One-way Imaging With an Aberrated Reference Beam

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One-way imaging with an aberrated reference beam

David Statman, Jason C. Puth, Christopher E. Sunderman, and Bruce W. Liby

We demonstrate image reconstruction by way of four-wave mixing for a signal that makes a single passthrough a distorting medium. Both the signal and the reference originate from the same source andpropagate through the same disturbance. In the four-wave mixing process, phase aberrations aresubtracted out. A read beam from a second source reconstructs the undistorted image, yielding a phaseconjugate without a double pass through the disturbance. Good reconstructed image fidelity is demon-strated for static distortions if the phase mismatch between the signal and the reference beams isminimized. If there is significant phase mismatch, reconstructed image fidelity is poor. We show thatthis technique can also be used to measure the autocorrelation function of the disturbance when themismatch between the signal and the reference in the four-wave mixing scheme is varied. © 2000Optical Society of America

OCIS codes: 100.3010, 190.4360, 190.5040.

1. Introduction

Nonlinear optics has shown promise for the develop-ment of one-way imaging technologies. Several re-searchers have demonstrated methods by whichnonlinear optics can be used to correct accumulateddistortions in optical signals. These distortions areusually the result of aberrations introduced throughatmospheric turbulence.

To date, only two general methods have been pro-posed to accomplish one-way imaging with nonlinearoptics. In the first method, the disturbance is re-corded in the nonlinear optical medium. Either asecond reference beam or the image, either of whichhas passed through the disturbance, is mixed with anundistorted reference. This method has been suc-cessful. In 1982, Fischer et al.1 demonstrated real-ime one-way image reconstruction by sending aecond reference beam through the disturbance andixing that reference with an undistorted reference.he signal was scattered from this dynamic hologramuch that the scattered signal was phase conjugate tohe distorted reference. As the signal backpropa-ated through the disturbance, the distortion it had

D. Statman [email protected]!, J. C. Puth, and C. E. Sunder-an are with the Department of Physics, Allegheny College, Mead-

ille, Pennsylvania 16335. B. W. Liby is with the Department ofhysics, Manhattan College, Riverdale, New York 10471.Received 30 August 1999; revised manuscript received 28 Feb-

uary 2000.0003-6935y00y183044-06$15.00y0© 2000 Optical Society of America

3044 APPLIED OPTICS y Vol. 39, No. 18 y 20 June 2000

acquired by scattering with four-wave mixing wasremoved. In 1991, Kramer et al.,2 and then in 1995Chakmakjian et al.,3 demonstrated one-way imagingby passing the image signal and a read beam col-linearly through a disturbance. In this case, the dis-torted image was recorded in the nonlinear mediumbecause that signal was mixed with an undistortedreference. The distortions in the read beam wereremoved upon scattering from the dynamic hologram.The result was the production of an undistorted im-age. In 1993, Sun and Moharam4 accomplished one-way image reconstruction using two photorefractivecrystals. With the first crystal, they accomplishedtwo-beam coupling between the distorted input sig-nal and a reference beam. Both phase and ampli-tude information were transferred from the inputsignal to the reference. The second crystal was thenused as a phase conjugator for this two-beam coupledoutput. The phase conjugate was sent back to thefirst crystal as a read beam from the dynamic holo-gram written in the first two-beam coupling process.The final output had all phase aberrations sub-tracted. One issue concerning this method, how-ever, is that the undisturbed reference must becoherent with one of the distorted beams. In 1983,Ikeda et al.5 found a way around this requirement.With a point reflector in the observation plane, asmall part of the signal was propagated backwardthrough the turbulent medium as the second refer-ence beam.

