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===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 [[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.
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