volume holographic imaging in transmission...

V

A

1

Astadcptwu

or

cfoins

coo

iea

2

olume holographic imaging in transmission geometry

rnab Sinha, Wenyang Sun, Tina Shih, and George Barbastathis

We address the performance of transmission geometry volume holograms as depth-selective imagingelements. We consider two simple implementations using holograms recorded with spherical and planebeams. We derive the point-spread function �PSF� of these systems using volume diffraction theory anduse the PSF to estimate depth resolution. Furthermore, we show that appropriately designed objectiveoptics can significantly improve the depth resolution or the working distance of plane-wave referenceholographic imaging systems. These results are confirmed experimentally and demonstrated for objectswith millimeter axial features, imaged from the 5- to 50-cm range. © 2004 Optical Society of America

OCIS codes: 090.7330, 110.0110, 110.6770.

3ica3emjaiila

ittdfisGpfmrTcpmt

itii

. Introduction

computational imaging system receives emitted orcattered radiation from an object of interest andransforms this radiation optically before capturing its an electronic signal on a photoelectric detector oretector array. This signal is then digitally pro-essed to recover object attributes such as, for exam-le, spatial structure and spectral composition. Inhis paper we concentrate on computational imaginghere the optical transformations are effected by vol-me holograms,1 a special class of optical elements.We focus on the recovery of spatial structure from

bjects of interest, so we classify imaging systems ofelevance:

�1� Two-dimensional �2-D� imaging systems typi-ally recover only the 2-D brightness or intensity in-ormation about an object as shown in Fig. 1�a�. Inther words, a 2-D image is of the form I�x, y�, anntensity distribution defined over the lateral coordi-ates. Photographic cameras and traditional micro-copes are examples of 2-D imaging systems.�2� Three-dimensional �3-D� imaging systems re-

over the entire 3-D intensity information about thebject, as shown in Fig. 1�b�. For 3-D imaging, thebject is required be at least translucent. Thus, a

The authors are with the Department of Mechanical Engineer-ng, Massachusetts Institute of Technology, 77 Massachusetts Av-nue, Room 3-466, Cambridge, Massachusetts 02139. The e-mailddress for A. Sinha is [email protected] 4 April 2003; revised manuscript received 28 October

003; accepted 5 November 2003.0003-6935�04�071533-19$15.00�0© 2004 Optical Society of America

-D image is of the form I�x, y, z�, i.e., a complete 3-Dntensity distribution. Fluorescence confocal mi-roscopy2 and optical coherence tomography,3 are ex-mples of 3-D imaging methods. Another class of-D imaging systems is referred to as tomographic,.g., such as x-ray computerized tomography andagnetic resonance imaging.4 In the latter, the ob-

ect information is retrieved as a set of projections,nd radon transform inversion techniques map thenformation back to native object coordinates. Typ-cally, 3-D imaging systems require scanning in ateast one dimension to obtain the complete 3-D im-ge.�3� Two and One-Half-Dimensional �2 1⁄2-D� imag-

ng systems,5 also referred to as profilometers, areypically used for reflective objects. The profilome-er returns a height map z�x, y� representing theistance of a particular point on the surface from axed base as shown in Fig. 1�c�. Confocal micro-copes6 and interferometers such as the Twyman–reen and the Mach–Zehnder are common opticalrofilometers. Heterodyne and white-light inter-erometers can yield accuracy as good as a few nano-eters in the axial z direction, but their lateral

esolution is limited typically to a few micrometers.he recently developed methods of atomic force mi-roscopy7 and scanning tunneling microscopy8 extendrofilometry to nanoscale accuracies in all three di-ensions. Most profilometers also require scanning

o recover a complete height map.

Gabor originally proposed use of holography as anmaging method to recover both the amplitude andhe phase of light coming from an object9 with thentent of using the phase to store and reconstruct 3-Dnformation about the object. A Gabor or Leith–

1 March 2004 � Vol. 43, No. 7 � APPLIED OPTICS 1533

UsrBu2itptsche

cVwdctTlmVd

nh

tctiatvict

btsesnhirsa

reFplttF2mcWbTed

mViroirbgtwcbisb

Vs

Ftrmd

F��

1

patnieks10 hologram is recorded in a thin photosen-itive material as the interference pattern between aeference beam and the light scattered by the object.oth analog11 and digital12,13 holograms have beensed extensively for 3-D and 2 1⁄2-D imaging. Figure�a� is a simplified holographic imaging setup; auxil-ary optical elements such as lenses have been omit-ed. It is possible to recover the amplitude anderform optical slicing of the object when we probehe hologram optically by scanning, or digitally ashown in Fig. 2�b�. Analog holography requires re-ording a fresh hologram for each object. Digitalolography requires a deconvolution operation forach frame recorded by the camera.In this paper we discuss a different imaging prin-

iple that we call volume holographic imaging �VHI�.HI can be used for both 3-D and 2 1⁄2-D imaging, bute discuss only 2 1⁄2-D imaging in this paper. VHI isifferent from traditional holographic imaging be-ause �1� A VHI system incorporates at least onehick holographic element, a volume hologram �VH�.he VH �which is also referred to as the volume ho-

ographic lens� acts as depth-selective imaging ele-ent to achieve 3-D or 2 1⁄2-D imaging. �2� A singleH lens can be used to image arbitrary objects on aigital camera. Thus there is no need to record a

ig. 1. �a� Two-dimensional imaging system cannot discriminatehe distance between two objects; �b� a 3-D imaging system canecover depth information by scanning; �c� a 2 1⁄2-D imaging systemaps the height of a reflective surface to an intensity on the

etector.

ig. 2. Simplified schematics of traditional holographic imaging:a� recording and �b� readout by scanning. Schematics of VHI:c� making VH lens and �d� readout.

534 APPLIED OPTICS � Vol. 43, No. 7 � 1 March 2004

ew hologram for each object as in the case of analogolographic imaging.The image, i.e., the 3-D or 2 1⁄2-D spatial structure of

he object, is recovered from the intensity data re-orded on the camera. This computational aspect ofhe VHI approach can be particularly simple, basicallyn the form of a map between intensity measurementsnd spatial coordinates. More-sophisticated compu-ational methods that use overconstraining and decon-olutions can also be applied to VHI, as we have shownn preliminary research.14 In this paper, however, weoncentrate on the physical transformations of the op-ical field realized by VHI elements.

Volume holographic gratings were first introducedy van Heerden.1 Since then, the properties of thesehick diffractive elements have been extensivelytudied.15,16 VHs have been used in several subar-as of optical information processing, namely, datatorage,15,17,18 optical interconnects,19 and artificialeural networks.20 Use of VHs as imaging elementsas only been proposed recently21 and demonstrated

n the context of a confocal microscope with a VHeplacing the pinhole,22 a volume holographic tele-cope,23 and a real-time �scan-free� hyperspectral im-ging instrument.24

A VH is created when the interference pattern ofeference and signal beams are recorded within thentire volume of a thick photosensitive material.25

or example, Fig. 2�c� depicts a VH recorded by aoint-source reference and plane-wave signal. Ho-ograms that are sufficiently thick26 are sensitive tohe nature of the illumination and diffract only whenhe input field matches the recording conditions.or an arbitrary object illumination as shown in Fig.�d�, the VH diffracts only the so-called Bragg-atched portions of the incident field and thus one

an see only a part of the entire object on the detector.e exploit this property in VHI to resolve depth am-

iguities and thus obtain 2 1⁄2-D and 3-D images.he entire object information is recovered when sev-ral holograms are scanned or multiplexed to probeifferent portions of the object simultaneously.24

To understand how VHs can serve as imaging ele-ents, we first briefly review diffraction from 3-DHs qualitatively. A hologram is 3-D if it diffracts

n the Bragg regime �as opposed to the Raman–Nathegime of thin holograms26�. An important propertyf the Bragg regime that is useful for VHI applications the phenomenon of Bragg selectivity. A VH isecorded as usual by use of two mutually coherenteams, the reference and the signal. The holo-raphic material thickness must exceed a certainhreshold, which depends on the fringe spacing andavelength of the interference pattern.26 The most

ommon choice for a reference beam is a plane wave,ut we also consider spherical wave reference beamsn this paper. For the implementation of VHI, it isufficient to record holograms with plane-wave signaleams.To understand Bragg selectivity, suppose that a

H is recorded with a certain reference beam and aignal beam. After recording is complete, we probe

trw

e3r

fpoennoB

srtmwolFacffitTri

tfotaofpjfoVstn

pipttiis

ctfatq2tiapbpWsdtatri

Fmmdgd

Fmbc

he hologram with a third beam. Depending on theelationship of the probe beam to the reference beam,e can obtain three types of diffraction:

�1� Bragg matched, where the probe beam is anxact replica of the reference beam, as shown in Fig.�b�. In this case, the full diffraction efficiency of theecorded hologram is recovered.

