TRANSVERSE ON-RESONANCE RESHAPING IN COHERENT PROBE AND PUMP SOLITONS ASYMPTOTIC EVOLUTION IN A THREE-LEVEL SYSTEM

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TRANSVERSE ON-RESONANCE RESHAPING IN COHERENT PROBE AND PUMP SOLITONS ASYMPTOTIC EVOLUTION IN A THREE-LEVEL

SYSTEM

F. Mattar

To cite this version:

F. Mattar. TRANSVERSE ON-RESONANCE RESHAPING IN COHERENT PROBE AND PUMP

SOLITONS ASYMPTOTIC EVOLUTION IN A THREE-LEVEL SYSTEM. Journal de Physique

Colloques, 1988, 49 (C2), pp.C2-463-C2-468. �10.1051/jphyscol:19882110�. �jpa-00227621�

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JOURNAL DE PHYSIQUE

Colloque C 2 , Suppl6ment a u n 0 6 , Tome 49, j u i n 1 9 8 8

TRANSVERSE ON-RESONANCE RESHAPING IN COHERENT PROBE AND PUMP SOLITONS ASYMPTOTIC EVOLUTION IN A THREE-LEVEL SYSTEM(')

F.P. MATTAR

Department of Physics, New York University, New York, NY 10003, U.S.A. and George R. Harrison Spectroscopy Laboratory,

Massachusetts Institute of Technology, Cambridge, M A 02139, U.S.A.

Rdsum6

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~'gtude d'un transfert dl;nergie en rgsonance entre une pompe et une sonde durant leur propagation dans un milieu

>

trois nivaux est rapportge. La longeur d'onde de la pompe est plus courte que celle de la sonde. Nous consid6rons une gvolution assymptotique o; l'effet de la nonlinearit; du milieu est cornpens; par celui de la diffraction. Le formulisme semi-classique est adopt;. Dans le but d'optimiser l'extraction bi-chromatique de ll;nergie les vitesses de groupe sont Ggales pour garantir le recouvrement des deux impulsions. Le ph&nomsne de la diffraction, donc des effets transverses, est essentiel pour 1'6volution des deux solitons. Le soliton associ6 L la sonde apparaEt tout d'abord suivi de celui associ6,; la pompe. Un dia- gramme de phase dgmontre le charactsre 5 valeur multiple nonlineaire de l'interaction.

Abstract

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The coherent frequency conversion between on-resonance pump and probe beams is examined in the physical situation where the medium nonlinearity balances the linear diffraction for each transition. Without diffraction neither soliton can occur. The paraxial-Maxwell Bloch formalism is adopted. To achieve optimally the energy transfer the pulse velocities must be equal to maintain maximum overlap throughout the propagation. The sequential evolution of probe soliton then pump soliton is reported in conjunction with pump depletion diffraction rigorously accounted for. Phase diagram of the pump and probe areas exhibit multi-value features.

I

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INTRODUCTION

The coherent on-resonance copropagation of two beams in a three-level system /I/ leads in the asymptotic regime to solitons. Previous coherent propagation analysis have included either two-field three-level in the uniform-plane-wave regime / 2 / or transverse effects in one-field two-level / 3 / with diffraction accounted for but never both. This study reports simultaneous treatment of both effects. The strongly overlapped pulse have considerably different input area. The pump area is 2n while the probe is .021~. Their ratio is 6. The pump evolves as a Self-Induced-Transparency (SIT) pulse /4/ while the probe is similar to superfluorescence emission 151. The probe builds up as the first order in 6 whereas its feedback onto the pump is of second order in 6. As soon as the probe grows the pulses loose their independence. The nonlinear-like matter equations are solved numerically in a self-consistent manner.

This research was motivated by swept-gain superradiance asymptotic evolution /6,7/, by obser- vation /8/ and theoretical analyses /8,9/ of a Raman pump soliton. Diffraction is identified as a key element in soliton formation and as the source of self-phase modulation and transverse energy flux /lo/. For long interaction lengths the probe (pr) and pump (p) beams (aDrll>>aD!L>>l), the calculations show a clear sequence of events: the probe experiences an initial buildup, gain saturation, and then an area stabilization which implies the creation of a soliton. During this time the pump depletes continuously. Further propagation results in probe decay and destruction of its soliton concurrent with a pump recovery similar to the anomalous Raman pump-depletion reversal

181.

