Editing Ionization (section)

==Strong field approximation for the ionization rate==
The calculations of PPT are done in the '''E'''-gauge, meaning that the laser field is taken as electromagnetic waves. The ionization rate can also be calculated in '''A'''-gauge, which emphasizes the particle nature of light (absorbing multiple photons during ionization). This approach was adopted by Krainov model<ref name="Krainov1997">{{cite journal |last1=Krainov |first1=Vladimir P. |title=Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions |journal=Journal of the Optical Society of America B |volume=14 |issue=2 |date=1997 |page=425 |issn=0740-3224 |doi=10.1364/JOSAB.14.000425|bibcode=1997JOSAB..14..425K }}</ref> based on the earlier works of Faisal<ref name="Faisal1973">{{cite journal |last1=Faisal |first1=F. H. M. |title=Multiple absorption of laser photons by atoms |journal=Journal of Physics B: Atomic and Molecular Physics |volume=6 |issue=4 |date=1973 |pages=L89–L92 |issn=0022-3700 |doi=10.1088/0022-3700/6/4/011|bibcode=1973JPhB....6L..89F }}</ref> and Reiss.<ref name="Reiss1980">{{cite journal |last1=Reiss |first1=Howard |title=Effect of an intense electromagnetic field on a weakly bound system |journal=Physical Review A |volume=22 |issue=5 |date=1980 |pages=1786–1813 |issn=0556-2791 |doi=10.1103/PhysRevA.22.1786 |bibcode=1980PhRvA..22.1786R }}</ref> The resulting rate is given by

:<math>W_{KRA} =
  \sum_{n=N}^{\infty} 2 \pi \omega^2 p \left(n - n_\mathrm{osc}\right)^2 \int \mathrm{d}\Omega
  \left|FT \left(I_{KAR} \Psi \left(\mathbf{r}\right)\right)\right|^2
  J_n^2 \left(n_f, \frac{n_\mathrm{osc}}{2}\right)
</math>

where:

*<math> n_{i} =E_i/\omega,</math>
*<math>n_\mathrm{osc}=U_{p}/ \omega </math> with <math>U_p</math> being the ponderomotive energy,
*<math>N=[n_i + n_\mathrm{osc}] </math> is the minimum number of photons necessary to ionize the atom, 
*<math>J_{n}(u,v)</math> is the double Bessel function,
*<math>p=\sqrt{ 2 \omega (n-n_\mathrm{osc}- n_i)},</math>  
*<math display="inline">n_{f}=2 \sqrt { n_\mathrm{osc} / \omega} p \cos(\theta)</math> with <math> \theta</math> the angle between the momentum of the electron, '''p''', and the electric field of the laser, '''F''', 
* ''FT'' is the three-dimensional Fourier transform, and 
* <math> I_{KAR}=\left(\frac {2 Z^2}{n^2 F r}\right)^n </math> incorporates the Coulomb correction in the SFA model.

===Population trapping===
In calculating the rate of MPI of atoms only transitions to the continuum states are considered. Such an approximation is acceptable as long as there is no multiphoton resonance between the ground state and some excited states. However, in real situation of interaction with pulsed lasers, during the evolution of laser intensity, due to different Stark shift of the ground and excited states there is a possibility that some excited state go into multiphoton resonance with the ground state. Within the dressed atom picture, the ground state dressed by <math>m</math> photons and the resonant state undergo an avoided crossing at the resonance intensity <math>I_r</math>. The minimum distance, <math>V_m</math>, at the avoided crossing is proportional to the generalized Rabi frequency, <math>\Gamma(t) =\Gamma_m I(t)^{m/2}</math> coupling the two states. According to Story et al.,<ref name="StoryDuncan1994">{{cite journal |last1=Story |first1=J. |last2=Duncan |first2=D. |last3=Gallagher |first3=T. |title=Landau-Zener treatment of intensity-tuned multiphoton resonances of potassium |journal=Physical Review A |volume=50 |issue=2 |date=1994 |pages=1607–1617 |issn=1050-2947 |doi=10.1103/PhysRevA.50.1607 |pmid=9911054 |bibcode=1994PhRvA..50.1607S }}</ref> the probability of remaining in the ground state, <math>P_g</math>, is given by

:<math>P_g = \exp\left(-\frac{2\pi W_m^2}{\mathrm{d}W/\mathrm{d}t}\right)</math>

where <math>W</math> is the time-dependent energy difference between the two dressed states. In interaction with a short pulse, if the dynamic resonance is reached in the rising or the falling part of the pulse, the population practically remains in the ground state and the effect of multiphoton resonances may be neglected. However, if the states go onto resonance at the peak of the pulse, where <math>\mathrm{d}W/\mathrm{d}t = 0</math>, then the excited state is populated. After being populated, since the ionization potential of the excited state is small, it is expected that the electron will be instantly ionized.

