249

Chapter 3

FROM MULTIPHOTON TO TUNNEL IONIZATION

S. L. CHIN Canada Research Chair (Senior)

Center for Optics, Photonics and Laser & Department of Physics, Engineering Physics and Optics

Laval University, Quebec City, Quebec, G1K 7P4, Canada

slchin@phy.ulaval.ca

The story of the development of multiphton ionization to tunnel ionization is told briefly from the experimental point of view. It stresses the physical idea of tunneling ionization and the practical use of some tunneling formulae to fit experimental data. ADK formula is not universally valid in spite of its popularity.

Contents 1. Introduction 250

2. Laser Induced Breakdown 251

3. Multiphoton Ionization of Atoms (1964–1978) 253

4. A Persistent Challenge in the 1960s and 1970s: Tunnel Ionization 254

5. Some Surprises (1979–1989) 254

6. Tunnel Ionization using Long and Short Wavelength Lasers 261

7. Which Tunnel Ionization Formula to Use? 263

Conclusion 269

Acknowledgments 269

References 269

250

1. Introduction

This is a personal account of the progress of high laser field physics in atoms and molecules. The author has experienced the entire development of this field of research from the very beginning of Q-switched laser induced breakdown of gases, multiphoton ionization (MPI) and tunnel ionization of atoms and molecules that continues into the current state of the control of molecules in ultrafast intense laser field, atto-second laser science, etc. The purpose of this review is to tell a brief story of the development from laser induced breakdown through multiphoton ionization to tunnel ionization. Emphasis will be given to: (1) the physics of tunnel ionization based on our experimental results using a CO2 laser; (2) the applicability of the concept of tunneling to ionization of atoms and molecules using intense femtosecond near i.r. laser pulses. We shall begin discussing the physics of laser-induced breakdown of gases in the early 1960s. The development of multiphoton ionization follows. There was a seemingly conceptual bottleneck towards the end of the 1970s when multiphoton ionization physics was thoroughly studied. While laser technology has raised the intensity higher and higher in the laboratory so that strong field effect should be observable, many did not yet accept the concept of tunnel ionization even though it has been proposed by Keldysh in 1964.36 One reason was probably because there was so far no experimental evidence of tunnel ionization. The author and his co-workers first confirmed experimentally the existence of tunnel ionization in the mid-1980s using a CO2 laser. The concept of tunnel ionization eventually becomes a doorway concept in modern strong field physics and chemistry. Meanwhile, many surprisingly new and unexpected high field phenomena were discovered. They include above threshold ionization (ATI), high order harmonic generation and multiple ionization. The latter naturally led to the question of sequential and non-sequential ionization. In molecules, many new concepts were proposed and confirmed experimentally using intense femtosecond laser pulses: bond softening and hardening, above threshold dissociation, charge resonance enhanced ionization, Coulomb explosion, etc. Most of these phenomena are based on tunnel ionization as the first step of the physical processes. The author will not go into any of the details of these phenomena except for ATI and multiple ionization. Parallel to the modern development of high laser field physics and chemistry, the propagation of intense femtosecond laser pulses in transparent optical media leads to self-focusing and filamentation. Filament as long as km piercing through the atmosphere was indicated by new experiments. Essentially, a qualitative picture is the following. A powerful near i.r (we have in mind the most popular 800 nm Ti-sapphire laser) femtosecond laser pulse self-transforms into a self-organizing light bullet (Chin et al.22) becoming a white light laser

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pulse whose spectral width covers the whole visible region while propagating in many kinds of optical media (air, water, glass, etc.). The broad spectrum extends into the i.r. to about 4 microns in the case of air. This would mean that the original amplitude and phase of the laser pulse, after propagating through any optical medium including air, would be changed dramatically due to self-phase modulation and self-steepening. No interaction system in any practical experiment would be able to avoid this eventual modification of the laser’s amplitude and phase when the pulse propagates through at least the optical window of the system. No systematic study on this effect is yet published while preliminary experiments in our laboratory show some important consequences, in particular, that due to self-focusing, pulse splitting, intensity clamping, etc. while the pulse propagates through the window of the interaction system. This propagation effect on strong field interaction will not be discussed. Interested readers on the physics of propagation are referred to the following papers and references therein (Chin et al.21; Kandidov et al.35) Much of the story to be told is based on the author’s personal experience and is thus rather subjective. Nevertheless, he tries as much as possible to give a just account of some of the important milestones in experimental strong laser field physics.

