Benedict, Mihály

Gyémánt, Iván Károly

English

Benedict Mihály

Gyémánt Iván Károly

University of Szeged

ON THE INTERACTION OF AN ULTRASHORT LIGHT PULSE WITH A THIN RESONANT MEDIUM II By M. G. BENEDICT and I. GYÉMÁNT Institute of Theoretical Physics, Attila József University, Szeged (Received 20th December, 1984) An optically thin, two level atomic system interacts with a pulse of resonant electromagnetic field. The effects of relaxation on the transmitted and reflected waves are investigated. As a general - rule damping of the polarization reduces reflexion and strengthens the transmission. In a previous paper [1] we have investigated the transmission and reflexion in the resonant interaction of an ultrashort light pulse with a thin film, containing two level atoms. In this paper we report results for a model including damping as well. The interaction is described by the modified Bloch equations [1, 2, 3] as where the damping of the polarization and of the inversion have been introduced. The coefficient }>| of the damping of the inversion is usually much less than that of the polarization, y||<scyx and we can simply set Vn=0. In the case of exact resonance the Bloch vector component v characterizes the polarization, and w is the difference between the populations of the upper and lower levels. In Eqs. (la, b) time is measured in units TR=kc/(2nco0p2ri) where co0 is the transition fre-quency, p is the induced dipole moment, n is the surface dipole density, and ¿—pEexTiJh is the dimensionless amplitude of the external field. In our system of units the transmitted field is S+v, while the reflected wave is v. In [1] damping effects have been ignored, i.e. the characteristic time T2 of the relaxation of polarization has been assumed to be much longer then the unit of time This relation, however, is not valid generally for materials used in coherent optical experiments, because either the transition dipole moment is small or the concentra-tion of the active atoms is not large enough. Nevertheless, in solids with high dipole density and with allowed dipole transition T r < T2 can be achieved. In [1] we have shown, that in the absence of damping a pulse-like excitation of the form V = {S + V)W— yLV\ W = — (S + V)V— 7||((M'+1), (la,b) T2 = TR/y± >>tR. (2) S - <?0sech— t (3) T l* 696 M. G. BENEDICT and I. GYÉMÁNT INTERACTION OF AN ULTRASHORT LIGHT PULSE WITH A THIN RESONANT MEDIUM II 697 yields very different behaviour for the transmitted and reflected amplitudes in the cases of A<(x+l)n and A >(r + l)7t, where r is the duration xt of the pulse in xR units, i.e. T=T(/Tk, and A is the pulse area: A = &0x. In the first case, A<(x+l)n, a complete inversion cannot be achieved and reflexion dominates, while in the second case w reaches +1, and then the system goes back to its ground state. Let us turn our attention to the solution of Eqs. (la, b). If we have a pulse-like excitation, the stationary values for v is 0 and for w is —1, but as we choose