Information erasing in the phenomenon of long-lived photonecho
N. N. Akhmediev
Institute of Physical Problems, Moscow 103460, USSR
Received February 20, 1990; accepted April 22, 1990
The possibility of erasing information in an arbitrary cell of an optical memory device based on the phenomenon oflong-lived photon echo is theoretically considered. Optimal conditions for a maximal number of write-erase cyclesin a given cell are obtained, and an experimental setup for observing this effect is proposed.
The phenomenon of long-lived photon echo (LPE) hasrecently attracted much attention.'- 8 This intereststems from the possible application of this phenome-non in high-capacity optical memory devices9'10
(OMD's). The LPE phenomenon allows us to realizepractically all functions of on-line nonarchival OMD's:high-capacity, high-speed read-write functions, etc.Moreover, it allows us to write images,4 to obtain phaseconjugation at readout,5 and to write and read com-posite pulse envelopes.6' 7"10
At present many problems crucial to the construc-tion of an operating device have been solved. Thepossibility of multiple information readout withoutrefresh has been shown theoretically" and experimen-tally.7"12 Resonant media with storage times of theorder of tens of hours and more have been investigat-ed.6'7 The Eu3 + impurity ions in different host crys-tals are used as resonant media in such experiments.These investigations give more hope of creatingOMD's based on the LPE phenomenon. Neverthe-less, up to now it has not been clear whether it ispossible to erase information selectively in an arbi-trary memory cell. The only method of erasing infor-mation considered to date is to heat the sample totemperatures above that of liquid helium. But it isclear that this will erase the entire volume of a reso-nant medium. It was pointed out for the first time inRef. 13 that it is in principle possible to erase informa-tion with random access by using LPE signal suppres-sion.
In this Letter the effect of LPE signal suppression isshown in more detail, and a simple technique for ran-dom access information erasing is shown. First, sev-eral leading considerations are cited. The effect ofaccumulated photon echo has been investigated.' 4"15
The essence of this effect is the writing of informationin a resonant multilevel medium as a frequency modu-lation of population by a continuous sequence of weakpulse pairs that appear successively, separated arbi-trarily in time. Obviously, all pairs of pulses must beidentical to each other up to the phase difference be-tween pulse carriers. In this case the population mod-ulation in the ground state will increase slowly aftereach pair of pulses, and the LPE signal intensity willalso increase up to an experimentally observable level.
The question arises as to whether the echo will be
accumulated in all cases. Is it possible that the echosignal will be suppressed rather than gained by thenext pair of pulses? It can easily be shown that thedepth of frequency modulation of the population willdecrease after the influence of the next pair of pulses ifthe phase difference between carriers of these pulses isopposite to the phase difference of the previous pair ofpulses. In this case the contribution to the frequencymodulation of the ground state by this pair will beopposite in phase to the frequency modulation createdby the previous pairs. As a result, the frequency mod-ulation of the population will decrease or even disap-pear completely and the echo signal from the mediumwill be suppressed.
In the case of the usual LPE (in contrast to theaccumulated photon echo), more-powerful pulses areused so that a single pair of pulses can create thepopulation modulation deeply enough to permit ob-servation of the LPE signal. It is clear that the infor-mation written by some pair of strong pulses can alsobe suppressed by an additional pair of pulses irradiat-ing the same region of the resonant media. To showthis, the simplest model of a resonant multilevel medi-um in which the LPE effect exists is used. This is ahost crystal with organic impurity molecules having along-lived triplet state.3 The scheme of energy levelsis shown in Fig. 1, in which I1) and 12) are the resonantlevels of energy transition and 13) is a long-lived tripletstate. The spin sublevels of 13) will be neglected.
For calculational convenience the method of solvingthe density-matrix equation that is given in Ref. 3 and
12> -
hw
I1>
k. ..k 23
, 21
T
k31
TI
Fig. 1. Scheme of energy levels used in the calculations.
applied in Ref. 11 is used to analyze the effect ofmultiple information readout. The method consistsof transforming from the density-matrix equation tothe Liouville equation, the solution of which allows usto express the excitation by short pulses as well asrelaxation processes simply as the action of a linearoperator on a column vector consisting of density-matrix elements that we are interested in. For brevitythe notation and the approximations, and the Liou-ville operators, of Ref. 3 are used. This allows us towrite out the final formulas immediately.
