Optial transistor ation by nonlinear oupling of stimulated emission and oherent sattering David L. Andrews * and David S. Bradshaw Nanostrutures and Photomoleular Systems, Shool of Chemistry, University of East Anglia, Norwih NR4 7TJ, United Kingdom ABSTRACT In the pursuit of improved platforms for omputing, ommuniations and internet onnetivity, all-optial systems offer exellent prospets for a speed and fidelity of data transmission that will greatly surpass onventional eletronis, alongside the antiipated benefits of redued energy loss. With a diverse range of soures and fiber optial onnetions already in prodution, muh urrent effort is being devoted towards forging optial omponents for signal swithing, suh as an all-optial transistor. Ahievement of the desired harateristis for any pratiable devie an be expeted to depend ruially on the engagement of a strongly nonlinear optial response. The innovative sheme proposed in the present work is based upon a third-order nonlinearity its effet enhaned by stimulated emission operating within a system designed to exploit the highly nonlinear response observed at the threshold for laser emission. Here, stimulated emission is strongly driven by oupling to the oherent sattering of a signal input beam whose optial frequeny is purposely off-set from resonane. An eletrodynamial analysis of the all-optial oupling proess shows that the signal beam an signifiantly modify the kinetis of emission, and so lead to a dramatially enhaned output of resonant radiation. The underlying nonlinear optial mehanism is analyzed, model alulations are performed for realizable three-level laser systems, and the results exhibited graphially. The advantages of implementing this all-optial transistor sheme, ompared to several previously envisaged proposals, are then outlined. Keywords: optial swithing, optial transistor, all-optial proessing, laser ation, nonlinear optis, stimulated emission, quantum eletrodynamis, oherent sattering * david.andrews@physis.org 1. INTRODUCTION The advantages of all-optial systems for information proessing, omputing, ommuniations and internet onnetion are familiar and well-rehearsed. When ompared to urrent eletroni implementations, numerous antiipated benefits an be identified inluding redued energy losses, and a greatly inreased speed and fidelity of data transmission. As the optial ommuniations industry gears up for the seond deade of a millennium already transformed by informatis, muh effort is being devoted towards devising new optial omponents suh as an all-optial transistor. Like its eletroni anteedent, the underlying priniple of suh a devie is to effet the swithing or amplifiation of a soure, under the ontrol of a data signal input. Amongst many novel shemes for the realization of suh a devie, one moleular devie based on saturated absorption has very reently been proposed by Hwang et al. 1 Other, all-optial swithing systems that have reently been proposed are based on eletromagneti indued transpareny, -4 the optial Kerr effet, 5,6 nonlinear transmission through oupling with surfae plasmons, 7-9 and beam filament rotation by appliation of a signal beam. 10,11 A partiular ommonality of priniple an be found in eah of these; the engagement of a strongly nonlinear optial response. However, most suh shemes are tailored for appliation to a speifi ombination of material and optial system. The sheme that is proposed in the present work is based upon a reently identified third-order optial nonlinearity, its effet enhaned by stimulated emission. This mehanism is brought into play in a system that is designed to exploit the highly nonlinear response of a system at the threshold for laser emission. In ontrast to the work of Hwang et al., the role of the probe beam, whose optial frequeny is purposely off-set from resonane, is to passively engage by oherent forward sattering with resonant stimulated emission. Detailed analysis shows that this beam, ating as the input signal, an modify the kinetis of emission and so lead to an enhaned output.,13 Signifiantly, optially ontrolled
fluoresene is a proess that is not limited to operation with any one material; with a judiious hoie of signal optial frequeny, operation of the mehanism is viable in any suitably nonlinear medium. Moreover the symmetry onditions for the existene of the appropriate nonlinearity are not unduly restritive. One the priniple is proven, a variety of new optial data handling omponents an be antiipated to emerge. The following results of alulations, for three-level laser systems, highlight the signifiant potential for devie development.. LASER-MODIFIED EMISSION First, we onsider the general priniple of laser-modified, optially ontrolled emission. In general, a single matterphoton interation is responsible for spontaneous emission, as depited in Fig. 1(a), and the standard theory for the proess is aordingly developed through first-order time-dependent perturbation theory. Even-order perturbative terms vanish, and higher order odd-rank orretion terms are insignifiant when no other light is present they simply denote self-energy orretions. However this is no longer the ase when the eletronially exited fluorophore is subjeted to a throughput of off-resonant, pulsed laser light. Then, signifiant non-zero orretions an arise even at relatively modest intensity levels it being assumed that the wavelength of the latter is hosen to prelude stimulated emission or any exitation of the material to higher eletroni levels. With a probe laser detuned to a region where the system is transparent, although in onsequene there an be no net absorption or stimulated emission, elasti forwardsattering events will our, photons being annihilated and reated into the same radiation mode. The throughout light, whih thus emerges