2. Laser Induced Breakdown Soon after its invention around 1963, the Q-switched ruby laser has created some huge surprises. It could generate hot plasmas when focused onto solid targets. This sparked the beginning of the construction of big national laser laboratories to do laser plasma research with the aim of achieving laser induced nuclear fusion in many countries. (The author will not talk about this event in this paper.) In the meantime, Meyerand and Haught43 observed that a Q-switched ruby laser pulse could even generate a spark (electrical discharge) in air. This so-called laser induced breakdown of air (gases) started the era of multiphoton physics. It was recognized that the spark involved the strong ionization of the air molecules. The surprise at that time was that the laser photon energy from a Q-switched ruby laser (1.78 eV) was much smaller than the ionization potentials of air molecules (oxygen: 12.1 eV and nitrogen: 15.6 eV). It was inconceivable that there could be any photoelectric effect by absorbing one photon and ionizing the molecules. The idea of multiphoton absorption was thus developed even though the theory of two-photon absorption was already published in 1931 by Goeppert-Mayer.33 After a lot of extensive theoretical and experimental work in many laboratories, it was finally proposed that optical breakdown of gases followed essentially a three-step process (Fig. 1). (1) Multiphoton ionization of impurity molecules with low ionization potentials would easily provide a few free electrons with low initial kinetic energy in the focal volume at the front part of the pulse.12 (2) The free electrons in the

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strong laser field could absorb n photons (n = 0, 1, 2, 3, …) while colliding (scattering) with a much heavier particle (atom, molecule or ion). The heavy particle is to conserve momentum during the interaction. This process is called inverse Bremsstrahlung or free–free transition. (3) After one or more inverse Bremsstrahlung processes, the free electron would acquire a kinetic energy Ee higher than the ionization potential of the gas molecule/atom. Subsequent collision would give rise to the ejection of an extra electron from the molecule/atom. This would result in two low energy electrons. They would undergo the same processes as before each giving rise to two electrons, and so on until the gas is fully ionized. This is called cascade or avalanche ionization; i.e. breakdown.44

Fig.1. Visualization of laser induced breakdown. Many experiments were performed to verify the cascade ionization process.44 The fundamental process of the initiation of breakdown by the MPI of impurities was proved experimentally by the author12 whereas the free–free transition which is at the center of cascade ionization was observed by Weingartshofer.51,52 The initiation process of cascade breakdown, namely MPI, had its own physical interest because it touches upon a question of fundamental importance, i.e. the absorption of more than one photon by an atom or a molecule.

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3. Multiphoton Ionization of Atoms (1964–1978) There are already many papers and books describing the physics of MPI, including the current series of books. (See for example Chin and Lambropoulos,15 Delone and Krainov23 and references therein). Essentially, the time dependent perturbation theory can explain all the experimental observations using ns and ps laser pulses. The author would like to point out a few historical milestones in the pursuit of the understanding of MPI physics.

1963–1964: Keldysh36 published the paper on the theory of ionization of atoms in a strong laser field. He coined a parameter γ that separates the regimes of MPI and tunnel ionization where

02ωγ EF

= (in atomic units)

where ω is the laser frequency, F the laser electric field strength and E0, the ionization potential of the atom. It was concluded that when γ << 1, tunnel ionization prevails while when γ >> 1, it is a multiphoton effect. The multiphoton effect means that the physics of interaction can be explained by the perturbation theory (Chin and Lambropoulos15 and references therein).

