Computation of nonlinear radiation trapping in a saturated cesium vapor

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15 October

1995 OPTICS COMMUNICATIONS

OpticsCommunications120 (1995) 149-154

ELSEVIER

Computation of nonlinear radiation trapping in a saturated cesium vapor A.F. Molisch a, M. Allegrini b, B.P. Oehry a, W. Schupita a, G. Magerl a aInstitutfiir Nachrichtentechnik

und Hochfrequenztechnik, Technische Vniversittit Wien GuJhausstraJe Z/389, A-1040 Wien. Austria b Vnitri INFM, Dipartimento di Fisica, Universitci di Pisa, Piazza Torricelli 2, I-56126, Pisa, Italy and Dipartimento di Fisica della Materia, Geojisica e Fisica dell’ Ambiente, Vniversith di Messinn, Salita Sperone 31, I-Sant’ Agato, Italy

Received24 April 1995

Abstract We analyze the temporal evolution of the excited-state distribution in a cesium vapor excited by a strong laser pulse of arbitrary duration. We model this situation with equations that include radiation trapping, saturation, and optical pumping. Both the strength and the temporal behavior of the emergent radiation after the laser is switched off depend on the duration of the laser pulse and on the diameter of the laser beam. The decay time constant of the emergent radiation varies from less than the natural lifetime to several hundred times the natural lifetime for cesium densities varying in the range 20-100°C.

F’45

1. Introduction

z3

The study of the resonance lines of alkali atoms, especially sodium and cesium, has been a subject of spectroscopic research for almost a century [ 11. However, a seemingly simple problem, the excitation of a cesium vapor with a strong’ laser pulse, has not yet been given a satisfactory theoretical treatment. This problem is not only of theoretical interest, but is also relevant for measurements of collision cross sections and of atomic lifetimes of cesium atoms. When exciting a Cs vapor with a laser pulse, optical pumping and radiation trapping takes place; when the pulse is strong, we additionally get saturation effects. Each of these three effects has been considered for itself [ 2-51, but, up to now, not the combination of all three. In this paper, we give a full quantitative description of the Cs-excitation problem, taking into account all three ’ When we say “strong” laser pulse, we mean that the laser intensity is much larger than the saturation intensity I,; I, is a few W/cm’. 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD10030-4018(95)00461-O

..

laser tuned to this transition

6s 112

Fig. 1. Partial energy level scheme of cesium (hypefine-splitting not to scale).

phenomena. Our treatment is valid for arbitrary durations of the exciting laser pulse, and we will show that the duration of the pulse actually has a strong influence on the behavior of the vapor. The relevant part of the energy level scheme of cesium is shown in Fig. 1. The hyperfine splitting (hfs) of the ground state 6s,,, is about 9.2 GHz [6], i.e.

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120 (1995) 149-154

is spectrally much narrower than the hfs-splitting of the 6s,,, state and is tuned to the 6s, ,2, F = 3 + 6p,,, transition.

Lasea: 2. Theory

Fig. 2. Geometry

of the problem.

much larger than the Doppler width (about 0.4 GHz at room temperature), while the hfs of the 6p,,, state is much smaller than the Doppler width, and can thus be neglected for our purposes’. We also neglect collisional effects like fine-structure-changing collisions or energy pooling. We can thus compute with a model atom that has only three relevant states and where all lines are pure Doppler lines. Collisional intermixing between the two hfs levels of the ground state is extremely small and can be neglected over the considered time scale ( l&1000 ns) [7]. For the mathematical formulation, we make the following assumptions: (i) The geometry we consider is a cylindrical cell illuminated by a laser along the z-axis, see Fig. 2. For a laser beam radius r,,,,, we assume a constant laser intensity within r,,,, and zero outside. The laser is strong enough to completely saturate the region 0 < r < rlaner.Strong saturation at the Cs resonant wavelength requires pulse energies of just a few mJ. Due to the saturation, the laser is not attenuated very much as it passes through the vapor. Hence, the excitation is uniform along the z-axis. We assume that the cell length is larger than the diameter. It has been shown [ 81 that this condition is sufficient to approximate the cell as an infinite cylinder. (ii) The frequency of a spontaneously emitted photon is statistically independent of the frequency of an absorbed photon (complete frequency redistribution). ( iii) Particle diffusion is not relevant. (iv) The side walls of the cylinder are completely transparent. (v) The flight time of the photons is negligibly short as compared to the natural lifetime of the excited-state atoms. The validity of assumptions (ii)-(v) is discussed e.g. in Ref. [ 51. (vi) The laser ’ One could of course think of cases where the hyperfine splitting of the 6pxj2 state becomes important, namely when the pump laser is tuned to the F= 4 + F= 5 transition; in that case the optical pumping will be considerably smaller than for no hyperfine splitting. We will henceforth not deal with these rather rare cases and neglect hyperfine sphttmg of the 6p,,, state.

