Phase-conjugation in the three-level resonant medium

Optica Applicata, Vol. X I I I , No. 4, 1983

Phase-conjugation in the three-level resonant medium*

Research Institute o f Condensed M edia Physics, Yerevan State University, K ievian Street la , 375028 Yerevan, USSR.

The nondegenerate phase-conjugation through four-wave mixing in potassium vapour has been theoretically and experim entally considered. The investigation o f spectral dependence o f th e phase-conjugate reflectivity shows that in the spectrum o f conjugate w ave there appear ranges o f am plification near the characteristic seven frequencies. Tw o o f these ranges occu r because o f coherent interactions between unpopulated levels o f doublet and are shifted relatively to the pum p frequency on the value o f doublet splitting.

1 . Introduction

Eecently, the phenomenon of optical phase-conjugation different nonlinear media has drawn a considerable interest (see, e.g., [1] and the references given there). Phase-conjugation is usually investigated through degenerate four-wave mixing. The results of such experiments in resonant gases are represented, for instance, in papers [2, 3]. However, the investigation of phase-conjugation through nondegenerate four-wave mixing is of particular interest. This process has been considered theoretically in papers [4, 5], where the nonlinear medium is modelled as a two-level atomic system. Since excited state of alkali metal atoms has doublet structure in theoretical investigation of light resonant in­ teraction the third level should be taken into account.

In this paper the nondegenerate phase-conjugation in potassium vapour has been considered both theoretically and experimentally. The theory of nondegenerate phase-conjugation in the three-level resonant medium is devel­ oped. It is shown, that taking account of the excited level doublet structure changes qualitatively the spectral composition of a conjugated wave. There occur novel spectral ranges of effective phase-conjugation, which are considerab­ ly changed by pump frequency tuning.

The paper presents experimental results obtained from the observations of nondegenerate phase-conjugation in four-wave mixing in potassium vapour. Hovel spectral ranges of the signal wave conjugation have been detected.

* This paper has been presented at the European Optical Conference (EOC’ 83), May 30-June 4, 1983, in R ydzyna, Poland.

348 Y. M. Arutuntan et al.

2 . Theory

Consider the interaction of four plane linearly polarized waves — two counter-propagating pump waves JE1 and E 3 and two counter-propagating signal waves E 3 and E t with three-level resonance medium (1 ,2 are excited levels, 3 is ground level). The angle between pump and signal directions of propagation exceeds largely the parametric interaction phase-matching angle for each wave separately. Pump waves are monochromatic with frequency co1 and signal waves are quasimonochromatic with frequency spectrum o}t . B y solving the equation of motion for the density matrix g with phenomenological relaxation terms in a steady-state approximation, the four-wave induced polarization has been calculated. In the third order approximation, after spatial averaging over the pump wave oscillation, we get the coupled set of two following equations for slowly varying Fourier's components of weak wave amplitudes E 3 and E t

dE3 ( co)

+ A 1(co)E3(co) = A 3(co)E1E 2Ef(2co1 — co) dE* ( 2 « ! — co)

dz ' i - / dz

+A*(2co1 — co)El(2co1—co) = A*(2co1 — œ)E*E\E3{(o)

+ q [ ^ (

È

V —---1

2nNco1 |d13|2

£ = l^l; ( l E ^ A IE2Iz), yik is half-width of transition cTi 7 " U2

line, (yi)-1 — atomic excited state life-time, = a», — co{3 is resonance detuning between pump wave frequency aq and frequency coi3 of the atomic transition 3 - * i (i = 1 , 2 ) , A = o)13 — co23 is a value of doublet splitting, dik is dipole matrix element, N — atom density, rj = |d23|2/ |d13|2.