1960s–1980s: In the 1960s till the early 1980s, only the condition of γ >> 1 was satisfied in all experiments using long (ns or sub-ns) pulses. The following lists the various contributions to MPI physics from the experimental point of view.

– Delone’s first experiment on MPI (1964–1965).49

– Mainfray’s major contribution to the field from the beginning of MPI soon after Delone et al.’s work till his retirement in 1996–1997: this includes MPI physics, resonance effect, coherence effect, pulse length effect42 leading to ATI and HHG (see later).

– Chin and Isenor’s13 observation and interpretation of the saturation of ionization in the center of the focal volume11: this was not a trivial conclusion because it was difficult to conceive that such a high order nonlinear process could have reached a probability of ionization of unity (i.e. saturation) so that the neutral atoms in the most intense part (usually center) of the focal volume are depleted. The consequence is the now famous slope of 3/2 in the log–log plot of ion yield vs laser peak intensity. This later conclusion was due to Cervenon and Isenor.10

– The author’s first experiment on the MPI of polyatomic molecules14: it was natural to extend multiphoton ionization to molecules and indeed both ionization and fragmentation were observed. Chemists later expanded into the now well-known branch of laser chemistry.

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– Weingartshofer first made the direct observation of free-free transition51: this was a very important experiment demonstrating the fundamental interaction process of the absorption/emission of one or more entire photons by a free electron in the vicinity of a much heavier particle (atom, molecule or ion) in the presence of a strong radiation field. In this process, the conservation of momentum through the implication of the much heavier particle is the key to inducing the absorption or emission of photons. This is distinct from the Compton process in which an electron is scattered by a photon resulting in the change of kinetic energy of the electron by a fraction of the photon energy while the photon’s frequency is changed. The consequence of many studies in the field of MPI towards the end of the 1970s led to applications such as REMPI (resonance MPI), laser isotope separation and in a general sense, laser chemistry. At that time, it was believed that the fundamental physics of MPI was well-understood, for e.g. the In law of MPI, the theoretically expected coherence effect on MPI, resonance effect, saturation, pulse length effect, etc. (cf. Chin and Lambropoulos15 and references therein). So, what could be new?

4. A Persistent Challenge in the 1960s and 1970s: Tunnel Ionization

Is tunnel ionization possible in the field of high optical frequency? Can one experimentally reach the condition of γ << 1? Keldysh’s paper certainly has inspired a huge amount of discussion right from the beginning of the MPI era. It inspired also the theoretical papers written by Faisal26 and Reiss48 which initially were not accepted as being valid physically by many MPI practitioners. (The trio was later referred to as the KFR theory.) There had been attempts to observe experimentally tunnel ionization of atoms by the French Saclay group in the 1970s. But the results were negative (Lompré et al.41). This reinforced the thinking that tunnel ionization, though possible in the case of DC field ionization, would be too difficult to be observed if not totally impossible. An early account of the attempt to observe tunnel ionization is given by the author in Chin and Lambropoulos,15 Chapter 1.

5. Some Surprises (1979–1989) Similar to many other scientific advancement, it was mostly, if not always, by accident and not by design that the advancement of the field of MPI was developed. The following gives an account of some of the most important developments.

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5.1. Above threshold ionization (ATI) observed by the Saclay group led by Mainfray