For a mathematical description of the vapor, we need the rate equations for the involved atomic levels: the 6s 1/Z, F= 3 level (henceforth called level a), the 6s l/29 F = 4 level (level b), and the 6p,,, level (level c).

State-c atoms are created (i) by the absorption of laser radiation by state-a atoms, (ii) by reabsorption of c-a fluorescence radiation, and (iii) by the reabsorption of c-b fluorescence radiation. Atoms leave state c via natural decay and stimulated emission. The latter is taken into account by introducing an “effective” absorption coefficient that describes the difference between absorption and stimulated emission (see below). By the balance between creating and destructing processes, we get the rate equation for state-c atoms

P

n,(r’,

+ -

7

t)GaC(n,, no r, r’, f)r’ dr’

0

n,(r’,t)Gbc(n,,n,,r,r’,t)r’dr’+E(r,t), 0

(1) where E(r, t) is the external excitation by the laser beam. p is the ca:cb branching ratio, P=A,,I (A,, +A,,). The kernel function G”” denotes the probability that a photon of the a-c transition that is emitted at point r’ is reabsorbed at point r [ 93,

Gac(na,n,, r, r’, t> 27l

m

=x C 4?r 111 0

Xexp

=

ko(n,, n,, r, r)e(x) 2’ + 1-2+ i2 - 2rr’cos( $)

-m--m

s

k&n,, n,, s, t) ds

1

drdzd$.

(2)

A. F. Molisch et al. /Optics Communications 120 (1995) 149-154

Here, x is the normalized frequency, k(x) is the lineshape, and C, is the lineshape normalization factor, C, = 1l/k(x) dr. An integration path S leads from point (r, 0, 0) to point (r’, cp, z), where (r, cp, z) are the coordinates in a cylindrical coordinate system. The effective center-of-line absorption coefficient k, (including stimulated emission) is

n,(t+bt)

anb(r, t) ----=-

1-P

at

7

1-P

n,(r, f>-- 7

R X

q(r’,

t)Gbc(nb,

nc, r, r’, t)r’ dr’ .

(4)

0

Since the Cs density, N,O,,must be constant throughout the cell, the rate equation for state-a atoms is simply n,(r, t) =N,,, -n,(r,

t) -n,(r,

1) .

(5)

Eqs. ( 1)-( 5) are a complete mathematical description of the problem. We thus have to solve a system of nonlinear, coupled integro-differential equations. We proceed by approximating n,, nbr and n, as a sum of coaxial cylindrical shells, or, in a mathematical formulation, as a sum of rectangular pulse functions p( r - rk) of width A with unknown amplitudes nk( t), n(r,

0

=

C

nk(t>p(r-rk)

.

(6)

k

The pulse function p(r- rk) is 1 for rk-A/2< r < rk + A 12 and 0 elsewhere. We use a forward-differencing scheme to take the time-dependence into account. The timesteps must be small enough that b does not change appreciably during one timestep. This forward-differencing scheme is quite similar to the propagator function method recently proposed in Ref. [ 10,111. With these approximations, we get for the cstate atoms a linear system of algebraic equations of the basic form

c

AE(t

which can be solved by standard means. The matrix elements Akm are r,,,+A/2

km

state-c atoms decayreabsorption of c-b

nk(t) -p

(7)

A=’

weights of levels a (T is the absorption

$

n,

(3) where g, and g, are the statistical and c, N,,, is the Cs density, and cross section of the a-c transition. Atoms in state b are created by ing to state b, and destroyed by photons. The rate equation is then

=nk(t) -

151

=

,

Gac(na, n,, rk, r’, t) rr dr’ .

(8)

rm-Al2

If we insert the kernel into Eq. (8), we see that the direct evaluation of the matrix elements requires fivefold integration, leading to prohibitive computer time requirements. Reducing the matrix elements to single integrals requires appreciably large efforts in analytical and numerical treatment of the problem; details are given elsewhere [9]. The actual evaluation was then done on a VAX 4000 workstation. Depending on the number of timesteps and pulse functions, evaluation times for the whole problem were between half a minute and two hours (typically, 20 pulse functions and 100 timesteps require a few minutes CPU time).

