Temporal reversal of picosecond optical pulses byholographic phase conjugation
Cecile Joubert, Marie L. Roblin, and Roger Grousson
We theoretically discuss the use of holography to achieve the phase conjugation of temporal signals. Thereversal of a particular asymmetric pulse envelope is experimentally demonstrated. The holographicmaterial is a photorefractive crystal used in a four-wave mixing configuration.
1. Introduction
In the present work, we achieve a temporal phaseconjugation mirror using a two-step holographic pro-cess. In a previous work,' we studied temporal signalprocessing by the same holographic method using thephase filtering properties of the hologram to realize thecompression of an optical pulse.
The interest of phase conjugation for chirp compen-sation achievement has been pointed out by severalauthors. They theoretically analyzed the applicationof nonlinear optical techniques (typically nearly de-generate four-wave mixing) to the time reversal ofoptical pulses. Marburger2 and Yariv et al. 3 assumeda cw pumping, and Miller4 assumed short pulse pump-ing. They consider nonlinear media with a responsetime much faster than the pulse duration. Their con-ditions are quite different from holography where themedium is sensitive to the time integrated intensity.They conclude that time reversal effects can be onlyexpected for particular geometrical configurations24
or phase law representable by the quadratic term of aTaylor series.3 But none of them gave any experimen-tal evidence of such a phenomenon.
Let us also mention the promising experiment usingphotochemical hole burning, which can be consideredas a direct holographic recording in the frequency do-main.5 Apart from this experiment which is per-formed with unusual materials and at low tempera-ture, we give here to our knowledge, the first
The authors are with Universite Paris VII, Groupe de Physiquedes Solides de l'Ecole Normale Superieure, Tour 23,2 Place Jussieu,Paris CEDEX 05, France.
experimental evidence of time reversal of pulse enve-lopes.
First, we compare phase conjugation for spatial andtemporal signals and then explain the principle of ourexperiment.
Let us consider a monochromatic optical beam prop-agating in the z-direction through a linear lossless dis-torting medium (Fig. 1). Before entering the medium,the plane wave is described by
E0(r,t) = 1/2 expi(wt - kz)] + c.c.1,
where c.c. stands for complex conjugate.The wave after the medium is described by
El(r,t) = /2 4f(r) exp[i(cot - kz)] + c.c.)
= /2A(r) exp(iwt) + c.c.],
withA 1(r) = P(r) exp(-ikz).
The dependence of on r reflects the spatial distor-tion introduced by the medium.
Yariv and many other authors6 7have demonstratedthe ability to generate by holographic and nonlinearmethods the complex conjugate field of E1(r,t) at someposition z = zo. They studied the properties of such afield described by
E2(r,t) = Re[A(r) exp(iwt)]
= Re[I*(r) exp[i(wt + kz)]}
z <ZO
The E2 conjugate of E1 propagates along z in the oppo-site direction with a conjugate amplitude; i.e., a phaseconjugation mirror is performed at z = zo.
To obtain E2 from E1, one takes only the complexconjugate of the spatial part leaving the factor exp(iwt)intact, which is equivalent to leaving the spatial partintact and to reverse the sign of t:
Therefore, E2 is related to E1 by time reversal, but it isnot possible to show experimentally this temporal re-version with a monochromatic wave.
To appreciate the main interest of the conjugatedfield E2, one may consider its propagation through thedistorting medium in the reverse direction. It re-emerges with the original unspoiled properties of E1.Such a healing of spatial distortion is shown in Fig. 1.
Let us consider now the spectral phase conjugationof an optical pulse. The temporal signal describing apulse of frequency spectrum amplitude a(v - v) isgiven by
EO(t) = /2[fa(v - vO) exp(i2rvt)dv + c.c.].
using the variable A = v - vo (vo being the mean fre-quency) it becomes
where AO(t) is the finite duration envelope of the pulse.In the initial pulse, a(v - Po) is assumed to be a realfunction. In this case the pulse is as short as possibleand is called Fourier transform limited. Every phasefactor depending on v in the spectrum amplitude yieldsan increased duration of the pulse.
After passing through a spectral dephasing medium,the frequency spectrum is a(v - vo) exp[iI'(v)], and thesignal is described by
where Al(t) = fa(y) exp[ib(,g)] exp(i21rgut)dg is themodified temporal envelope.
