Strip algebra: Difference between revisions
en>David Eppstein →Applications: this needs help, but it is not a stub |
en>Khazar2 m clean up, typos fixed: conection → connection using AWB |
||
| Line 1: | Line 1: | ||
[ | {{lead missing|date=February 2012}} | ||
== Perturbative Harmonic Generation == | |||
Perturbative '''Harmonic Generation''' is a process whereby laser light of frequency ''ω'' and photon energy ''ħω'' can be used to generate new frequencies of light. The newly generated frequencies are integer multiples ''nħω'' of the original light's frequency. This process was first discovered in 1961 by | |||
Franken et al.,<ref>P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. | |||
7, 118 (1961).</ref> using a [[ruby laser]], with crystalline [[quartz]] as the [[nonlinear optics|nonlinear medium]]. | |||
[[Harmonic]] generation in [[dielectric|dielectric solids]] is well understood and | |||
extensively used in modern [[Laser science|laser physics]] (see [[second harmonic generation]]). In 1967 New et al. observed the first third harmonic | |||
generation in a gas.<ref>G. H. C. New and J. F. Ward, Phys. Rev. Lett. 19, 556 (1967).</ref> In monatomic gases it is only | |||
possible to produce odd numbered harmonics for reasons of | |||
symmetry. Harmonic generation in the perturbative (weak field) | |||
regime is characterised by [[Vanish at infinity|rapidly decreasing]] efficiency with | |||
increasing harmonic order and harmonics up to the 11th order have | |||
been observed under these conditions | |||
.<ref>J. Wildenauer, Journal of [[Applied physics|Applied Physics]] 62, 41 (1987).</ref> This behaviour can be | |||
understood by considering an atom absorbing ''n'' [[photons]] | |||
then emitting a single high energy photon. The probability of | |||
absorbing ''n'' photons decreases as ''n'' increases, explaining the rapid | |||
decrease in the initial harmonic intensities. | |||
== High Harmonic Generation (HHG)== | |||
The first High Harmonic Generation (HHG) was observed in 1977 in interaction of intense CO<sub>2</sub> laser pulses with [[plasma (physics)|plasma]] generated from solid targets.<ref>N. H. Burnett et al., Appl. Phys. Lett., vol. 31, pp. 172–174, 1977.</ref> HHG in gases, far more widespread in application today, was first observed by McPherson and colleagues in 1987,<ref>A. McPherson et al, JOSA B 4, 595 (1987).</ref> and later by Ferray et al. in 1988,<ref>M. Ferray et al., Journal of Physics B-Atomic Molecular and Optical | |||
Physics 21, L31 (1988).</ref> with surprising results: the high | |||
harmonics were found to decrease in intensity at low orders, as expected, | |||
but then were observed to form a plateau, with the intensity of the harmonics | |||
remaining approximately constant over many orders.<ref>X. F. Li, A. L'Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, Physical | |||
Review A 39, 5751 (1989).</ref> | |||
Plateau harmonics spanning hundreds of eV have been measured which | |||
extend into the [[X-ray|soft x-ray]] regime.<ref>J. Seres et al., Nature 433, 596 (2005).</ref> This plateau ends abruptly at a position called | |||
the High Harmonic Cut-off. | |||
== Properties of High Harmonics == | |||
High harmonics have a number of interesting properties. They are a | |||
tunable table-top source of XUV/Soft X-rays, synchronised with the | |||
driving laser and produced with the same repetition rate. The | |||
harmonic cut-off varies linearly with increasing laser intensity up | |||
until the saturation intensity I<sub>sat</sub> where harmonic generation | |||
stops.<ref>T. Brabec and F. Krausz, [[Reviews of Modern Physics]] 72, 545 (2000).</ref> The saturation intensity can be | |||
increased by changing the atomic species to lighter [[noble gas]]es | |||
but these have a lower conversion efficiency so there is a balance | |||
to be found depending on the photon energies required. | |||
High harmonic generation strongly depends on the driving laser | |||
field and as a result the harmonics have similar temporal and | |||
spatial coherence properties.<ref>A. L'Huillier, K. J. Schafer, and K. C. Kulander, Journal of Physics B Atomic | |||
Molecular and [[Optical physics|Optical Physics]] 24, 3315 (1991).</ref> High harmonics are often generated | |||
with pulse durations shorter than that of the driving laser. This | |||
is due to phase matching and [[ionization]]. Often harmonics are only | |||
produced in a very small temporal window when the phase matching | |||
condition is met. Depletion of | |||
the generating media due to ionization also means that harmonic | |||
