Electronic heterodyne recording and processing of optical
holograms using phase modulated reference waves
A. J. Decker, Yoh-Han Pao, and P. C. Claspy
The use of a phase-modulated reference wave for the electronic heterodyne recording and processing of a ho-
logram is described. Heterodyne recording is used to eliminate the self-interference terms of a hologram
and to create a Leith-Upatnieks hologram with coaxial object and reference waves. Phase modulation is also
shown to be the foundation of a multiple-view hologram system. When combined with hologram scale
transformations, heterodyne recording is the key to general optical processing. Spatial filtering is treated
as an example.
1. Introduction
A system that records optical holograms electroni-cally using a phase modulated reference wave has been
developed. This system is potentially useful in optical
processing.' In addition to overcoming the difficultobstacles of low resolution holography, the system isconvenient for applying the phase mask often requiredin optical processing.
The feasibility of transmitting a hologram via TV was
first proven by Enloe et al. 2 The key problem was thelow resolution of the TV camera tube. Although theuseful information in an object wave is coded in a slow
variation of magnitude and phase, holography is truelyconvenient only if the different diffraction orders in the
reconstruction can be separated spatially. This sepa-ration requires an off-axis reference beam,3 and thespatial resolution required is increased by the maximum
spatial frequency of the reference beam. It was recog-nized, however, that these requirements are unnecessary
if angle modulation is performed on the reference beam.
Then, the signals generated from the cross-interferenceterms and the self-interference terms of the hologramcan be placed in different bands and the self-interfer-ence terms eliminated by bandpass filtering. Fur-thermore, the hologram assumes the form of a hologram
recorded with an off-axis reference wave even though
the original object and reference waves are coaxial. The
potential usefulness of angle modulation was proven by
the heterodyne scanner method 4 5 in which a reference
The authors are with Case Western Reserve University, Depart-
ment of Electrical Engineering & Applied Physics, Cleveland, Ohio
wave is frequency modulated by reflection from a vi-
brating mirror, is focused to a spot on a photodetector,and is scanned over the photodetector. A more con-
venient and versatile method is to phase -modulate thereference wave and to record the hologram with animage dissector. This method was mentioned byMacovski in his thesis.6
This report describes how phase modulation can beused to overcome the obstacles of low resolution ho-lography. We show how a hologram can be constructedthat does not have the self-interference terms but doeshave the phase of a hologram recorded with an off-axisreference wave. Phase modulation is shown to be thekey to multiple-view holography. A wide-angle holo-gram can be constructed from a series of narrow-angleholograms.
We show how a phase mask is easily applied to a ho-
logram. Since the paraxial approximation is applicable
to an electronically recorded hologram, the phase maskcan be combined with scale transformations to effectoptical processing. Spatial filtering is presented as anexample.
II. Holography with Phase Modulation
An arrangement for recording low resolution holo-
grams of semitransparent objects is shown in Fig. 1.
The hologram is recorded by an image dissector (or any
scanned detector with an instantaneous response), andthe reference wave is phase modulated by an elec-trooptic modulator.7 The object and reference wavesare written in real notation as
U0(X,Y) = O(x,y) cos[wt - O(xy),1
Ur(x,y) = s(x,y) cost - r(x,y) + a sincOHt].
The circular frequency of the light wave is denoted byco, and a phase-modulation term 6 sinwHt has been in-troduced in the phase of the reference wave. By per-
This time-varying pattern must be scanned by ascanning aperture having a finite area, and the signalmust be convolved with an impulse response functionof finite width. The finite durations of scan lines, fields,and frames must also be inserted. These details9 addlittle other than algebraic complexity and are handledas follows. An upper limit is specified for the spatialresolution of the detector. If Ar is the width of theimpulse response function, we specify that 1/(Ar) >>NfH, where N is the integer defining the highest har-monic of interest of the frequency fH of phase modula-tion. Upon scanning, each spatial-frequency pair (fx,fy)contributes a corresponding time-frequency compo-nent
f = Vxfx + Vyfy, (4)
In that case, each time-frequency component generatedby scanning corresponds to a unique, x-directed spa-tial-frequency component.