unhanged, nonetheless passively engages by nonlinear oupling with the spontaneous emission, and the net effet is to modify the transition moment for eletroni deay. This mehanism, represented in Fig. 1(b), entails three matter-photon interations, i.e. third-order perturbation theory. (a) (b) Fig. 1. (a) Energy level representation for spontaneous emission. Eletroni states (and their vibrational manifolds) are signified by boxes, the wavy line is the radiative emission ( ) and the blak vertial arrow the deay transition: 1 and are the eletroni ground and exited states, respetively, the blak dot symbolizing a single matter-photon interation. (b) Same emission but engaging an off-resonant laser beam ( ) denoted by the horizontal dashed arrow; the open dot symbolizes two matter-photon interation (i.e. elasti forward-sattering). The radiant intensity of emission, I (power per unit solid angle), whih follows from the Fermi Golden Rule rate 14 multiplied by the energy of an emission photon, k, 15 is now determined from I = k' M (1) + M (3) M (1) and M (3) are the quantum amplitudes for the first- and third-order interation proesses, respetively, 3 and the density of radiation states is k V 8 d, where d signifies an element of solid angle for the emission. 16 The effets to be onsidered below now depend on the relative signs of the first- and third-order amplitudes; a ommon sign leads to emission enhanement, opposite signs its suppression. To proeed, the following is found for a given emission polarization;
4 k I + I ;, e e e e e e 8 0 i j i j 0 i j k l ijk l I 0 e iejekelemen ijk lmn 4 ;, ;,, (1) where ijk is a transition hyperpolarizability (nonlinear suseptibility) tensor, e and orrespond to a probe laser photon, and I is the irradiane of the laser probe. Furthermore, the normal deay transition dipole moment is designated by the shorthand notation 1 in whih 1 and denote the states of levels E 1 and E, respetively. In equation (1), the Einstein implied summation onvention for repeated (Cartesian) indies is deployed. The initial term on the right-hand side of (1) orresponds to spontaneous emission the usual one-photon transition, intrinsi to the system and independent of the probe laser beam while the last term signifies oupling of the elastially forward sattered probe beam with the spontaneous emission, overall a three-photon event. The seond term, linear in I, represents a quantum interferene of these two onurrent proesses. In general, it may be assumed that the leading term in (1) is non-zero, and the seond term a leading orretion. The key parameter within equation (1) is the nonlinear transition (inelasti) suseptibility, ijk, mediating radiative deay of the moleular exited state in onsequene of whih, the first frequeny parameter, registering the moleular deay, differs from the sum of those whih follow in the argument of. The expliit form of this tensor is determined from well-attested and reported methods, 15,17-19 and the result is given by: 1s 1 sr r s sr r i j i j k k ;, = ijk r s r Es Er Es E r s r1 s 1s sr r 1s sr r j i k k i j E E Es Er s r 1s sr r 1s sr r j k i k j i E E Es s r E r, () where is the signal beam frequeny, and the transition moments are defined in the same manner as ; r and s are intermediate states, and E xy = E x E y is an energy differene between two states. In passing, we observe that the nonlinear mehanism may alternatively be interpreted in terms of a dressing of the moleular states by the throughput beam, manifest in a modifiation to the E E 1 transition moment; although a different derivation method ensues, the same expression as equation () will emerge. The third-rank polar tensor () has non-vanishing elements if the produt of the initial and final state symmetries spans one or more omponents of suitable symmetry. In fat, any transition that is eletri-dipole allowed will also support a non-vanishing 3. OPTICAL TRANSISTOR ACTION We now fous onsideration on a typial three-level laser system optially pumped within a miroavity. The kinetis of emission are primarily determined by a pump rate R p driving population from the ground state E 0 into a metastable upper level E, lasing ation from E into E 1, and ultrafast relaxation from E 1 (Fig. ). Following Siegman, 0 the rate equations orresponding to the temporal behavior of the avity photon number, n, and the E population, N, are as follows: dn K n 1 N n, (3) dt 3
dn dt R nkn N. (4) p Here, the population of E 1 is assumed to be vanishingly small; K denotes the oupling oeffiient for the laser transition, whilst and signify the avity and population deay rate, respetively. Under steady state onditions, equations (3) and (4) may be solved to give the result; 1 p p 4 p R g p R g p R n. (5) where g rad and K rad p (in whih rad is the radiative deay rate and p is the number of resonant avity modes). The relaxation from E into E 1 is not entirely radiative, i.e. rad, sine non-radiative relaxation proesses (the exitation of lattie phonons et.) arise. For present alulational purposes, given that the level E deay rate subsumes (but is dominated by) the rate of radiative deay, a value of 5/4 is to be assumed for g in the absene of the offresonant input signal onsidered below. Proeeding from equation (5), employing typial values p = 10 10 8 and 10 s -1, the familiar vertial limb in avity photons at laser threshold emerges, as graphially illustrated by Fig. 3 (solid line). All-optial ontrol of suh a pumped ative medium may be ahieved by nonlinear optial engagement of the laser emission with stimulated elasti forward sattering of off-resonant (signal) laser pulses, effeting a modifiation to the dipole transition moment for the E E 1 laser transition. Fig.. Energy level diagram for the three-level laser system: blak solid arrows denote eletroni transitions, the wavy line emission ( ), and the dashed arrow, the off-resonant probe beam ( ). Blak and open dots symbolize single and dual matter-photon interations, respetively. Disrete energy levels are depited for simpliity. Returning to equation (), r and s now equate to either 0, 1 or in the three-level system of Fig. exept when preluded in ertain summations, as indiated in (). We may also suppose, and that these frequenies determine an offset, E E0, that is a small fration of the energy for a typial eletroni transition. Under these onditions the fourth term of equation () is dominant, delivering the following: 4