generation is mainly confined to the [[signal edge|leading edge]] of the driving | |||
pulse.<ref>K. J. Schafer and K. C. Kulander, [[Physical Review Letters]] 78, 638 (1997).</ref> | |||
High harmonics are emitted co-linearly with the driving laser and | |||
can have a very tight angular confinement, sometimes with less | |||
divergence than that of the fundamental field and near Gaussian | |||
beam profiles.<ref>J. W. G. Tisch et al., Physical Review A 49, R28 (1994).</ref> | |||
== Semi-classical approach to describe HHG== | |||
The maximum photon energy producible with high harmonic generation is given by the cut-off of the harmonic plateau. This can be calculated [[classical mechanics|classically]] by examining the | |||
maximum energy the ionized electron can gain in the [[electric field]] of the laser. The cut-off energy is given by, | |||
<math> E_{max}=I_p+3.17U_p </math> | |||
where U<sub>p</sub> is the [[ponderomotive energy]] from the laser field and I<sub>p</sub> is the | |||
[[ionization potential]]. | |||
This derivation of the cut-off energy is derived from a | |||
semi-classical calculation. The electron is initially treated quantum | |||
mechanically as it [[tunnel ionization|tunnel ionizes]] from the parent atom, but then its subsequent dynamics | |||
are treated classically. The electron is assumed to be born into | |||
the [[vacuum]] with zero initial velocity, and to be subsequently accelerated | |||
by the laser beam's [[electric field]]. | |||
[[Image:ThreeStep.png|thumb|alt=Illustration of the semi-classical three-step model of HHG.|The three-step model.]] | |||
Half an optical cycle after ionization, the electron will reverse direction as the | |||
electric field changes, and will accelerate back towards the parent nucleus. Upon returning to the parent nucleus it can then emit [[bremsstrahlung]]-like radiation during a recombination | |||
process with the atom as it returns to its [[Stationary state|ground state]]. | |||
This description has become known as the recollisional model of high harmonic generation | |||
.<ref>P. B. Corkum, Physical Review Letters 71, 1994 (1993).</ref> | |||
Some interesting limits on the HHG process which are explained by this model show that HHG will only occur if | |||
the driving laser field is linearly polarised. Ellipticity on the [[Laser|laser beam]] | |||
causes the returning electron to miss the parent nucleus. Quantum | |||
mechanically, the overlap of the returning electron wavepacket | |||
with the nuclear wavepacket is reduced. This has been observed | |||
experimentally, where the intensity of harmonics decreases rapidly | |||
with increasing ellipticity.<ref>P. Dietrich, N. H. Burnett, M. Ivanov, and P. B. Corkum, Physical Review | |||
A 50, R3585 (1994).</ref> Another effect which limits the | |||
intensity of the driving laser is the [[Lorentz force]]. At | |||
intensities above 10<sup>16</sup> Wcm<sup>−2</sup> the magnetic component of | |||
the laser pulse, which is ignored in weak field optics, can become | |||
strong enough to deflect the returning electron. This will cause | |||
it to 'miss' the parent nucleus and hence prevent HHG. | |||
==See also== | |||
*[[Nonlinear optics]] | |||
*[[Photoionization]] | |||
== References == | |||
{{reflist|2}} | |||
[[Category:Nonlinear optics]] | |||
[[Category:Laser science]] | |||
Latest revision as of 04:24, 10 July 2013
Perturbative Harmonic Generation
Perturbative Harmonic Generation is a process whereby laser light of frequency ω and photon energy ħω can be used to generate new frequencies of light. The newly generated frequencies are integer multiples nħω of the original light's frequency. This process was first discovered in 1961 by Franken et al.,[1] using a ruby laser, with crystalline quartz as the nonlinear medium.
Harmonic generation in dielectric solids is well understood and extensively used in modern laser physics (see second harmonic generation). In 1967 New et al. observed the first third harmonic generation in a gas.[2] In monatomic gases it is only possible to produce odd numbered harmonics for reasons of symmetry. Harmonic generation in the perturbative (weak field) regime is characterised by rapidly decreasing efficiency with increasing harmonic order and harmonics up to the 11th order have been observed under these conditions .[3] This behaviour can be understood by considering an atom absorbing n photons then emitting a single high energy photon. The probability of absorbing n photons decreases as n increases, explaining the rapid decrease in the initial harmonic intensities.