From the obtained set of Eqs. (1) it is seen that the Fourier's component of the wave E 3 at frequency co is coupled with the Fourier's component of the wave E l at frequency 2 co1— co according to the energy conservation law. Coeffi­ cient J-i(a)) characterizes resonant medium susceptibility at frequency co, A s(co) is parametric coupling between waves. From the formula (la) it is seen that A 1(co) has five resonance poles corresponding to the five processes of the waves interaction with the three-level atom: 1) absorption at the frequency co13 (the pole co = — Q 2), 2) absorption at the frequency eo23 (the pole co = co1 —Q3),

i Qi bi Л 2 Г -2у13 t(2y13- y j ) Ί η Γ Δ - iy13 - ίγ23 ΐ(γ13+ γ23- у 12) Ц [<*и12 Г ει У13 l ß 6L yxß j ( ß j - ß 3)2 J + ß 7L ß j ß 5 ( Й 1 - Й 6)2 Jj Л ^ М 0 ! -ß« + ηΩΊ Ω3) (ß j — Ϊγ13) (Ωχ ß 5) (Û1 Ω-f) 2 «2+^23 ” - , £ Ш —2^23 , i(2y23- n ) · + 7 L Ух^ 2 { “^2 ß 3) ] 1 Г ^ + гу13 + гу2з г(у13 + у23- у 12) Ц |^23|2 Г ηΩΊ J ß6 L Ω Μ . + (ß2- ß 4)2 J j Ü2ß 2 L (ß22 — ^ 3) (Ω% — ty23) - ß 3) (£ - +( ß2 — ß4 ) ( ß2 — ß3 O« 1 — ß,l) ( Ωα — ße J

3 tyj 2i£ Г yi ~ 2yia . ^ ( П - 2У2з) 1 l Ą i Ą - f l / ß7(ß2- ß 3)2 J 2гИ13|2 h2 Г η - 2У13 , r;2(y1- 2y23) I L ß4 (ß3 — ß 6) (ß3 — ßj) ß2(ß3- ß 7(ß3- ß 2) J 4 / H - t y12 trçf У13 + У28- У 12 Ωε(Ωπ ßi)2 * 1*^231 У12 — У13 У23 Й2 ß2(ß4- ß e) ( ß4- ß 2) 5 - J + t y12 г ^ Пз + Угз~ У12-12 / S ß7( ß j - ß5)2 l^23|2 У12 . У13 У23 Ä2 ß j ( ß5- ß 7) ( ß ä - ß j ) 6 - e j + t y13 - |Дц12_Ä2 i (ß j ty13) (ßg ß 3) V " (Ω3 — ß 2) [ ß3 ~b t (У12 У13 У23)] J 7 - ε2 + ty23 — I«**

. { _____

350 V. M. Arutuntan et al.

3) Rayleigh light scattering at the frequency oq (the pole o> = oq — od3), 4) stimulated light scattering at the frequency oq — A (the pole to = aq — f34), and 5) stimulated scattering at the frequency aq + A (the pole to — oq — Q s).

The fourth and fifth stimulated processes of light scattering coincide by means of frequency with the Stokes and anti-Stokes frequencies of stimulated electronic Raman scattering (SERS) (between doublet levels populated by the light excitation). These resonance processes, however, differ from SERS essen­ tially, which does not occur in the first nonlinear approximation. The main difference of these processes is that they disappear in the absence of atomic collisions. In fact, the relaxation rates (for instance, y 13) can be separated into a radiative part ( l/2 y 1) and a term proportional to the pressure (yf3), so that a4,5 ~ ( y i 2 — 7 x3 — y2a) = (Y12 ~ Yu — yfD· This equality shows that the fourth and fifth resonances disappear if only radiative processes are taken into account. For the first time those new resonances have been suggested by Bloembergen in four-wave mixing processes and investigated in [6-8].

Parametric coupling coefficient A 2(co) contains seven resonance poles co = « 1 — Qf (i — 1 , . . . , 7), where sixth and seventh poles are three-photon scattering frequencies in the process of which an atom is excited absorbing two photons and eliminating one signal photon [9].

Such a complicated dispersive dependence of resonant susceptibility J.1(co) and parametric coupling coefficient A 2(to) essentially affects phase-con­ jugated wave spectral pattern. B y neglecting the pump intensity depletion and solving the set of Eqs. (1) with the boundary conditions 'E3(z = 0) = JES0,

= l) = 0, we obtain phase-conjugate reflectivity R = \Et (z = 0)|2/|F3O|a,

2 (

2

where

R =

2A*(2to1 —

)E\EI

2m coshwi + [A 4 ( to) + A * (2oq — to) ]sinh ml

m

{1I4[A1(

) + A*(2

1-

> ) Y - A 2(

)A*2(2

1-

)\E1\*\E3\*}1I2.