further ?n=0, w= — 1 is not achieved actually, while v=0 will be essentially when $ turns to 0. Results of numerical integration are presented in Figs. 1—9, where the time dependence of the inversion (w) and of the exciting (e), transmitted (t) and reflected (r) waves are shown. The scales are fixed by the maximum of the exciting pulse sech t/x switched on at t= —5.5. Namely its maximum posi-tion is at t=0, while for the inversion along the vertical scale goes from —1 to 1. Fig. 9. A=nj2, r = 0.25, y± = 5 698 M. G. BENEDICT and I. GYÉMÁNT When the reflected wave is small still in the undamped case, e.g. for A = 2n, т=0.25, the introduction of damping will cause a little effect, as demonstrated in Figs. 1 and 2 for y_L = l and y± = 5, respectively. The reflected and transmitted waves are essentially the same as without damping, the only difference is that the inversion does not reach +1 and later does not fall back to —1; the atomic system dissipates the energy of the excitation. Damping of the tail of the reflected wave can be observed when l/y± ~т. For still greater y± the reflected wave will completely vanish. The introduction of damping may change the situation radically only if v is relatively large in the undamped case, which is valid for long and/or weak pulses [1]. The situation is shown in Figs. 3—6 for a 2n pulse of duration Tí=4tr, with 7_i_=0, 0.2, l.,5., respectively. As grows the reflected wave diminishes and trans-mission is getting closer to the exciting pulse. The same can be observed for а тг/2 pulse with т(=0.25тя (Figs. 1—9) though here the effect is not so drastic, but at y± = 5 the picture is very similar to the previous case (cp. Figs. 6 and 9). As it is expected, for pulses of duration much less then tR the relaxation with characteristic time T2 < rR will effect mainly the tails of the transmitted and reflected waves — the superradiant part; the initial stimulated part will be changed only when Г2<tj . In experiments it is easy to meet the condition т г >Г 2 >т л and to realize a small area exciting pulse, when a relatively strong reflexion can be expected. However, the more interesting case of strong reflexion of large area pulses requires the condition which seems more difficult to ensure experimentally. References [1[ Benedict, M. G., I. Gyémánt: Acta Phys. et Chem. Szeged 30,115 (1984). [2] Allen, L„ J. Eberly: Optical Resonance and Two Level Atoms, Wiley, N. Y., (1975). [3] Маликов P. Ф., В. А. Малышев, E. Д. Трифонов: В. сб. Теория кооперативных когерентных эффектов в излучении, JI., (1980). [4] Рупасов В. И., В. И. Юдсон: Квант, электр. 9 2179 (1982). О ВЗАИМОДЕЙСТВИИ УЛЬТРАКОРОТКОГО СВЕТОВОГО ИМПУЛЬСА С ТОНКИМ РЕЗОНАНСНЫМ СЛОЕМ М. Г. Бенедикт и И. Дьемант Оптически тонкая двухуровневая атомная система взаимодействует с импульсом резо-нансного электромагнитного поля. Исследуется влияние релаксации на пропущенную и отраженную волны. Как правило, затухание поляризации уменьшает отражение и усиляет пропускание. 