The second method involves the subtraction of thedisturbance from the signal while the dynamic holo-gram is written with a scheme demonstrated for

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static holograms by Goodman et al. In this case,both the signal and a reference are passed throughthe same distorting medium. In 1988, MacDonald et

l.7 demonstrated one-way imaging using thismethod. As long as the correlation length of thedistortion was longer than the separation betweenthe two beams, the distortion could easily be sub-tracted by use of four-wave mixing. A third beamfrom a separate source was then used to read thecorrected image from the nonlinear medium. In1990, Kramer et al.8 tried to improve on the work ofMacDonald et al. by copropagating cross-polarizedsignal and reference beams. They had hoped toavoid the issues of correlation length discussed byGoodman et al. They could not, however, correct forslow or static aberrations. Apparently Kramer et al.had been able to remove dynamic distortions by usingthe nonlinear medium as a novelty filter rather thanby subtraction of phase information. This was dem-onstrated by Zhang et al.9

In this paper we demonstrate one-way image re-construction using a reference beam that traversesthe same disturbance as the signal for a static aber-ration. Building on the scheme of Kramer et al.,8 weshow that the analysis by Goodman et al.6 cannot beignored. Critical to high-fidelity image reconstruc-tion is an appraisal of the correlation length of thedisturbance. Beam overlap within the nonlinear op-tical medium must consider the impact of the corre-lation length of the disturbance. In Section 2theoretical analysis of the system is provided. InSection 3, the experiment is described and the resultsare shown. A summary is given in Section 4.

2. Theoretical Analysis

Consider two beams, a signal and a reference beam,whose fields are given by u1~x, y! and u2, respectively,coincident on a disturbance from the plane desig-nated by S1, as shown in Fig. 1. The reference beamu2 contains no spatial information in the S1 plane.In the plane, designated by S2, the field of each beamis given by the Huygens–Fresnel principle10:

u1~h, j! 5exp~ikz!

ilz ** u1~x, y!K~h, j, x, y!dxdy, (1)

u2~h, j! 5 u0

exp~ikz!

ilz ** K~h, j, x, y!dxdy, (2)

where K~h, j, x, y! is the point-spread propagationfunction through the disturbance given by4,8

K~h, j, x, y! 5 K0 expH2ik2z

@~x 2 h!2 1 ~y 2 j!2#

1 i * c~h, j, x, y!dpJ , (3)

and it is assumed that the distance z is much largerhan ~x2 1 y2!1y2 and ~h2 1 j2!1y2. c~h, j, x, y! is the

phase that is accumulated over path p because of the

berrations. From Eqs. ~1!–~3!, it is clear that atny point ~h, j!, both u1~h, j! and u2~h, j! must have

the same accumulated phase.If a beam splitter, as shown in Fig. 2, separates the

signal and reference beams by a distance x0, we canrewrite Eqs. ~1!–~3! with the corresponding x and ycoordinates primed for the reference beam. At theS2 plane the interference pattern is governed by themutual intensity function J~h, j!, which, because thewaves are coherent, is simply the product of the fieldsat the points under consideration. In the Fraun-hofer limit the mutual intensity function is given by10

J~h, j! 5K0

2

l2z2 **S1

**S19

u2 u1*~x, y!expH2ik2z

@~r92 2 r2!

1 2~x9 2 x 1 x0!h 1 2~y9 2 y!j#J3 expHiF* c~h, j, x9, y9!dp

2 * c~h, j, x, y!dpGJdxdydx9dy9, (4)

Fig. 1. Two beams, input signal beam 1 and reference beam 2,propagate through a disturbance and are recorded in plane S2.

lane S2 can either be holographic film or a nonlinear opticalaterial.

Fig. 2. Input and signal beams, 1 and 2, respectively, are sepa-rated before having traversed the disturbance and are recorded atS2. Read beam 3 propagates counter to reference beam 2. Theimage signal is beam 4. When the distance between beams 1 and2 is less than the correlation length of the disturbance, beam 4 atS10 is nearly phase conjugate to beam 1 at S1.

20 June 2000 y Vol. 39, No. 18 y APPLIED OPTICS 3045

2 2 1y2

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~v

di

dp

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where r 5 $@x 1 ~x0y2!# 1 y % and r9 5 $@x9 2~x0y2!#2 1 y92%1y2, respectively. Equation ~4! is thetandard expression for double-wave interference.n four-wave mixing, the two write beams are giveny u1~h, j! and u2~h, j!, respectively. If a read beam,

u3~h, j!, given by

u3~h, j! 5 2u2*exp~2ikz!

ilz ** K0*

3 expHik2z

@~x 2 h!2 1 ~y 2 j!2#Jdxdy (5)

is incident on a dynamic hologram written by thewrite beams, the output signal will be

u4~x0, y0! 5 uu2u2K0

3

il3z3 exp~2ikz! **S2

**S19

**S1

u1*~x, y!