High Harmonic Generation (HHG)
The first High Harmonic Generation (HHG) was observed in 1977 in interaction of intense CO2 laser pulses with plasma generated from solid targets.[4] HHG in gases, far more widespread in application today, was first observed by McPherson and colleagues in 1987,[5] and later by Ferray et al. in 1988,[6] with surprising results: the high harmonics were found to decrease in intensity at low orders, as expected, but then were observed to form a plateau, with the intensity of the harmonics remaining approximately constant over many orders.[7] Plateau harmonics spanning hundreds of eV have been measured which extend into the soft x-ray regime.[8] This plateau ends abruptly at a position called the High Harmonic Cut-off.
Properties of High Harmonics
High harmonics have a number of interesting properties. They are a tunable table-top source of XUV/Soft X-rays, synchronised with the driving laser and produced with the same repetition rate. The harmonic cut-off varies linearly with increasing laser intensity up until the saturation intensity Isat where harmonic generation stops.[9] The saturation intensity can be increased by changing the atomic species to lighter noble gases but these have a lower conversion efficiency so there is a balance to be found depending on the photon energies required.
High harmonic generation strongly depends on the driving laser field and as a result the harmonics have similar temporal and spatial coherence properties.[10] High harmonics are often generated with pulse durations shorter than that of the driving laser. This is due to phase matching and ionization. Often harmonics are only produced in a very small temporal window when the phase matching condition is met. Depletion of the generating media due to ionization also means that harmonic generation is mainly confined to the leading edge of the driving pulse.[11]
High harmonics are emitted co-linearly with the driving laser and can have a very tight angular confinement, sometimes with less divergence than that of the fundamental field and near Gaussian beam profiles.[12]
Semi-classical approach to describe HHG
The maximum photon energy producible with high harmonic generation is given by the cut-off of the harmonic plateau. This can be calculated classically by examining the maximum energy the ionized electron can gain in the electric field of the laser. The cut-off energy is given by,
where Up is the ponderomotive energy from the laser field and Ip is the ionization potential.
This derivation of the cut-off energy is derived from a semi-classical calculation. The electron is initially treated quantum mechanically as it tunnel ionizes from the parent atom, but then its subsequent dynamics are treated classically. The electron is assumed to be born into the vacuum with zero initial velocity, and to be subsequently accelerated by the laser beam's electric field.
Half an optical cycle after ionization, the electron will reverse direction as the electric field changes, and will accelerate back towards the parent nucleus. Upon returning to the parent nucleus it can then emit bremsstrahlung-like radiation during a recombination process with the atom as it returns to its ground state. This description has become known as the recollisional model of high harmonic generation .[13]
Some interesting limits on the HHG process which are explained by this model show that HHG will only occur if the driving laser field is linearly polarised. Ellipticity on the laser beam causes the returning electron to miss the parent nucleus. Quantum mechanically, the overlap of the returning electron wavepacket with the nuclear wavepacket is reduced. This has been observed experimentally, where the intensity of harmonics decreases rapidly with increasing ellipticity.[14] Another effect which limits the intensity of the driving laser is the Lorentz force. At intensities above 1016 Wcm−2 the magnetic component of the laser pulse, which is ignored in weak field optics, can become strong enough to deflect the returning electron. This will cause it to 'miss' the parent nucleus and hence prevent HHG.
See also
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
- ↑ G. H. C. New and J. F. Ward, Phys. Rev. Lett. 19, 556 (1967).
- ↑ J. Wildenauer, Journal of Applied Physics 62, 41 (1987).
- ↑ N. H. Burnett et al., Appl. Phys. Lett., vol. 31, pp. 172–174, 1977.
- ↑ A. McPherson et al, JOSA B 4, 595 (1987).
- ↑ M. Ferray et al., Journal of Physics B-Atomic Molecular and Optical Physics 21, L31 (1988).
- ↑ X. F. Li, A. L'Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, Physical Review A 39, 5751 (1989).
- ↑ J. Seres et al., Nature 433, 596 (2005).
- ↑ T. Brabec and F. Krausz, Reviews of Modern Physics 72, 545 (2000).
- ↑ A. L'Huillier, K. J. Schafer, and K. C. Kulander, Journal of Physics B Atomic Molecular and Optical Physics 24, 3315 (1991).
- ↑ K. J. Schafer and K. C. Kulander, Physical Review Letters 78, 638 (1997).
- ↑ J. W. G. Tisch et al., Physical Review A 49, R28 (1994).
- ↑ P. B. Corkum, Physical Review Letters 71, 1994 (1993).
- ↑ P. Dietrich, N. H. Burnett, M. Ivanov, and P. B. Corkum, Physical Review A 50, R3585 (1994).