In our problem we can operate with a column vectorconsisting of only five elements, p = Jpln, P12, P21, P22,
P331. Let us consider a resonant system in thermody-namic equilibrium with a host crystal before excita-tion by an optical field, such that the single nonzerodensity-matrix element is Pl = 1. (We assume thatthe time separation -r, between the writing pulses isshort relative to the longitudinal relaxation time T,and comparable with or shorter than the transverserelaxation time T2; see Fig. 2.) In this case the diago-nal elements of the density matrix at the momentimmediately after the influence of pair of pulses 1 and2 are of the form
pil = 1/2[(l + cos s01 cos f 2) - sin pl sin 'P2
X exp(-r/T 2 )cos(Akwr + AO5. + AO2T.)],
P22 = '/2[(1 + COS (P1 cos 'P2) + sin (,l sin (P2
X exp(-r/T 2)cos(Akwr + A,,, + AQ-r)],
(1)
(2)
where (,p and 'P2 are the areas'6 of the first and secondpulses, respectively, AO is the frequency deviationfrom the center of the inhomogeneous line, k, and k2are the wave vectors, Xl and 02 are the phases of thetwo pulses, Akw = k, - k2, and AO, = -l - 02, in whichthe subscript w denotes the relevance of correspondingvalues to the pair of writing pulses.
The forms of the nondiagonal elements of the densi-ty matrix are not essential because they are equal tozero after a time interval tw >> T2 after the secondpulse. Moreover, we consider time intervals tw >> T,because we are interested in information storage fortimes longer than the time of pure optical relaxation.In this case the matrix element P22 = 0 as well. For theground state we have
p1l = pll + [1 - 1 exp(-h3 1tw)]P2 2
= 1 - exp(-kh3tw)(1 - cos S cos 'P2)2
- exp(-k 3 ltw)sin Sal sin 'P2 exp(-i-/T2 )2
X cos(Akwr + Abw + AOrw), (3)
where a k23/(k31 + k23 - k3l), and the decay rateconstants kij are shown in Fig. 1.
One can see from Eq. (3) that the population of theground state is frequency and spatially modulated.This grating is stored during a period tw- kh31- andleads to generation of an echo signal3 after illumina-tion of the medium by a third (readout) pulse [Fig.2(a)]. As mentioned above, the grating can be sup-
1 2(a) ri M
(b)
R SI-
1'."i~ ~ _ I I__ n t
1 2 3 4 R
v t:<C-~ I +- t- > e14-C4 I<- t - --- ~ I I t
Fig. 2. Sequence of pulses (a) at information writing andreadout and (b) at information erasing. 1 and 2 are thewriting pulses, 3 and 4 are the erasing pulses, R is the read-out pulse, and S is the echo signal.
pressed by heating the sample to temperatures higherthan that of liquid helium. Heating leads to a sharpincrease in the decay rate constant k3 l and hence to thereturn of the resonant system from the triplet state tothe ground state, yielding Pl1 = 1. However, in thiscase the information will be lost in the entire crystal.