�2� Bragg mismatched, where the probe beam dif-ers from the reference beam. For example, therobe beam could have a different angle of incidencer wavelength15 or location �for a spherical refer-nce27�. For a Bragg-mismatched probe, which isot degenerate �see item below�, the hologram doesot diffract at all or at least the diffraction efficiencybtained from the hologram is much weaker than theragg-matched case. This is shown in Fig. 3�c�.�3� Bragg degenerate diffraction is obtained for a

pecial class of probe beams that differ from theeference beam yet still give rise to strong diffrac-ion of a magnitude comparable to the Bragg-atched case. For VHs recorded with a plane-ave reference, degenerate diffraction can bebtained by either a combined change in wave-ength and angle of a plane-wave probe, as shown inig. 3�d�,28,29 or by the tilting of the probe beam indirection along the orientation of the fringes that

onstitute the VH.15,25 Generally, degenerate dif-raction occurs if the probe beam can be obtainedrom the reference beam by a transformation that isnvariant with respect to the orientation of the in-erference pattern that constitutes the hologram.he degeneracy property is exploited extensively toeduce the scanning required for VHI, as we discussn Subsections 2.B and 3.B.

ig. 3. Bragg selectivity of VHs: �a� recording, �b� Bragg-atched readout with a replica of the reference beam, �c� Bragg-ismatched readout results in a weak diffracted field, �d� a Bragg

egenerate beam yields a strong diffracted beam. The Bragg de-enerate beam is of a different wavelength and is incident at aifferent reference angle governed by Bragg selectivity.

As mentioned above, in this paper we aim to usehe Bragg selectivity of volume holographic gratingsor depth-selective imaging systems with reflectivebjects. The general geometry of the imaging sys-em is shown in Fig. 4. When light scattered fromn object contains a component that Bragg matchesr is Bragg degenerate for the hologram, some dif-raction is obtained and so it is possible to detect theresence of Bragg-matched and Bragg degenerate ob-ect illumination by means of monitoring the dif-racted field. On the other hand, portions of thebject that are Bragg mismatched are invisible to theHI system. Thus we can perform optical slicing,imilar to a confocal microscope, for example, usinghe Bragg diffraction as the depth-sensitive compo-ent in the system.VHI systems can function under both active and

assive illumination. Active illumination systemsncorporate the light for illuminating the object as aart of the imaging system. Typically, the illumina-ion is monochromatic laser light. Passive illumina-ion systems on the other hand rely on ambientllumination to provide the light necessary for imag-ng purposes. In this paper we restrict the discus-ion to active, monochromatic illumination.A major portion of this paper is devoted to the

haracterization of image quality that can be ob-ained from VHI systems in terms of the point-spreadunction �PSF� in different geometries. In the liter-ture, there is some confusion about the definition ofhe PSF. When the imaging system can be ade-uately modeled as a linear transformation from a-D object �input� space to a 2-D image �output� space,he PSF is interpreted as the impulse response of themaging system, i.e., the intensity pattern observedt the output of an imaging system in response to aoint-source input. This definition can, in principle,e used for 2 1⁄2-D or 3-D imaging systems as well,rovided that the output space is defined accordingly.e are more interested in the response of the VHI

ystem measured on a single-pixel detector or a 2-Detector array �camera�. Thus we have to deal withhe definition of the PSF when the output space haslower dimension than the input space. To resolve

his problem, we interpret the PSF as the intensityesponse of the VHI system to displacement of thenput point source �the probe source� away from a

ig. 4. Schematic of the active VHI system. A CCD cameraonitors the diffracted beam to detect whether the light scattered

y the object contains any Bragg-matched or Bragg degenerateomponents.


rfiscfrt

otVtttdTafmtcmf

mbyFPtshttasipatCt

2R

A

Fo

waacwo

tyagradflettwdciyppe

Ntsi

Ar

waormpElta

Fr

1

eference location. The reference location is speci-ed during the recording step of the VH lens. Aimilar situation occurs in confocal microscopy, ex-ept then the reference location is specified by theocus of the illumination beam, which is also ar-anged to be conjugate of the pinhole used in front ofhe detector.

For example, the depth resolution of the system isbtained from the PSF as follows: We first obtainhe intensity measured at the detector plane of theHI system as a function of axial probe displacement,

heoretically or experimentally. Then we estimatehe extent of the PSF that tells us roughly how farwo objects can be placed apart in the longitudinalirection and still be resolved by our imaging system.his definition ignores the noise present in the im-ging system, but still provides an adequate measureor comparison of alternative systems �a system with

ore extended PSF will perform worse than an al-ernative with narrower PSF under identical noiseonditions�. We elected to use the full width at half-aximum �FWHM� of the PSF, denoted as �zFWHM,

or comparisons of resolution.We examine transmission geometry VHI imple-entations recorded with spherical and plane-wave

eams in Sections 2 and 3, respectively. The anal-sis sequence for each case proceeds as follows:irst we use volume diffraction theory to derive theSF; then we estimate the functional dependence of

he PSF on basic parameters of the imaging systemuch as numerical aperture, working distance, andologram thickness; we discuss the role of magnifiers,elescopes, or other optical elements that can be usedo shape the field produced by the object before itrrives at the hologram; and we conclude the discus-ion of each VHI implementation with experimentalmaging results and verification of the theoreticalredictions. The two implementations �sphericalnd plane wave� are shown to have several similari-ies but also significant performance differences.omparisons are given throughout Sections 2, 3, and

he concluding Section 4.

. Volume Holographic Imaging with Spherical Waveeference Beams

. Derivation and Properties of the Diffracted Field

igure 5�a� shows the recording setup for VHI by usef a spherical reference �SR VHI�. In this analysis

ig. 5. Spherical wave reference VHI: �a� recording and �b�eadout �imaging�.


e use a coordinate system centered on the hologram,s shown in Fig. 5. Thus all wavelengths and anglesre measured inside the holographic medium. Theorresponding free-space values can be obtainedhen we multiply with the average refractive index n

f the medium.The VH is the stored 3-D interference pattern of

he point-source reference emanating at rf � xfx �

fy � zfz and the plane-wave signal inclined at anngle �s �� 1 with respect to the z axis. The holo-raphic medium is disk shaped with thickness L andadius R. The origin of the SR is the reference deptht which the holographic imaging system operatesuring readout. As shown in Fig. 5�b�, the field dif-racted by the VH is Fourier transformed �FT� by aens of focal length F, assumed to have infinite lateralxtent. The FT diffracted field is observed on a de-ector or a detector array �e.g., CCD or complemen-ary metal-oxide semiconductor camera�. Becausee use the �zFWHM of the PSF as a measure of theepth resolution of the system, our first objective is toalculate the longitudinal PSF. Therefore we ideal-ze the object as a point source located at rp � xpx �

py � zpz and explicitly calculate the diffracted fieldroduced on the detector by this point source. In thearaxial approximation, the reference beam can bexpressed as

Ef�r� � exp�i2�z � zf

� i�

� x � xf�2 � � y � yf�

2

� z � zf�� .

(1)

ote that here and in the sequel we neglect a term ofhe form 1��z zf � because it varies with z muchlower than the exponential term. The signal beams expressed in the paraxial approximation as

Es�r� � exp�i2��1 ��s

2

2 � z

� i2��s

x� . (2)

fter recording is complete, the index modulationecorded in the hologram is

��r� � Ef*�r� Es�r�, (3)

here the asterisk denotes complex conjugate. Thectual interference pattern is given by �Ef � Es�

2, butut of the four resulting product terms only the one inelation �3� results in significant diffraction; the re-aining three terms are Bragg mismatched. Sup-

ose that the hologram is illuminated by a probe fieldp�r�. Because we know the refractive-index modu-

ation of the VH from relation �3�, we can calculatehe light diffracted by the VH in response to Ep�r�pproximately by

Ed�r � � �� Ep�r��r�G�r � r�d3r, (4)

wf

aoapetifitTr5aba

Tw�wwpRc

Tb

T

abttaaevats

wspcywRit

I

WytEegfi

tw

FVDBd

here � is the diffraction efficiency, G�r� is Green’sunction for free space

G�r � r� �exp�ik�r � r��

�r � r�� exp�i2�

z � z

� i�� x � x�2 � � y � y�2

� z � z� � , (5)

nd r � �x , y � is the position vector on the exit facef the hologram. The result gives the diffracted fields the superposition of fields scattered by individualoint radiators such that the radiation generated atach point is proportional �in amplitude and phase� tohe product of the illuminating field and the localndex modulation. This approximation, known asrst-order Born scattering, neglects rediffraction ofhe scattered field as it propagates through the VH.he approximation is valid only if the hologram iselatively weak �in practice, it works well for � up to0% or higher, depending on the specific modulationnd thickness of the hologram�. The spherical probeeam emanating at rp is expressed in the paraxialpproximation as

Ep�r� � exp�i2�z � zp

� i�

� x � xp�2 � � y � yp�

2

� z � zp�� .