Towards the second half of the 1970s, Mainfray realized that the experimental study of MPI by measuring only the ion yield is already mature in the sense that not many new things could be observed (Mainfray, private communication). Since in the interaction of a strong radiation field with atoms, the product of the interaction consists of ions, electrons and photons, Mainfray believed that one should try to look at the electrons and the photons. He led his group to do exactly this and the unexpected phenomena of above threshold ionization or ATI1 and high order harmonic generation (HHG) were discovered.40 The observation of the absorption of many more photons by the ionized electron in the continuum (ATI) was initially counter intuitive. How could an electron in the continuum (i.e. free electron) absorb photons? Should it not be similar to the Compton effect where entire photon could not be absorbed? Very soon, it was recognized that the concept of an electron in the continuum has to be carefully defined. The electron raised into the continuum by MPI is not yet free. MPI leads to an almost instantaneous excitation (in the limit of the uncertainty principle) of a bound electron to the continuum. However, geometrically speaking, it is very similar to the Frank–Condon transition in molecular absorption of radiation. The electron that is excited into the energy continuum energetically speaking is still in its original vicinity of the nucleus. Moreover, it is bathed in the strong electromagnetic field (i.e. in a dressed state or Volkov state). When it starts to move away from this original position towards infinity, the electron would have the probability to absorb more photons in the strong laser field very similar to free-free transition. In fact, it was described by Kupersztych37 as half-inverse Bremsstrahlung or half free–free transition because the electron starts to move out from around the vicinity of the nucleus. Hence, its kinetic energy spectrum shows up as peaks spaced by one photon energy very similar to the experimental results of free–free transition done by Weingartshofer.51 It turns out later that this phenomenon of ATI is intimately related to non-sequential ionization and high order harmonic generation. The semi-classical electron re-scattering model in the tunneling regime (Kuchiev 1986, Corkum 1994, Kulander et al. 1993, van Linden van den Heuvell 1988, Gallagher 1988) could explain a lot of observation todate. (See review by DiMauro and Agostini.24) Today, ATI is put to excellent use in the detection of the absolute phase of intense few cycle laser pulses.45,9

5.2. Double and multiple ionization Double ionization in MPI of rare earth atoms (two electron systems) was first observed by Aleksakhin et al.2,3 In the era when multiphoton ionization of one

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single electron was considered highly improbable, the ejection of two electrons was certainly a rarity. It was in this work that the question of how the two electrons were ejected whether sequentially or simultaneously (non-sequential ionization) was first discussed. Similar work by Feldmann et al.30 followed. Later, in the French Saclay group led by Mainfray, the first sign of non-sequential ionization of Xe2+ was observed by L’Huillier et al.39 They also observed multiply-charged ions of up to Xe4+. It turned out that so long as the laser intensity keeps increasing with the advancement of the laser technology, one can almost indefinitely increase the multiply-charged state of an ion and no limit is in sight yet. Thus, later in the 1980s, up to Xe6+ could be measured using a CO2 laser.17 Using the newly developed TW level CPA laser in the laboratory of Mourou at the University of Rochester, the author and co-workers were able to measure up to Xe12+ (Fig. 2). (See also Augst et al.5,6) The most recent experiment from the JAERI Laboratory in Japan observed up to Xe26+ using 100TW femtosecond Ti-sapphire laser pulses.53

Xe → Xe (8+ 7+ 6+ 5+ 4+ 3+ 2+ 1+)Time-of-flight mass spectrum (unpublished)

The following publications were based upon such spectra: S. Augst et al,PRL, 63,2212-2215(1989); S. Augst et al, JOSA-B, 8,858-867(1991).

1st result on multiple ionization of Xe using the 1st T3 laser by S. Augst, D. Strickland, S.L. Chin, J. H. Eberly & G. Mourou (1988, Rochester, NY)

Even Xe (12+)

Fig. 2. Using the first T3 laser at 1.06 microns wavelength in the University of Rochester’s Laboratory for Laser Energetics to ionize Xe resulting in multiply-charged states of up to Xe12+.

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5.3. High order harmonic generation The TW level CPA laser used in the Saclay laboratory led by Mainfray produced another surprise; i.e. the now well-known higher order harmonics generation was observed. The physics of the relatively very efficient generation of high harmonics is again based upon the re-scattering theory (Kuchiev 1986, Corkum 1994, Kulander et al. 1993) and will not be discussed further here.

