JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Single-Sideband Holography*

OLOF BRYNGDAHL AND ADOLF LOHMANN

IBM Research Laboratory, San Jose, California 95114(Received 8 November 1967)

In making in-line holograms of amplitude objects with a strong background, the single-sideband techniquecan be used to improve the quality of the reconstruction. The main advantage of this method is the sup-pression of the twin image. Modifications of the same technique are also presented for making hologramsof complex objects and objects with a weak background.INDEX HEADINGS: Holography; Image formation; Interference.

THE original in-line holography of Gabor' sufferedTsomewhat from the presence of the unsharp twinimage. It is now customary to avoid the overlappingtwin image by means of the off-line technique developedby Leith and Upatnieks.2 Historically, between thesetwo approaches stands the single-sideband technique'which was suggested as a means for eliminating thetwin image from holograms made with an in-line refer-ence wave. The single-sideband technique is performedby spatial filtering with knife edges in planes where thespatial-frequency spectra of the object and the hologramare displayed. Somewhere between the source andhologram, a lens has to be introduced to obtain aFraunhofer diffraction pattern of the object. The pro-cedure then consists of removing half of the spatial-frequency spectrum from the transmitted signal in re-cording the hologram and also a corresponding half inthe reconstruction process. In this way, each pointscatterer in the object will cause only one half of azone-plate pattern in the hologram plane. Then, inreconstruction, either the real or the virtual image canbe filtered out with a half-plane mask, since their cor-responding frequency components will be located inopposite sides of the Fraunhofer diffraction plane.

The procedure is applicable for real objects with astrong background because, in this case, the frequencyspectra are symmetrical and, therefore, redundant. Wewill also show how the single-sideband technique can bemodified so that holograms of other objects, such astransparent lines on an opaque background or phaseobjects, can be made.

In the first part of this paper, we will present thetheory of single-sideband holography, followed by thedescription of some verifying experiments.

THEORY OF SINGLE-SIDEBAND HOLOGRAPHY

First, we will present the theory of coherent single-sideband image formation which then will be extendedto the case where ordinary one-step image formation isreplaced by two-step holographic image formation.

* Paper presented at Detroit meeting of the Optical Society,October, 1967, [J. Opt. Soc. Am. 57, 1419A (1967)].

1 D. Gabor, Nature 161, 777 (1948).2E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123

(1962).3 A. Lohmann, Opt. Acta 3, 97 (1956).

We assume the complex amplitude u0(x0 ,y0 ) of theobject to be real and to have a strong background

1to(Xo,yo) uo,*(X0,y0); i!o(P,4) = f!0*(.-V r-J1.) (1)

110(X0,y0))1 + Ao(Xoyo); Au02<<1;

uio(v,,u) = 3(v,,) + Afto(v,,) . (2)

xo, yo are the coordinates of the object, x, y of the image,and v, u of the frequency plane. A Fourier transforma-tion is indicated by a tilde symbol and a complex con-jugate by an asterisk; 6 is the Dirac-delta function.

We assume the optical system of Fig. 1. This systemis also used later for recording the single-sidebandhologram. The object is illuminated by a plane wave,and the image is formed by a telecentric system. Othersetups would be acceptable too, but then some unin-teresting phase factors would complicate the analysis.In the Fraunhofer diffraction plane, where the spatial-frequency spectrum ilo(v,,) of the object is displayed,a single-sideband filter, a knife edge, eliminates half ofthe spectrum (v<0); hence, the complex amplitude inthe image plane will be

+A +A1u/(X) dvf go(vP.) exp[2iri(xv+yA)]dA

4A +A

- 2-J XA -o(V) exp[27ri(xv+yA)]dvd,4, (3)o A

where we have used the fact foib f(v)3(v)dv= 2f(°). Thesize, A, of the aperture is expected to be sufficientlylarge so that it does not influence the process of imageformation.

In general, the single-sideband image utt might bequite unlike the object ito, but assuming that the objecthas the properties (1) and (2), the image irradiance

It2, as shown in the following derivation is propor-tional to the object.