On the interaction of an ultrashort light pulse with a thin resonant medium 2.

ON THE INTERACTION OF AN ULTRASHORT LIGHT PULSE WITH A THIN RESONANT MEDIUM By M. G. BENEDICT and I. GYÉMÁNT Institute of Theoretical Physics, Attila József University, Szeged (Received May 20,1984) An optically thin, two level atomic system interacts with a pulse of resonant electromagnetic field. The transmitted and reflected waves are investigated for the case of exact resonance. The amplitudes and the phases of the secondary fields strongly depend on the area and on the width of the exciting pulse. The resonant interaction of an ultrashort light pulse with a two level atomic system gives rise to several interesting pheomena [1]. Among them, however, only little attention has been paid to the reflected wave. If the pulse is ultrashort and relaxation effects can be ignored, then the interac-tion of the two level system with the light field can be described by the optical Bloch equations. Using the rotating wave approximation one has [1] « = — Av, v = Au+j-Ew, w = --j-Ev. (1) Here u, v and w are the components of the Bloch vector, E is the slowly varying ampli-tude of the field acting on the atoms, p is the transition dipole moment beeing paral-lel to the linearly polarized field, and A is the detuning of the resonance frequency from the carrier frequency of the field. If the medium is optically thin, then system (1) is usually solved for u, v and w regarding E as a given function of time, namely the incoming field. The effective field strength however consists of two parts. One is the external exciting field and the other is the field originating from the radiating dipoles themselves. The effects of this second field has been investigated in [2] and [3] for the case of superradiation when there is no external excitaition but the system is in the upper unstable state. We restrict our considerations to the case of exact resonance A =0. In this case the effective field is: E = t E n + 2 / ( 2 ) where Eex is the external excitation and the second term is the secondary field origi-nating from the thin medium with a surface dipole density n. Substituting this expres-116 M. G. BENEDICT and I. GYÉMÁNT sion into (1) with A =0, and going over to a time scale t r = tic/lncjop^n we have i> = (g+v)w, w = ~(£+v)v (3) where S = ^ E e x t R is the dimensionless amplitude of the external field. Introducing the Bloch angle with v = — sin 9, w = — cos 6 we get the single equation 6 + sin 6 = S (4) which has been obtained in [4] in another way. We note that xR is the superfluores-cence time [3] and in the case of a solid material it falls into the nanosecond range. The reflected wave is proportional to v= — sin 9 while the transmitted wave is the sum of the incoming and the forward scattered wave S+v=6, which is identical with the effective field. Eq. (4) has been solved analytically in [4] for a square pulse, and approximate considerations were made for exciting pulses of smooth envelope. Here we shall present the results of numerical solutions of eq. (4) for smooth pulses of the form 8 = sech { ~ to , s 0 = — , T T7T where A is the pulse area. We have performed calculations for Gaussian and Lorent-zian pulses as well but except for one special case, which will be discussed below, there are no qualitative differences in the evolution of the process. In Figs. 1-8 are shown the time dependence of the inversion (w) and of the exciting (e), transmitted (/) and reflected (r) waves. The scales are fixed by the maximum of the exciting pulse, namely its position is at t0=5.5z, its height is while for the inver-sion the vertical scale goes from — 1 to +1. Initially the two level system is in its ground state, 6=0, w= — 1. It is seen in all cases, that at the beginning of the excitation 6 is increasing in time, the transmitted wave is positive, which means that it is in same phase as the incoming field. At the same time —sin 9 is negative, i.e. the reflected wave is in the opposite phase. The subsequent evolution of the process however depends on both the area of the in-coming pulse and its steepness. When the area is small, e.g. for a 7t/2 pulse, and the pulse width INTERACTION OF AN ULTRASHORT LIGHT PULSE WITH A THIN RESONANT MEDIUM 117 Fig. 3. A =4JI, T = 4 is large, e.g. x=4, 6 increases only slowly, 6 remains small, so that sin 6 = 8, and the reflected wave will be nearly equal to the exciting wave (Fig. 1). The transmission of the medium turns to be small, the reflexion will dominate. The weak transmitted wave exhibits two small peaks at symmetric positions to the maximum of the exciting pulse. Be-tween these peaks the transmitted wave changes its phase though the system remains close to its ground state, w % — 1 during the whole process. Figs. 2 and 3 show the effects of 2n and 4n secant hyperbolic pulses with t—4, respectively. In these cases the evolving of the fields is similar to that of rc/2 pulses but with a more pronounced increase in the inversion. Note however that the behaviour of the Bloch angle is far from what is expected when neglect-ing the self field. A TT/2 pulse of a much smaller width, 1=0.25 shows a somewhat different