The other method of echo-signal suppession pro-posed here is to write new information with a phase offrequency-modulated population opposite to that ofthe existing information, thus superimposing two fre-quency population modulations that have oppositephases such that the full modulation disappears.Suppose that the radiation of a new pair of pulsesidentical to the first one [Fig. 2(b)] but with a phasedifference Acke = AIb, + nr impinges upon the samespatial memory cell. It can easily be shown that, un-der the approximations used above, the influence ofthis pair and the optical relaxation during the time te> T7 after the pulses mathematically are equivalent tothe factorization of density-matrix element Pl1 withrespect to the value x, which is
x = 1 - exp(-k 31tQ)(1 - cos 'P3 COS 'P4)2
- exp(-k 3it,)Sin ~03 sin 'P4 exp(-T/IT2)2
X cos(Aker + Ache + Are), (4)
where 'P3 and 04 are the areas of the erasing pulses, te isthe time interval after the erasing pair, and the sub-script e relates to the parameters of the erasing pair.The second nonzero matrix element p33 has no influ-ence on the echo signal and is not written here. Thus,after irradiating the sample by an erasing pair ofpulses, the matrix element Pi, takes the form
The echo signal after irradiation by the readoutpulse (denoted by R in Fig. 2) is defined by termsproportional to cos(Akwr + A,,, + AQwr,) and cos(Aker+ AObe + AMTe). The conditions of erasing are identi-cal to the conditions of writing, namely, 'Pj = 'P3, 'P2 ='P4, Akw = Ake, and -r = -e, but the phase differencesbetween the carriers for writing and erasing pulsesdiffer by 7r: AOke = AO,,W + 7r. We take the relaxationfactors exp(-k 3 itJ) and exp(-k 3 lte) to be equal tounity because of the condition te << k 31 -'. In thiscase the terms in Eq. (5) with frequency modulationare canceled. The remaining terms do not depend onfrequency and have no influence on the echo signal.Thus, by irradiating the given memory cell by a pair ofpulses identical to the writing pair but with a phasedifference of 7r, one can suppress the LPE signal total-ly.
Strictly speaking, the procedure described is notliterally erasing because recurrence to the initial stateof the cell with plu = 1, which was the case before anyirradiation, does not occur. After the erasing proce-dure the probability of finding the system in theground state does not equal unity but differs from it byC _-1/2(2 - cos 'P1 cos P2 -cos 'P3 COS 'P4) owing to thetriplet-state bottleneck effect. Hole burning in theinhomogeneous line occurs with a width equal to thespectral width of the laser light. Hence, the secondpair of pulses paints over the information in the mem-ory cell rather than erasing it. Therefore, for largerelaxation times k 3 1-' it is possible to write and eraseinformation in a particular memory-cell-limited num-ber of times. The number of cycles depends on therelation between k2l and k23 and on the areas of thewriting and erasing pulses.
As an example, consider the case in which the areasof all pulses are equal to each other. In this case C _ Asin2 s0j. If 1 - 0.1 (Ref. 9) and so - 0.1, then thenumber of write-erase cycles is of the order of 103. Bychoosing the most suitable resonance molecules andoptimizing the pulse parameters, we can increase thenumber of write-erase cycles up to 105 or more.Hence, the described erasing process can be used wellin OMD's. Thus information erasing in OMD's basedon the effect of LPE can be done by simply introduc-ing some kind of additional device for monitoring the
L
1127
I8 6
L7
L-3
I _EE~~
Fig. 3. Proposed experimental setup for observation ofLPE and information erasing from a given memory cell. 1,The writing laser; 2, the erasing laser; 3, the resonant medi-um; 4, the optical delay line; 5, the electro-optical crystal; 6,the photodetector; 7, 8, the beam splitters.
phase delay in the optical path of one of the pulses inthe pair.
For experimental observation of the erase processthe setup shown in Fig. 3 can be used. In this scheme1 is a writing laser. The pulse produced by this laser issplit off by beam splitter 7, producing two pulses prop-agating along two different optical paths. One of thepulses irradiates the resonance medium directly aftertransmission through beam splitter 7, whereas the sec-ond pulse irradiates the sample after transmissionthrough the electro-optic crystal 5 and optical delayline 4. Electro-optical crystal 5 with transparent elec-trodes permits monitoring of the phase of this delayedpulse. For simplicity, the deflectors and focusing ele-ments are not shown.
During the writing of information the voltage on theelectrodes of crystal 5 is absent. This corresponds to adefinite phase difference between the pulses. If oneneeds to erase the information in a given memory cellone must apply the fixed voltage on crystal 5, whichgives an additional phase difference of 7r between thepulses. Then the influence of the second pair ofpulses on resonance medium 3 erases the informationwritten by the first pair. The erasing of, e.g., writtenimages and pulses of complex form can be done in thesame way. In these cases the erasing pair of pulsesmust also be equal to the writing pair, with the excep-tion of the phase shift r between carriers.
The author thanks D. Heatley for valuable discus-sions and help in the manuscript preparation.
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