5.4. Tunnel ionization As mentioned above, tunnel ionization was considered by many to be almost impossible before the mid-1980s. The author’s first experimental observation of the existence of tunnel ionization of Xe atoms using a CO2 laser in 1985 was indeed a surprise.16 The experimental condition satisfied Keldysh’s condition of γ << 1 and the results are unambiguous. More surprising was the observation of Xe2+ and Xe3+ but now under the condition of γ < 1. Preceding this observation, Farkas and the author have already observed tunnel ejection of electrons from a gold surface28,29 using a CO2 laser. Meanwhile, Delone and co-workers in the Soviet Union were also attempting the same experiment. Unfortunately, while they were preparing the experiment, the author had already published the first observation. Since then, they did not complete the experiments. (The above was described to the author by one of his co-workers, Dr. F A Ilkov, who was responsible for setting up the experiment.) Later, they did the calculation by adopting an improved form of the Keldysh formula that was due to Perelomov et al.46 for the ionization of a hydrogen atom. The ionization rate of an arbitrary state of a hydrogen atom is given by:

( )

−

=

−−

F

F

F

F

F

FEmlfCw

mn

ln 3

2exp

2

π3

, 012

02

1

00, (1)

where ( ) 23

00 2EF = and 0E = ionization potential. The factor

0π3

F

F is a result

of averaging the rate over one cycle of the field. n

F

F2

02

is the long range effect

of the Coulomb potential; Cn,l and f (l, m) contain in formation of the initial

atomic state.

The scope of the above formula was extended by Ammosov, Delone and Krainov4 by replacing the hydrogen n,l by the effective quantum numbers n* and

258

l* and approximating the coefficient Cn,l by a simple expression using the quasi-classical approximation for n* >> 1. The resulting formula became a simpler one. In the original paper, there were a number of misprints. The following is the corrected ADK formula which was due to Ilkov, Decker and Chin34:

( )

−

−−+

=

−

Fn

Z

Fnn

eZ

n

l

n

Zew

n

3*

32

3**

3

*3*

2

3

2exp

12

4

12

12

3π3

23*

23

(2)

e = 0.71828 …, Z is the ionic charge, F the electric field strength of the laser; l the orbital angular momentum quantum number, and n* the effective principal number =

02EZ .

This formula remains valid down to n* ~ 1. This allowed the modified formula (1) to be applied to more complex atomic/ionic systems. Equation (2) is the result of this development for the case of m = 0 where m is the magnetic quantum number, the other terms with m = ±1, ±2, ±3, … being negligibly small in comparison to the case when F0/F >> 1. The above formulae are valid for linear polarization. The ADK4 formula (Eq. (2)) fits very well the author’s first experimental observation of tunnel ionization of Xe atoms.16 The author’s continued experimental study of tunnel ionization using the CO2 laser (including the first observation of tunnel ionization of molecules18) led to significant advancement in the understanding of the process. For example, simple molecules with no i.r. vibrational resonance behave like atoms. In fact, all atoms and non-resonant molecules can be tunnel ionized by the CO2 laser radiation and are very well described by the ADK formula. For example, the ADK formula (Eq. (2)) fits nicely the experimental data obtained in the author’s laboratory using a CO2 laser50 to ionize Xe, Kr, Ar, H2, O2, N2, CO, CO2, and NO. Figures 3 and 4 show two examples of tunnel ionization, one of multiple ionization of Xe (Fig. 3) and one of Kr as compared to CO (with almost equal ionization potentials, 13.99 eV for Kr and 14.01 eV for CO) (Fig. 4). The fit of the ADK formula without using any adjustable parameter is excellent. We note that they are all under the condition of γ < 1/2 instead of γ << 1 as required by the Keldysh theory (Walsh, et al.50 and references there-in). The reason is explained in what follows. We recall first the definition of γ 36

02ωγ EF

= (in atomic units) (3)

It was derived from the following consideration. Referring to Fig. 5, the ease or difficulty of tunneling can be expressed as the ratio between the equivalent classical time it takes for the electron to tunnel out the potential barrier while the