(6)

he diffracted field at the detector plane is obtainedhen we substitute Eqs. �1�, �2�, and �6� and relation

3� in Eq. �4� and perform a Fourier transformationith conjugate coordinates �x , y � � �x��F, y��F�here �x�, y�� are the coordinates on the detectorlane. The details of the calculation are given inef. 25 �pp. 38–42�. The result after we omit someonstant phase factors is

Ed� x�, y�� 2�R2�� L�2

L�2

exp�i�C� z��

� ��2�A� z� R2, 2�B� z� R�dz. (7)

he coefficients A�z�, Bx�z�, By�z�, and C�z� are giveny

A� z� �1

� z � zf��

1� z � zp�

, (8)

Bx� z� � xp

� z � zp��

xf

� z � zf��

x�

F�

�s

, (9)

By� z� � yp

� z � zp��

yf

� z � zf��

y�

F, (10)

B� z� � �Bx� z�2 � By� z�2�1�2, (11)

C� z� �xp

2 � yp2

� z � zp��

xf2 � yf

2

� z � zf�� x�2 � y�2

F2 ��s

2

�z.

(12)

he function

��u, v� � �0

1

exp�i2

u�2�J0�v��d� (13)

lso occurs in the calculation of the 3-D light distri-ution near the focus of a lens. It is the Hankelransform of the radial quadratic chirp exp�iu�2�2�runcated to 0 � � � 1. We compute this integral asseries expansion given in Ref. 30, although faster

pproximate methods have been suggested in the lit-rature.31 Equation �7� describes the action of theolume holographic filter as an imaging element withspherical wave reference. We obtain the PSF of

he VH by calculating the total diffracted field inten-ity

�Ed� x�, y�; xp, yp, zp��2,

hich is a function of the location of the probing pointource and the observation location at the detectorlane. For simplicity, we consider a hologram re-orded and probed with on-axis point sources, i.e., xf,f, xp, yp � 0. Equation �7� is further simplifiedhen we set zp zf � � and assume L�zp �� 1.eplacing these in Eqs. �8�–�12� and carrying out the

ntegration, we obtain an approximate expression forhe normalized diffracted field intensity:

I� x�, y�; ��

b��s F, 0; �� 2�R2�

zf2 ,

2�RF

�� x� � �F�2

� y�2�1�2�sinc� x�2 � y�2 � ��F�2�L2F2 �2

.

(14)

e chose the normalization constant Ib � I�x� � �sF,� � 0� because the diffracted intensity peaks at de-ector location �x�, y�� sF, 0�, as can be seen fromqs. �9� and �10�. Equation �14� offers an intuitivexplanation for the behavior of this volume holo-raphic optical element. The observed diffractedeld contains two contributions, shown in Fig. 6:

�1� The ��,�� term shown in Fig. 6�a� is similar tohe impulse response of a lens with defocus �,30 butith an important difference: The equivalent nu-

ig. 6. Intensity pattern observed on the detector for sphericalHI for a �NA� of 0.07, �s � 12° �0.21 rad�, and � � 6 mm. �a�iffraction pattern caused by axial defocus, �b� crescent-shapedragg filter of a SR hologram, �c� diffracted pattern observed on theetector.


mrwliw

TtcigGi

ttt

ThrrnoddTdSTmcc

fhjilwcrmdld

tvsmpcfgdfd

iwli

Tz

ctoszsttocscdac7

B

Tioea

b

Ffpf

1

erical aperture �NA� is R�zf in the longitudinal di-ection and R�F in the lateral. For a normal lens, itould have been R�zf for both the lateral and the

ongitudinal directions. Because we are primarilynterested in the depth selectivity of the VHI system,e define

�NA� �Rzf

. (15)

his definition is also intuitively satisfactory fromhe geometry of the system. For large enough defo-us �, the ��,�� term represents an almost uniformlylluminated disk �see Fig. 6�a��, in agreement witheometrical optics. The disk is centered at theaussian image point �x�, y�� sF, 0� where the

ntensity peaks, as noted above.�2� The sinc�� term shown in Fig. 6�b� peaks along

he degeneracy circle, which in this geometry is cen-ered at �0, 0� and has radius F�s. The thickness ofhe degeneracy circle is

�r� �2FL�s

. (16)

his term originates from the Bragg selectivity of theologram, and in the SR VHI system it serves toeject out-of-focus light. It can be seen that the filterejects light only in mostly one lateral �x� directionear the Gaussian image point; points along thether �y� direction are mapped along the arc of theegeneracy circle. This behavior is evidence of the yegeneracy of the VH, as mentioned in Section 1.he circular shape of the locus of the degeneracy isue to the curvature of the fringes that constitute theR VH.he degeneracy circle due to the sinc� term acts as aask superimposed on the defocused disk image be-

ause of the ��,�� term. Together they result in therescentlike shape of Fig. 6�c�.

The pinhole in a confocal microscope performs aunction similar to the sinc�� filter imposed by theologram. Both reject out-of-focus light from an ob-

ect. However, the rejection mechanism is differentn the two cases: The pinhole operates as a hardimiter, whereas the hologram is a matched filterith respect to the wave front of a point source lo-

ated at the reference depth. An information theo-etic comparison of the two systems32 shows that theatched filter can provide superior image quality

espite the limited diffraction efficiency � � 1 for aine-shaped source oriented along the z axis and un-er the assumption of Gaussian noise statistics.An alternative point of view of the above observa-

ions is that depth selectivity results from the shiftariance of the VH.33 Indeed, the SR hologram hashift selectivity27; i.e., the diffracted field is Braggismatched when the spherical wave probe is dis-

laced in the direction xx, perpendicular to the re-orded fringes, for any value of � �both in and out ofocus�. On the other hand, the response of the holo-ram to a probe displaced by �y in the other lateralirection is qualitatively different. If the probe is inocus, i.e., rp � �0, �y, zf �, the point image is simplyisplaced to the location y� � F� �z . That is, the
y f

n-focus system is shift invariant in the ˆy direction,ithin a magnification factor F�zf. If the probe is

ocated out of focus at rp � �0, �y, zf � ��, the responses

I� x�, y��

Ib� �2�R2�

zf2 ,

2�RF �� x� � �s F�2

� �y� �F�y

zf�2�1�2�2

� sinc� x�2 � y�2 � F2��s2 � �y

2�zf2��L

2F2 �2

.

(17)

his represents a crescent of radius F��s2 � �y

2�f2�1�2, still centered at �0, 0� as in the on-axis case.The above discussion can be easily extended to the

ase in which the input field consists of a superposi-ion of mutually incoherent point sources arrangedn a plane perpendicular to the z axis. When theource plane coincides with the reference plane, zp �

f for all the sources, as shown in Fig. 7�a�, only theources with xp � 0 are visible to the hologram, andhey generate a line of point images oriented alonghe y� axis of the detector plane. The horizontal co-rdinate is x� � �sF for all the point images. In thease of a plane out of focus, zp � zf � �, the visibleources still lie on the xp � 0 line, but each source isreating its own crescent image on the detector plane,isplaced and with a different radius as discussedbove. The image is the superposition of intensitiesorresponding to these crescents, as shown in Fig.�b�.

. Depth Resolution

he presence of a masked defocus term in Eq. �14�ndicates that SR VHI can map depth to the intensityf the diffracted field. We describe two methods forxploiting this property to obtain 2 1⁄2-D or 3-D im-ges:

�1� Method 1. The reflective object is illuminatedy focused illumination as shown in Fig. 8�a�. A

ig. 7. VHI is a shift-variant imaging system; the observed dif-racted field changes with a change in the spatial location of theoint source: �a� mutually incoherent point sources with no de-ocus, �b� mutually incoherent point sources with defocus �.

ltsajle

FapPIa1ic

b8eC

lnedoaetiTpop9odp

ipta�pwicd

o

FtC

FS

FdBVc

arge-area detector is placed at the Fourier plane ofhe FT lens and the entire diffracted power is mea-ured. Thus the object shape is obtained one voxelt a time. Complete 3-D scanning recovers the ob-ect in its entirety. In this case, we calculate theongitudinal PSF by integrating Eq. �14� over thentire detector area:

Id��

�

I� x�, y��dx�dy�. (18)

igure 9 shows theoretical and experimental plots of Ids a function of defocus � for the case of a single probeoint with an integrating detector. We normalize theSF to the Bragg-matched diffracted intensity I0 �

d�0�. In both theory and experiment, we used a �NA�pproximating 0.07, R � 3.5 mm, zf � 50 mm, and �s �2° inside the hologram. The hologram was recordedn a 2-mm-thick, 0.03% Fe-doped LiNbO3 crystal. Itan be seen that �zFWHM � 800 �m for this case.

