Bistability in multiphoton ionization with recombination
Howard C. Baker
Department of Physics, Berea College, Berea, Kentucky 40404
Lloyd Armstrong, Jr.
Department of Physics, Johns Hopkins University, Baltimore, Maryland 21218
Received March 30,1981
The steady-state degree of ionization produced by multiphoton processes under bistable conditions is investigated.We show that the presence of recombination leads to a bistable steady state, with the degree of ionization abruptlyincreasing as the incident laser intensity Ii is increased beyond a certain critical value Iii and abruptly decreasingas the intensity is decreased below a second critical value 42 < Iil. Effects of dispersion on the bistability charac-teristics of the steady state are considered.
New effects on multiphoton ionization rates arisingfrom cooperative atomic behavior in a bistable systemwere recently predicted.' Some novel effects on mul-tiphoton ionization and excitation line shapes were infact observed recently,2' 3 and these effects have beensuccessfully accounted for in terms of collective atomicbehavior. 4
In this Letter we extend the analysis of Ref. 1 to ex-plore for possible additional effects that may arise inassociation with recombination and dispersion. Weconsider the ionization of atoms in a ring cavity withmirrors. A laser beam L of angular frequency coo istuned close to a selected bound-bound resonance-transition frequency Co0 of the atoms in the cavity. Asecond laser beam L' of frequency co' > wo and intensityIi' probes the cavity and can ionize atoms resonantlyexcited by L. We require that multiphoton ionizationby L be negligible and that there be a resonant-cavitymode xc near coo. We allow for the possibility thatfree-cavity electrons will recombine with the ions atsome rate that, along with the ionization rate that is dueto L', will determine a steady-state degree of ionization.The steady-state line shape will depend not only on theincident intensity Ii of L but also on the values of thedetunings b5a ==a w- co0 and &, = coc - oo.
The model that we apply is an extension of themean-field model of Bonifacio and Lugiato. 5 The in-tensity Ii is taken to be an adjustable quantity, as arethe detunings 6a and 6,. The atoms are near-reson-antly driven by L at the Rabi frequency between thelower state Ig), called the ground state, and the selectedexcited state l a). We assume that, because of relativelyrapid redistribution in the energy of freed electronsbecause of thermalization, free-free transitions, etc.,stimulated radiative recombination will be negligible.The recombination rate a to the state I a) and the rate3 to the state 1g) thus will be essentially independent
of laser intensities. The beam L' ionizes atoms directlyfrom the state I a) with rate 'Ya, an adjustable quantityvarying approximately linearly with intensity Ii'.
Following Bates et al.,6 we assume that the net rateof recombination to the state Ia) is given by the functionaNeNI, where Ne is the number of free electronspresent and NJ is the number of ions present. Let Nobe the (large, constant) number of neutral atoms plusions and N be the number of ionized atoms; then N1 =NO - N. Further, because of confinement of the pho-toelectrons, we set Ne = No - N, and thus the net re-combination rate to the state I a ) may be written as a(No- N)2 . Similarly, the recombination to the state 1g)occurs at the net rate 3(No - N)2 .
A realistic estimate of the separate rates a and : isdifficult to make. The rates depend on a large numberof parameters, such as the specific element being ion-ized, the way in which the photoelectrons are beingthermalized (if in fact they are, before recombination),and collision rates. We find below, however, that forthe most interesting situations the results depend pre-dominantly on the sum a + /3, that is, on the total re-combination rate.
If the laser beam does not fill the cavity, one mustdeal also with a difficult diffusion problem for atomsand electrons. An implicit assumption of the mean-field model is, however, that the laser does fill the cavity.Moreover, there will be additional effects, such as col-lisional excitation and ionization, such as have beenobserved by Lucatorto and McIlrath 7 and by Skinner. 8
However, rapid collisional excitation has been observedonly in gases considerably more dense than those nor-mally used for bistability experiments and thus isprobably of minor importance to the problem consid-ered here. We believe in any case that the gross fea-tures of the experiment are sufficiently well modeledthat effects predicted herein would not be washed outby corrections and could be readily observed.
We define the following quantities:
F I(1) = yI(11) + Ya/ 2 , (1)
where y II () are the natural damping rates of the diag-onal (off-diagonal) density-matrix elements arising from
all incoherent relaxation processes. Whenever thespontaneous radiation rate To is dominant, then -y yo/2 and y o yo. R is the coefficient of reflectivity of
the mirror, and I is the length of the cavity. The cavityradiation-loss rate K iS then given by c(1-R)/l. We letE, and Ei be the electric fields produced by photons inthe resonant-cavity mode of frequency xc and in theinjected field L, respectively. We let A represent theatomic-transition dipole moment and V the volume ofthe cavity; the effective atom-field-coupling constantA is then
A= (27rwo)1/2 (2)
The population inversion A is defined by
A = 1/2 (Ng-Na.), (3)
and S is the complex macroscopic polarization of thegas.