259

potential is bent down. This ratio is called γ. Since the potential is bent down during half a cycle of the field oscillation, the ratio γ can be expressed as

L

T

ττ

cyclelaser

timetunnelling

21

21

≡≡γ (4)

where τT ≡ tunneling time = classical time of flight of an electron through a potential barrier, and τL ≡ period of laser field oscillation ≡ 1/ω. Note that ω is simply the field oscillation frequency and not the circular frequency. We can calculate the classical time of flight of an electron through the potential barrier shown in Fig. 5 according to Keldysh36:

τT = "tunneling distance" / "tunneling speed" .

Fig. 3. CO2 laser tunnel ionization of Xe. The ADK tunneling formula is used to fit the data without any adjustable parameter.

Fig. 4. CO2 laser tunnel ionization of Kr and CO. Both have similar ionization potentials. The total overlap of the ion yield indicates that CO behaves as if it were an atom. The ADK tunneling formula is used to fit the data without any adjustable parameter.

260

rroRc

-ro

|V(r)|<Fr

E

|V(r)|<Fr|V(r)|>Fr

V(r)

-Fr V(r)-Fr

-E i

tunnel ionization

Fig. 5. Schematic diagram of tunnel ionization.

Assume that the electron in the classical Coulomb potential well moves back and forth along a one-dimensional axis r (Fig. 5) with a kinetic energy equal to the ionization potential E0 before the external field is applied. Its speed of motion ν is thus

2

1

02

=

m

Ev (5)

where m is the electron mass. When the external field is applied, the electron would "pass" through the potential barrier with this speed through the "tunneling distance" Rc assuming that the electron starts to move out of the barrier ar r = 0.

Rc = E0 / eF

where e is the electron charge, and F is the field strength.

∴ ( ) 2/10

21

21

2ττ

τmE

eFL

vR

L

Tc ω===γ

261

or in atomic units (e = m = 1)

( ) 2/102

ωγ EF

=

which is Eq. (3). The experimentally found condition of γ <1/2 means that the tunneling time is less than ¼ of the laser field’s period (see Eq. (4)). This is reasonable because as shown in Fig. 5, the Coulomb potential of the electron is bent down on one side of the Coulomb potential well during ½ of a period and during most of this half period, the barrier is too large for the electron to go through. Obviously, the concept of tunnel ionization does not take into account the influence of the intermediate states although the latter does play a role as in the case of the theory of multiphoton ionization based upon the perturbation theory (Geltman,31 Gold and Bebb32; more references can be found in Chin and Lambropoulos15). To explain this, one recognizes that the coupling between the ground state and any excited state of an atom or molecule (meaning the existence of an excited state) would take some time to develop. This time is at least one cycle of the field oscillation from the quantum mechanical point of view because the transition (coupling) probability is obtained by integrating the transition matrix over time over at least one cycle of the laser oscillation. In a time less than one cycle of oscillation, such a coupling could be negligible or meaningless. Since the experimentally observed condition for tunnel ionization to occur is γ < ½; i.e. the tunneling time is smaller than ¼ of the field oscillation, there is no need to consider the influence of intermediate states.20 The concept of tunnel ionization was finally accepted by most in the strong laser field community since 1993 as was evidenced by the transformation of the mentality during the 1993 International Conference on Multiphoton Processes held in Quebec City, Canada.25

6. Tunnel Ionization using Long and Short Wavelength Lasers

It is recognized by now that using nanosecond CO2 laser pulses (wavelength around 10 microns, one photon energy around 0.1 eV), ionization of atoms and molecules is always due to tunneling.16,18,50 There is practically no chance for multiphoton ionization to occur in any significant way because multiphoton ionization of any atom or molecule would imply the simultaneous absorption of nearly or more than 100 photons. This would require, in principle, a huge intensity. Before this happens, tunneling takes over. Thus, using the CO2 laser, there is an advantage to observe "clean" tunnel ionization. In the case of using near i.r. or visible wavelength (photon energy around 1 to 4 eV) laser pulses with long duration (nanosecond to tens of picoseconds), the