�2� Method 2. The reflective object is illuminatedy extended monochromatic light as shown in Fig.�b�. The y degeneracy allows the VH to image anntire strip along the degenerate y direction. ACD camera is placed at the Fourier plane of the FT

ig. 8. VHI methods: �a� 3-D scanning with an integrating de-ector and focused illumination, �b� exploiting y degeneracy with aCD and 2-D scanning with extended illumination.

ig. 9. Theoretical and experimental plots of longitudinal PSF forR VHI.

ens to capture the strip. As a result, only 2-D scan-ing is required to recover the entire object. How-ver, from Eq. �17�, it is obvious that each Braggegenerate point in the y direction gives rise to anut-of-focus crescent. These crescents overlap fordjacent degenerate points, thus degrading the lat-ral resolution. One way to compensate is to adap-ively estimate the size of the crescent and deconvolven a way similar to depth-from-defocus techniques.34

he starting point for this type of estimation is theercentage of power contained in a spot of given sizen the detector plane. This is given in Fig. 10, whichlots the radii of the circles confining 95%, 99%, and9.9% of the total diffracted energy for various valuesf defocus. The complete derivation of the adaptiveeconvolution scheme is beyond the scope of this pa-er.Returning to the point-by-point method 1 by an

ntegrating detector, which appears to be the mostractical, we now estimate the depth resolution ashe �zFWHM from the PSF. The most interestingspect of this calculation is the dependence of thezFWHM on the distance zf between the referencelane and the optical system, in other words theorking distance of the imaging system. The scal-

ng factor in the defocus term of Eq. �14� and therescent width term of Eq. �16� suggests a depen-ence of the form

�zFWHM�SR� �GSRzf

2

R2L, (19)

r, in terms of the �NA� of the system,

�zFWHM�SR� �GSR

�NA�2L. (20)

ig. 10. Computed contour lines giving the fraction of the totaliffracted power that falls within small circles centered at theragg-matched point versus selected values of the defocus term.alues corresponding to a defocus equal to the FWHM are indi-ated by the dotted–dashed lines.


Isfam

Fto

C

TdSiltmatbitoc

btdedi

TdLt

Wpaia

LcT

Hoo

Fatet

IaiTm

a�

Fo

1

n Eqs. �19� and �20� the factor GSR depends on theignal beam angle �s only. Its value is determinedrom the integration of Eq. �18� and is given in Fig. 11s a function of �s. The result matches well theodel

GSR �18.2

�s. (21)

or the value �s � 12 used in the experiment, theheoretical value is GSR � 14, in agreement with ourbservation.

. Design of Objective Optics

he result of Eq. �19� states that the depth resolutionegrades quadratically with working distance.uch behavior is common in image-forming ranging

nstruments. To obtain a good depth resolution atong working distances, i.e., remote objects, we needo use large apertures or thicker holograms; bothethods are impractical and expensive. One way to

ccomplish this is by use of objective optics. We usehis term to refer to optical elements that are insertedetween the object and the hologram to transform thencoming field and thus improve the working dis-ance or the resolution. For example, in Ref. 23 ourbjective optics was simply a telescope. Here we dis-uss use of objective optics in more general terms.

Consider the two SR VHI systems shown in Fig. 12,oth of which have the same working distance d. Inhe system of Fig. 12�a�, the field from the object isirectly incident on the hologram; therefore the ref-rence distance for this case must be zf � d, and theepth resolution in the vicinity of d is given, accord-ng to Eq. �19�, by

�zFWHM�SR� �GSRd2

R2L. (22)

he objective optics in Fig. 12�b� create an interme-iate image of the object in front of the hologram.et m1 denote the lateral magnification of the objec-ive optics and z � denote the new reference distance.

Fig. 11. Dependence of GSR on �s.

f


e assume that zf� is chosen such that the diffractedower is maximum when the object is located exactlyt d. Now suppose that the object is displaced by �n the longitudinal direction. The intermediate im-ge is displaced by

�� m12�. (23)

et a denote the size of the entrance pupil of theombination of the objective optics and hologram.he effective �NA� of the optical system is then

�NA�obj �ad

. (24)

owever, because the angular magnification of thebjective optics is 1�m1, the �NA� of the field incidentn the hologram is

�NA� ��NA�obj

m1�

Rzf�

. (25)

rom Eqs. �23�, �25�, and �20� we can see that thepparent gain �or loss� in �NA� due to the magnifica-ion of the objective optics is exactly balanced by anqual loss �or gain� in longitudinal displacement ofhe intermediate image. Therefore the resolution is

�zFWHM�SR� �GSR

�NA�obj2L

�GSRd2

a2L. (26)

t appears that there is no “free lunch” in this case,nd the objective optics did not help improve themage quality. However, we did gain one thing:he hologram aperture required to achieve perfor-ance as in Eq. �26� is given by

R � �m1�a (27)

ccording to Eq. �25�. By using demagnifying optics�m � � 1� we can utilize a smaller hologram with the

ig. 12. SR VHI systems �a� without objective optics and �b� withbjective optics in between the hologram and the object.

1

s1ptid

�d

Ih

D

Fd�pochtfYr

ehtipomo

fceiTsamamip

tahlAuWN2It1

phwamailutt

Fq

Fot

Fip

ystem of Fig. 12�b� compared with the system of Fig.2�a�. In most implementations, this is desirable inractice. This was the case in Ref. 23 where theelescope was acting as a demagnifier. The theoret-cal and experimental PSF dependence on workingistance d is given in Fig. 13.It is worthwhile to mention that a confocal system

CF� with objective optics would also exhibit a similarependence:

�zFWHM�CF� �GCFr2

d2 . (28)

n Eq. �28�, GCF is a factor that depends on the pin-ole radius of the confocal system.35

. Experimental Results

igure 14 compares the theoretical and experimentaliffraction patterns captured on a camera for defocus� 4 mm. The VH was recorded with a reference

oint source located zf � 50 mm away from the centerf the hologram and a plane-wave signal beam in-lined at angle �s � 12° inside the hologram. Theologram radius was R � 3.5 mm, which correspondso a �NA� approximating 0.07. The FT lens had aocal length F � 63.9 mm. A frequency-doubled Nd:AG cw laser � � 532 nm� was used for both theecording and the imaging in this and all subsequent

ig. 13. Depth resolution �zFWHM of a SR VHI system degradesuadratically with working distance �after Ref. 23�.

ig. 14. �a� Theoretical and �b� experimental diffracted patternsn a CCD for SR VHI with a point-source displaced � � 4 mm fromhe Bragg match.

xperiments. The typical diffraction efficiency of theologram was � � 5%. We used a Jai CV235 indus-rial CCD camera to capture the diffraction patternsn all experiments. The experimental degeneracyattern matches well with the theoretical predictionsf curvature, spot size, and thickness. The experi-ental strip is slightly thicker because of saturation

f the pixels on the CCD.Three experimental diffraction patterns obtained

rom point sources with different yp were added in-oherently and are shown combined in Fig. 15. Thisxperiment was intended to emulate the incoherentmaging case discussed at the end of Subsection 2.A.he experimental setup was identical to the one de-cribed in the previous paragraph. The in-focus im-ges obtained with � � 0 and �y � 1.5, 0, and 1.5m are shown in Fig. 15�a� and the out-of-focus im-

ges for � � 4 mm and the same lateral displace-ents are shown in Fig. 15�b�. The crescent radii

ncrease with �y, in agreement with the theoreticalrediction.Figure 16 presents SR VHI of an aluminum artifact

hat we fabricated specifically for this purpose. Thertifact consisted of the letters MIT at differenteights as a staircase with a 2-mm step size. The

etter M was the tallest letter and T was the shortest.computer-aided design rendering from the model

sed to machine the artifact is given in Fig. 16�a�.e performed 2-D scanning using two orthogonalewport CMA-25CCCL actuators and a Newport832C powermeter, both controlled through Nationalnstruments’ LabVIEW software. The experimen-al setup was identical to the one described for Fig.4.Figure 16�b� shows a point-by-point scan with a

ixel size of 100 �m2 of the artifact at the referenceeight of the letter M �i.e., the surface of this letteras Bragg matched during the scan�. The letters Ind T are consequently weaker because they wereismatched. To quantify the Bragg mismatch, we

veraged numerically the intensities obtained exper-mentally over the entire bright part of the matchedetter M and compared it with the corresponding val-es for the mismatched letters I and T. We denotehese values as PM, PI, and PT, respectively. Fromhe experiment we found P � �5.82 � 2.7� � 107,

ig. 15. Observed diffracted fields on a CCD for three mutuallyncoherent points: �a� points at the Bragg-matched plane and �b�oints at a defocused plane � � 4 mm.

M


P1

atTfsds

3R

A

Fnlmgls

i

Ti

ltiFaam

Ttwct

VowtjrgibTtIBlin

ftod

Fclww

FB

1

I � �1.35 � 0.67� � 107, and PT � �0.79 � 0.25� �07. The ratios

PI

PM� 0.23 � 0.22,

PT

PM� 0.13 � 0.07

re slightly higher compared with the estimates fromhe PSF calculation and the experiments of Fig. 9.his is because point scanning is susceptible to sur-

ace irregularities on the artifact. At local roughpots, most of the scattered light does not reach theetector. This effect is evident in the high values oftandard deviation for PM, PI, and PT.