In terms of dimensionless amplitudes y and x,
2yEiv =h[P1 'r1l(1-R)12 '
X dyrnamrl e i/2q a
the Maxwell-Block dynamical equations are
S = (rLrll)1/2X - (rF + i6)S,
A = -(f 17 j)112 (XS* + X*S) - ril (s2
(No-N) 2± = - a) 2
£=-2A 2S(p pjl/2S - K(X - y) -i5X
0
N = Ya (A - + (a - -)(No- )2 .
We wish to investigate the stationary-state situEand accordingly set the time derivatives to zero ir(6). The resulting equations can be most easilypressed in terms of the dimensionless intensitiesy2 andIt =1x12:
,qIt = (1 -F)2[Q +j2i (1 -&)j
X |^Ya F2- 1F)2|,
IF = It [(1 + X1)2 + (0c - X2)2 1.
In Eqs. (7), the additional variables
A = & = aN0 , Noo, F = N/No,N0'
(4)
of Bonifacio and Lugiato.5 The susceptibilities X1,2 aregiven by the expressions
X1 1 + IFE+ (1 F)2| (10a)
X2 = -7 X1|+ F (1 F)2| (10b)
where we define dimensionless detuning parameters 0,6 c/K and 0°a ba/P1The basic Eqs. (7a) and (7b) may be solved simulta-
neously to obtain F(Ii), the un-ionized fraction of thesample, as a function of the input laser intensity Ii at coo,with the intensity Ii' at w' being held constant. Onecharacteristic of the solution is easily seen from Eq. (7a),however. Because Tya/P < 2, the numerator is alwayspositive. Therefore, in order that It be positive, thedenominator must always be positive. This sets aminimum allowed value to F, i.e., F Ž Fmin, where Fminis fixed by the relation
Fmin 2(&+ ) 1(l -Fmin)2 -
2Yaa (1It is noteworthy that this relation is independent of £4
(5) and Oa-It is convenient here to consider some typical values
of the parameters in Eqs. (7a) and (7b). The rate To is(6a) characteristically around 108/sec, and the ionization rate
about 10 hi' (sec-1), when Ii' is in watts per squarecentimeter. For pressures around 10-4 Torr, corre-sponding to densities No/V _ 1012/cm3 and tempera-tures T $ 103 K, a reasonable value for Q is 5 X 103/
(6b) sec.6 ' 8 Thus, for Ii' < 106 W/cM 2, P 1 y- /2 and 'llyo; Ya/r 1l << 1, and (I3 - 5)/rF << 1; and the second term
(6c) in Eq. (7a) may safely be ignored. That is, Eqs. (7a) and(7b) depend only on the total recombination rate 5.That this limit also corresponds to high steady-state
(6d) values of ionization can be seen by using Eq. (11). For¢ = 5 X 10 3/sec, an intensity Ii' of only 2 X 104 W/cm 2
ition leads to 95% ionization; 2 X 105 W/cm 2 leads to 99.5%iEq.I ex-Ii =
$ &+g =?IP2/(]P2 +ba2 ) (8)have been introduced, and the bistability coefficient Bis defined as
N = 0 A2 (9)
In the limit y,, -> 0, B is just the bistability coefficient
I.
150
(7a)100
(7b)
50
0
0
Fig. 1. The relationship between Q and Ii as a function ofdimensionless detunings ft and O,. The numerical resultsshow only slight change under interchange of (6t, 0,) - (0,,;a).
August 1981 / Vol. 6, No. 8 / OPTICS LETTERS 359
1.0
Q
-°2 j t
I t
j~li2 lI I
too
Fig. 2. Illustration of hysteresis between Ii and Q associatedwith Ca = °c = 0 in Fig. 1. Arrows indicate whether loci werereached by increasing or by decreasing Ii.
ionization. Larger values of Ii' lead essentially tocomplete ionization.