Suppose that the reference wave has a negligiblespatial bandwidth and that the maximum spatial fre-quency contributed by the object wave is fmx. Uponscanning, the first few terms of Eq. (3) generate signalsthat occupy the bands
[O, 2fmx Vx ]
[H fmxVx fH + fxVx
[2fH fmx Vx, 2fH + fmx Vx .Clearly, there is unqualified separation of the signalsif
fH > 3fmxVx. (6)
Equation (6) is determined by the fact that the self-interference term of Io in Eq. (3) contributes a maxi-mum frequency 2 Vxfmx. If the detector does not re-solve spatial frequencies exceeding Jinx, the requirementof Eq. (6) can be relaxed. The signals are then sepa-rated provided that
fH > 2fmxVx. (7)
The signal generated from I in Eq. (3) carries thehologram in either sideband of the carrier H. If thelower sideband is selected, the useful signal is givenby
The signal i(t) can be used to write a hologram on a
cathode-ray display, and that display can be photo-graphed. At unity magnification, the amplitudetransmittance of this hologram is given by
T(x,y) = IB + y2J1 (6)s(x,y)O(x,y)
*cos [(xy) - r(x,y) -OHX (9)
where IB is a bias level, and -y is a proportionality con-
stant. The phase wHx/Vx is equivalent to the phasethat would have resulted had an off-axis reference wave
been used with a propagation angle 0 given by
(10)sinO = [(fH)/(Vx)Ix,
where X is the wavelength of the light used to record and
reconstruct the hologram. The waves reconstructedfrom that hologram are shown schematically in Fig.
2.
Ill. Multiple-View Holography
In principle, it should be possible to replicate a higher
resolution hologram than can be recorded. In recording
a hologram electronically, a high density of resolutionelements is required. However, the hologram can be
replicated on a large display with a low element density
and can be reduced in size photographically. The way
to take advantage of the additional output resolutionis to transmit several narrow-angle views of an objectand to synthesize a single wide-angle hologram fromthese views.
Each view corresponds to a different direction of the
axis of the reference beam. The axis of the referencebeam can be moved by tilting beam splitter 2 in Fig. 1.
A different harmonic of WH is selected for each view.
The modulation index 5 is chosen to emphasize that
harmonic. If the reference wave is collimated, a signal
corresponding to a particular view is given by
IN = ON(XY) CoS[ON(XY) + kx sinO - NWHt, (11)
where the substitution x = Vxt has not been made ex-
plicitly and where k is the wavenumber of the light used
to record the hologram. The subscript N emphasizesthat only the object wave about a particular viewing
direction leads to a resolvable interference pattern.
The angle ON defines the direction of the axis of the
reference beam for view N.The presence of kx sinON cancels an identical offset
in the phase qN(x,y) and permits a low resolution re-cording of the corresponding view. However, thepresence of NWHX/Vx = NWHt makes it possible toreinsert that offset. The result is the same as if the axis
of the reference beam had never been moved.Since each view covers the recordable spatial band-
pass of the detector, the choice of fH according to Eq.(7) insures separation of the views. Replicating themultiple-view hologram consists of making a linear re-cording where the amplitude transmittance of that re-cording is given by
T(X,Y) = IB + 'Y E IN. (12)N
If the final recording film does not have enough dynamic
range, the holograms can be recorded on separate pieces
of film that are assembled in a film strip. That strip can
be moved in front of a replication beam so that eachview is presented about 10 times/sec.
IV. Optical Processing
If an arbitrary phase modulation a(t) is added tobsinWHt, the first three terms of Eq. (3) become
Upon scanning, time t is associated with position(x,y); hence, an arbitrary phase mask can be applied to
the hologram. However, the presence of a(t) in thephase increases the bandwidth associated with eachsignal. This increase in bandwidth can be compensatedby reducing the scan velocity Vx. The required re-strictions are estimated as follows.
The phase is to be controllable within the range [- r< a(t) < 7r]. The maximum frequency at which thephase is likely to be varied is Vxf mx. Hence, theworst-case time variation of a(t) is given by
a(t) = r sin(2irVxfmxx).
With a modulation index -r, at least the first four har-monics of Vxfmx must be expected. Hence, a sidebandmust cover at least the range 5VX.jf. If the full reso-lution of the detector is to be used, the scan velocityshould be reduced by at least a factor of 5.
