Rainbow holography of 3-D diffuse objects with HOE/MEHL

Guo-Guang Mu, Zhao-Qi Wang, and Ke-Mei Wang

A new technique of one-step rainbow holography with a multiexposure holographic lens (MEHL) is described.The synthesization of the slitlike function in the object wave can be performed with the specially fabricatedholographic lens. The MEHL can also be applied to the oriented speckle encoding of images in white-lightoptical image processing. Successful experimental results are presented.

I. Introduction

Several methods of the one-step rainbow holograph-ic process based on synthesization of the slitlike func-tion in the object wave have been proposed. Grover etal.1 demonstrated that a slitlike function can be syn-thesized in the focal plane of the imaging lens by trans-lating 2-D transparent objects in the transverse direc-tion during recording. Later, Shan et al. 2 extendedthe technique to record 3-D diffuse objects. Morerecently, Beauregard and Lessard3 put forward anoth-er technique of rainbow holography with no slit for 3-Ddiffuse stationary objects. Instead of translating theobjects, a small transverse motion is imparted to theimaging lens to produce the desired sinc function mod-ulation in the object wave.

In this paper, we propose a new method of one-steprainbow holography with a special multiexposure holo-graphic lens (MEHL). This technique is character-ized by the following:

(1) The process of holographic recording is withoutany slit and the translation of the object or the imaginglens is not needed.

(2) It can be applied to 3-D diffuse objects as well as2-D objects; the 2-D reconstructed image has the sin-gle-eye depth perception.

(3) The MEHL can also be used in other applica-tions such as in the oriented speckle encoding in white-light optical image processing.

II. Multiexposure Holographic Lens Formation

A special holographic lens, MEHL, should be madefirst and from which the slitlike function can be syn-thesized in the object wave during the fabrication ofrainbow holograms. The configuration for making theMEHL is shown schematically in Fig. 1. A laser beamis split into two beams. The photosensitive plate isplaced at the proper position. To get a convergingwavefront, an imaging lens L is introduced. TheMEHL is fabricated using the oblique plane referencewave. The plate is displaced along the y axis with acertain distance e for each exposure. The numericalaperture of the MEHL is equal to that of the imaginglens. If the MEHL is used as the imaging lens in acoherent optical system and an impulse function (xo

xoyo - y) is in the object plane, the complex lightamplitude distribution in the image plane is given bythe Fresnel approximation:

N-1

Ui(Xiyi;x0,y0) - C E f.ff O(X - xo'yO - o)n=O

X exp{ z [(X - 0)2 + (y - o)2]}

X exp{ Xf [x + ( - ne)21} exp{3Z

X [(x, -x1 )2 + (y - y1)i]JdxodYodxldyI

p[-i (X + [ 2+) ]1XZO X~i L I

(1)X = xi+ xo;Yi +z Y - ne

The authors are with Nankai University, Institute of ModernOptics, Tianjin, China.

Received 13 July 1987.0003-6935/88/020321-03$02.00/0.©O 1988 Optical Society of America.

where (xo,yo), (xl,yi), and (xi,yi) are coordinates in theobject, lens, and image planes, respectively; N is thenumber of exposure times; zo is the object distance; ziis the image distance; and C is the appropriate complexconstant.

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7Z0 so- i_ - | - Zf

Yl YhHP

HP

K3

Fig. 1. Optical configuration for making a multiexposure holo-

graphic lens (MEHL): BS, beam splitter; Ml, M2 , and M3, mirrors;BE, and BE2 , beam enlargers; Ll and L2 , lenses; L, imaging lens; HP,

holographic recording plate.

Ill. Rainbow Holographic Process with the MEHL

A schematic of the rainbow holographic process withthe MEHL is shown in Fig. 2. A 3-D diffuse objectU(xo,yozo) is illuminated by a coherent light and im-aged through the MEHL on a photosensitive plate.The complex light amplitude distribution in the Fouri-er plane of the MEHL is given by

Uf(xf,yf) = C>3 fE f U(xoyo) exp{-z [(x1 -

+ (y - yo)2)} exp{ -i [Xl + (y - n)2]}

X eXp{Xf [(X - x1)2 + (yf - yl)2]}dxdydxdy

= C eXp[Xf (f + y)(- )]

Nx 1 (i2rzoyfne) (2)

Fig. 2. Optical setup for the fabrication of a rainbow hologram withthe MEHL: BS, beam splitter; M, mirror; BE1 and BE2 , beam

enlargers; L, lens; HP, holographic recording plate.

AYf > fVmax.

Substituting Eqs. (4) and (5) into Eq. (6), we havefollowing:

2XfeSD.