Let us consider that in some region of space near zo,for example (Fig. 2), we somehow generate a field E2 (t)whose spectrum is the conjugate spectrum of El(t), a(v- vo) exp[-i4(v)], propagating in the opposite direc-tion along z.
The temporal signal is now given by
E2(t) = /2 [fa(v-v 0 )
X exp[-iM(v)] exp(i2rvt)dv + c.c.] for z < zo,
1) ikz 4) e ikz 2) T(,)e-ikz 3) .(r)e ikz
I medium
K <-iW _, dephasing
z=O
phase
conjugation
mirror
z=zo
Fig. 1. Plane wave 1) incident on a distorting medium and emerg-ing with a phase distortion T(r) 2). The wave reflected by theconjugation mirror 3) traverses the medium in reverse. The waveobtained at the exit of the medium 4) is identical to the incident wave
1).
1) EO(t) 4) E(t)
FO(v)=a(v) a(v)
) _> <1I
2) E1 (t) 3) E2 (t)=E1 (-t)
F1(v)=a(v)e F1*(v)=a(v)e -iD(v)
spectralphase
distorting / 1medium >
spectralphase
conjugationmirror
z=O z=zo
Fig. 2. Pulse Eo(t) 1) incident on a spectral phase distorting medi-um and emerging withaspectral phase distortion (v) 2). The pulsereflected by the spectral phase conjugation mirror 3) has a spectralphase 'I(v) and traverses the medium in reverse. The pulse ob-tained at the exit of the medium is identical to the incident pulse 1).
and from Eqs. (1) and (2) it becomes E2(t) = E1 (-t).The key point to appreciate is that changing the sign
of the spectral phase of a pulse El(t) is equivalent togenerating the temporal reversed pulse E1 (-t). Thisresult can be easily verified with asymmetric pulseenvelopes.
Now if E2(t) passes again through the dephasor, thephase '(v) previously introduced in the spectrum issuppressed and the pulse recovers its initial shape.
The holographic method we used to perform such aspectral phase conjugation mirror is schematicallyshown in Fig. 3 and is identical to a four-wave mixingsetup. A Fourier transform limited pulse EO(t) (pump1) and the same perturbated by a phase law El(t)(probe) interfere in a photorefractive medium, which
Fig. 3. Holographic setup identical to a four-wave mixing configu-ration. The object pulse E1(t) (probe) interferes with the referencepulse EO(t) (pumpl). The hologram is read out by the referencepulse propagating in reverse (pump2) and reconstructs the conju-
gate pulse E2 (t).
records the volume hologram. The hologram is readout by EO(t) propagating in the opposite direction(pump 2). The reconstructed pulse E2(t) is the conju-gate of El(t).
We first explain the theoretical model used to de-scribe the holographic process; then we apply thismodel to our configuration and complete it by a thor-ough study of the dephasor. Finally we compare thequantitative theoretical expectations and the experi-mental results.
II. Theoretical Analysis of the Holographic Process
A. Recording
The holographic medium (here a photorefractivecrystal) records spatial variations of the refractive in-dex proportional to the intensity of the incident light.Intensity variations are induced inside the crystal bythe interference between the incident beam E0 and thespectrally dephased beam E1.
Fo(v) is the complex amplitude of the frequencyspectrum of E0, whose modulus is a(v - v). Theincident pulse is assumed to be Fourier transform lim-ited so Fo(v) = a(v - vo), F1 (v) = Fo(v) exp[i4'(v)J is thespectral complex amplitude after crossing the depha-sor responsible for (v). We used a Gires Tournoisinterferometer (GTI).8 Let &o,&i be the unit vectors ofpropagation of the two interfering beams, and r a pointof the holographic medium. The temporal signals at r= 0 are
[Eo(t) and El(t) denote now the complex amplitudes ofthe signals].
The complex amplitude of the resulting interferencepattern at r is given by
E(t,r) = E0 (t v )+ E1 ( - )
where v = C/n is the light velocity in the holographicmedium, and n is the mean refractive index assumed tobe a constant in the useful range of frequencies. Therefractive index variation is proportional to the timeintegrated intensity
An(r) f IEO + E1I2dt.