character: the solution be-comes asymmetric. The first peak of the transmitted amplitude is stronger and gets closer to the maximum of the exciting pulse, while the second peak in the opposite phase becomes smaller. The reflected wave is getting weaker too and its maximum is de-layed (Fig. 4). Now if we have a short pulse of area n the picture is changing further (Fig. 5). The maximum of the amplitude of the transmitted wave takes place at the centre of the excitation, the reflected wave has a long smooth tail of small amplitude and after the decay of the incoming pulse the transmitted wave goes over into the same form as the reflected wave. This part of the transmission can be regarded as a forward scattered wave, which has to be the same as the backward wave. The in-version grows over zero, and the pulse evolving after the excitation can be regarded as superradiation from a not fully inverted state. This process is strengthened by the tail of the excitation. Fig. 4. A = n/1, r=0.25 Fig. 5. A = n, T=0.25 118 M. G. BENEDICT and I. GYÉMÁNT Fig. 6. A = 2n, r=0.25 l \ J, K \ 7 wj Fig. 7. A= 1.571 T=0.5 Fig. 8. A=4n t=2 For short 2it pulses the situat-ion will be the opposite to the weak and long excitation case. Now the transmitted wave will be in the same phase during the whole process, and its single maximum will somewhat delayed. The weak reflected wave is double peaked and changes its -phase at the centre of the incoming pulse, while the inversion behaves as usual, the Bloch vector is turned around by 2n. (Fig. 6). There is an interesting limiting case, when an analytical solution for Eq. (4) can be obtained. If &= sech t — ta then 0 = 2 arctg e( '-'°) / , r T a solution which corresponds the initial condition 0(0) = The numerical is to = 2 arctge~'o/t. results show however that essentially the same solution can be obtained if 0(0)=0 (Fig. 7). In the numerical procedure the exciting pulse was set up with /0=5.5T. In this case 6 tends to n as the excitation decays (with other forms of excitations this could be achieved only by cutting them off when 9 gets equal to n). The area of the exciting pulse is ( t + l)7i which shows that in order to reach complete inversion one must have a pulse area larger than n, and this is the more so, the wider is the incoming pulse. The reflected and the transmitted waves are in opposite phase both of them having the same form as the exciting pulse, and the ratio of their amplitudes turns to be r, i.e. independent of time. Of course the 9 = n value is not stable, the system will radiate spontaneously according to the sech (t—O law in both directions after a small perturbation. The characteristic time of this superradiating pulse is xR not depending on the excitation. For pulses of area larger than 2n also a different behaviour can be obtained for different widths. For a An pulse and T=4 see Fig. 3. For T=2 however the inversion may increase once again after having decayed to — 1. Accordingly the reflected wave \n! INTERACTION OF AN ULTRASHORT LIGHT PULSE WITH A THIN RESONANT MEDIUM 119 exhibits further oscillations. What is more interesting the transmitted pulse may be higher than the excitation, while it is narrower than the incoming pulse, thus showing up the effect of pulse compression (Fig. 8.) For still shorter pulses the situation will be close to the case when the self field is neglected, though in all cases this self field turns back the system into its ground state. These examples clearly show that the behaviour of the transmitted and reflected waves are very different for wide and short pulses of the same area. Beside the varia-tions of the amplitudes of the secondary waves the changes in the phases also deserve attention. We note that measurements of similar phase jumps for choherent ampli-fication, predicted at first time in [5] have been reported only very recently [6]. The results of our calculations show, at least in principle, the possibility to control the amplitude and phase of a coherent light pulse. References [1] Allen L., J. H. Eberly: Optical Resonance and Two-Level Atoms, Wiley, New York (1975). [2] de Costa L. С. Т., G. A. P. Munguia: Phys. Rev. A14, 1745 (1976). [3] Трифонов E. Д., А. И. Зайцев, P. Ф. Маликов: ЖЭТФ 76, 65 (1979). [4] Рупасов В. И., В. И. Юдсон: Квант, элект. 9, 2179 (1982). [5] Burnham D. Е„ R. Y. Chiao: Phys. Rev. 188, 667 (1969). [6] Varnavsky, О. P., A. N. Kirkin, A. M. Leontovich, A. M. Mozharovsky: Opt. Comm. 49, 71 (1984). О ВЗАИМОДЕЙСТВИИ УЛЬТРАКОРОТКОГО СВЕТОВОГО ИМПУЛЬСА С ТОНКИМ РЕЗОНАНСНЫМ СЛОЕМ М. Г. Бенедикт, и И. Демант Оптически тонкая двухуровневая атомная система взаимодействует с резонансным электромагнитным импульсом. Исследуются пропущенная и отраженная волны в случае точного резонанса. Амплитуды и фазы во вторичных полях сильно зависят от площади и ширины возбуждающего импульса. ь 

On the interaction of an ultrashort light pulse with a thin resonant medium

http://acta.bibl.u-szeged.hu/24241/1/phys_chem_030_fasc_003_004_115-119.pdf

On the interaction of an ultrashort light pulse with a thin resonant medium 2.

Abstract

Similar works

Full text

Available Versions

University of Szeged

University of Szeged