This is followed by pump area stabilization, or 'pump soliton' assuming a sufficient optical thickness for the pump. We stress the orderly evolution of these events. The probe experiences pulse compression and beam blooming. The pump beam narrows.

For a probe soliton to evolve, the accumulated diffraction loss r E has to balance out the nonlinear amplification gt,. This condition requires a near-unity gain-Fresnel number F =g/u>l

g with g the gain g=a(cz ); a the Beer's length; z the pulse length and c=nr2/A where r is the

P P P P

beam width; otherwise, transverse self-lensing instabilities predominate and the beam quality

(l)~upported by ARO. AFOSR and NSF

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882110

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C2-464 JOURNAL DE PHYSIQUE

collapses. In addition several conditions must be satisfied for each transition: T2<<tc (with

t being the cooperation time) ; a=g/T2 ; z~=(cK)-' (zE is an escape time associated with the diffraction length K-') ; z = K T ~ / ~ = T ~ / F ~ ; and Z;=T,/~C (i. e., zz=zEzs). This asymptotic evolution into a solitary wave with a stabilized pulse-area /lo/ is equivalent to beam self- trapping /11/.

The soliton is defined as a pulse for which the time-integrated-area (THETA), time-integrated- energy f luence (ENER)

,

radially-integrated energy output power (OPOWR)

,

effective temporal length zeff (TAUEFF) and effective radial width peff (RHOEFF) do not vary appreciably over several optical thicknesses in the asymptotic regime.

2 - THREE-LEVEL ASYMPTOTIC CALCULATIONS

While achieving the balance between transition gain and diffraction loss for the probe, one finds that a considerable long cell is needed. The pump experiences a large optical thickness (a L > 7 ) . An SIT pulse on-axis fluence would have been enhanced by seven. However, in this

P

three-level analysis no enhancement occurs due to energy leakage to probe. Initially, the on-axis pump energy experiences an enhancement of 1.69; subsequently it subdues and depletes, its trend to self-focusing is quenched by the probe buildup. The probe displays the same z- independent on-axis area stabilization as its two-level counterpart /lo/. Its area reaches a peak of about 5n and then decays down to an asymptotic value of 47~ after some minor oscillations. The probe pulse area in normalized Rabi frequency units is twice as large as the pump.

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4 The probe to pump fluence ratio is eight while it was initially 1x10

.

The on-axis asymptotic stabilization of pump and probe areas is shown in graph a of Fig.1 while its three-dimensional distribution is shown in graph b. The temporal and transverse reshaping dynamics associated with the output power is shown in Fig. 2. Pump depletion is clearly exhibited. The peak probe output power is larger than that of the pump, while its pulse length is significantly shorter. Figure 3 shows that z ~ defined in terms of its ~ ~ , output power, and p eff, defined in terms of its fluence, saturate and stabilize over an extended optical thickness.

For the first time the interplay of pump depletion, diffraction

(no

self-similar radial profile) and on-axis asymptotic probe area stabilization has been demonstrated rigorously.

For a longer interaction region, appropriate relative phase variations take place. The pre- stage of anomalous pump recovery is set up. Probe stabilization ceases. The probe depletes its energy to the pump. The pump shows a depletion reversal then displays an asymptotic area- stabilization. The accumulated nonlinear action experienced by the pump has been compensated by its diffraction. A pump soliton is thus obtained at the pump frequency. This calculation is the on-resonance equivalent of the Raman soliton 181. The probe beam continues to be broader than the pump. The on-axis pump soliton area is smaller .45n than its initial '2n' value. It is in contrast to what: evolves for the probe area: from 0.0211 to 3.3111. This represents a significant gain for the probe.