It follows from this formula, that reflectivity sharply increases near the seven specific frequencies represented in Table. Five of them can be predicted by the linear phase-matching condition. For the illustration these frequencies are written in linear approximation (far off resonance line)

ojh) = f = Wl — Q 0,0) = cq -j- Q

(3) ft)*4) = aq — Q _ , ft>d) = <y1 Q_

where

Q± = ~ ^ 1,2) {2c(I + i?) + eJ + e 2 ± v '[e ? -e i + g c ( l - j ?)]2 + 4j?g2c2}1/2.

Phase-conjugation in the there-level resonant medium 351

atom density N . These seven phase-conjugate ranges correspond to four-photon processes, diagrams of which are presented in Figs. la -d .

The phase-conjugate reflectivity dependence on probing signal frequency is shown in Fig. 2. It is seen from the given curve that phase-conjugation reflec­ tivity resonantly increases in the above-mentioned spectral ranges oscillating at the same time, blear co(6) = oq — i24 and oj<7) = 0)1 —QB the phase-conjugation reflectivity is by several orders of magnitude smaller than that in the range

Fig. 1. The diagram o f four-photon processes on a three-level atom . T w o photons with frequency co1 (from the counter-propagating pump waves) are absorbed, and the other tw o are radiated w ith the follow ing frequencies: (a) b oth photons o f the same frequency ojd), (b) a p h oton w ith frequency co<2> close to the atom ic absorption resonance 3 -> l frequency, the second one with frequency co<3> close to the same transition three-photon scattering, (c)a p h o ­ ton with frequency co<4> close to the atom ic transition resonance 3-*2, and the second one with frequency co<5> close to the same transition three-photon scattering, (d) a p h oton with the fourth stim ulated scattering process frequency co<6> = oq — Di , the other one w ith the fifth stimulated scattering process frequency ah7> = co1 — coB

Fig. 2. The lg ~R(co)/Tl(a)<l'>) vs. signal detuning at the dimensionless parameter o f pump intensity |/(e2-t- y\3) + 5 x 10~2, yl3 = y23 = 1/4 y 12 = l O ^ c m '1, A = 58 cm 4 , 1 = 1 cm , jy _ i()i4 cm -3( £i = _ i o cm -1. T he breaks o f resonance poles are due to the fa ct that the pum p w ave saturation was not taken in to account

352 V. M. Aetjtttnyan et al.

co(1>-co(5>. It is caused by the mismatch for resonant frequencies of co(6) and co(7) at the given geometry.

The numerical estimates of B show that for all conjugation ranges, simulta­ neous observations of large pump wave detunings (of order of a few tens of cm-1) are required, so that the pumping laser of high intensity becomes necessary.

3 . Experiment

The scheme of experimental setup is shown in Figure 3. The tunable dye laser 2 with two-stage amplifier pumped by a Q-switch ruby laser 1 (with intensity !=» 30 M W cm-2) has been used as a driving oscillator. A grating (600 grooves / mm) with near grazing incident beam was used as the dye laser intracavity selector. The dye laser generation intensity was about 2 M W cm-2, the pulse duration was about 30 ns. The radiation wavelength of the bandwidth of about

Fig. 3. The schem e o f experim ental setu p: 1 — ruby laser, 2, 3 — dye lasers, 4 — half- -transmitting m irror, 5, 5 ' — reflecting m irrors, potassium vapour cell, 7, 8 — spheric lenses, 9, 10 — cylindric lenses, 11-15 — prism s, 16-20 — beam splitters, 21 — diaphragm, 22 — spectrograph, 23 — potassium lamp

0.5 cm“ 1 was tuned near both D lines of potassium doublet. To provide the counter-propagating reference pump waves the laser radiation 2 was directed into triangular interferometer, the latter consists of a half-transmitting mirror 4 and the mirrors 5, 5 ' with 0.98 reflectivity at the wavelength 767 nm. As a signal beam we used the radiation of the laser 3 with wide spectrum overlapping both the potassium lines focussed into 4 cm long potassium cell by means of lens 7 ( / = 120 cm). For synchronization of lasers 2 and 3 the latter was pumped by the portion of the radiation of laser 1. The angle between reference and signal waves exceeds that of parametric scattering of each pump wave and was ~ 4° [10]. During the experiment the temperature of medium was modified from 200° C to 300° C, so that potassium vapour density was 1.3 -1014 cm~3- 1 0 16 cm*3.