The spatially modulated part of An(r), which is theholographic term H(r), is easily shown to be
H(r) = JFO(v1)FI(v1) exp[-i2rv, (&O - &1)r d1 (3)
or
H(r) = exp[i27rvo (&
X JAO (t &oA*r) (t - tr) (4)
According to Eq. (4) the fringe pattern recordedinside the crystal can be interpreted as a phase gratingof spatial frequency zo/v, with its rules perpendicularto (&o - &1)r (see Fig. 4). The visibility curve
V(X) = JAO (t - -)A(t)dt, (5)
where x = (0 - al)r is the cross-correlation of thetemporal envelope amplitudes of the two pulses. Thisvisibility curve directly results from the impulsive na-ture of the interfering waves and marks the differencebetween impulsive and monochromatic holography.Recording the entire curve of visibility is necessary to
EO(t)
a(v-vo)E1(t)
a(v-vo)e i4(V)HOLOGRAM
Fig. 4. Visibility curve V(x) corresponding to the spatial indexmodulation An(x) recorded in the photorefractive crystal by the
interference of two pulses.
store all the information in opposition to classical ho-lography where any part of the hologram allows thereconstruction of the object. The expression for V(x)can also be written as
V(x) = a2(,u) exp[i(ih)] exp(2ire x)d.
It clearly appears that the phase law has been recordedby modulation of a spatial grating.
Note that all the information about 4'(v) is concen-trated along the direction (o - &1), so the transversalintensity variations of the light beam disturb very littlethe index grating visibility law in the configurationperformed here, where the angle between &o and & isvery small (a few degrees).
B. Reconstruction
Let us consider a spatially plane wave propagating ina direction & with a frequency spectrum amplitudeF(v) illuminating the hologram. The temporal signalat r inside the hologram is
E t- -)= F()ex1 2i7rv t- v)dv,
if the incident wave is not depleted by the diffractionprocess. We shall assume that in this case the holo-graphic term H(r) acts as a multiplicative factor for theincident amplitude at each point r of the hologram.The temporal signal obtained in a direction &' is givenby the integral
E'(t,) = JE (t -- + r dr-
Infinite limits can be used for the integral bound-aries because the whole intensity curve has been re-corded in the holographic material. To determine thetemporal and spatial properties of the reconstructedbeam we consider its frequency spectrum F'(v,&'):
F'(v,&') = fE'(ta') exp(-i2rvt)dt
= F(V)JH(r) exp i27rv (& - ]dr.
Using Eq. (3) for H(r) we obtain the Fourier spec-trum amplitude in direction &':
In our particular case the hologram is illuminated bythe reading pulse EO(t) propagating in the direction-&o. Thus we have to take & =-&o and F(v) = Fo(v) tocalculate the frequency spectrum and the direction ofpropagation of the reconstructed beam. We obtain inthe direction -&, an optical pulse with a frequencyspectrum F2 (v), such as
F2 (V) = F0 (v)F0 (v)FY(v),
that is,
F2 (v) = a3(v - O) exp[-i4'(v)], (7)
which corresponds to a temporal signal
A(t) = fF2(v) exp(i2irvt)dv.
This theoretical analysis shows(1) the expected spectral phase reversal and(2) an additional unwanted effect: a change of the
frequency spectrum modulus, which becomes a 3 (v -
vo) in the conjugate beam.Therefore, the temporal inversion described by (2)
for a single change of the phase sign cannot be exactlyobtained. We shall take into account the spectralamplitude variation for a quantitative comparison be-tween the temporal envelopes of the object and recon-structed pulses. But it is theoretically possible tosuppress this amplitude holographic effect and thusperform an exact temporal reversion by using a fre-quency spectrum much broader in the pumps than inthe signal to be reversed. It is difficult to do so, and inthe present experimental study all frequency spec-trums are identical.
111. Experimental Principles
The main goal of this work is to verify as quantita-tively as possible the theoretical predictions presented
in the last section for reconstructed pulse temporalenvelopes. Therefore, we have to check
the healing of the conjugated pulse passing againthrough the dephasor, and
the temporal inversion of the conjugate pulse enve-lope.
Neither the phase law 4(v) nor the temporal enve-lope shape is directly measureable. Thus the differentpulses will be characterized by their amplitude correla-tion functions with a reference pulse, which is theincident pulse E0 (Sec. III.A). The different steps ofthe quantitative verification are as follows:
(1) Introduction of a known phase law D(v) by aninterferometric process using the GTI, which is de-scribed in Sec. III.A. The shape and amplitude corre-lation function for the resulting pulse can be easilycalculated and compared to experimental curves.
