On the fundamentals of optical scanning holography.pdf

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 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/76/8?ver=pdfcov Published by the American Association of Physics Teachers

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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.

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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.


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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.


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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.


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