Gabor first invented holography in 1947 to correct thespherical aberration of the electronic microscope.1–3
The main essence of holography is to record a wavefront on photographic film and then regenerate thewave field from the recording. Many researchersfollowed Gabor as the pioneers of this significantfield.4–8 Later on, as a consequence of the develop-ment of synthetic aperture radar �SAR�, anotherwave of innovation arrived in holography: the qual-ity of recorded images was significantly improved,and the field expanded to include broad-spectrumrecording, volume holography, holographic stereo-grams, and multiplex holograms.9–13
In this paper we present a novel technique for re-constructing a three-dimensional �3-D� image with aspatially partially incoherent broad spectral source.The recording may be done with light that is close tobeing spatially coherent and a broad-spectrumsource. The reconstruction is done with light that isclose to being spatially incoherent and a broad-spectrum source. The 3-D imaging is obtained notby the recording of the phase �and thus it is not truly3-D information� but by the creating of a stereoscopiceffect between two or more recorded images. Thusin this implementation the hologram is not used anymore for recording wave fronts. The incoherence of
the reconstructed source is adapted to the requiredangular separation, which allows the observer toview a different image in each one of his eyes. Acolor coding of the stereoscopic images will furtherincrease the separation between the two images inthe eyes of the observer. Some theoretical investi-gations of the idea are presented in Sections 2 and 3,and the experimental results are illustrated in Sec-tion 4. The paper is concluded in Section 5.
2. Theoretical Discussion
A. Jointly Partially Coherent Sources
Let us assume two light sources that are jointly par-tially coherent. If a hologram of them is recorded,the obtained pattern will be the following,
I� x� � �u� x��2 � �R� x��2
� 2 Re�R*� x��u� x, t�exp�i�0 t���, (1)
where �. . .� stands for the ensemble average, Re�. . .�is the real part taking operation, and �0 is the tem-poral radial frequency of the reference source R.Choosing R to be a tilted plane wave and assumingtemporal–spatial variable separability for u,
R� x� � exp��2ix0�
u� x, t� � u� x�u�t� � u� x�exp��i��t� , (2)
results in
I� x� � �u� x��2 � 1 � 2u� x�� cos�20 x � ��,
� exp�i�� � �u�t�exp�i�0 t��
�1T �
0
T
exp��i��t� exp�i�0 t�dt, (3)
The authors are with the Department of Electrical Engineeringand Computer Science, University of Michigan, Ann Arbor, Mich-igan 48109-2122. Z. Zalevsky’s e-mail address is [email protected].
Received 12 February 2001; revised manuscript received 8 Au-gust 2001.
where T is the averaging time period. For instance,for totally jointly coherent sources one obtains
� � 1, � � 0, (4)
for totally incoherent sources one has
� � 0, (5)
and for partial joint coherence one has
0 � � � 1. (6)
B. Stereoscopic Imaging
Conventional holography tends to reconstruct thewave front of the recorded object, and during thereconstruction it is as if the object is actually there.However, to see the third dimension, reconstructionof the wave front is not necessary. The 3-D imagingmay be obtained for spatially incoherent illuminationas well. For instance, daylight is spatially incoher-ent, and yet one may see the third dimension. Thisis achieved because of the stereoscopic effect. Theposition of human eyes is such that they see objectswith a slight angular difference that allows the esti-mation of the third dimension. For instance, withtwo cameras as depicted in Fig. 1, range estimation tothe object may be obtained,
x1 � ��X2
� X0� fR
,
x2 � � ��X2
� X0� fR
, (7)
where f is the focal length of the lenses, R is the dis-tance to the object’s point, x1 and x2 are the coordinatesin the detector’s plane in which the object was de-tected, �X is the separation distance between the twocameras, and X0 is the lateral position of the object.Equations �7� assumed that the object is far enough sothat the distance between the imaging lens and thedetection plane is approximately equal to the focallength. Solving the equations yields
R ��Xf
x1 � x2, (8)
X0 �x1 � x2
x2 � x1��X
2 � . (9)
C. Three-Dimensional Sensing
To obtain 3-D sensing a sequence of double recordingsis required. Each double recording will contain thefollowing information,
I� x� � �u1� x��2 � �u2� x��2 � 2
� 2��u1� x�cos�21 x � �1�
� u2� x�cos�22 x � �2� , (10)
where 1 and 2 are the spatial directions correspond-ing to the angular difference between the two eyes ofthe observer. u1 and u2 are the two images to beseen by the left and right eyes to obtain the stereo-scopic effect. �1 � 2��2 is the diffraction order atwhich the reconstruction is to be obtained.
It is obvious that those spatial frequencies are to beadapted to the observation distance. Assuming thatthe distance is R, then
��1 � 2� R � �s, (11)
where �s is the distance between the eyes ��7 cm�and � is the wavelength.
In the reconstruction the hologram is to be illumi-nated with a spatially incoherent light, and thus thereconstruction in the first diffraction order will yield�u1�x��2 for the left eye and �u2�x��2 for the right eye.