. Volume Holographic Imaging with Plane-Waveeference Beams

. Derivation and Properties of the Diffracted Field

igure 17�a� is the recording setup for VHI with pla-ar reference �PR VHI�. The collimating objective

ens ensures that the reference beam is approxi-ately a plane wave when it is incident on the holo-

ram. Thus the front focal length of the objectiveens is the reference depth at which the PR VHIystem operates.During recording, the reference beam is a normally

ncident plane wave:

Ef�r� � exp�i2�z� . (29)

he hologram and the FT lens are assumed to havenfinite lateral extent. L is the thickness of the ho-

ig. 16. SR VHI of the fabricated letters MIT. �a� The actualomputer-aided design rendering of the object. �b� A volume ho-ographic image of the object obtained by a complete lateral scanith the surface of the letter M placed at a Bragg-matched location,hich consequently appears to be bright.


ogram, f is the focal length of the collimating objec-ive lens, and a is its radius. Thus the �NA� of themaging system equals a�f. The focal length of theT lens is F. The signal beam is a plane wave prop-gating at angle �s �� 1 with respect to the z axis, justs in the case of SR VHI. In the paraxial approxi-ation, the signal beam is expressed as

Es�r� � exp�i2��1 ��s

2

2 � z

� i2��s

x� . (30)

he VH is the recorded 3-D interference pattern ofhe reference and signal beams and it is stored as aeak modulation ��r� � �Ef � Es�

2 of the dielectriconstant. The Bragg-matched term of the modula-ion is given by

��r� � exp�i2�

�x�s � z�s

2

2 �� . (31)

Figure 17�b� shows the readout procedure for PRHI. A probe point source is placed in front of thebjective lens at coordinates rp � �xp, yp, zp�. Firste consider the case in which the probe is placed at

he Bragg-matched location �0, 0, f �. Then the ob-ective lens collimates the probe field. As a result, aeplica of the reference beam is incident on the holo-ram. Because the probe is exactly Bragg matchedn this case, the VH diffracts a replica of the signaleam, i.e., a plane wave propagating in direction �s.he Fourier lens placed behind the hologram focuseshis diffracted plane wave onto the detector surface.f instead the probe is axially displaced by � from theragg-matched location, the objective lens can no

onger collimate the probe field. Instead, the fieldncident on the hologram is a spherical wave origi-ating at

zp� �f � f � ��

��

f 2

�(32)

or � �� f. Only the on-axis Fourier component ofhis spherical probe is Bragg matched; therefore theverall intensity diffracted from the hologram is re-uced.

ig. 17. Plane-wave reference VHI schematic: �a� recording, �b�ragg-matched readout, �c� Bragg-mismatched readout.

pomitaC

ssIimc3uefid

wppBppwtr

sW

Wtit

TFt

li

wpdh�tVtrpabprm

tftnIichld

ssmdaodqrt

FVDod

We now derive the dependence of the diffractedower on the probe displacement �. According to thebservation in the previous paragraph, it is mathe-atically convenient to express the defocused spher-

cal probe as well as the diffracted field in terms ofheir plane-wave components with wave vectors kpnd kd, respectively. The transfer function �Ref. 25,hap. 1�

A�kp, kd� � S ��

��r� � exp�i�kp � kd� � r�d3r

(33)

pecifies the spatial spectrum of the hologram re-ponse to a single spatial frequency in the input field.n Eq. �33�, S is a constant determined by the polar-zation and index modulation; our analysis will nor-

alize the diffracted intensities, and so this factoran be ignored. A�kp, kd� can be interpreted as the-D Fourier transform of the dielectric constant mod-lation �� evaluated at kp kd. In the more gen-ral case in which the spatial spectrum of the probeeld is given by Ep�kp�, the spatial spectrum of theiffracted field is obtained from Eq. �33� as

Ed�kd� � �� Ep�kp�A�kp, kd�dkpxdkpy, (34)

here kpx and kpy are the x and y components of therobe wave vector, respectively, whereas the z com-onent is given by the Bragg constraint �kp� � 2��.ecause the detector is located at a Fourier transformlane, the diffracted field as a function of the detectorlane coordinates �x�, y�� is obtained from Eq. �34�hen we substitute kdx � 2�x��F, kdy � 2�y��F for

he xx and yy components of the wave vector kd,espectively.

In the specific case of interest, where the probe is apherical wave, we express its spatial spectrum usingeyl’s identity:

Ep�kpx, kpy� � exp�izp��kpx � kpy�2

2�k� � . (35)

e evaluate the integral Eq. �33� using Eq. �31� andhen substitute the result of the integral and Eq. �35�n Eq. �34� to obtain the diffracted field on the detec-or plane:

Ed� x�, y�� expi�zp�

��x�

F� �s�2

� �y�

F�2��sinc�L sin �s

�x�

F� �s�� .

(36)

his function is almost zero outside a disk of radiusa��f2 centered at x� � �sF. This disk represents

he geometric image of the aperture of the objective

ens on the detector plane. Therefore the intensitys expressed approximately as

I� x�, y��

Ib� circ�� x� � �s F�2 � y�2

Fa��f 2 �1�2

� sinc2�L sin �s

�x�

F� �s�� , (37)

here Ib � I�x� � �sF, y� � 0� is the peak intensityroduced by the probe. The approximation neglectsiffraction ripples at the disk edges; these ripplesave negligible effect in practice. Comparing Eqs.37� and �14� suggests that the origin of depth selec-ivity is similar in the PR and SR implementations ofHI. In the PR case, the diffraction pattern con-

ains two contributions, shown in Fig. 18: �1� a disk,epresented by the circ�.� function, whose radius isroportional to the defocus �; and �2� a slit orientedlong the x �Bragg-selective� direction, representedy the sinc2�.� function, whose width is inversely pro-ortional to the hologram thickness L. This termejects out-of-focus light because of the Bragg mis-atch.There are also some significant differences: �1� In

he PR case, the disk radius grows linearly with de-ocus �, whereas this dependence was quadratic inhe SR case. We belabor this point further in con-ection with the resolution of the two methods. �2�n the PR case, the shape of the Bragg degeneratellumination on the detector plane is a straight slit,onsistent with the planar fringes recorded in theologram. On the other hand, the fringes in SR ho-

ograms are curved leading to the crescent-shapedegeneracy.PR VHI is also a shift-variant imaging system33

imilar to SR VHI. The PR hologram has angularelectivity15; i.e., the diffracted field is Bragg mis-atched when the probe source is displaced in the

irection x, perpendicular to the recorded fringes, forny value of �x �both in and out of focus�. On thether hand, the response of the hologram to a probeisplaced by �y in the other lateral direction �y� isualitatively different. If the probe is in focus, i.e.,p � �0, �y, f �, the point image is simply displaced tohe location y� � F� �f. That is, the in-focus system

ig. 18. Intensity pattern observed on the detector for sphericalHI for a �NA� of 0.07, �s � 12° �0.21 rad�, and � � 8 mm. �a�iffraction pattern caused by axial defocus and the finite aperture

f the collimating lens, �b� straight Bragg slit of a PR hologram, �c�iffracted pattern observed on the detector.

y


iI�

TsGbEiocchice

B

Wsnpftpp

Sdtc

�e

op1a2By

stl

ti

wN

IPtSbtsttdnm

Fcss

1

s shift invariant within a magnification factor F�f.f the probe is located out of focus at rp � �0, �y, f ��, the response is

I� x�, y��

Ib� circ�� x� � �s F�2 � � y� � F�y�f �2�1�2

Fa��f 2 �� sinc2�L sin �s

�x�

F� �s�� . (38)

his represents a disk masked by the Bragg-electivity slit similar to Eq. �37�, centered at theaussian image point �0, F�y�f �. The center woulde located at �0, 0� for an on-axis probe �y � 0. Fromq. �38� we can see that PR VHI is completely shift

nvariant in the y direction for both in-focus and out-f-focus probe points. This is because SR VHI has arescent-shaped degeneracy curve whose radiushanges with lateral displacement �y. On the otherand, PR VHI has a straight-line degeneracy, which

s independent of lateral displacement �y. We dis-uss the effect of this lateral shift invariance at thend of Subsection 3.B.

. Depth Resolution

e now calculate the longitudinal PSF of the PRystem with an integrating detector and 3-D scan-ing �method 1 of Subsection 2.B�. The calculationroceeds as follows. First we integrate the dif-racted intensity of Eq. �37� with respect to the detec-or coordinates �x�, y��. Then we normalize to theeak Bragg-matched power I0 received when therobe is in focus. The result is

Id

I0�

1� �

0

2�

d� �0

1

d�� sinc2�aL sin �s�

f 2 � sin �� .

(39)

o far, all the derivations were performed by angles,istances, and a wavelength of light measured insidehe holographic material of refractive index n. Theorrected PSF for quantities measured in air is

Id

I0�

1� �

0

2�

d� �0

1

d�� sinc2�aL sin �s�

nf 2 � sin �� .