3. Results and discussion With the above method, we can analyze the temporal behavior of the excited atoms and of the emergent radiation. It turns out that the combination of trapping, optical pumping and saturation leads to some interesting effects, which we will show and explain physically in this section. Most notably, the strength and the temporal dependence of the emergent radiation depends on the duration of the laser pulse. Let us first consider the pump phase, i.e. the time when the laser pulse is on. Fig. 3 shows the densities A,, fii, and rs,, of atoms in state a, b, and c averaged over the cell, at the end of the pulse as a function of the pulse duration. First, let the radius of the laser beam be small compared to the cell radius. We see that for very long pulses, ii, at the end of the excitation is smaller than for very short pulses (a very short pulse means much shorter than the natural lifetime 7). This effect is a consequence of the optical pumping. However, we also see that A, goes through a maximum, and this requires more explanation of the

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lo”

s >. .= g 10’ $ $I lOA e p! a loa 0

1OT

202

302

4OT

507

Time t Fig. 3. Average densities ti,, fir,, and A, at the end of a strong laser pulse as a function of the pulse duration. Parameters: opacity k$ = 10, laser beam radius rlasW= 0.1. R.

involved geometry. We distinguish between two regions of the cell: the primary-excitation region ) where Cs atoms can be excited directly by (r&l_ser) 7 where atoms can be excited only by fluorescence radiation from the primary region. In these two regions, there are two competing processes: (i) In the primary-excitation region, the optical pumping leads to a decrease of n, with time. n, is tightly coupled to n, because of the strong excitation, so that n, will also decrease with time. (ii) In the secondaryexcitation region, radiation trapping, i.e. the reabsorption of fluorescence photons from the primary-excitation region and the secondary-excitation region, determines n,. If we just had a two-level problem, then the absorption of fluorescence from the primary-excitation region would lead to a continuous increase in upper-state atoms until steady-state is achieved ( “accumulation” of excitation). In the Cs problem, however, the primary fluorescence becomes weaker with time, because n, in the primary-excitation region becomes smaller with time because of the optical pumping (see above). Thus, n, in the secondary-excitation region will decrease after some time 3. The two competing effects, the accumulation of excitation, and the decrease of primary fluorescence, cause n, in the

120 (1995) 149-154

secondary-excitation region to go through a maximum. This can also be clearly seen in Fig. 4, which depicts the spatial distribution of n, as a function of time. We thus have two regions, the core with decreasing n,, and the outer region where n, first increases and then decreases. It is obvious that depending on the relative importance of these two processes, the average density 3, either decreases monotonously, or goes through a maximum. The relative importance of the two processes depends on the opacity, on the size of the excited region, and on the duration of the pulse. In our example, the two competing processes cause n, to go through a maximum, see Fig. 3. Fig. 5 shows ii,(t) and ff,( t) when we excite the whole vapor cell. In that case, we have no secondaryexcitation region that can reabsorb fluorescence, and thus no effects that tend to increase n, with time. As we increase the duration of the pulse, A, decreases monotonously. Generally, the maximum in fi, appears only when the secondary region is both geometrically large enough (as compared to the primary excitation region) and it is optically thick enough to absorb an appreciable number of photons reemitted in the primary region. The duration of the exciting pulse also influences the shape of the time decay of the excited atoms and of the

11

“I

0.2 ,ii

‘I

1 I!

0.1 /

Ilill!lillll~

primary excitation region

-

152 3This effect is intensified by the fact that we also have (albeit weaker) optical pumping in the secondary-excitation region, so that the number of absorbers in this region becomes smaller with time. This of course leads to a decrease in secondary excitation.

TirnZ Fig. 4. Temporal evolution of n,(r, t) during a strong laser pulse. Parameters as in Fig. 3.

A.F. Molisch et al. /Optics

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120 (1995) 149-154

153

Pulse duration 0.1 Z

Pulse duratlon 50 ‘C

0

Time after pulse

1cF

0

1OT

207

30-c

4OT

507

ST

107

1%

2cn

2%

Time after pulse

Fig. 6. Temporal evolution of the emergent fluorescence radiation after a strong short laser pulse is switched off. Parameters: !@ = 50, r ,_ = R, duration of the laser pulse = 0.1.~ (left) and 50.7 (right).