262

rising and the falling parts of the laser pulse would contribute significantly to MPI even if the peak intensity is higher than that which satisfies the tunnel ionization condition. The total ion yield due to tunnel ionization by the peak intensity would thus be masked by this MPI event at the rising/falling parts of the pulse. If the long pulse’s peak intensity is much higher than the intensity for tunnel ionization (i.e. γ << 1), the integrated effect of MPI at the rising part of the pulse would be enough to almost totally ionize the atom. Thus, before the pulse’s intensity reaches the regime of tunnel ionization, there are no more neutral atoms left to interact with the high intensity part of the pulse. The ion that is left over would have a higher ionization potential; i.e. γ >> 1, and MPI would prevail again. This probably was the reason why using a 30 ps, 1 micron wavelength laser pulse whose peak intensity gave a γ value less than 1, the French Saclay group did not observe any clear tunneling signal in the ionization of helium.41 Using the long (ns) CO2 laser pulse, the rising/falling parts would not contribute to any significant MPI process because it requires more than 100 photons absorption and the probability is next to zero. When the pulse intensity reaches the peak region where the intensity is high enough to satisfy the condition of γ < 1/2, tunnel ionization turns on/off "suddenly" as if the pulse had very short rise/fall times. This leads to the question as to whether one could ever observe tunnel ionization experimentally using near i.r. to near uv laser pulses. The answer lies in the success of the CO2 laser as mentioned in the preceding paragraph; i.e. if the rise/fall times of the near i.r. to near u.v.laser pulses are fast enough so that MPI in the rising/falling parts of the pulse is negligible because of the low intensity, tunnel ionization would be observable at the peak region. With this thinking, one could also conclude that the shorter the wavelength is the faster the rise time should be for the neglect of MPI at the rising/falling parts of the pulse. The above prediction is confirmed in recent years. Using femtosecond laser pulses in the near i.r. to the visible regime, tunnel ionization seems to dominate all the interaction physics of atomic or molecular ionization. Since about 1993, almost all femtosecond laser (near i.r. to near uv) ionization work uses one form or another of the tunneling theories to explain/fit the experimental results. Indeed, almost all experimental work on the interaction of intense femtosecond lasers (from near i.r. to near uv) with atoms and molecules leading to non-sequential ionization, fragmentation, high order harmonic generation, above threshold ionization, atto-second laser pulse generation, etc. start with the tunnel ionization of the species before discussing the subsequent physics of interaction. Tunnel ionization becomes the initiating process in all intense femtosecond laser interaction physics that involves the ejection of electrons from atoms or molecules; i.e. tunnel ionization becomes the doorway concept in modern ultrafast intense laser science.