(6)

the

(7)

In the rainbow holographic recording plane, locatedat a distance zfh from the focal plane, the complex lightdistribution coming from the diffuse object can beobtained again by the Fresnel approximation:

Uh(xhyh) = C fs Uf(xfyf) expI± i [(Xh -Xf)~XZfh

+ (Yh - yf)2]}dxfdYf. (8)

Together with a reference wave of unit complex ampli-tude Ur(XhYh), the intensity on the holographic re-cording plane is given by

I(Xhlyh) = lUh(Xhyh)12 + IUr(XhYh)12 + Uh(xh,yh)Ur(xh,yh)

where (xf,yf) are the coordinates in the back focalplane, and Fxy denotes the Fourier transform. Wenote that the summation in the above equation repre-sents a modulation function that is characterized by asequence of narrow periodic pulses. The width of thepulse is approximately

W 2Xf2 (3)Nezo

The distance between two pulses can be determined by

AYf = Ef . (4)

We know that the highest spatial frequency Vmax of adiffuse object in the back focal plane of the lens is

Dmax - D z (5)

where D is the lens aperture. Thus to get only thecentral order pulse function in the back focal plane, thedisplacement parameter e should satisfy the condition

+ Uh(XhYh)Ur(XhYh)- (9)

On reconstruction with the conjugate beam of theoriginal reference wave, an interesting diffraction wavethat is proportional to the fourth term in Eq. (9) will betransmitted. The reconstructed complex light distri-bution in the plane at the distance of Zfh from thehologram is given by

Rfh(xfyf) = C SIlUr(Xhyh) 2h(Xh,Yh) exp{ i [(Xh xd2

+ (Yh - yf)2]}dxhdyh

(10)

which is the conjugate light field of the original objectfiltered by the slitlike function of the central orderpulse. When we view through the slit image, a pseu-doscopic image of the object will be observed. Need-less to say, a rainbow holographic reconstruction willbe obtained under white-light illumination.

322 APPLIED OPTICS / Vol. 27, No. 2 / 15 January 1988

I

= C' Uf (xf'yd)

Fig. 3. Black-and-white photograph of a reconstructed virtual im-age.

(a) (b)Fig. 4. Retrieving images extracted from the encoded specklegramby spatially filtering (a) the longitudinal set of Fourier spectra and

(b) the horizontal set of Fourier spectra.

IV. Image Encoding with the MEHL in White-LightImage Processing

Yu and Ruterbusch have proposed a technique thatutilizes coherent speckles modulated by a slit to en-code color images onto black-and-white film for colorimage retrieval with a white-light processor.4 Thistechnique can be improved by using the MEHL. Theimage encoding process with the MEHL is as follows:Two diffuse objects illuminated by coherent light aresequentially imaged onto a photographic plate by theMEHL. With respect to two recordings, the photo-sensitive plate should first be oriented in one directionand then rotated by 90°. This would give us a spatial-ly encoded transparency in which two encoded imageshave speckles elongated in orthogonal directions [seeEq. (1)]. To retrieve original images, we insert theencoded specklegram in the input plane of a white-light processor. The corresponding Fourier spectrawill be distributed in confined directions perpendicu-lar to these elongated speckles. By spatially filteringeach set of Fourier spectra, the original images can beretrieved in the output plane.

V. Experimental Results

In making the MEHL, a He-Ne laser is used. Theaperture of the imaging lens is 30 mm and the focaldistance is 300 mm. The displacement parameter of eis chosen to be 10 gm, which satisfies the condition ofEq. (7). The number of exposures is 7. Thus thewidth of the slitlike pulse, according to Eq. (3), is -3mm. The fabricated MEHL is then used for rainbowholography of the 3-D diffuse object described in Sec.III. The black-and-white photograph of the rainbowvirtual image, reconstructed with a conjugate refer-ence beam of white light, is presented in Fig. 3.

For experimental demonstration of image encodingwith the MEHL described in Sec. IV, two images wereencoded onto one photographic film with orthogonalelongated speckles. Then the encoded specklegram isinserted in the white-light processor. Figures 4(a) and(b) show the retrieving original images by spatial filter-ing each set of Fourier spectra.

VI. Conclusions

We have fabricated a special holographic opticalelement, MEHL, and successfully used it for rainbowholography with no slit, and for image encoding inwhite-light image processing. Although we only dem-onstrated the images stored with the MEHL and theextraction of each image from the multiplex speckle-gram in a white-light processor, the MEHL can cer-tainly be applied to various white-light image process-ing applications such as archival storage of color films4and white-light stereoprojection through Fourier spec-trum polarization.5

This paper has been accepted by the 1987 Interna-tional Conference on Lasers, China.

References1. C. P. Grover, R. A. Lessard, and R. Tremblay, "Lensless One-Step

Rainbow Holography Using a Synthesized Masking Slit," Appl.Opt. 22, 3300 (1983).

2. Q-Z. Shan, Q-C. Chen, and H. Chen, "One-Step Rainbow Holog-raphy of Diffuse 3-D Objects," Appl. Opt. 22, 3902 (1983).

3. A. Beauregard and R. A. Lessard, "Rainbow Holography of 3-DStationary Objects with No Slit," Appl. Opt. 23, 3095 (1984).

4. F. T. S. Yu and P. H. Ruterbusch, "Color-Image Retrieval fromCoherent Speckles by White-Light Processing," Appl. Opt. 21,3300 (1982).

5. G-G. Mu, Z-Q. Wang, F-X. Wu, and N-Y. Zang, "White-LightStereoprojection with Modulated Speckle Encoding," Acta Phys.Sin. 31, 1547 (1982).

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