Time t Fig. 5. Average density n,, fit,. and & at the end of a strong laser pulse as a function of the pulse duration. Parameters: k$= 10, rlaser= R.

emergent fluorescence radiation after the excitation is switched off. Let us consider the case where the laser excites the whole vessel. We have seen above that if we have a long exciting pulse, and thus much optical pumping, then both n, and n, will be small, and rib will be large, at the start of the decay phase. At this time, transition c-b will be strongly trapped, while the transition c-a will be completely transparent -the effective absorption coefficient being zero because of the saturation. Because both n, and n, are small, the opacity of transition c-a will stay small even if all state-c atoms decay to state a 4. Thus, the opacities of both transition c-a and transition c-b will stay constant throughout the decay process; the former will be very small, while the latter will be very large. We thus have a classical 3level trapping problem with a branching ratio of about 50%, with one rather weakly and one strongly trapped transition [ 121. Hence, n, will decay with a time constant that is approximately the decay time constant of the transparent transition. Since the opacities do not change appreciably during the decay, this is also true for the emergent radiation The situation is completely different when we have only a short exciting pulse. At the beginning of the decay phase, transition c-a is completely transparent because of the strong saturation. However, there is little 4 Actually, most atoms in state c will decay to state a, since the transition b is strongly trapped and thus has a low effective decay rate.

optical pumping during the pump phase (since it is so short). The number of atoms in state b is thus not affected by the excitation pulse, while the ratio of atoms in states a and c is determined by the ratio of their statistical weights. Thus, we have approximately n,:n,:nc = 49:351:224 = 2:14:9. With increasing time, many of the state-c atoms will decay to state a, so that the effective opacity of transition c-a increases to considerable values. The opacity of transition b will increase only slightly from its already high value. In the beginning, nC will thus decay with a time constant that is almost equal to the time constant of the c-a transition (as in the above case). At later times, both transitions c-a and c-b are rather opaque, so that the decay time constant is almost g,,r at late times. g, is the trapping factor for the non-saturated case [5], it can easily reach 10-1000 in typical experimental situations. There is thus a strong change in the decay time constant. Since the number of excited atoms decreases and the number of absorbers increases, the decay of the emergent radiation can even be faster than rin the beginning. This situation can only occur when the excitation pulse is short. Fig. 6 give computed examples both for short and long pulses. The above effects will be much less pronounced when only a small part of the vapor cell is excited.

4. Summary and conclusion Summarizing, we have given a mathematical description of a cesium vapor excited by a strong laser pulse of arbitrary duration. Our computation method is based on approximating the spatial distribution of

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excited atoms by a set of orthonormal basis functions, and on a forward-finite differencing scheme for the temporal dependence. Our method allows computations to be done within reasonable CPU time. We modelled the effects of optical pumping, of saturation, and of radiation trapping. We showed that due to a combination of these three effects, the duration of the exciting pulse has an influence both on the strength and on the temporal behavior of the emergent radiation after the excitation is switched off.

References [ I ] A.G. Mitchell and M.W. Zemansky, Resonance radiation and excited atoms (Cambridge University Press, Cambridge, 1961).

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[2] A. Yariv,OpticalElectronics (CBS CollegePublishing, 1985). [ 31 A. Comey, Atomic and Laser Spectroscopy (Oxford University Press, Oxford, 1977). 141 T. Stacewicz, T. Kotowski, P. Wiewor and J. Chorazy, Optics Comm. 100 ( 1993) 99. [5] A.F. Molisch, B.P. Oehry and G. Magerl, J. Quantum Spectrosc. Radiat. Transfer 48 ( 1992) 377. [6] E. Arimondo, M. Inguscio and P. Violino, Rev. Mod. Phys. 49 (1977) 31. [ 71 F.A. Franz, Phys. Rev. 148 (1966) 82. [81 L.W. Avery, L.L. House and A. Skumanich, J. Quantum Spectrosc. Radiat. Transfer 9 (1969) 519. [9] A.F. Molisch, B.P. Oehry, W. Schupita and G. Magerl, Optics Comm. 118 (1995) 520. [lo] J.E. Lawler, G.J. Parker and W.N.G. Hitchon, J. Quantum Spectrosc. Radiat. Transfer 49 ( 1993) 627. [ 111 G.J. Parker, W.N.G. Hitchon and J.E. Lawler, J. Phys. B 26 (1993) 4643. [ 121 A.F. Molisch, B.P. Oehry, W. Schupita and G. Magerl, Optics Comm. 90 (1992) 245.