3 expH2ikz

@~x 2 x0!h 1 ~y 2 y0!j#J3 expHiF* c~h, j, x9, y9!dp

2 * c~h, j, x, y!dpGJdxdydx9dy9dhdj.

(6)

When

* c~h, j, x, y!dp < * c~h, j, x9, y9!dp (7)

for the corresponding pair of points ~x, y! and ~x9, y9!,the effects of the distortions are subtracted out of theoutput signal. In this case, u4~x0, y0! at the S10 planeis the diffraction-limited phase-conjugate image ofthe input signal u1~x, y! at the S1 plane—without

aving a double pass through the disturbance.ithin the limits imposed by the aperture at S2, the

image should be faithfully reconstructed.In Goodman et al.’s experiment,6 approximation ~7!

was satisfied by guaranteeing that the distance x0was much less than the correlation length of the dis-turbance. When approximation ~7! is not satisfied,output signal 4 will be degraded by an amount cor-responding to the phase mismatch given in Eq. ~6!.

Ideally one would like the distance x0 to be zero.However, this does not facilitate four-wave mixingschemes. To get around this, one can separate sig-nal beam 1 and reference beam 2 after having prop-agated through the aberration, as shown in Fig. 3.Planes S1 and S19 are virtual planes, each traversingequivalent virtual disturbances. Four-wave mixingis again accomplished by our sending read beam 3counter to reference beam 2 and given by Eq. ~5!where the x and y coordinates were primed. Outputsignal beam 4 will again be in the direction counter toinput signal beam 1. This scheme, however, does


not eliminate the requirement given by approxima-tion ~7!. It can be seen from Fig. 3 that Eqs. ~4! and6! are valid for this scheme as well. In this case, theirtual planes S1 and S19 for the input signal and

reference beams, respectively, must be equivalentwith respect to the disturbance. To accomplish this,the z and z9 axes need to be coincident at S2 so thatthe points ~x, y! and ~x9, y9! correspond and the phase

istributions of beams 1 and 2, respectively, overlapn the S2 plane. If the overlap is good, then beam 4

will contain the reconstructed image minus the dis-tortions, and u4~x0, y0! at the S10 plane will be the

iffraction-limited phase-conjugate image of the in-ut signal u1~x, y! at the S1 plane. On the other

hand, if the phase mismatch is such that the distancebetween the z and z9 axes is greater than the corre-lation length of the disturbance, then the recon-structed image fidelity will be poor.

An important result of this method, however, isthat it can provide a means of determining the auto-correlation function of the disturbance:

g~x0! 5 **S2

**S19

**S1

expHiF* c~h, j, x9, y9!dp

2 * c~h, j, x, y!dpGJdxdydx9dy9dhdj. (8)

By deliberately mismatching the phase, we canmeasure this property of the distortion as a functionof distance x0. If the input signal u1~x, y! has no

Fig. 3. Input and signal beams, 1 and 2, respectively, are sepa-rated after having traversed the disturbance and are recorded atS2. Read beam 3 propagates counter to reference beam 2. Theimage signal is beam 4. When the phase overlap between beams1 and 2 is good, beam 4 at S10 is phase conjugate to beam 1 at S1.

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spatial information, then in the limit of a largeaperture in the S2 plane, the output signal at S10 isiven by

u4 5 u1*uu2u2K0

3

il3z3 exp~2ikz! **S2

**S19

**S1

3 expHiF* c~h, j, x9, y9!dp 2 * c~h, j, x, y!dpGJ3 expH2ik

z@~x 2 x0!h 1 ~y 2 y0!j#J

3 dxdydx9dy9dhdj

< u1*uu2u2K0

3

il3z3 exp~2ikz!g~x0!. (9)

In practice, Eq. ~6!, approximation ~7!, and Eqs. ~8!and ~9! suggest that, for a Gaussian input signal u1~e.g., a TEM00 laser beam!, the relative beam qualityof the reconstructed signal u4 ~defined as the inten-sity of the reconstructed signal that has been focusedthrough a diffraction-limited aperture and normal-ized by that of the input signal! is a reasonable mea-sure of the autocorrelation function g~x0! because it isproportional to the square of g~x0!.