(2) Determination of the analytic expression of theamplitude correlation functions of the reconstructedpulse after and before passing again through the GTI(Sec. III.C) using the theoretical results obtained inSec. II.
(3) Realization and readout of the hologram; com-parison of the theoretical and experimental amplitudecorrelation curves.
A. Characterization of the Pulse Envelopes
Practically, the amplitude correlation function oftwo optical pulses can be obtained as the visibilityfunction of the interference pattern produced by thetwo pulses to be correlated (see Sec. II.B). This isachieved in the following way:
The optical beam corresponding to Eo is collimatedon a moving mirror MO, whose displacement gives avariable optical path . After reflection on a beamsplitter B1, it is adjusted in exact coincidence with thebeam corresponding to the studied pulse (Fig. 5). Thetwo coincident parallel beams are focused by a lens Lon a small hole (50 ALm) placed before a photodiode.The amplitude of the interference signal for some giv-en is
The photodiode integrates with respect to time theintensity IS(tb)J2. It generates a signal whose modu-lated part with respect to is
IMM = V(a) cos[2irvO5 + (PJ,
where V(6) is the modulus of
JAl(t)Ao (t - b dt,
which is the expected amplitude correlation functionof the pulse envelopes.
The diode is connected to an analogic electronicdevice. This system enables us to obtain V(W, which isdirectly drawn by a plotter.
It is well known that such a method is only sensitiveto the spectral phase difference between the two pulsesto be correlated. In particular, the amplitude autocor-
relation function of a given pulse does not depend onits spectral phase law. It gives information only aboutits coherence time and not about its actual duration.But in the same way the holographic process reversesonly the phase difference. To achieve a real phaseconjugation mirror, it is necessary to use Fourier trans-form limited pulses as pump beams.
The same reference pulse E0 is used for holographicrecording and amplitude correlation analysis. Itsoriginal phase does not matter, and the method is verysuitable for demonstrating the holographic effect ofphase reversal, whatever the incident pulse phase.
B. Achieving a Particular Phase Law: the Gires TournoisInterferometer
1. Principle of the GTIThe GTI8 is a multiple beam interferometer com-
monly used to achieve linear chirp compensation inpulse compression experiments. It is used here inquite different conditions.
Its principle is illustrated in Fig. 6. Two parallelplates, or, in our setup, the two faces of a parallel plate,have reflection coefficients r2 = 1 and r, = r, respec-tively. For a parallel monochromatic wave reflectedon such a device, the complex amplitude at infinityresults from the coherent addition of the beams succes-sively reflected on the two parallel faces. The backface being totally reflective, the modulus is alwaysequal to 1. But the phase varies with the frequency vor the angle of incidence. The phase law 4(v) inducedby a GTI for a given angle of incidence 0 (which will betaken equal to 0) is
exp[i4(v)] = -r + t2 exp(iap)[1 + r exp(iXp) + r exp(2iso) + ... ],
(8)
where so = 27rvr, r = (2ne cos0)/C, t2 = 1 - r2, n is the re-fractive index, and e is the thickness of the plate. Weused as the GTI a parallel plate of silica 200 /m thick,which induces a delay r of -2 ps for a normally incidentbeam. Figure 7 shows the derivative of this phase law.Its periodicity is Apo = 1hr.
The effect of such a phase law on a particular pulsedepends on the spectrum width and mean frequencyvo. If the GTI is used for linear chirp compensation,the spectral width of the pulse to be corrected has to besmaller than the frequency interval 3v correspondingto the linear part of the curve and <<Avo. The proper-ties of the frequency spectrum amplitude of our pulseshown in Fig. 7(b) are derived from the experimental
a (- .vo)
(b)
_Av
VO V
Fig. 7. (a) Derivative of the GTI phase law and (b) frequencyspectrum of the pulse.
amplitude autocorrelation curve shown in Fig. 8(a).By comparing it to a theoretical Gaussian, one canadmit a Gaussian analytic representation for the pulseenvelope.
