x2 + y2F0x,y ;z02
= OTFoshkx,ky ;z0F0x,y ;z02 , A5
where we have used the identity Fg1x ,y g2x ,y=Fg1x ,yFg2x ,y to
obtain the second equality inEq. A5. The first term of the right
side of Eq. A5 can beshown to give
OTFoshkx,ky ;z0 = exp iz02k0 kx2 + ky2 , A6which is the optical
transfer function of optical scanningholography.8 For a given
planar intensity object,0x ,y ;z02 we calculate the complex
hologram accordingto Eq. A5, and aside from a constant, the sine-
and cosine-holograms can be obtained by taking the real and
imaginarypart of the spatial domain of Eq. A5; that is,
Hsinx,y = ReF1OTFoshkx,ky ;z0F0x,y ;z02A7a
andHcosx,y = ImF1OTFoshkx,ky ;z0F0x,y ;z02 ,
A7b
where F1 denotes the inverse Fourier transform operation.
Acomplex hologram can then be constructed according to Eq.29 to
obtain Hc x ,y. A reconstruction of the hologram at aplane located
z0 from the hologram is calculated according to
Hanyx,y hx,y ;z0 , A8
where Hanyx ,y represents the sine-hologram, the
cosine-hologram, or the complex hologram. We implement the
re-construction in the frequency domain and write Eq. A8 as
F1FHanyx,yFhx,y ;z0 . A9It can be shown that
Fhx,y ;z0 = exp ik0z0exp iz02k0 kx2 + ky2= OTFoshkx,ky ;z0*,
A10
where we have neglected the constant phase termexpik0z0 to
obtain the last step. Hence, the reconstructionformula given by Eq.
A9 can be written as
Fig. 7. a Original planar intensity object, b sine-hologram of
a, and ccosine-hologram of a. The holograms are based on Eq. A7
with z0 /2k0=2.0.
Fig. 8. a Reconstruction of Fig. 7b, b reconstruction of Fig.
7c, andc reconstruction of the complex hologram constructed
according to Eq.A3. These reconstructions are based on Eq. A11 with
z0 /2k0=2.0. Notethat twin-image noise exists in a and b.
744 744Am. J. Phys., Vol. 76, No. 8, August 2008 Ting-Chung Poon
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-
Reconstruction = F1FHanyx,yOTFoshkx,ky ;z0* .A11
In summary, we have formulated optical scanning holog-raphy in
the frequency domain. To construct holograms, weincorporate Eq. A6
into Eqs. A7 and 29. For reconstruc-tion we use Eq. A11.
In Fig. 7 we show the original planar intensity object andits
sine- and cosine-holograms. Both of the holograms basedon Eq. A7
are generated using z0 /2k0=2, which is propor-tional to the
recording distance z0. In Fig. 8 we show thereconstruction of the
sine-hologram, the cosine-hologram,and the complex hologram
constructed based on Eq. A3.These reconstructions are based on Eq.
A11 with z0 /2k0=2. In Fig. 9, we show reconstruction in different
transverseplanes using z0 /2k0=1.95. Note that these
reconstructionsillustrate defocusing of the original object.
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Fig. 9. a Reconstruction of Fig. 7b, b reconstruction of Fig.
7c, andc reconstruction of the complex hologram constructed
according to Eq.A3. These reconstructions are based on Eq. A11 with
z0 /2k0=1.95. Notethat twin-image noise exists in a and b. These
reconstructions are gener-ally out of focus.
745 745Am. J. Phys., Vol. 76, No. 8, August 2008 Ting-Chung Poon
This article is copyrighted as indicated in the article. Reuse of
AAPT content is subject to the terms at:
http://scitation.aip.org/termsconditions. Downloaded to IP:
163.180.57.41 On: Sun, 19 Apr 2015 12:19:24