In 1992, de Boer and Muller <ref>{{cite journal| doi=10.1103/PhysRevLett.68.2747| title=Observation of large populations in excited states after short-pulse multiphoton ionization |date=1992|last1=De Boer|first1=M. |last2=Muller|first2=H. |journal=Physical Review Letters |volume=68 |issue=18 |pages=2747–2750|pmid=10045482| bibcode=1992PhRvL..68.2747D }}</ref> showed that Xe atoms subjected to short laser pulses could survive in the highly excited states 4f, 5f, and 6f. These states were believed to have been excited by the dynamic Stark shift of the levels into multiphoton resonance with the field during the rising part of the laser pulse. Subsequent evolution of the laser pulse did not completely ionize these states, leaving behind some highly excited atoms. We shall refer to this phenomenon as "population trapping".

[[File:Lambda type population trapping.png|right|thumb|Schematic presentation of lambda type population trapping. G is the ground state of the atom. 1 and 2 are two degenerate excited states. After the population is transferred to the states due to multiphoton resonance, these states are coupled through continuum c and the population is trapped in the superposition of these states.]]

We mention the theoretical calculation that incomplete ionization occurs whenever there is parallel resonant excitation into a common level with ionization loss.<ref>{{cite journal |last1=Hioe |first1=F. T. |last2=Carrol |first2=C. E. |doi=10.1103/PhysRevA.37.3000| title=Coherent population trapping in N-level quantum systems| date=1988| journal=Physical Review A|volume=37| issue=8| pages=3000–3005 |pmid=9900034| bibcode=1988PhRvA..37.3000H }}</ref> We consider a state such as 6f of Xe which consists of 7 quasi-degnerate levels in the range of the laser bandwidth. These levels along with the continuum constitute a lambda system. The mechanism of the lambda type trapping is schematically presented in figure. At the rising part of the pulse (a) the excited state (with two degenerate levels 1 and 2) are not in multiphoton resonance with the ground state. The electron is ionized through multiphoton coupling with the continuum. As the intensity of the pulse is increased the excited state and the continuum are shifted in energy due to the Stark shift. At the peak of the pulse (b) the excited states go into multiphoton resonance with the ground state. As the intensity starts to decrease (c), the two state are coupled through continuum and the population is trapped in a coherent superposition of the two states. Under subsequent action of the same pulse, due to interference in the transition amplitudes of the lambda system, the field cannot ionize the population completely and a fraction of the population will be trapped in a coherent superposition of the quasi degenerate levels. According to this explanation the states with higher angular momentum – with more sublevels – would have a higher probability of trapping the population. In general the strength of the trapping will be determined by the strength of the two photon coupling between the quasi-degenerate levels via the continuum. In 1996, using a very stable laser and by minimizing the masking effects of the focal region expansion with increasing intensity, Talebpour et al.<ref>{{cite journal |last1=Talebpour |first1=A. |last2=Chien |first2=C. Y. |last3=Chin |first3=S. L. |doi=10.1088/0953-4075/29/23/015 |title=Population trapping in rare gases| date=1996|journal=Journal of Physics B: Atomic, Molecular and Optical Physics |volume=29| issue=23|page=5725 |bibcode=1996JPhB...29.5725T |s2cid=250757252 }}</ref> observed structures on the curves of singly charged ions of Xe, Kr and Ar. These structures were attributed to electron trapping in the strong laser field. A more unambiguous demonstration of population trapping has been reported by T. Morishita and [[Chii-Dong Lin|C. D. Lin]].<ref name="MorishitaLin2013">{{cite journal |last1=Morishita |first1=Toru |last2=Lin |first2=C. D. |title=Photoelectron spectra and high Rydberg states of lithium generated by intense lasers in the over-the-barrier ionization regime |journal=Physical Review A |volume=87 |issue=6 |page=63405 |date=2013 |issn=1050-2947 |doi=10.1103/PhysRevA.87.063405|bibcode=2013PhRvA..87f3405M |hdl=2097/16373 |url=http://krex.k-state.edu/dspace/bitstream/2097/16373/1/LinPhysRevA2013.pdf |hdl-access=free }}</ref>