Fig. 3. Plot of log n, where n is the number of avity photons, against the pumping rate, Rp, for absent (solid line) and present signal beam; example irradianes of the latter are 10 11 W m - (dashed urve) and 4 10 11 W m - (dotted). Horizontal arrow illustrates movement of the lasing threshold to the left for inreasing laser intensities. The vertial dotted line represents a onstant Rp at whih an introdution of the signal beam produes above-threshold operation (denoted by the upper pair of horizontal dotted lines). 3 e ieje ;,. (6) k ijk E E To seure a more quantitative assessment of the generi sheme, it is expedient to assume that the relevant transition dipole moment omponents have broadly similar diretion and magnitude the latter now simply represented as. In alulations on speifi systems, this approximation an of ourse be surrendered for greater auray. It should be observed that both fators in the denominator of (6) have negative values, so that the resulting suseptibility is always positive and in onsequene leads to enhaned emission; under other onditions the suseptibility omponents may assume a negative value, representative of redued emission. On insertion of equation (6) into (1), typial values of I may be alulated for various signal beam intensities. As indiated in Setion, it is the seond term of (1) (linear in I) that represents the leading orretion. With this in mind, the degree of enhanement (or in other ases any suppression) of the emission an be measured by taking the ratio of the seond term against the first in equation (1); the orresponding parameter may be approximated as: I E E+ 0. (7) Returning to equation (1), it is lear that the variable g will be affeted by introdution of the input signal beam, sine the radiative deay rate, rad, and population deay rate,, both thereby suffer hange (but to differing degrees); the nonradiative deay rate, nr, an be assumed to be onstant. By simple manipulation, an expression for g is given by; g I 1Y I 1 Y I Y I, (8) 5
where YI rad I I and nr I 1 YI. With the previous ondition that g = 5/4 for I = 0, and adopting indiative values = 16 10-30 C m, E = 10-0 J and = 10-19 J, insertion of (8) into (5) generates the results exhibited in Fig. 3. It is learly evident that transistor ation with respet to the signal beam ours. For a onstant pumping rate at a level indiated by the dotted vertial line, the system operates below threshold when no signal laser present; however on the introdution of an off-resonant beam with an irradiane approahing 10 11 W m -, the devie output limbs by fourteen orders of magnitude, rising to sixteen if the signal input is doubled. 4. DISCUSSION It is interesting to ompare the sheme set forth above with the three-level proposal reently offered by Hwang et al. 1 These authors devised a system based on the variable transmission of a tunable probe, whih is a ontinuous-wave narrow bandwidth beam whose optial frequeny is sanned aross a 0-1 transition (Fig. 3). When the probe beam is oinident in time with ultrashort pulses from a dye laser ating as pump, their frequeny loked onto the 0- transition, the probe absorption loses intensity as the pump intensity is inreased (optial bleahing) eventually being observed to marginally inrease the on-resonane throughput intensity. The authors have interpreted this enhanement of the probe as amplifiation; logially the effet derives from stimulated emission of the transition between the two lower levels, level 1 being populated by rapid deay from level. A deay hannel from level 1 into a vibrationally exited sub-level of the ground state affords an optial output that an be registered against zero bakground. Fig. 3. Energy level sheme for the three-level laser system of Hwang et al.: double-headed arrow denotes eletroni transition due to a ontinuous-wave (and resonant) probe beam, upward arrow represents a pumping dye laser transition, downward arrow is a Stokes-shifted deay hannel, and dashed arrows denote non-radiative ultrafast relaxation (UR). In ontrast, the all-optial transistor system we propose offers several apparent advantages. First and foremost, offresonant ativation of laser emission is effeted by a throughput beam whih operates entirely passively; the latter beam experienes no loss or gain of intensity, and the optial output that is fully delivered into a zero bakground hannel. The reported mehanism offers ultrafast response with high repetition rate, high effiieny, and a straightforward experimental setup. Moreover, it is based on a priniple that is not limited to operation with any one speifi material; with judiious hoie of signal optial frequeny, the mehanism is viable in any suitably nonlinear medium. Our analysis paves a new pathway for all-optial swithing and amplifiation. The realization of a system suitable for implementing this mehanism is an entiing goal, whose ahievement will require the identifiation of systems for whih the key tensor parameters an be optimized; all the neessary theory is now delivered. ACKNOWLEDGEMENTS The authors are grateful to the Leverhulme Trust for finanial assistane. 6
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