If At is the halfwidth at l/e height, this analyticformula is
exp[ (t)2]
for the pulse envelope, and consequently,
e[ (t )2]
for its autocorrelation function drawn in Fig. 8(a) (Wi= 2l2At). Thus At can be directly measured on thiscurve and is equal to 0.7 ps. We can deduce Avi byAvi(2At) = 4/7r, using the Gaussian function propertiesand the ratio
Avi 2r
AvO irAt
Thus Avi is equal to -2Avo. It is not possible to repre-sent as usual the phase law by the polynomial develop-ment of powers of (v - v0) 2 .
2. Temporal Envelope and Amplitude CorrelationFunction of the Pulse after Reflection on the GTI
The complex amplitude of the pulse envelope can beobtained as the Fourier transform of the complex fre-quency spectrum amplitude, which is a(v - vo)exp[i-I'(v)] after impinging on the GTI. In this case itis easier to calculate directly the reflected pulse tempo-ral amplitude as a sum of successively reflected pulses:
where Eo(t) is the Fourier transform of a(v - vo) and soo= 27rvor is a phase parameter tunable between 0 and 27rby adjusting the mean frequency vo, i.e., by translatingthe frequency spectrum relative to the phase law func-tion.
Fig. 8. Theoretical and experimental correlation of the (a) incident, (b) GTI reflected, (c) inversed, (d) healed pulse.
The resonant temporal envelope Al(t) is
Al(t) = -rAe(t) + t2 exp(iso)A(t - r)
X [1 + r exp(iq')A(t - 2 r)
+ r2 exp(i2<0)A,(t - 3r) + ... II. (10)
AO(t) is the initial temporal envelope whose analyticexpression is
eXp[ (t) 2]
To calculate the correlation function of A1(t) withAO(t), let us recall that the Fourier transform of thecross-correlation function is the product of their re-spective spectrum amplitudes, i.e., a2(v - vo)exp[i'1(v)].
Thus Ao(t) has to be replaced in Eq. (10) by
x[_ (Tt _
which is the Fourier transform of a2(V - o).Nevertheless, using the only possible adjustment of
po and the given values for the reflection coefficients,the experimental curve cannot be correctly fitted.Mistaken values of the reflection coefficients andlosses by diffusion on the surface can explain thisdiscrepancy. To obtain a better agreement with theexperimental curves we replace r by r r2 inside the
brackets, t2 by t 2 * r2, and adjust the parameters. Thebest fit, which corresponds to r2 = 0.45 and r2 = 0.904,is drawn in Fig. 8(b).
The remaining differences between theoretical andexperimental curves are not larger than for the auto-correlation curves drawn in Fig. 8(a). They can bejustified by experimental incertainties. Of course, thesame values of the adjustable parameters are used forcalculating the reconstructed pulse amplitude correla-tion function.
C. Determination of the Reconstructed Pulse AmplitudeCorrelation Function
The frequency spectrum amplitude of the conjugatepulse is a3( - vo) exp[-i4Ž(v)], as in Eq. (7). TheFourier transform of its cross-correlation with the ref-erence pulse is a 4(p - vo) exp[-i4'(v)].
We shall use Eq. (10) by replacing AO(t) by
exp[- (t)]the Fourier transform of a 4(V - vo) [see Fig. 8(c)].
Concerning the reconstructed pulse healed by pass-ing again through the GTI, the amplitude autocorrela-tion is simply
exp[_ (t)]
[see Fig. 8(d)]. In our previous work about pulse com-
Fig. 9. General experimental apparatus. Bo splits the incidentbeam. The part sent to the moving mirror Mo is the reference beamfor pulse analysis and is directed to D by means of B 5. B2 extractsthe reading beam which is directed toHbyMl and M2. B3 separatesthe reference beam for hologram recording which is directed to H byM3. B4 splits the incident (or reconstructed) beam. After collima-tion on M4 (or G) the reflected beam can be directed either to H (for
holographic recording) or D by B5 (for analysis purpose).
Fig. 10. Input pulse analysis. I0 is the initial pulse reflected byM4,and I, is the pulse spectrally dephased by reflection on G. They are
directed to D by B5 and interfere with Io coming from Mo.
H
Fig. 11. Hologram recording. Io and I, interfere at H.
pression 1 the same result is obtained for the directlycorrected pulse.
Obviously, the ratio between the width at le ofcurves Wh and wi in Figs. 8(d) and (a), respectively, istheoretically equal to 12.