+A +A

Iul 42 +2 f Aftoexp(.. )dvdA0 -A

+,.W|i~oexp(... )dvd,u4 (4x)

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SINGLE-SIDEBAND HOLOGRAPHY

jf } t =+aj± A o*(vu) exp[-2ri(xv+ypi)]dvdA

+A +A

=J J AR0(-V,-A)f -A

X exp[- 2ri(xv+yiu)]dvdAu

=JJAulo(P'J)AX exp[+ 2ri(xv+ygu)]dvd . (5)

In processing the formulae (5), we made use of thereality symmetry Auio*(v,,)=AFio(-v,-p) as stated inEq. (1), and then changed variables (v,IA) -v-

Equation (5) tells us that the conjugate complex term inEq. (4) is the sideband which is missing in the imageamplitude ul, but which is regenerated by the modulus-square operation in calculating the image irradiance

I U| 2. In other words, we have shown thatI ui(X,y) I 2 = 4u0

2(X0,y o). (6)

Now we combine the single-sideband theory withholography, again assuming a real object (1) with astrong background (2). The first step is, again, theelimination of half of the spectrum (v<0). Next wecalculate the complex amplitude uH(x,y) in the plane,where we want to record the hologram. This plane mightbe at a distance z from the image plane,

+A +A

1UH(XY) = I LA UO(V,,)OA

Xexp2iri(xv+y,4+ [1_-X2 (v2 +b 2)] 1z/X))dvd,4. (7)

This is indeed the complex amplitude at plane z,because it satisfies the wave equation and becomes UI

when z goes to zero. On the hologram, the irradiance

I I2 is recorded,

UH12= - exp(27riz/X)+ Aiioexp( .. )dvdiut2

4+ exp(-27riz/X)ff.. dvdAu

+2 exp(+2iriz/X) f.. dvdMI (8)

By performing the steps of Eq. (5), we can show that..... * dvdA}* contains only the left sideband (v<0),whereas JJ * -*- dvdji contains the right sideband (v> 0).

The photographic development of the hologram hasto be performed with gamma of -2 so that the ampli-tude transmittance is proportional to the exposure (8).The hologram is inserted into plane z and illuminatedby a plane wave exp(2iriz/X). A lens performs a Fouriertransform, yielding a display of the spatial frequency

spectrum of I u 12. A knife edge removes the left side-band (v<0), which means the elimination of the lastterm in Eq. (8). Virtually, this is the same as if thecomplex amplitude transmittance in plane z where4+2 exp(-27riz/X) *ff ... dvdg4. If we illuminate thisvirtual amplitude transmittance by a plane waveexp(21riz/X), the complex amplitude will be

+A +A

I exp(27riz/X)+ 2 f Aio(Vy)

[1 -A 2V+.2]Xexp(27ri(xv~y1.z+ [1-z2(v 2 ± 2 )]Z/X)}dvdi. (9)

Now, moving virtually to plane z=0, we get from Eq.(9) the same expression as in Eq. (3) which describedthe image amplitude of one-step single-sideband imageformation

4A +A

+2 A X io(V,A)O A

Xexp[2iri(xv+yp.)]dvdpu= lu(x,y). (10)

This plane, z = 0, can be observed with a lens system,for example a telecentric system, which follows thehologram, and which contains the knife edge in theFraunhofer diffraction plane (see Fig. 1). The irradianceI uI 12 in the plane of observation is proportional to theobject irradiance, as already proven in Eqs. (5) and (6).The elimination of the left sideband (v<0) during re-construction is equivalent to the elimination of thetwin image, that otherwise would appear at plane 2z.