(40)

Note that in the SR case there was no residual nntering the free-space formula.�Figure 19 shows theoretical and experimental plots

f the PSF as a function of defocus � for a �NA� ap-roximating 0.07, a � 3.5 mm, f � 50.2 mm, and �s �2° inside the hologram. In both the experimentnd the simulation, the hologram was recorded in a-mm-thick Fe-doped LiNbO3 crystal �n � 2.2�.oth the experimental and the theoretical curvesield a �zFWHM � 1.7 mm.The trend of longitudinal resolution �zFWHM ver-

us the working distance d is computed directly fromhe PSF expression. In this case, we select the focalength of the objective lens such that f � d, and from


he scaling factors in the argument of the integrandn Eq. �40� we find

�zFWHM �GPRf 2

aL, (41)

here GPR is a factor that depends linearly on �s.umerical regression on Eq. �40� yields

GPR �5.34

�s. (42)

t is possible to exploit the straight-line degeneracy ofR VHI and use a CCD camera to acquire informa-ion one slit at a time in a way similar to method 2 ofubsection 2.B. If a reflective object is illuminatedy extended monochromatic light as shown in Fig. 20,he y degeneracy allows the PR VH to image an entiretrip along the degenerate y direction. As men-ioned above, PR VHI is completely shift invariant inhe ˆy direction unlike SR VHI, which results is aifferent crescent for each degenerate point. Weow examine two situations and point out whichethod �SR or PR VHI� would prove more beneficial.

Fig. 19. Longitudinal PSF for PR VHI.

ig. 20. VHI when the y degeneracy is exploited by use of aamera at the detector plane. �a� Line scanning of an object withmall surface features, �b� line scanning of an object with largeurface features.

sasbboOcsrism

fifsdosBP

C

Wocb�vwSi�o

2sToT�

w

tif�otpT

i

Fw

Ic

�ptcteeS

tctbtbtst

wfpLd

aV

Tt

Fos

�1� The reflective object shown in Fig. 20�a� hasurface features of size �z � �zFWHM. In this case,

SR VHI system would result in an image witheveral weak crescent-shaped defocus blurs and aright Bragg-matched slit. This image can be easilye processed �e.g., by deconvolution or iterative meth-ds� to give an accurate representation of the surface.n the other hand, the PR VHI system finds it diffi-

ult to distinguish between the surfaces because themaller surface features and the straight degeneracyesults in the slit of the out-of-focus surface appear-ng almost as bright as the slit of the Bragg-matchedurface. Hence a SR VHI system that employsethod 2 is preferred in this case.�2� The reflective object shown in Fig. 20�b� has sur-

ace features �z � �zFWHM. In this case, the SR VHImage consists of a thick band of crescents �arisingrom the large defocus blur� and the Bragg-matchedlit. This band makes further image processing andeconvolution susceptible to noise artifacts. On thether hand, the large surface features ensure that thelit image out-of-focus surface is much fainter than theragg-matched slit. As a result, it is preferable to useR VHI with method 2 in this case.

. Design of the Objective Optics

e now revert back to our discussion on the resolutionf PR VHI employing an integrating detector and dis-uss whether it is possible to improve depth resolutiony appropriate choice of objective optics. From Eq.55� we observe that the �zFWHM of the PR VHI systemaries quadratically with f but only inverse linearlyith a. This is a significant difference compared withR VHI systems, where the dependence on the phys-

cal aperture R of the hologram was inverse quadraticsee Eq. �19��. Below, we show how to design objectiveptics for the PR case to exploit this property.Consider the objective optical system shown in Fig.

1. The first and second principal planes of thisystem are denoted as PP1 and PP2, respectively.he working distance d is measured between thebject and the entrance pupil of the objective optics.he radius of the entrance pupil is a. The effective

NA� of the optical system is

�NA�obj �ad

, (43)

hich is of course identical to Eq. �24�.

The purpose of the objective optics is to illuminatehe hologram with a collimated beam when the objects in focus, i.e., located exactly at d. Therefore theront focal length �FFL� must be equal to d. LetEFL� denote the effective focal length of the objectiveptics and r the radius of the collimated beam. Inhe simplest case, the objective optics is a single thinositive lens, as described in Subsections 3.A and 3.B.hen �EFL� � �FFL� � f � d and r � a.For an out-of-focus object, Eq. �41� can be rewritten

n terms of �EFL� and r as

�zFWHM�PR� �GPR�EFL�2

rL. (44)

rom Fig. 21, we can see that r � �NA�obj �EFL�,hich on substitution in Eq. �44� yields

�zFWHM�PR� �r

NAobj2L

. (45)

n terms of the physical apertures, this expressionan be rewritten as

�zFWHM�PR� �GPRrd2

a2L. (46)

This indicates that the quadratic degradation ofzFWHM�PR� with increasing d can be offset when thehysical size r of the collimated beam illuminatinghe hologram is reduced. Equivalently, this is ac-omplished when the objective optics is designed sohat PP1 is as close to the probe as possible. Anxample of a standard optical system with this prop-rty is a telephoto lens, which is analyzed in detail inubsection 3.D.A side effect of making r arbitrarily small is that

he ray bundle exiting the objective optics wouldease to behave like a plane wave. To make surehat this does not happen, we need to place a loweround rmin on r. A reasonable method to calculatehe lower bound is to ensure that the spread of the rayundle due to diffraction beyond the exit pupil andhroughout the length L of the hologram remains amall fraction of r itself. This leads to a condition ofhe form36

rmin � c�L, (47)

here c is selected depending on the amount of dif-raction that we are willing to tolerate. For exam-le, in the midvisible range of wavelengths and for

� 2 mm, rmin � 100 �m ensures less than 5%iffraction.Replacing Eq. �47� in Eq. �46� we find that the best

chievable depth resolution �zFWHM�PR�opt for PRHI varies as

�zFWHM�PR�opt

GPRrmind2

a2L. (48)

herefore the resolution of the optimized PR VHI sys-em still degrades quadratically with working distance

ig. 21. Schematic for the design of an objective optical system tobtain high depth resolution at large working distances in VHIystems.


dcw

Vrste�ti�

ttoadmofwtbts

DR

Tsot

Tw

lbss

Tbmshiutp

uda05a

Foptics, �b� PR VHI schematic with objective optics.

1

, but the objective optics provide a favorable �reduced�onstant of proportionality by a factor rmin�a comparedith the value without objective optics.It is also possible to design objective optics for PR

HI such that the resolution degrades with d at aate slower than quadratic, e.g., linear or even con-tant. This is accomplished when we use zoom op-ics in the objective to vary r as function of d. Forxample, if we maintain r � d1 then we obtainzFWHM � d. However, because of the constraint on

he minimum allowable r, the subquadratic behaviors possible only if we are willing to toleratezFWHM�PR� � �zFWHM�PR�opt.Before we conclude this subsection, it is worthwhile

o summarize the different effects that objective op-ics have in the SR and PR cases. In the SR case, thebjective optics permit use of a smaller hologram forgiven �NA�obj without any inherent gain in longitu-inal resolution. In the PR case, the objective opticsove the effective focal distance much closer to the

bject than the actual working distance d. There-ore resolution is improved by as much as rmin�a,here a is the physical aperture of the objective op-

ics and rmin, the minimum allowable collimatedeam diameter. In Subsection 3.D we quantifyhese differences for the specific case of a PR VHIystem implemented with a telephoto lens.

. Design of Telephoto Objective Optics for Planareference Volume Holographic Imaging Systems

o analyze objective optical systems, we consider aystem similar to the one depicted in Fig. 22 and usef the generalized ray-tracing matrix representa-ion37:

M � �M11 M12

M M � . (49)
21 22

he key parameters to determine the resolution andorking distance are then given by

M12 � 1

�EFL�, (50)

M11 ��FFL�

�EFL�. (51)

A telephoto system comprises two lenses of focalengths f1, f2, with f2 � 0. The lenses are separatedy distance t. For the matrix elements M11 and M12pecified by Eqs. �50� and �51�, the focal lengths areelected as

f1 �M11t

M11 � 1 � M12t, (52)

f2 �t

1 � M11. (53)

he separation t is selected to locate PP2 immediatelyehind the second lens �otherwise the rmin require-ent becomes more stringent.� PP2 is exactly on the

econd lens if t � f1, so in practice we select a slightlyigher value for t. It should be noted that the same

mprovement in resolution can be accomplished byse of a demagnifying telescope or other optical sys-ems; however, the telephoto is one of the most com-act and widely available implementations.Figure 23 shows that a telephoto system can be

sed to increase the working distance without anyegradation in the �zFWHM for PR VHI. The stand-lone PR VHI system had a �NA�obj approximating.08 by use of a collimating objective lens with d � f �0 mm and radius a � 4 mm to record the hologramnd � � 12°. The telephoto system was comprised

ig. 22. Appropriately designed objective optics can improve the �zFWHM of a PR VHI system. �a� PR VHI schematic without objective

s

ospf1�rc1st5

fct�adEe

TttttFdrosra

sr

ptamwso

Tf

E

Wut

FwPp�

F�

Ft

f two lenses with f1 � 250 mm and f2 � 25 mmeparated by a distance t � 500 mm. The object waslaced at a working distance d � �FFL� � 500 mm inront of the positive lens that had an aperture a �2.7 mm. Thus, for the telephoto system, theNA�obj approximated 0.025. The actual size of theecorded hologram was r � 2 mm and �s � 12°. Itan be seen that both PSFs have the same �zFWHM �mm despite the fact that the stand-alone PR VHI

ystem has a working distance d � 50 mm and theelephoto PR VHI system has a working distance d �00 mm.Figure 24 compares the optimum depth resolution

or a PR VHI system with a SR VHI system and aonfocal �CF� system as a function of working dis-ance d on a log–log scale. We substitute the valuess � 13.6°, L � 2 mm, a � 12.7 mm, rmin � 100 �m,nd pinhole radius 1.22d�4a in the expressions for aepth resolution for each of these systems given byqs. �46�, �26�, and �28�, respectively. The resultingquations are