IV. Experimental Achievement
The source is a dye laser pumped by an argon-ionlaser. The dye is rhodamine 110, and a two-platebirefringent Lyot filter is put inside the cavity. Apulse coherence time of about a picosecond is generat-ed with an output power of 10 mW at an 80-MHzrepetition rate. The average pump power is -700mW. The wavelength range is 530-550 nm, suitablefor the good sensibility of the holographic material.The Lyot filter allows adjusting the mean frequency ofthe pulse and consequently the phase (po of Eq. (10) tochange the shape of the GTI reflected pulse.
The holographic material is an iron-doped lithiumniobate crystal (LiNbO3:Fe). This photorefractivematerial records the intensity modulation produced byinterference as refractive index variations. The crys-tal has a thickness of 6 mm and is cut perpendicular tothe optical axis. In this configuration, the main trans-port process is the diffusion of the photoelectrons gen-erated by the sinusoidal spatial variations of light in-tensity. (The spatial interfringe is -0.1 Mm.)
The experimental setup is shown in Figs. 9-12.Mirrors are denoted by Mo,M,M 2,.. ., and beam split-ters by B1,B2, .. . ,G is the Gires Tournois interferome-
Fig. 12. Readout and reconstructed pulse analysis. Io is the read-ing pulse, I2 is the reconstructed pulse. After collimation on M4, I2 issent to D by B5. After reflection on B4 and collimation on G, I2 ishealed and becomes I3, which is sent to D by B4 and B5 I and I3
interfere with Io coming from Mo.
ter, H is the holographic crystal, and D is the analysisdevice. 10,11. . ., denote pulses of different shape.
Figure 9 shows the complete apparatus. Bo splitsthe incident beam. The very small part sent to themoving mirror Mo is the reference beam for pulse anal-ysis. It is directed to D by means of Bl. B2, a halfbeam splitter cube, extracts the reading beam which isdirected to H by Ml and M2. B3 separates the refer-ence beam for hologram recording, which is directed toH by M3. The three mirrors, M1,M2,M3, permit ad-justment of the parallelism and lateral superpositionof the counterpropagating beams (pumps or reference
and reading beams of the holographic process). B4splits the incident, or reconstructed, beam. After col-limation on M 4 (or G) the reflected beam can be direct-ed to either H (for holographic recording) or D by B5(for analysis purposes).
Figures 10-12 specify the different steps of the ex-periment. Figure 10 shows the analysis of the inputpulses Io and I,. Io is the initial pulse reflected by M4and I, the pulse spectrally dephased by the GTI. Theexperimental amplitude correlation curves are shownin Figs. 8(a) and (b). Figure 11 shows the hologramrecording. I is the pulse coming from G. G has to beadjusted perpendicular to the incident beam, b(v) be-ing strongly dependent on the angle of incidence. Ac-curate adjustments of B3 and M 3 are needed to achievethe exact coincidence of the beams inside the crystal.Of course, adjustments of the optical path lengths arenecessary to localize the visibility curve inside thecrystal with an accuracy of a few millimeters. Figure12 concerns the readout process and the reconstructedpulse analysis. Io is the reading pulse, and I2 is thereconstructed pulse. After collimation on G, I2 ishealed and becomes 13, which is analyzed. The experi-mental curve is shown in Fig. 8(d). After collimationon M4, the time reversed pulse I2 is sent to the analysisdevice: the experimental curve is shown in Fig. 8(c).
V. Discussion
The main results are shown on Figs. 8. Owing to theasymmetric shape of the pulse envelope, the compari-son between Figs. 8(b) and (c) illustrates well the tem-poral reversal (already noticed in Ref. 9). But themore important and new result is the good quantita-tive agreement between theoretical and experimentalcurves [see Figs. 8(c) and (d)]. The validity of thetheoretical expression for the reconstructed pulse fre-quency spectrum is thus demonstrated. The differ-ence between the reconstructed pulse shape and theexact temporal reversal of the initial disturbed pulseshape is entirely explained by the change in the spec-tral amplitude at the reconstruction. Furthermore,the perfect healing of the pulse after the return tripinto the GTI and the good value of the ratio betweenthe Gaussians widths wi and Wh of Figs. 8(d) and (a)(very close to the theoretical value 2) attests to thecorrect inversion of the spectral phase law.