The phase mask can be combined with some addi-tional operations for convenient optical processing.The low-resolution nature of electronic recording as-sures the validity of the paraxial approximation. Then,the axial distances and sizes of the real and virtual im-ages can be varied by changing the scale of the hologramand by varying the curvatures of the reference and re-construction waves.1 0 11
Suppose that the input to the system is a 2-D trans-parency having an amplitude transmittance b(xi,yi).Let the reference wave be a spherical wave originatingfrom an- axially located point source at a distance db
from the hologram. Let the reconstruction wave orig-inate at a distance dr from the hologram. Let thewavenumbers of the sources used for recording and re-construction be k and kB, respectively. A scale trans-
formation is performed on the transmittance T(x,y) ofthe hologram such that the final copy of the hologramhas the transmittance T(x/a,y/a), where a is the scaleparameter. The transparency is to be located at a dis-tance S from the recording plane of the hologram. Thevirtual and real images are to be located at distances Svand SR from the final copy of the hologram. The signconvention is that real images appear at positive dis-tances from the hologram, and virtual images appear at
Reconstruction Process: Onlthe Real-Image Output is hown
Fig. 3. Filtered output in the real image with a virtual image atinfinity.
negative distances from the hologram.By using a paraxial approximation of one of the dif-
fraction integrals, we can show that the quantities de-fined above are connected by the equations
db= 2SVSR (15)
SR + SV((2)1/2 ( k )1/2 SVSR )1/2 ( S 1/2t$}~~~~~~~~ thg (S -) SR (16)
A negative value for db or d implies that the associatedwave converges to a focus in back of the hologram. Thevirtual and real images experience magnifications Mvand mR, respectively. These magnifications are givenby
k Svv=kE Sa'
k SR (17)MR =--.
k SaNote that S must be negative and SR must be pos-
itive for the magnification to be positive.A convenient lensless spatial-filtering operation is
now possible. The transfer function of the filter is givenby
H(x,y) = expUa(x,y)], (18)
where t in a(t) has been replaced by the pair (x,y) sinceposition is uniquely associated with time. A problemin performing lensless transforms is to eliminatespherical phase factors at both the object and the outputplanes. Spherical phase factors can be eliminated bychoosing the proper object illumination, and the scale
transformations increase the possible number ofchoices. In the case of the real image, the proper illu-mination is the spherical wave.
E(xi,yi) = exp [ - 1 (1 S MR2) (X12 + yi2)J. (19)
If the output is sought from the virtual image, theproper illumination is the wave
E(xi,yi) = exp [- - (1+ -S Mv 2 ) (xi2+ yi 2)].L2S\ k SV /
(20)
If the proper illumination is chosen for the real image,we can show that the output is proportional to
fHkBSR
IR= Rb* YR ) h*(aXRayR),MR MR/
where 0 denotes the convolution operation and whereh(XR,yR), the impulse response function, is obtainedfrom the Fourier transformation of H(x,y) as given byEq. (18).
As an example, let the virtual-image output be atinfinity (Sv - ). Select SR = S and kB = k. FromEq. (15), db =-2S, and the reconstruction wave con-verges in back of the output plane. From Eq. (16), a =21/2(1 - S/dr) 1 12 . Note that MR = 1/a. As determinedfrom Eq. (19), choosing a < 1 will make it possible to usea diverging spherical wave for illumination. Select dr= 2S so that a = 0.707. Then
E(x,,yi) = exp [S (Xi2 + yi2)
and the input transparency is illuminated by a divergingspherical wave originating at a distance 2S in front ofthe hologram recording plane. These recording andreconstruction operations are summarized in Fig. 3.
V. Conclusions
In recording a hologram electronically, electroopticphase modulation can overcome the restrictions oflow-resolution recording. The self-interference termscan be eliminated, and a Leith-Upatnieks hologram canbe created from a hologram recorded with coaxial objectand reference waves. The fact that electronically re-corded holograms are confined to narrow-angle fieldsof view is not a restriction provided that the multiple-view feature of phase modulation is employed.
These features are not only interesting but also po-tentially useful. Phase modulation is the key to generaloptical processing. A phase mask is easily applied, andscale transformations can be used to control the locationand scale of the output. The system described shouldbe readily interfaceable with other data processingsystems. Hologram mappings' more complex than thesimple scale transformation should be possible.
This work has been supported by a grant from NASALewis Research Center. A. J. Decker is on leave of ab-sence from NASA Lewis Research Center.