�zFWHM�PR�opt � 1.5 � 106d2,

�zFWHM�SR� � 20.45 � 106d2,

�zFWHM�CF� � 3.50 � 106d2.

hus the PR VHI system has better resolution thanhe confocal system that in turn has better resolutionhan the SR VHI system. In addition, it is possibleo design zoom objective optics for a PR VHI systemo operate along the lines �1�, �2�, or �3� as shown inig. 24. Operating along �1� achieves the optimalepth resolution. However, if we are willing to sac-ifice some depth resolution, the PR VHI system canperate anywhere between lines �3� and �1�. In �2�,ome depth resolution is sacrificed to ensure that theesolution degrades linearly with increasing d � d*,nd in �3� more depth resolution is sacrificed to en-

ig. 23. Appropriately designed telephoto system can improveorking distance d without any degradation in the resolution.SFs for a stand-alone PR VHI system �dashed curve� and a tele-hoto PR VHI system �solid curve� show that both systems havezFWHM � 1 mm for d � 50 mm and d � 500 mm, respectively.
ure that the depth resolution stays constant over the
ange 0 � d*.For the telephoto system, the hologram must be

laced at the second principal plane PP2 of the sys-em. When we image using method 2 �line scan�, andditional consideration for the size r of the colli-ated beam is that it acts as a field stop. To seehy, consider Fig. 25 where the hologram is placed

uboptimally at a distance b behind PP2. The fieldf view �FOV� of this telephoto system is

�FOV� �2r�EFL�

b � L. (54)

hus it is beneficial to select b as small as possible toully utilize the parallelism of method 2.

. Experimental Results

e numerically integrated Eq. �40� for different val-es of a, L, and f and we fitted the numerical data tohe equation

�zFWHM �GPRf 2

aL, (55)

ig. 24. Plots of �zFWHM versus object distance d for PR VHIsolid lines�, SR VHI �dashed line�, confocal systems �dotted line�.

ig. 25. Calculation of the FOV for the telephoto PR VHI system;he chief ray makes an angle � with the optical axis.


w�lsvpsbw

tdcag�fepi

isFnaoysL

ootam

F�p�a

Foa

Fmisisis

1

here GPR is a factor that depends linearly on �s. Fors � 13.6° �signal beam angle inside the hologram�, aeast-squares fit yielded GPR � 9.3. We performed aeries of experiments to determine the experimentalalue of GPR in the identical geometry and found ex-erimental GPR � 10.09 as shown in Fig. 26. Themall disagreement with the theory can be explainedy the aberrations in our experimental system thatere not accounted for in our paraxial theory.Figure 27 compares the theoretical and experimen-

al diffraction patterns captured on a camera for aisplacement of � � 8 mm from a Bragg match. Aollimating objective lens of focal length f � 50.2 mmnd radius a � 3.5 mm was used to record the holo-ram. The signal beam was inclined at an angles � 12° inside the hologram and the FT lens had aocal length F � 63.9 mm. We can see that thexperimentally and theoretically predicted diffractedatterns match well. Again, pixel saturation resultsn a slightly thicker experimental degeneracy line.

ig. 26. �a� Theoretical �solid line� and experimental �asterisks�zFWHM versus a for fixed f and L, confirming the inversely pro-ortional relationship to the �NA� in this case. �b� Theoreticalsolid line� and experimental �asterisks� �zFWHM versus f for fixed

and L, confirming the quadratic dependence on f.


Figure 28 presents line scan �method 2� PR VHImages of the MIT artifact that was described in Sub-ection 2.D. The artifact can be seen in Fig. 16�a�.or the imaging experiments, we employed 2-D scan-ing using two orthogonal Newport CMA-25CCCLctuators and a CCD camera to acquire images of thebject one slit at a time by exploiting the straight-linedegeneracy of PR VHI. The entire experimental

etup for image acquisition was controlled with MAT-AB. The experimental setup was identical to thene described in Fig. 27. Figure 28�a� is a PR imagef the object with the surface of the letter M located athe Bragg-matched height. As a result, the letter Mppears brightest. The letters I and T are 2 and 4m away from the Bragg-matched location and thus

ig. 27. �a� Theoretical and �b� experimental diffracted patternsn a CCD for PR VHI with a point-source displaced � � 8 mm fromBragg match.

ig. 28. PR VH images of the fabricated letters MIT placed 50.2m away from the entrance pupil of the system. �a� a PR VH

mage of the object obtained by a one-dimensional scan with theurface of the letter M placed at a Bragg-matched location, �b� anmage of the object obtained by a one-dimensional scan with theurface of the letter I placed at a Bragg-matched location, �c� anmage of the object obtained by a one-dimensional scan with theurface of the letter T placed at a Bragg-matched location.

aVtaVtMtattiat

wc2lo5acFa5ioe

rdttm

4

Wcocbtr

FM�slslsl

ppear progressively darker. Figure 28�b� is the PRH image with the surface of the letter I located at

he Bragg-matched height. Note that the letters Mnd T appear equally dark. Figure 28�c� is the PRH image with the surface of the letter T located at

he Bragg-matched height; note that the letters I andnow again appear progressively darker. To quan-

ify the Bragg mismatch, we define Pi, j to be theverage intensity of the letter i when the letter j is athe Bragg-matched location. The average values ofhe intensity are given in Table 1. The correspond-ng ratios are given in Table 2, which are in goodgreement with the estimates from the PSF calcula-ion and experiments �Fig. 19�.

Finally, Fig. 29 presents PR VHI of the same objectith use of telephoto objective optics. The telephoto

onsisted of a thin positive lens of focal length f1 �50 mm separated from a thin negative lens of focalength f2 � 25 mm by a distance t � 500 mm. Thebject was placed at a working distance d � �FFL� �00 mm in front of the positive lens that had anperture a � 12.7 mm. The actual size of the re-orded hologram was r � 2 mm. From the PSF ofig. 23, we can see that the collector optics allow us tochieve a �zFWHM � 1 mm for an object located D �00 mm away. Figure 29 shows a progression ofmages referenced at the three characteristic heightsf the artifact similar to Fig. 28. The average pow-rs and power ratios are given in Tables 3 and 4,

Table 1. Measured Intensity Values for the Stand-Alone PR VHI �a.u.�

Measured Intensity

PM,M � 10.6 � 0.13PI,M � 2.05 � 0.28PT,M � 1.89 � 0.27PM,I � 2.51 � 0.25PI,I � 9.21 � 0.14PT,I � 2.18 � 0.29

PM,T � 1.80 � 0.24PI,T � 2.05 � 0.31PT,T � 9.59 � 0.13

Table 2. Ratios of Intensity Values Calculated from Table 1

Measured Intensity

PI,M

PM,M� 0.19 � 0.05

PT,M

PM,M� 0.17 � 0.03

PM,I

PI,I� 0.27 � 0.04

PT,I

PI,I� 0.24 � 0.07

PM,T

PT,T� 0.19 � 0.05

PI,T

PT,T� 0.21 � 0.04

espectively. Again, good agreement with the PSFata of Fig. 23 is observed. The corresponding ra-ios, given in Table 4, are also in good agreement withhe estimates from the PSF calculation and experi-ents �Fig. 23�.

. Discussion and Conclusions

e have presented the comparative analysis andharacterization of two alternative implementationsf VHI using transmission geometry holograms re-orded with spherical and plane-wave referenceeams. The common trends observed in both sys-ems with respect to depth resolution are summa-ized as follows:

ig. 29. PR VH images by collector optics of the fabricated lettersIT placed 500 mm away from the entrance pupil of the system.

a� A PR VHI image of the object obtained by a one-dimensionalcan with the surface of the letter M placed at a Bragg-matchedocation, �b� an image of the object obtained by a one-dimensionalcan with the surface of the letter I placed at a Bragg-matchedocation, �c� an image of the object obtained by a one-dimensionalcan with the surface of the letter T placed at a Bragg-matchedocation.

Table 3. Measured Intensity Values for PR VHI with the TelephotoSystem �a.u.�

Measured Intensity

PM,M � 10.3 � 0.14PI,M � 1.96 � 0.35PT,M � 1.55 � 0.3PM,I � 2.06 � 0.28PI,I � 9.83 � 0.16PT,I � 2.00 � 0.33

PM,T � 1.50 � 0.24PI,T � 1.89 � 0.35PT,T � 9.40 � 0.15


n

d

B

Ta

ww

ilso

rcjt

det

c

IwHqd�

tWqr

opcoRmw

ccwmdtircbspbimerdYtermtcocopp

cpTsCaeY

R

1

�1� inverse linear dependence on hologram thick-ess L,�2� quadratic dependence on the working distance

,�3� inverse quadratic dependence on the �NA�, and�4� reduced scanning time by the exploitation of the

ragg degeneracies of the VHs.

he most important differences of the two systemsre the following:

�1� In the SR case, the depth resolution dependenceith the physical aperture is inverse quadratic,hereas in the PR case it is linear.�2� In the SR case, use of objective optics does not

mprove resolution; however, the aperture of the ho-ogram required for a given resolution is smaller for aystem with demagnifying objective optics than with-ut.�3� In the PR case, the objective optics improve

esolution if the principal plane of the objective opti-al system can be placed sufficiently close to the ob-ect without degrading the quality of the beam exitinghe objective optics.