===Non-sequential multiple ionization===
The phenomenon of non-sequential ionization (NSI) of atoms exposed to intense laser fields has been a subject of many theoretical and experimental studies since 1983. The pioneering work began with the observation of a "knee" structure on the Xe<sup>2+</sup> ion signal versus intensity curve by L’Huillier et al.<ref>{{cite journal |last1=L’Huillier |first1=A. |last2=Lompre |first2=L. A. |last3=Mainfray |first3=G. |last4=Manus |first4=C. |doi=10.1103/PhysRevA.27.2503|title=Multiply charged ions induced by multiphoton absorption in rare gases at 0.53 μm|date=1983|journal=Physical Review A|volume=27|issue=5|page=2503|bibcode=1983PhRvA..27.2503L }}</ref> From the experimental point of view, the NS double ionization refers to processes which somehow enhance the rate of production of doubly charged ions by a huge factor at intensities below the saturation intensity of the singly charged ion. Many, on the other hand, prefer to define the NSI as a process by which two electrons are ionized nearly simultaneously. This definition implies that apart from the sequential channel <math> A+L -> A^+ + L -> A^{++} </math> there is another channel <math> A+L-> A^{++} </math> which is the main contribution to the production of doubly charged ions at lower intensities. The first observation of triple NSI in [[argon]] interacting with a 1&nbsp;[[μm]] laser was reported by Augst et al.<ref>{{cite journal |last1=Augst |first1=S. |last2=Talebpour |first2=A. |last3=Chin |first3=S. L. |last4=Beaudoin |first4=Y. |last5=Chaker |first5=M. |doi=10.1103/PhysRevA.52.R917|title=Nonsequential triple ionization of argon atoms in a high-intensity laser field|date=1995|journal=Physical Review A|volume=52|issue=2|pages=R917–R919|pmid=9912436|bibcode=1995PhRvA..52..917A }}</ref> Later, systematically studying the NSI of all rare gas atoms, the quadruple NSI of Xe was observed.<ref>{{cite journal|doi=10.1088/0953-4075/31/6/008|title=Non-sequential multiple ionization of rare gas atoms in a Ti:Sapphire laser field |date=1998 |last1=Larochelle |first1=S. |last2=Talebpour |first2=A. |last3=Chin |first3=S. L. |journal=Journal of Physics B: Atomic, Molecular and Optical Physics|volume=31|issue=6|page=1201|bibcode=1998JPhB...31.1201L |s2cid=250747225 }}</ref> The most important conclusion of this study was the observation of the following relation between the rate of NSI to any charge state and the rate of tunnel ionization (predicted by the ADK formula) to the previous charge states; 
:<math> W_{NS}(A^{n+})= \sum_{i=1}^{n-1} \alpha_n\left(\lambda\right) W_{ADK}\left(A^{i+}\right)</math>

where <math>W_{ADK}\left(A^{i+}\right)</math> is the rate of quasi-static tunneling to i'th charge state and <math>\alpha_n(\lambda)</math> are some constants depending on the wavelength of the laser (but not on the pulse duration).

Two models have been proposed to explain the non-sequential ionization; the shake-off model and electron re-scattering model. The shake-off (SO) model, first proposed by Fittinghoff et al.,<ref>{{cite journal |last1=Fittinghoff |first1=D. N. |last2=Bolton |first2=P. R. |last3=Chang |first3=B. |last4=Kulander |first4=K. C. |doi=10.1103/PhysRevLett.69.2642 |title=Observation of nonsequential double ionization of helium with optical tunneling|date=1992|journal=Physical Review Letters|volume=69|issue=18|pages=2642–2645|pmid=10046547|bibcode=1992PhRvL..69.2642F |url=https://zenodo.org/record/1233895 }}</ref> is adopted from the field of ionization of atoms by X rays and electron projectiles where the SO process is one of the major mechanisms responsible for the multiple ionization of atoms. The SO model describes the NSI process as a mechanism where one electron is ionized by the laser field and the departure of this electron is so rapid that the remaining electrons do not have enough time to adjust themselves to the new energy states. Therefore, there is a certain probability that, after the ionization of the first electron, a second electron is excited to states with higher energy (shake-up) or even ionized (shake-off). We should mention that, until now, there has been no quantitative calculation based on the SO model, and the model is still qualitative.