So far, our theory was based on the two assumptions(1) and (2), which required the object to be real and tocontain a strong background. It is easy, at least intheory, to avoid the background requirement bymodifying the spatial filter in the recording process,

F(v,M)=0 (if v<0); =1 (if v=IL=0);

= 1/10 (if v> 0) . (11)

In this way, the right sideband (v> 0) is reducedsufficiently in comparison to the background wavecoming from v = =0, so that we are again permitted toomit the quadratic term when calculating the imageirradiance [Eq. (4)]. The reduction factor X as shownin Eq. (11) is, of course, only an example, which has tobe adapted to any particular object.

Finally, we want to prove that even the assumptionof reality (1) is not necessary. Now, the two sidebands(v>0;v<0) are not redundant; hence, we need twoindependent single-sideband procedures in parallel, inorder to reconstruct both sidebands. In essence, thesetup contains a beamsplitter behind the object, twoknife-edge systems, two holograms, two reconstructionsystems with opposite knife edges, and a beamsplitterclose to the image plane, for combining both beams. Thebeam which is responsible for the right sideband (v> 0)

621M~ay 1968

0. BRYNGDAHL AND A. LOH MIANN Vl 5

Recording

H

M

Reconstruction

FIG. 1. Optical setup used for recording and reconstrsingle-sideband holograms. Ixor notation see tex

yields a hologram exposure,

1 f+Af +AlTIR?,J2= -exp(2iri!,/X)+I A ziexp(*

2 A

1 exp(-2iriz/X)ff... dvdi

+ 2 exp(+2wriz/X) I f... M *

This is formally the same as Eq. (8). Again we show that{ .* **dvd.u} * contains only negative frequencies(v<0), which can be blocked out by a knife edge in thereconstruction process. But the important differencefrom the former proof is that we do not make use ofthe reality symmetry (1) u We onlychange variables (VIA) (-v ,-A),

..If dvdlt

A, +A1= ffJ Aif0 (v,8i)

AR0,

Xexp{-27ri(xv+yyz+ [1-XI(V2+A,2)]iz/X)}dvd,4

I IX Aio*( vL exp{+27ri(xv+yA

- [1- 2(v2 +u2 )]iz/X)}didAu. (13)

Whatever Ai-1o(-v,-A) may be, the essential fact isthat Eq. (13) contains only those spatial-frequencyterms exp[27ri(xv+yA)], for which v<0. Hence, thisterm can be eliminated by means of a knife edge in theFraunhofer plane of the reconstruction process.

Knife edges are also located in the other beam fromthe object, which eliminate the right sideband, (v>0)in both the recording and reconstruction. After the twobeams are combined, the final image amplitude is

~+4-+A

Xexp[2iri(x.v+yg)]dvdi= ltuo(x,y). (14)

Other solutions for eliminating the twin image inin-line holography by taking two holograms simul-

H taneously have been described earlier.4

EXPERIMENTAL RESULTS

I. Holograms of Real Objects with Strong Background

We have found that the single-sideband techniqueworks well for real objects with a strong background.The optical setup used is shown in Fig. 1. A telescopic

R svstem with two identical lenses L1 and L2 (1= 135 mm)ucting was used and a knife-edge mask was placed in M. Theit object 0 was illuminated with parallel light from a

He-Ne laser and the knife edge was adjusted to shieldoff almost half of the diffraction pattern in M. The

2 hologram was recorded in a plane H behind the lens) )dvdy L2. In reconstruction, the hologram was placed in

(12)

(a) (b)

(C) (d)FIG. 2. Holograms and reconstructions of an object with a

strong background. Diameter of circle in object 7 mm; (a) showsa single-sideband hologram and (b) its reconstruction; (c) and (d)show the corresponding conventional ones.

I W. L. Bragg and G. L. Rogers, Nature 167, 190 (1951);G. I,. Rogers, Proc. Roy. Soc. (Edinburgh) A64, II, 209 (1956);D. Gabor and WV. P. Goss, J. Opt. Soc. Am. 56, 849 (1966).

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SINGLE-SIDEBAND HOLOGRAPHY

front of Li with the knife edge shielding the oppositehalf of the plane M-this gave the reconstruction of thevirtual image in R. If the half-plane mask is in the sameposition as in recording process, the real image is ob-tained. Of course, the technique works as well with onlyone lens, which may be placed either before or after theobject. The telescopic arrangement was chosen onlyfor convenience, in order to make the longitudinal loca-tion of the hologram not critical.