263

7. Which Tunnel Ionization Formula to Use? The above analysis shows that when using very short laser pulses, it is legitimate to use the tunneling concept to describe the phenomenon of ionizing atoms and molecules. We now ask the question: which model or which tunneling formula should one use? Most people seem to favor the ADK formula without analyzing the suitability. The authors like to discuss this from the experimental point of view. We have done many careful experiments in recent years on the tunnel ionization of atoms and molecules using intense 200 fs Ti-sapphire laser pulses. We have fitted our experimental results using various models of tunnel ionization. The ADK formula certainly could not fit most of the experimental results. In fact, strictly speaking, it fits only the results of ionization of helium. Two theories excellently fit our and many other experimental results. One is what we called the ppt theory (Perelomov, Popov and Terent’ev46) and the other is the S-matrix theory developed by Faisal and Becker.27 Figures 6–10 show our experimental results on the tunnel ionization of all rare gas atoms using 200 fs Ti-sapphire laser pulses (800 nm central wavelength).38 The ion yield is plotted against both the peak intensity of the laser pulse (lower horizontal scale) and the corresponding values of γ (upper horizontal scale). Several theories were used to fit the experimental results. These include the strong field approximation (SFA)48 theory, the ADK formula, the ppt model and Szoke’s theory.47 One can see that the ADK formula fits only in the range where γ < 0.5 in the cases of He and Ne; i.e. in the higher intensity ranges. At the lower intensity ranges where γ > 0.5, there is a deviation from the experimental data as we should expect. Now, in the cases of the other rare gas atoms, namely, Ar, Kr and Xe, the fit of the ADK formula is way off since before γ < 0.5 is reached, saturation of the ion yield is reached. In this saturation region, no interaction physics can be extracted even though the ADK curve approaches the experimental data in the saturation region. The SFA and the Szoke theories do not fit our experimental data either. Only the ppt theory fits all the experimental data very well. The reason why the ppt theory fits well the experimental results seems to come from its inclusion of the Coulomb field effect on the outgoing electron during ionization. Becker and Faisal8 used their S-matrix theory27 to fit our experimental data and the fit was excellent. From the experimental point of view, we could conclude that whenever one wants to compare experimental data with the ADK theory, one must make sure that the tunnel ionization condition of γ < 0.5 is satisfied. Otherwise, the fit could only be considered as a qualitative guide. Many experimental data of this type contain a lot of fluctuation (i.e. not as smooth as our data) and the dynamic range of the yield spans over only 2–3 orders of magnitude, sometimes including the saturation region. When such huge fluctuation points over such a small dynamic range are fitted by the ADK formula in the region one or two orders of

264

magnitudes below the saturation point, the fit would look "not bad" as can be imagined from the fit with the Xe or Kr data in Figs. 6 and 7.

Fig.6. Tunnel ionization of Xe using a Ti-sapphire laser pulse at 800 nm/200 fs. Only the ppt theory fits well the experimental data. ADK as well as other known theories with analytical formulae could not fit the data.

265

Fig. 7. Tunnel ionization of Kr using a Ti-sapphire laser pulse at 800 nm/200 fs. Only the ppt theory fits well the experimental data. ADK as well as other known theories with analytical formulae could not fit the data.

266

Fig. 8. Tunnel ionization of Ar using a Ti-sapphire laser pulse at 800 nm/200 fs. Only the ppt theory fits well the experimental data. ADK as well as other known theories with analytical formulae could not fit the data.

267

Fig. 9. Tunnel ionization of Ne using a Ti-sapphire laser pulse at 800 nm/200 fs. Only the ppt theory fits well the experimental data. ADK formula starts to get closer to the experimental data while other known theories with analytical formulae could not fit the data.

268

Fig. 10. Tunnel ionization of He using a Ti-sapphire laser pulse at 800 nm/200 fs. Both the ppt theory and the ADK formula fit well the experimental data. No other known theories with analytical formulae could fit the data.

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Conclusion The author has given a brief historical account on the experimental progress of high field physics which is now taken up by chemistry extensively. The story starts from the observation of laser induced breakdown to the extensive study of the physics of multiphoton ionization. The theory of tunnel ionization was predicted at the same time. However, the reality of the experiments prohibited the observation of tunnel ionization easily. It takes a delay of more than 20 years before such a phenomenon was confirmed in our laboratory using a CO2 laser. The advancement of laser technology permits experimentalists to use femtosecond laser to ionize atoms and molecules and the short duration condition of the laser pulse favors tunnel ionization. It was pointed out that care should be taken in the use of tunnel ionization formula to fit with experimental data. The ADK formula is good only under the condition of γ < ½ . The better theories would be the ppt theory and/or the S-matrix theory of Faisal and Becker. Acknowledgment This work was supported by NSERC, FQRNT (FCAR) and the Canada Research Chairs. References 1. Agostini, P, Fabre, F., Mainfray, G, Petite, G and Rahman, N K, 1979,

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