The electron rescattering model was independently developed by Kuchiev,<ref>[http://www.jetpletters.ac.ru/ps/1241/article_18763.pdf]{{cite journal |last=Kuchiev |first=M. Yu |date=1987 |title=Atomic antenna |journal=Soviet Phys. JETP Lett. |volume=45 |pages=404–406}}</ref> Schafer ''et al'',<ref>{{cite journal |last1=Schafer |first1=K. J. |last2=Yang |first2=B. |last3=DiMauro |first3=L.F. |last4=Kulander |first4=K.C.|doi=10.1103/PhysRevLett.70.1599|pmid=10053336 |title=Above threshold ionization beyond the high harmonic cutoff|date=1992|journal=Physical Review Letters|volume=70|issue=11|pages=1599–1602|bibcode=1993PhRvL..70.1599S }}</ref> Corkum,<ref>{{cite journal |last=Corkum |first=P. B. | doi=10.1103/PhysRevLett.71.1994 |title=Plasma perspective on strong field multiphoton ionization|date=1993 |journal=Physical Review Letters |volume=71 |issue=13 |pages=1994–1997 |pmid=10054556 |bibcode=1993PhRvL..71.1994C|s2cid=29947935 |url=https://nrc-publications.canada.ca/eng/view/accepted/?id=c16dde3a-0d05-4437-bca6-3292fdd9d9ff }}</ref> Becker and Faisal<ref name="BeckerFaisal1996">{{cite journal |last1=Becker |first1=Andreas |last2=Faisal |first2=Farhad H M |title=Mechanism of laser-induced double ionization of helium |journal=Journal of Physics B: Atomic, Molecular and Optical Physics |volume=29 |issue=6 |date=1996 |pages=L197–L202 |issn=0953-4075 |doi=10.1088/0953-4075/29/6/005|bibcode=1996JPhB...29L.197B |s2cid=250808704 }}</ref> and Faisal and Becker.<ref>[https://pub.uni-bielefeld.de/publication/1627967]{{cite journal |last1=Faisal |first1=F. H. M. |last2=Becker |first2=A. |date=1997 |title=Nonsequential double ionization: Mechanism and model formula |journal=Laser Phys. |volume=7 |page=684}}</ref> The principal features of the model can be understood easily from Corkum's version. Corkum's model describes the NS ionization as a process whereby an electron is tunnel ionized. The electron then interacts with the laser field where it is accelerated away from the nuclear core. If the electron has been ionized at an appropriate phase of the field, it will pass by the position of the remaining ion half a cycle later, where it can free an additional electron by electron impact. Only half of the time the electron is released with the appropriate phase and the other half it never return to the nuclear core. The maximum kinetic energy that the returning electron can have is 3.17 times the ponderomotive potential (<math> U_p </math>) of the laser. Corkum's model places a cut-off limit on the minimum intensity (<math> U_p </math> is proportional to intensity) where ionization due to re-scattering can occur.

[[File:Kuchiev's model.png|thumb|upright=1.25|Feynman diagram for the process of double ionization in an atom through re-scattering mechanism]]
The re-scattering model in Kuchiev's version (Kuchiev's model) is quantum mechanical. The basic idea of the model is illustrated by Feynman diagrams in figure a. First both electrons are in the ground state of an atom. The lines marked a and b describe the corresponding atomic states. Then the electron a is ionized. The beginning of the ionization process is shown by the intersection with a sloped dashed line. where the MPI occurs. The propagation of the ionized electron in the laser field, during which it absorbs other photons (ATI), is shown by the full thick line. The collision of this electron with the parent atomic ion is shown by a vertical dotted line representing the Coulomb interaction between the electrons. The state marked with c describes the ion excitation to a discrete or continuum state. Figure b describes the exchange process. Kuchiev's model, contrary to Corkum's model, does not predict any threshold intensity for the occurrence of NS ionization.