In Fig. 1 we plot Ii versus the ionization Q = 1 - F forh' = 2 X 104 W/cm2 , t = 5 X 103 /sec, and B = 50 forvarious values of Ca and Oc. This value of B correspondsroughly to the bistability coefficient for Na at No =1012/cm3 , with R = 0.95. Let us consider specificallythe absorption case, C,, = °c = 0. What would be ob-served in an experiment in which Ii is first increased andthen decreased slowly enough that equilibrium isreached at each value of Ii is shown in Fig. 2. As Ii isincreased slowly to I, _ 81.2, the ionization increasesslowly to a value Qi _ 0.65. A slight further increasein Ii causes an abrupt jump in Q from Qi to the value Q2= 0.949. Further increases in Ii produce only smallincreases in Q, since Q saturates at 0.95 in this case. AsIi is slowly decreased from Ii > Iij, the value of Q de-creases slowly until Ii = h2 = 19.6, where Q = 0.941. Aslight further decrease in Ii will result in a large drop inthe equilibrium value of Q to the value 0.21. Thus thegas will experience a sudden large increase in ionizationas the incident intensity is increased beyond a criticalvalue. Having passed this value, the incident intensitycan be dropped by more than a factor of 4, with less than1% decrease in the fraction of atoms that are ionized.Decrease in incident intensity below Ii2 < lii results ina sudden and dramatic decrease in the ionization frac-tion.
General trends in behavior when dispersive effectsare present can also be seen in Fig. 1. Numerical resultsshow that there is a pronounced symmetry in effects ofcavity and atomic detuning, i.e., results are nearly in-dependent of which of Ca or 0c takes on a particularvalue. Noting this, and using Fig. 1, one sees that if oneof 0 a or 0C is small, then as the other is tuned away fromzero the values Ii, and Qi at which up-switching occurschange only slightly. The value Ii2 at which down-switching occurs, however, rises quite rapidly, whereasthe value of ionization Q2 decreases only slightly.
The most dramatic changes in the curves occur whenCa and 0c are kept nearly equal in sign and magnitude
and are tuned gradually away from zero. The thresholdintensities for bistable behavior are lowered consider-ably, with the value of Ii1 decreasing to 14% of its valuewith (9a = Cc = 0. For B = 50, bistable behavior is soonquenched for opposite-sign values of OCa and 0c becauseof an associated rapid rise in hi2. For Oc _n°a > 6, thebistable behavior is quenched because of an associatedrapid falling of Ii'. With either Ca < 6 or Cc < 6 andwith the other varying intermediately to zero, the re-sultant curves vary smoothly between the extremes. Inall cases there is a decrease in the value of Q2 relative tothat for zero detunings, which is quite large for Ca -c- 6. Also, the decrease in the value 1i2 when 0c - ,varies gradually from zero is not so prominent as the risein hi2 that occurs when only one of the 0's is increased.Finally, the magnitude of the jump Q2 - Qi is seen todecrease with dispersive contributions generally, beingsmallest at lower values of lil. The result with both 0'snegative is of course the same as with both 0's posi-tive.
For larger values of B, the qualitative features remainthe same, albeit with a wider range of variation of Ca and0, compatible with bistability. Conversely, decreasingB leads to a smaller range of bistable behavior; as Bvaries below 25 bistability soon vanishes for all valuesof dCa and Oc.
The steady-state degree of ionization produced in agas by two-photon ionization thus exhibits bistablebehavior. In particular, when carrying out a two-pho-ton ionization experiment at fairly high densities in aclosely tuned cavity, one could ionize the sample witha high-intensity pulse and subsequently maintain thationization with a laser of considerably lower intensity.Refinements of this effect might be used to carry outmeasurements of the total recombination rate 0 and,under proper conditions, of the separate rates & and/.
H. C. Baker is grateful to the National ScienceFoundation for the award of a science faculty fellowshipunder which at Johns Hopkins University this work wasinitiated. He also wishes to acknowledge the warmsupport of the physics faculty at Johns Hopkins duringhis visit here.
References
1. L. Armstrong, J. Phys. B 12, L719 (1979).2. R. N. Compton, J. C. Miller, A. E. Carter, and P. Kruit,
Chem. Phys. Lett. 71, 87 (1980).3. J. C. Miller, R. N. Compton, M. G. Payne, and W. R. Gar-
rett, Phys. Rev. Lett. 45, 114 (1980).4. M. G. Payne, W. R. Garrett, and H. C. Baker, Chem. Phys.
Lett. 75, 468 (1980).5. R. Bonifacio and L. A. Lugiato, Lett. Nuovo Cimento 21,
517 (1978).6. D. R. Bates, A. E. Kingston, and R. W. P. McWhirter, Proc.
R. Soc. London A627, 297 (1962).7. T. B. Lucatorto and J. J. McIlrath, Phys. Rev. Lett. 37,428
(1976).8. C. H. Skinner, J. Phys. B 13, 55 (1980).