The procedure described in here realizes a 3-D im-age from a certain observation direction. To obtaina wide angular observation region, in which the 3-Dreconstruction is obtained, one may repeat the above-described recording process for several angular as-pects.
D. Experimental Considerations
The technique previously suggested is to be applied toa broad-spectrum white source. To obtain the re-cording over a proper transversal and longitudinaldistance �a volume�, one needs to preserve sufficienttemporal and spatial coherence lengths. Otherwise,interference fringes will not appear. Because we aredealing with a broad-spectrum source, its temporalcoherent length is short, and thus it will yield a smalltransversal interference pattern. To solve this prob-lem, the grating interferometer setup seen in Fig. 2
may be used to obtain the recording. Here owing tothe gratings each wavelength is spatially separated,and a wavelength interferes only with itself; thusvery small temporal coherence length is required.
E. Multiple Exposure Technique
The usage of the experimental setup described in Fig.2 may allow performing the recording in spatiallyincoherent illumination as well. Let us assume thatwe have a spatially incoherent source. It is obviousthat the spatial incoherence is obtained owing to longtemporal averaging. If, for instance, the source is tobe observed over a short integration period, its coher-ence is improved. This happens because for shortaveraging the phase fluctuations do not manage tozero the third term of Eq. �1� �� � 0�. Because theholographic reconstruction should be in the first dif-fraction order, it is essential that � � 0. This may beensured by separating the recording into a set ofshort exposures instead of using a one long exposure.After the recording one will have
I� x� � �n
��un� x��2 � 1
� 2���n
un� x�cos�20 x � �n�� . (12)
The main problem with Eq. �12� is that becauseeach exposure results in a different phase factor �nthe summation of the last term will still be averagedto zero in the reconstruction stage. Thus to elimi-nate this phase factor, the various exposures must besummed incoherently. To do so, we will encode eachexposure with a different wavelength. The coding isto be done by addition of a spatial mask in plane 1 ofFig. 2. This mask will transmit and block part of thewavelengths. Each exposure will have its ownmask. The masks are constructed such that theyare orthogonal; i.e., no identical wavelength is ap-pearing in two masks.14 Note that the spectral re-sponse of the eye makes a color projection over threespectral response curves.15 Thus for properly de-signed spatial masks, despite the color coding of eachexposure, the spectral projection over the three spec-tral response curves of the eye will be equal for all theexposures. This means that the human eye will notfeel the color difference between the various expo-sures.
Thus, after recording the transmission function ofEq. �12� and illuminating the hologram with a broadsource, we obtain, in the first diffraction order,
Irec� x� � 2� �n
�un� x��2 (13)
because the different wavelengths are summed byintensities owing to their mutual incoherence. Notethat Irec is the reconstructed intensity in the firstdiffraction order. The technique suggested here isan alternative approach yielding the required sum ofintensities of Eq. �13� even when a spatially coherentillumination is used for the reconstruction.
3. Mathematical Investigation of the PartialIncoherence
Let us assume that one has the configuration de-scribed in Fig. 3�a�. We will denote by h the point-spread function of plane a, which also equals theinverse Fourier transform of the aperture of the firstlens �everything is computed in �f units, where � isthe wavelength and f is the focal length�. The finiteaperture of the imaging lens in Fig. 3�a� simulates thepartial incoherence of the illumination. The mutualintensity at that plane equals
J12� x1, x2� � �u2� x1�u2*� x2��
� �� �u1� x� 1�u1*� x� 2��h� x1 � x� 1�
� h*� x2 � x� 2�dx� 1dx� 2, (14)
which, assuming that the input illumination is anincoherent plane wave, yields
�u1� x� 1�u1*� x� 2�� � Iin�� x� 1 � x� 2�,
J12� x1, x2� � �� Iin�� x� 1 � x� 2�h� x1 � x� 1�
� h*� x2 � x� 2�dx� 1dx� 2,
� Iin � h� x1 � x� 1�h*� x2 � x� 1�dx� 1.
(15)
Passing this mutual intensity through a spatial dis-tribution of G�x� and a prism or a grating formulatedas exp�2ix�� yields
J1�2�� x1, x2� � G� x1�exp�2ix1��G*� x2�
� exp��2ix2��J12� x1, x2�. (16)
Fig. 3. Schematic sketch used for mathematical investigation.
Note that H is the aperture of the first lens in Fig.3�a�. Thus from Eq. �21� one may see that the outputdistribution is the convolution between the apertureof the lens and the intensity of the recorded holo-graphic information.