3. Experiment

A schematic of the experimental design is shown inFig. 4. The output from an argon-ion laser operat-ing at 100 mW was polarized horizontally. Part ofthis beam was split off to be used as the read beam.The path-length difference between the read and thewrite beams was significantly greater than the coher-ence length of the laser. This was done to simulatetwo independent read and write lasers. The writebeam was sent through a Mach–Zehnder interferom-eter, separating it into an input signal and referencebeam. Using a half-wave plate, we rotated the po-larization of the reference beam by 90°. The objectrecorded on the input signal beam consisted of hori-zontal bars of a U.S. Air Force resolution chart. Theinput and reference beams were recombined and co-propagated through a disturbance made of a glassslide that was inhomogeneously coated with oil. A30-cm lens, L-1, was placed at the receiver to collectthe light. Beyond that lens, the two beams wereseparated with a polarizing beam splitter. Polariza-tion of the reference beam was again rotated by 90°.Four-wave mixing was accomplished in BaTiO3:Rhplaced 60 cm beyond lens L-1. The input signal andthe reference beams wrote a dynamic hologram in thephotorefractive crystal, which was read by the readbeam. The output signal was imaged onto a CCDcamera with a second 30-cm lens, L-2, and recordedon a PC. The unreconstructed image was imagedonto a second CCD camera. Because of the opticalarrangement ~as well as a lack of space!, the unre-onstructed image was inverted with respect to the

bject. The phase-conjugate image was not in-erted. It should also be noted that this technique isimited in that in can only reconstruct signals inhich the disturbance does not rotate the polariza-

ion, as can be demonstrated when crinkled plastic issed for the aberration.To guarantee high-fidelity image reconstruction,

he z and z9 axes of the input signal and referenceeams, respectively, had to be properly aligned.rior to placing the object to be recorded in the signaleam, we placed a U.S. Air Force resolution chartfter the disturbance and 60 cm before lens L-1. Us-ng lens L-1, we aligned the images of the resolutionhart, now in both the reference and the input signaleams, both horizontally and vertically. By doingo, we nearly satisfied the requirement given by ap-roximation ~7!. The dynamic hologram in the Ba-iO3 crystal, however, is a volume hologram.

Therefore there are regions of poor phase overlapwithin the crystal. To minimize this effect, the an-gle between the two beams can be made as small aspractically possible. After doing so, we removed theU.S. Air Force resolution chart, and the object wasplaced in the object plane of the Mach–Zehnder in-terferometer.

As noted in Section 2, the image in the S2 plane isdiffraction limited by the aperture of the recordingmedium, in this case the BaTiO3 crystal. With a0.5-cm square aperture, the angular resolution of theimage is limited to approximately 0.1 mrad. For

Fig. 4. Experimental design. The output from an argon-ion la-ser was directed through a polarizer, P, and a beam splitter, BS.Propagating through the beam splitter, the light was separated byanother beam splitter into input signal beam 1 and reference beam2, which were cross polarized with a half-wave plate, ly2, and apolarizer. The signal and reference were recombined with mir-rors, M, and a polarizing beam splitter, PBS. The reference andsignal were copropagated through the disturbance. At the re-ceiver the input signal and reference were separated with a polar-izing beam splitter, the reference polarization was rotated 90° witha half-wave plate and a polarizer, and a dynamic hologram wasrecorded in BaTiO3:Rh. The disturbance was imaged into the

aTiO3:Rh with lens L-1. The unreconstructed image was re-corded with a CCD camera. Read beam 3 was directed counter toreference beam 2, and the output signal was imaged onto a CCDcamera with lens L-2.


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this experiment, then, the point-to-point resolutioncannot be better than approximately 30 mm.

Figure 5~a! shows the reconstructed image in theabsence of any disturbance. In Fig. 5~b! we show thedistorted image located at the image plane of lensL-1. The CCD camera was moved in and out to tryand get the best picture of the unreconstructed im-

Fig. 5. Images of horizontal bars: ~a! output image in the absenca disturbance.