�4� In the PR case, the working distance depen-ence can be reduced to subquadratic �e.g., linear orven constant�, but in these cases the resolution ob-ained is suboptimal.

�5� The degeneracy shape of SR holograms is arescent, whereas for PR it is a straight line.

n terms of resolution, the best system is PR VHIith optimized objective optics, as shown in Fig. 24.owever, there are other trade-offs related to theuality of the object surface and the shape of theegeneracy curves when we use line scanningmethod 2� to acquire depth.

Our analysis was restricted to the case of VHI sys-ems where the VH is the result of a single exposure.

hen several holograms are multiplexed with a se-uence of exposures on the same holographic mate-ial, it is possible to reduce scanning even more and

Table 4. Ratios of Intensity Values Calculated from Table 3

Measured Intensity

PI,M

PM,M� 0.19 � 0.07

PT,M

PM,M� 0.15 � 0.04

PM,I

PI,I� 0.21 � 0.04

PT,I

PI,I� 0.20 � 0.07

PM,T

PT,T� 0.16 � 0.05

PI,T

PT,T� 0.20 � 0.05


btain the 2 1⁄2-D or 3-D image even in real time,rovided that there are enough pixels on the 2-Damera to adequately sample the higher-dimensionalbject information. This method was employed inef. 24 to obtain hyperspectral and 3-D spatial infor-ation from a fluorescent object in real time, i.e.,ithout scanning.Our experimental tests of SR and PR VHI were

onducted using an artifact that was fabricated spe-ifically for the purposes of our characterization, ande presented the raw image data, i.e., the intensityeasurements directly as they were returned by the

etector or camera. It would also have been possibleo apply deconvolution techniques to further improvemage quality. This possibility is promising for twoeasons: first because the system PSF can be wellharacterized and second because the PSF itself cane fine-tuned to improve the deconvolution results byimple modifications to the holographic recordingrocess, e.g., with deliberately aberrated referenceeams or nonplanar signal beams. For example, anntriguing possibility would be to use a cubic phase

ask38 to modify the spherical reference beam forxtended depth of field, which might reduce the depthesolution but would enhance overall image qualityue to the uniform PSF over the entire object space.et another possibility, which has been experimen-

ally implemented in our laboratory,14 is to use sev-ral VHI systems collaboratively to improve depthesolution. The mathematical foundation of thisethod is the theory of overconstrained linear sys-

ems, where resolution improves by virtue of noiseancellation among the multiple measurements �inther words, a sophisticated form of averaging.� In-orporation of such capture and postprocessing meth-ds into systems tailored for real-life imagingroblems will bring forth the full advantages of com-utational VHI.

We are grateful to Stanley M. Jurga, Jr., for fabri-ating the MIT artifact and Kehan Tian, Robert Mur-hey, and Brian H. Miles for helpful discussions.his project was funded by the U.S. Air Force Re-earch Laboratory �Eglin Air Force Base� and theharles Stark Draper Laboratory. G. Barbastathislso acknowledges the support of the National Sci-nce Foundation through the CAREER �formerlyoung Investigator� Award.

eferences1. P. J. van Heerden, “Theory of optical information storage in

solids,” Appl. Opt. 2, 393–400 �1963�.2. C. J. R. Sheppard and C. J. Cogswell, “Three-dimensional im-

aging in confocal microscopy,” in Confocal Microscopy, T. Wil-son, ed. �Academic, San Diego, Calif., 1990�, Chap. 4, pp. 143–169.

3. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G.Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A.Puliafito, and J. G. Fujimoto, “Optical coherence tomography,”Science 254, 1178–1181 �1991�.

4. P. Grangeat, “Mathematical framework of cone beam 3D re-construction via the first derivative of the radon transform,” inLecture Notes in Mathematics, Vol. 1497 of Mathematical

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3

3

3

3

3

3

33

Methods in Tomography, G. T. Herman, A. K. Louis, and F.Natterer, eds. �Springer-Verlag, Berlin, 1990�.

5. D. Marr, Vision �Freeman, New York, 1982�.6. M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467 �19

December 1961�.7. G. Binnig, C. F. Quate, and C. Gerber, “Atomic force micro-

scope,” Phys. Rev. Lett. 56, 930–933 �1986�.8. G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, “7 � 7 recon-

struction on Si�111� resolved in real space,” Phys. Rev. Lett. 50,120–123 �1983�.

9. D. Gabor, “A new microscopic principle,” Nature �London� 161,777–779 �1948�.

0. E. Leith and J. Upatnieks, “Wavefront reconstruction andcommunication theory,” J. Opt. Soc. Am. 52, 1123–1134 �1962�.

1. J. Zhang, B. Tao, and J. Katz, “Three-dimensional velocitymeasurements using hybrid HPIV,” in Developments in LaserTechniques and Fluid Mechanics, R. J. Adrian, D. F. G. Durao,F. Durst, M. V. Heitor, M. Maeda, and J. H. Whitelaw, eds.�Springer, Berlin, Germany, 1997�.

2. T. Zhang and I. Yamaguchi, “Three-dimensional microscopywith phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 �1998�.

3. J. H. Milgram and W. Li, “Computational reconstruction ofimages from holograms,” Appl. Opt. 41, 853–864 �2002�.

4. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “N-ocularholographic 3d imaging,” in Proceedings of the OSA AnnualMeeting �Optical Society of America, Washington, D.C., 2002�,paper WD7.

5. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey,“Holographic data storage in three-dimensional media,” Appl.Opt. 5, 1303–1311 �1966�.

6. H. Kogelnik, “Coupled wave theory for thick hologram grat-ings,” Bell Syst. Tech. J. 48, 2909–2947 �1969�.

7. D. Psaltis and F. Mok, “Holographic memories,” Sci. Am. 273,70–76 �1995�.

8. J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Volume holo-graphic storage and retrieval of digital data,” Science 265,749–752 �1994�.

9. H. Lee, X.-G. Gu, and D. Psaltis, “Volume holographic inter-connections with maximal capacity and minimal cross talk,”J. Appl. Phys. 65, 2191–2194 �1989�.

0. D. Brady, S. Lin, X. G. Gu, and D. Psaltis, “Holography in

artificial neural networks,” Nature �London� 343, 325–330�1990�.

1. G. Barbastathis and D. J. Brady, “Multidimensional tomo-graphic imaging using volume holography,” Proc. IEEE 87,2098–2120 �1999�.

2. G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal mi-croscopy with a volume holographic filter,” Opt. Lett. 24, 811–813 �1999�.

3. A. Sinha and G. Barbastathis, “Volume holographic telescope,”Opt. Lett. 27, 1690–1692 �2002�.

4. W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectralimaging in three spatial dimensions,” Opt. Lett. 27, 854–856�2002�.

5. H. Coufal, D. Psaltis, and G. Sincerbox, eds., Holographic DataStorage �Springer, New York, 2000�.

6. P. Yeh, Introduction to Photorefractive Nonlinear Optics�Wiley, New York, 1993�.

7. G. Barbastathis, M. Levene, and D. Psaltis, “Shift multiplex-ing with spherical reference waves,” Appl. Opt. 35, 2403–2417�1996�.

8. G. Barbastathis and D. Psaltis, “Shift-multiplexed holographicmemory using the two-lambda method,” Opt. Lett. 21, 432–434 �1996�.

9. D. Psaltis, F. Mok, and H. Y.-S. Li, “Nonvolatile storage inphotorefractive crystals,” Opt. Lett. 19, 210–212 �1994�.

0. M. Born and E. Wolf, Principles of Optics, 7th ed. �Pergamon,Cambridge, U.K., 1998�.

1. H. Fisk-Johnson, “An improved method for computing a dis-crete Hankel transform,” Comput. Phys. Commun. 43, 181–202 �1987�.

2. G. Barbastathis and A. Sinha, “Information content of volumeholographic imaging,” Trends Biotechnol. 19, 383–392 �2001�.

3. A. Stein and G. Barbastathis, “Axial imaging necessitates lossof lateral shift invariance,” Appl. Opt. 41, 6055–6061 �2002�.

4. M. Subbarao and S. Gopal, “Depth from defocus: a spatialdomain approach,” Intl. J. Comput. Vision 13, 271–294 �1994�.

5. T. Wilson, ed., Confocal Microscopy �Academic, San Diego,Calif., 1990�.

6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.�McGraw-Hill, New York, 1996�.

7. M. V. Klein and T. E. Furtak, Optics �Wiley, New York, 1986�.8. E. R. Dowski and W. T. Cathey, “Extended depth of field

through wave-front coding,” Appl. Opt. 34, 1859–1866 �1994�.


volume holographic imaging in transmission...

Documents