The partial coherence of the source may be ex-pressed also in another manner. Instead of using animaging module as described in Fig. 3�a� one may usean incoherent source having a size of �x and a Fou-rier transformer �see Fig. 3�b� . For this case themutual intensity function of the output plane is
where u�x, t� is the size of the incoherent source,
�u�x0, t�u*�x0�, t�� � �rect� x0��x�
�x �2
�� x0 � x0��
�rect� x0��x�
�x2 �� x0 � x0��, (23)
and the final result is
J12� x, x�� �sinc��x� x � x��
�x. (24)
For this case using similar mathematics one obtains
Io��� � � rect� z��x�
�x 2
� g�� � � � z��2dz. (25)
Thus when an incoherent illumination is used thereconstructed image is blurred. The aim in our caseis to use such a partial incoherence so that the blur-ring will be small, and yet the depth feeling will beenhanced owing to the stereoscopic effect. Obvi-ously, the resolution is related to the angular direc-tions of the propagating light. For instance, let usassume that the blurring is such that the angulardirection is restricted so that the information reachesonly one eye and not both eyes �the hologram’s recon-struction is seen in a narrow angular observationrange�. Then the spatial resolution is almost un-damaged �the light covers the entire aperture of oneeye�, but the 3-D feeling is low �it is coming from onlyone eye�. Using the suggested technique, the 3-Dfeeling is reinforced because the stereoscopic effectwill redirect the angular information to both eyes.
Assuming now that one wishes to reconstruct thehologram from a distance of R, then the maximal
Fig. 4. Mach–Zhender interferometer used for the experimentalrecording.
angular spread allowed to avoid overlapping of thetwo stereoscopic images should be
�� � �s�R, (26)
where �s is the eye’s separation distance. For R �50 cm and �s � 7 cm we have �� � 8°.
To have some numerical feeling, the angularspread of information may be translated to the di-
mensions of the finest detail that may be seen, ac-cording to
�x � 2.44�f# �2.44�
N.A.�
2.44�0.5 �m�
4°���180� 18.3 �m.
(27)
Here 4° is the angular spread.To obtain a separation between the zero and the
first order of the hologram, it is obvious that 1 mustbe higher than 1��x. To obtain the required angularseparation between the two eyes one should have
��2 � 1� �8°
180. (28)
4. Experimental Results
The experimental recording of the hologram was per-formed with the Mach–Zehnder interferometer thatwas constructed as depicted in Fig. 4. The recordingwas done with a 10 mW red He–Ne laser. The re-construction distance was set to be 100 cm, and thus,according to the considerations of Sections 2 and 3,the angle between the reference beams of each expo-
Fig. 5. Reconstruction configuration.
Fig. 6. Experimental results. �a� The first stereoscopic reconstruction of target #1. �b� The second stereoscopic reconstruction of target#1. �c� The first stereoscopic reconstruction of target #2. �d� The second stereoscopic reconstruction of target #2.
sure was set to 4 deg. The target itself may also berotated between exposures to reinforce the stereo-scopic effect. For instance, we have rotated it by 4deg. An attenuator of 5.3% transmission was placedin front of the laser. The exposure time was set to 50ms. Two exposures recorded the two stereoscopicimages.
Schematically, the reconstruction of the hologramis to be performed according to Fig. 5. In practice, arotating diffuser was attached to the collimating lensof the laser beam, and the hologram was placed rightbehind the diffuser. The beam diameter just infront of the diffuser was �2 cm. This was the mostincoherent state of the beam that could practically beachieved. This incoherence did not disturb the re-construction of the holographic recording.
The obtained results may be seen in Fig. 6. Fig-ures 6�a� and 6�b� present the first set of recon-structed stereoscopic images, and Figs. 6�c� and 6�d�show another set. In both cases one may see theparallax effect between the two reconstructed imag-es; this parallax allows the 3-D vision for the ob-server. Figures 6�a� and 6�b� display a target havingtwo horizontal and one vertical bars. The separa-tion distance in the depth dimension between thebars was approximately 5 cm. Figures 6�c� and 6�d�display a resolution target. This is almost a planetarget. As one may see, the spatial blurring is notsevere.
An illumination of the holograms with white lightresulted with sufficient quality of reconstruction aswell. The reconstruction may be seen in Fig. 7. In-deed, a certain smear appears in the image owing tothe use of a broad-spectrum source while the record-ing was done with only a red light. To avoid that,one must use the setup in Fig. 2 in the recordingstage.
When the setup in Fig. 2 is used for the recordingof white-light holograms, coding white filters maybe inserted during the recording, as described inSubsection 2.E. The orthogonal coding for the twostereoscopic images may allow an improved discrim-ination between them if the observer wears a pair ofcompatible white glasses in which the filter for eacheye has a different orthogonal encoding. Note thatin this case the glasses will be white and no chromaticdifference will be observed between the two eyes.
5. Conclusions
This paper has demonstrated a novel approach forthe reconstruction of holograms with a spatially par-tially incoherent broad-spectrum source. The 3-Dimaging of the object is obtained not by recording thephase information �and thus it is not truly 3-D infor-mation� but by using a stereoscopic effect. Each re-cording contains two stereoscopic images that aremodulated with adequate spatial frequencies thatwill redirect the reconstructed pictures into the cor-responding eye of the observer. By recording a set ofstereoscopic images a wider angular aspect range ofobservation may become available. Experimentalresults validate the suggested approach.
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Fig. 7. Reconstruction of target #2 for a white-light illumination.