Fig. 6. Images of the number 5: ~a! reconstructed image in thebsence of a disturbance; ~b! distorted image with the number

inverted; ~c! reconstructed image with the disturbance; and ~d!, ~e!,and ~f ! reconstructed image when the input image and referenceare slightly misaligned in the BaTiO3 crystal, in order of increasingmisalignment.


age. Clearly the image fidelity was poor, as ex-pected. In Fig. 5~c! we show the reconstructedimage in the phase-conjugate image plane. Again,the CCD camera was moved in and out to get the bestpicture of the reconstructed image. It is clear thatmost of the distortions have been removed, as ex-pected.

According to Eq. ~6!, misalignment between writeeams 1 and 2, with respect to the disturbance,hould result in poor image reconstruction. To dem-nstrate the effects of misalignment, we recon-tructed the image of the number 5 in a sequence ofncreasing misalignments. In Fig. 6~a! we show theeconstructed image of the number 5 in the absence ofdisturbance, in Fig. 6~b! we show the distorted im-

ge, and in Fig. 6~c! we show the reconstructed imageith the disturbance. As already noted from Fig. 4,

he distorted image is inverted. In Figs. 6~d!–6~f !e show the reconstructed image when input signalnd reference beams 1 and 2, respectively, arelightly misaligned. Figures 6~d!–6~f ! are pre-ented in order of increasing misalignment. As isbvious, misalignment in the write beams can haveerious consequences on the fidelity of the recon-tructed image.

4. Discussion

The scheme described in this paper has several ad-vantages over previous schemes. Because the refer-ence and the input signal must be mutually coherentto write a dynamic hologram, both beams must prop-agate through the same disturbance. By our prop-agating collinearly, the ability to subtract thedisturbance is maximized, but not eliminated. How-ever, with careful alignment, good reconstructed im-age fidelity can be obtained. In addition, as shownby Kramer et al.,2 disturbances, which are faster thanthe response time of the nonlinear medium, will not

disturbance, ~b! distorted image, and ~c! reconstructed image with
e of a

pensation for high-spatial-frequency aberration correction,”
be recorded and therefore will not be seen by the readbeam.
In a future paper we will present measurements ofthe autocorrelation function of the disturbance usingthis technique.

We thank G. Charmaine Gilbreath of the RemoteSensing Division of the Naval Research Laboratoryfor suggesting this project, as well as her many help-ful discussions.

References1. B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Real-

time phase conjugate window for one-way optical field imagingthrough a distortion,” Appl. Phys. Lett. 41, 141–143 ~1982!.

2. M. A. Kramer, C. J. Wetterer, and T. Martinez, “One-wayimaging through an aberrator with spatially incoherent lightby using an optically addressed spatial light modulator,” Appl.Opt. 30, 3319–3323 ~1991!.

3. S. H. Chakmakjian, M. T. Gruneisen, K. Koch, M. A. Kramer,and V. Esch, “Time-multiplexed real-time one-way image com-

Appl. Opt. 34, 1076–1080 ~1995!.4. Y. Sun and M. G. Moharam, “Real-time image transmission

and interferometry through a distorting medium using twophase conjugators,” Appl. Opt. 32, 1954–1957 ~1993!.

5. O. Ikeda, T. Suzuki, and T. Sato, “Image transmission througha turbulent medium using a point reflector and four-wavemixing in BSO crystal,” Appl. Opt. 22, 2192–2195 ~1983!.

6. J. W. Goodman, W. H. Huntley, Jr., D. W. Jackson, and M.Lehman, “Wavefront-reconstruction imaging through randommedia,” Appl. Phys. Lett. 8, 311–313 ~1966!.

7. K. R. MacDonald, W. R. Tompkin, and R. W. Boyd, “Passiveone-way aberration correction using four-wave mixing,” Opt.Lett. 13, 485–487 ~1988!.

8. M. A. Kramer, T. G. Alley, D. R. Martinez, and L. P. Schelonka,“Effects of thick aberrators in one-way imaging schemes,”Appl. Opt. 29, 2576–2581 ~1990!.

9. J. Zhang, H. Wang, S. Yoshikado, and T. Aruga, “Image trans-mission through a thick dynamic distorter by the photorefrac-tive fanning effect,” Opt. Lett. 23, 585–587 ~1998!.

10. J. W. Goodman, Statistical Optics ~Wiley, New York, 1985!.


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One-way Imaging With an Aberrated Reference Beam