In holographic work with an in-line reference beam,the amplitude transmittance of the final hologram hasto be proportional to the recorded irradiance in thehologram plane. Thus, a two-step photographic proc-ess is necessary when negative material is used. Wehave used Kodak 649F plates throughout this work.They were processed in D-19 developer diluted 1:20with brush development (2.5 min) to give a gamma of1.4, resulting in a total gamma of -2 for the two-stepprocess.

Figures 2(a) and (b) show a single-sideband hologramand the corresponding reconstruction. For comparison,a conventional in-line hologram and its reconstructionmade under equal conditions are shown in Figs. 2(c)and (d). As can be seen from this figure, the knife edgewas in a vertical position; thus, the twin image fromdetails whose diffraction patterns are located alongthe edge cannot be eliminated [see horizontal bar ofcentral triangle in Fig. 2(b)]. Besides elimination of thedisturbing twin image, a cleaner background was ob-tained (only one half of each Fresnel zone-plate patternis recorded) with the single-sideband hologram whencompared to the conventional in-line type.

Experiments have confirmed that the technique alsoworks satisfactorily for 3-D objects.

II. Holograms of Real Objects with Weak Background

We show here that it is also possible to use the single-sideband technique for objects with a weak background(amplitude of diffracted light greater than amplitude ofthe background) if the diffracted light from the objectis attenuated.

(a) (b)FIG. 3. Hologram and reconstruction of an object without

background with single-sideband technique. Height of letters1.2 mm.

0L BS,

RP,

RP,

H

BS2

FIG. 4. Optical setup for recording and reconstruction ofholograms of complex objects using a single-sideband technique.For notation see text.

Some experiments have been performed using trans-parent letters on an opaque background as object. Inthe Fraunhofer diffraction plane of the object, a neutraldensity filter with a clear pinhole at the central spotwas introduced beside the knife edge. The filter with a10-a pinhole was produced photographically on a 649Fplate, and the knife edge was placed in contact with theemulsion just touching the edge of the pinhole. Theother parts of the optical arrangement were as indicatedin Fig. 1.

Figure 3(a) shows a hologram recorded through afilter of density 2 and Fig. 3(b) shows the correspondingreconstruction. The background level can be attenuatedin the reconstruction by placing a corresponding spot atthe location of the center of the diffraction pattern ofthe hologram.

III. Holograms of Phase Objects

The frequency spectrum of a complex object is ingeneral unsymmetrical and any restriction in this planewill have a detrimental effect on the transmitted in-formation. The single-sideband technique can, however,also be applied to this case if the light from the object issplit so that the same spectrum is displayed in twolaterally shifted positions. Opposite halves of theFraunhofer diffraction spectra are then shielded off andtwo holograms are obtained, each containing a particularpart of the total information about the object. In recon-struction, both holograms will give symmetrical spectraand the corresponding images, real or virtual, can be ex-tracted by masking respective halves of their spectra.The final reconstruction is obtained by interferometricsuperposition of the two wavefronts.

We have done some experiments with a Mach-Zehnder-type arrangement, which is illustrated inFig. 4. When recording the holograms, we illuminatedthe object 0 with a collimated coherent beam from theleft; the Fraunhofer diffraction spectrum was displayedin the rear focal plane of the lens L (f= 135 mm). Afterthe lens L, the light was divided into two paths, onestraight and the other laterally shifted by the beam-splitter BS, and the right-angle prism RP1. Identicalspectra of the object appeared in the planes Ml and M2.

623May 1968

0. BRYNGDAHL AND A. LOHMANN

(a)

(a) (b)

(b)

FIG. 5. Holograms of a fly's wing using (a) left and (I) rightsideband, respectively, in recording.