Kuchiev did not include the Coulomb effects on the dynamics of the ionized electron. This resulted in the underestimation of the double ionization rate by a huge factor. Obviously, in the approach of Becker and Faisal (which is equivalent to Kuchiev's model in spirit), this drawback does not exist. In fact, their model is more exact and does not suffer from the large number of approximations made by Kuchiev. Their calculation results perfectly fit with the experimental results of Walker et al.<ref>{{cite journal |last1=Walker |first1=B. |last2=Sheehy |first2=B. |last3=Dimauro |first3=L. F. |last4=Agostini |first4=P. |last5=Schafer |first5=K. J. |last6=Kulander |first6=K. C. |doi=10.1103/PhysRevLett.73.1227|title=Precision Measurement of Strong Field Double Ionization of Helium|date=1994|journal=Physical Review Letters|volume=73|issue=9|pages=1227–1230|pmid=10057657|bibcode=1994PhRvL..73.1227W }}</ref> Becker and Faisal<ref>{{cite journal|doi=10.1088/0953-4075/32/14/101|title=S-matrix analysis of ionization yields of noble gas atoms at the focus of Ti:sapphire laser pulses |date=1999 |last1=Becker |first1=A. |last2=Faisal |first2=F. H. M. |journal=Journal of Physics B: Atomic, Molecular and Optical Physics |volume=32 |issue=14 |pages=L335 |bibcode=1999JPhB...32L.335B |s2cid=250766534 }}</ref> have been able to fit the experimental results on the multiple NSI of rare gas atoms using their model. As a result, the electron re-scattering can be taken as the main mechanism for the occurrence of the NSI process.

===Multiphoton ionization of inner-valence electrons and fragmentation of polyatomic molecules===
The ionization of inner valence electrons are responsible for the fragmentation of polyatomic molecules in strong laser fields. According to a qualitative model<ref>{{cite journal|url=http://slchin-symposium.copl.ulaval.ca/MPublication/168_CPL_313_0789.pdf |doi=10.1016/S0009-2614(99)01075-1 |title=Multiphoton ionization of inner-valence electrons and fragmentation of ethylene in an intense Ti:sapphire laser pulse |date=1999 |last1=Talebpour |first1=A. |last2=Bandrauk |first2=A. D. |last3=Yang |first3=J |last4=Chin |first4=S. L. |journal=Chemical Physics Letters |volume=313 |issue=5–6 |page=789 |bibcode=1999CPL...313..789T |url-status=dead |archive-url=https://web.archive.org/web/20141121115050/http://slchin-symposium.copl.ulaval.ca/MPublication/168_CPL_313_0789.pdf |archive-date=November 21, 2014 }}</ref><ref>{{cite journal|doi=10.1088/0953-4075/33/21/307|title=Dissociative ionization of benzene in intense ultra-fast laser pulses|date=2000|last1=Talebpour|first1=A|last2=Bandrauk|first2=A D|last3=Vijayalakshmi|first3=K|last4=Chin|first4=S L|journal=Journal of Physics B: Atomic, Molecular and Optical Physics|volume=33|issue=21|page=4615|bibcode=2000JPhB...33.4615T |s2cid=250738396 }}</ref> the dissociation of the molecules occurs through a three-step mechanism:
*MPI of electrons from the inner orbitals of the molecule which results in a molecular ion in ro-vibrational levels of an excited electronic state;
*Rapid radiationless transition to the high-lying ro-vibrational levels of a lower electronic state; and
*Subsequent dissociation of the ion to different fragments through various fragmentation channels.

The short pulse induced molecular fragmentation may be used as an ion source for high performance mass spectroscopy. The selectivity provided by a short pulse based source is superior to that expected when using the conventional electron ionization based sources, in particular when the identification of optical isomers is required.<ref>{{cite journal|doi=10.1007/s00340-008-3038-y|title=Ultra-fast laser pulses provide an ion source for highly selective mass spectroscopy|journal=Applied Physics B|volume=91|issue=3–4|pages=579|year=2008 |last1=Mehdi Sharifi |first1=S. |last2=Talebpour |first2=A. |last3=Chin |first3=S. L.| bibcode=2008ApPhB..91..579M| s2cid=122546433}}</ref><ref name="PengPuskas2012">{{cite journal |last1=Peng |first1=Jiahui |last2=Puskas |first2=Noah |last3=Corkum |first3=Paul B. |last4=Rayner |first4=David M.|last5=Loboda |first5=Alexandre V. |title=High-Pressure Gas Phase Femtosecond Laser Ionization Mass Spectrometry |journal=Analytical Chemistry |volume=84 |issue=13 |date=2012 |pages=5633–5640 |issn=0003-2700 |doi=10.1021/ac300743k|pmid=22670784|s2cid=10780362 |url=https://nrc-publications.canada.ca/eng/view/accepted/?id=8c0691b7-362f-4e8b-bc8f-39e7fc7b6c62 }}</ref>