Our theoretical model is built on the two followingassumptions:
The refractive index variation is a linear function ofthe fringe intensity modulation; i.e., the index varia-tion has not reached its saturation value.
The reading beam is not depleted, i.e., the efficiency,and the refractive index variations are small.
Let us now specify our particular conditions of effi-ciency and refractive index variations. The efficiencycan be experimentally measured. To evaluate the cor-responding index variation we have to establish therelationship between efficiency and maximal refrac-tive index variation, taking into account the spatiallyvariable modulation of An.
Let us consider an elementary grating of thicknessAz. Assuming &° - &2 ,Az = 2Ax. After Kogelnik,10
the intensity efficiency for such a grating is
- th2 (7AnAz) (7rAnAz ) 2
This term is assumed to be very small. The contri-bution to the reconstructed signal of this elementarygrating illuminated by the reading beam is
rAn(x)Ax x A.E 2 = ex -2iirvo x )Ao (t + - exp2irvo (+ v
and for the whole hologram
E(t) = exp(2irvot) A fAn(x)A0 (t + )dx.
Let AN be the maximum refractive index variation
An(x) = ANV X
where V(x/v) is the normalized visibility function (see'Sec. I), which is for two identical pulses
A, (t - x )A,(t)dtfAo(t)12dt
Let A(t) be the reconstructed pulse envelope.experimental efficiency is obtained as
J A(t) 2 dt
JIAo(t)I2dt
butxA(t) = ANy + tr) V(r)dr withT =--
2X f ~~~~~~~V
Here -q can be calculated and found to be
(=rANvAt)2 ifr_) -
The
(11)
Figure 13 shows a sample of the experimental curvea. It is obvious that mechanical instabilities prevent aregular increase in efficiency with respect to the re-cording time exposure. Nevertheless, using the onlyregularly increasing parts of this experimental curve,we deduced a corrected efficiency curve (b). Curve cshows the corresponding AN derived from Eq. (11).
The maximum efficiency is 5 X 10-3 (<1%); thecorresponding AN is 1.2 X 10-4 with a useful exposuretime of 40 mn. Let us recall that the grating periodinside the crystal is -0.1 gm. Relative displacementof the fringe pattern with respect to the grating alreadyrecorded partially erases the hologram and can explainthe decreasing parts of curve a. Curve c shows analmost linear increase of AN with respect to the timeexposure, and we can reasonably confirm that the satu-ration value has not been reached.
Due to the mechanical instabilities and the low pow-er level of the dye, we could not obtain very high
Fig. 13. (a) Experimental, (b) corrected efficiency curve, and (c)
corresponding refractive index variations.
efficiency, and we could not reach the range where ourtheoretical model is no longer valid.
VI. Conclusion
We have shown the ability of holography to performphase conjugation for any spectral phase law c1(v).
Thus, compared with nonlinear optical techniques,holographic methods appear to give the more generalresponse to the problem of phase conjugation for tem-poral signals.
We have specified the conditions for achieving anexact time reversal of a picosecond pulse. The pumppulses have to be Fourier transform limited, and theirfrequency spectrum is broader than that of the probepulse. These theoretical conditions are not fulfilled inthe present work, but experimental discrepancies to aperfect time reversal are exactly predicted by our theo-retical model, thus attesting to its validity.
The main limitations of the holographic methodsconcern the duration of the pulse to be reversed. Itsmaximum duration is limited by the crystal dimen-sions, which have to be large enough to record thewhole visibility curve (-50 ps for a LiNbO3 crystal of 6-mm thickness and 2.5 refractive index). For veryshort pulses with a very broad frequency spectrum thelimitation results from the frequency range in whichthe holographic medium sensitivity remains constant.
The diffraction efficiency is always very small (b-a1%for a few picoseconds in our experiment). It is propor-tional to the square of the pulse duration for a fixedmaximum refractive index modulation. To performthe time reversal of very short pulses, it is necessary touse materials with higher electrooptic coefficients.
Due to the large buildup time of the LiNbO3 we hadto use a two-step holographic process. Our next workperforms the same experiment in real time holography
using a crystal of BSO. One can expect that the sametheoretical analysis remains valid, the response time ofthe BSO still being much larger than the pulse dura-tion.
The authors would like to acknowledge helpful dis-cussions with F. Gires and P. Lavallard.
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