The spectral composition of the phase-conjugated wave was recorded by spectrograph ISP-51 (reverse linear dispersion at wavelength 767 nm is 2 nm / /mm). The pump wave frequency was controlled during the experiment.

Phase-conjugation in the three-leuel resonant medium 353

4 . Results

The spectrograms of the conjugated wave at different temperatures of atomic potassium vapour are shown in Fig. 4. It is seen from Fig. 4a, b that at a tem­ perature of 200 °C (N ~ 1.3 -1014 c m -3) 3 ranges of conjugation near frequencies <u(1), co(2\ co(4) were detected irrespectively of detunings ex, e2. W ith the

in-^ 7699 7665

b

Fig. 4. Spectrograms o f the conjugated w ave at different tem peratures: T = 200°C (a, b), T = 284°C (c), and T = 256°C (d). I - conjugated wave spectrum , II — potassium lam p spectrum (Dv D 2 — lines), III — pum p w ave spectrum

354 Y. M. Abutunyan et al.

creasing temperature there appear six ranges of conjugation near the frequencies co(1)-co(4) and <o(6), co<7> with wide conjugate background (Fig. 4c). A photometric measurement near the frequency co(7) shows that at this frequency the reflectivity exceeds about 1.5 times the nearby background. That is in good accordance with theoretical calculations (Fig. 2). Such a negligible deflection from back­ ground is due to the fact that, at given geometry of the experiment, phase

matching condition is not fulfilled at frequencies o>(6) and <w(7), as mentioned above.

The observed temperature dependence of phase-conjugation process dynamics is caused by reflectivity dependence on N 2 in four-wave interaction. Thus, at low temperatures, i.e., at low densities, conjugation is observed only at the resonant ranges with dramatically large reflectivity. W ith further in­ crease of temperature (T = 330 °C, N ~ 1.1 -1016 cm-3) the efficiency decreases abruptly. It seems to be caused by broadening of atomic absorption lines and nonlinear absorption growth.

The difference in intensities of conjugate symmetric lines is represented in Fig. 4,. and is possibly connected with inhomogeneous frequency distribution of incident signal wave intensity. The absence of far line a>(5) near three-photon scattering frequency detection is due to the same reason. This fact is confirmed by experimental results shown in Fig. 4d. In this figure wide spectrum of inci­ dent signal wave overlaps only potassium D 2 line, moreover, three conjugation ranges were detected: one near the pump wave frequency co(1), and two symmertic ones near the frequency w(2) of resonant transition 4 $ 1/2-> 4P 3/2 and at frequency <o(3) of the same transition three-photon scattering.

Thus, the results obtained show that if the alkali metal vapour was used as a resonant medium not only phase-conjugate reflectivity can be obtained but also conjugated wave spectrum control can be made.

References

[1] Ducloy M., A dv. in Solid State Physics 22 (1982), 35.

[2 ] Bloom D . M., Liao P. F ., Economou N. P ., A ppl. Phys. L ett. 32 (1978), 813. [3] Ta n-no N .. Hoshimiya T ., Inaba H., IE E E J. Quant. Electron, qe-16 (1980), 147. [4 ] Nilson I., Yabiv A ., A ppl. Opt. 18 (1979), 143.

[5 ] F u T., Sargent M., I l l , Opt. L ett. 4 (1979), 366.

[6 ] Bloembebgen N .. Laser spectroscopy X V , Springer Series in Optical Sciences, Yol. 21, Springer-Verlag 1979, p. 340.

[7] Pbiok I., Bogdan A . R ., Dagenais M., Bloembeegen N., Phys. R ev. L ett. 46 (1981), 111

.

[8 ] Gbynbebg G., J. Phys. B : A t. Mol. Phys. 14 (1981), 2089.

[9 ] Abutunyan Y. M., Kanetsyan E. G., Chaltykyan V. 0 ., Zh. Eksp. Teor. Fiz. 59 (1970), 195 (Sov. Phys. J E T P 32 (1970), 108).

[10] Abutunyan V . M., Adonts G. G., Papazyan T. A., Saeejssyan S. M., Abzumanyan G. M ., Meliksetyan T . E ., Izv. A N Arm. SSR, Fizika 12 (1974), 338.