The conditions which support a pump soliton destroy those for the probe wave and vice versa;

we cannot have both at the same time. In Fig. 4b the pump and probe areas are plotted versus p

and rl side by side. The pump depletes while the probe grows. After reaching an absolute peak the probe saturates, ceases to grow, decreases and oscillates about a quasi-steady-state value until it stabilizes. We want to emphasize that only when appropriate relative phase variations evolve does the probe depart significantly from its asymptotic value, depletes and experiences radial distortion. The probe and pump area profiles experience different rates of change (growth or decay) along q. In the left-hand graph, the probe area displays a well-defined and near-center stabilizatlon as a solitary wave. The pump, in the right-hand graph, is going through reductions in the depletion rate. Its rate of absorption diminishes until it vanishes and then reverses itse1.f. The pump begins to recover while the probe depletes. Afterwards, the probe behavior becomes erratic while the pump depletion is interrupted. The pump anomalously recovers. Its area stabilizes. Its beam forms a channel of essentially constant width. Both beams have departed from their input profiles.

The on-axis probe and pump areas exhibit a non-coincidence of the stabilizations. The two solitary waves, first probe then pump, occur in sequence. The succession of probe amplification to absorption leading 01. probe area oscillations are drastic. The off-resonance theory leads to only a pump soliton while this on-resonance theory leads to both probe and pump solitons.

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Universal transmission characteristics of the probe versus pump area (Fig. 5) have been compiled as a function of the propagation distances q for specific radii p or q. These energy conversion transmission graphs exhibit a dti-value dependence since (a) the probe stabilizes after its build up while the pump continues to deplete; then, (b) the pump anomalously recovers and its area stabilizes, while the probe depletes, reversing its traditional role.

3

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CONCLUSION

We have calculated the effect of diffraction and pump dynamics on the coherent probe on- resonance amplification and Raman-like frequency conversion. Both probe and pump exhibit asymptotically a solitary wave. The probe soliton precedes the pump soliton. Each soliton occurs to the exclusion of the other. The nonlinear action must compensate the diffraction loss for each beam so that a soliton appears at each transition.

REFERENCES

/I/ F.P. Mattar, Physical Mechanism in Coherent On-Resonance Propagation of Two Beams in a Three-Level System, sub. Prog. Quantum Electronics.

/2/ M.J. Konopnicki & J.H. Eberly, Phys. Rev.

g ,

2567 (1981).

/3/ F.P. Mattar & M.C. Newstein, IEEE J. Quantum Electron. l3, 507 (1977).

/4/ S.L. McCall & E.L. Hahn, Phys. Rev.

183,

457 (1969).

151 M.S. Feld & J.C. MacGillivray, Contemporary Physics, 22, 299 (1981).

/6/ R. Bonifacio et al., Phys. Rev.

m,

2568 (1975).

/7/ C.M. Bowden & F.P. Mattar, SPIE 288, 364 (1981), &

369,

151 (1983).

/8/ J.L. Carlsten et al., SPIE Vol.

380,

201-207 (1983); K.J. Driihl et al., Phys. Rev. Lett.

51, 1171 (1983); and R.G. Wenzel et al., J. Stat. Phys. 39, 615; and 621 (1985).

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191 D.J. Kaup, Physica

E,

621 (1986).

/lo/ F.P. Mattar, et al., in Modeling & Simulation XVII, ed. W.G. Vogt & M.H. Mickle, (Instr.

Soc.Am., 1987) pp.1387-1558; F.P. Mattar et al., in Multiple-Photon Excitation & Dissoci- ation of Polyatomic Molecules, ed. C.D.Cantrel1, (Springer-Verlag, 1986) pp. 223-283.

/11/ H.A. Haus, Appl.Phys.Lett.8, 128 (1966) & Waves C Field in OptoElectronics (Prentice Hall, 1984); Progress in Quantum Electronics, Vol.

4,

ed. J.H. Sanders & S. Stenholm (Pergamon, 1975): (i) J.H. Marburger pp. 35-110 & (ii) Y.R. Shen pp. 1-34; Y.R. Shen, Principles of Nonlinear Optics, (Wiley, 1984); M. Lax et al., J.Appl.Phys.

51,

109(1985).

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Fig. 1

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Comparison vs. q of (a) on-axis pump & probe areas;

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Fig. 3

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Stabilization of probe effective pulse length and beam width.

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k

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Fig. 4

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(a) Isometric and (b) on-axis plots of pump and probe areas.

P A R A M E T R I C AREA P L O T ROTATED

PARAMETRIC AREA PLOT

Fig. 5

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Phase diagrams of pump & probe areas.