Opposite halves of these spectra were shielded off byknife edges adjusted to pass somewhat more than halfof the central spots of the spectra, giving a uniformbackground of proper irradiance. The lenses Li and L2(both f=156 mm) were placed with their front focalplanes in Ml and M2, respectively, and adjusted torecollimate the light. The holograms were recorded ona 649F plate large enough to record both beams. Anadjustable holder was used so that the final hologramscould be relocated in their original positions. Figures5(a) and (b) show the holograms made from the wingof a fly. They were located about 5-cm apart onthe plate.

In reconstruction, the same optical setup was used.The holograms were illuminated from the right withcollimated light. In order to facilitate the interferometricsuperposition of the two reconstructed wavefronts, aright-angle prism RP2 and a beam-splitter BS2 wereadjusted prior to the experiments, so that the outcomingwavefronts from the two interferometer arms were inphase and overlapping laterally. The collimated beamused for reconstruction was adjusted so that the twowavefronts after the lens L also overlapped preciselyand at the same time gave a uniform field. The re-constructions corresponding to the individual hologramsin Figs. 5(a) and (b) are shown in Fig. 6(a) and (b), re-spectively. Especially at the left edge of the wing, theeffect of the phase change in the object appears. Thefinal reconstruction-the interference of the wavefrontsin Figs. 6(a) and (b)-is shown in Fig. 6(c).

DISCUSSION

Sastisfactory results show that the single-sidebandtechniques can be used for making in-line holograms ofamplitude as well as phase objects. Besides eliminatingthe twin image, we obtain a cleaner background, com-pared to the conventional in-line holographic technique.

However, an optical system has to be used and certainrequirements must be met; good-quality optical com-ponents are necessary because a knife-edge test is per-formed. Furthermore, the aperture of the setup islimited to that of the optical system,

(C)

FIG. 6. Reconstructions of the holograms in Fig. 5. (a) re-construction of the fly's wing using left sideband and (b) usingright sideband. (c) shows the interferometric superposition of(a) and (b).

The amplitude objects corresponding to Figs. 2and 3 were photographic transparencies. These are, ingeneral, difficult to obtain without phase distortion byemulsion relief; therefore a knife-edge effect is visiblein reconstruction [see Fig. 2(b)]. Of course, emulsionrelief in the hologram also contributes to this effect;however, it was eliminated in later experiments byinserting the object as well as the hologram in a liquidgate. We also have tried pressing the emulsion againstan optical flat at the end of the drying process in orderto eliminate the relief, but considerable stray-light wasobtained because pressing created microstructure in thegelatin surface.

Instead of using a density filter in the modificationfor objects with weak background, a diffuser can alsobe used to cover half of the diffraction pattern (exceptfor the very center) in order to increase the ratio ofundiffracted to diffracted light. In this way a crypto-gram is obtained essentially like the one that is achievedwith a diffuser between the object and the hologram inan off-line technique,5 and the same ground glass used inmaking the hologram is also needed for reconstruction.Furthermore, using a diffuser gives a more radical en-coding (object no longer recognizable in the hologram).

The setup for making holograms of complex objectsmay be simplified by using a polarization interferome-ter, e.g., Wollaston prism as beam splitter. We mayalso mention that it is easy to apply Wolter's Minimum-strahlkennzeichnung,6 i.e., a X/2-shift between the twosidebands in our setup: the interferometer was adjustedto give a dark background in reconstruction.

In-line holography is advantageous for use in caseswhere the resolution of the recording material, ex-posure time, mechanical stability, or temporal coher-ence is limited. The single-sideband technique may beapplied to improve the quality of the reconstructedimage without reducing these advantages.

ACKNOWLEDGMENTS

The authors wish to express their sincere thanks toA. Nafarrate for his skillful assistance in performing theexperiments.

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 56, 523 (1966).H. Wolter Ann. Physik (6) 7, 341 (1950); A. Kastler, Rev.

Opt. 29, 307 (1950); S. Lowenthal and Y. Belvaux, Appl. Phys.Letters 11, 49 (1967).

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