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| '''Saturable absorption''' is a property of materials where the [[absorption (optics)|absorption]] of light decreases with increasing light [[Intensity (physics)|intensity]]. Most materials show some saturable absorption, but often only at very high optical intensities (close to the optical damage). At sufficiently high incident light intensity, atoms in the ground state of a saturable absorber material become excited into an upper energy state at such a rate that there is insufficient time for them to decay back to the ground state before the ground state becomes depleted, and the absorption subsequently saturates. Saturable absorbers are useful in [[Optical cavity|laser cavities]]. The key parameters for a saturable absorber are its [[wavelength]] range (where it absorbs), its dynamic response (how fast it recovers), and its saturation intensity and fluence (at what intensity or pulse energy it saturates). They are commonly used for passive [[Q-switching]].
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| ==Phenomenology of saturable absorption==
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| Within the simple model of saturated absorption, the relaxation rate of excitations does not depend on the intensity
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| Then, for the [[continuous-wave]] operation, the absorption rate (or simply absorption) <math>A</math> is determined by intensity <math>I</math>:
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| : <math>(1)~~ ~~ A= \frac{\alpha}{1+I/I_0}</math> | |
| where <math> \alpha </math> is linear absorption, and
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| <math>I_0</math> is saturation intensity.
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| These parameters are related with the [[concentration]] <math> N </math> of the active centers in the medium,
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| the [[effective cross-sections]] <math>\sigma </math> and the lifetime <math>\tau</math> of the excitations.<ref name="colin">{{cite journal
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| |author=S. Colin
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| |coauthors= E. Contesse, P. Le Boudec, G. Stephan and F. Sanchez
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| |title=Evidence of a saturable-absorption effect in heavily erbium-doped fibers
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| |journal=[[Optics Letters]]
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| |volume= 21
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| |issue=24
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| |pages= 1987–1989
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| |year=1996
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| |doi=10.1364/OL.21.001987
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| |pmid=19881868
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| |bibcode = 1996OptL...21.1987C
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| }}
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| </ref>
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| ==Relation with Wright Omega function==
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| [[File:WrightOmega.png|200px|tight|thumb|[[Wright Omega function]]]]
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| In the simplest geometry, when the rays of the absorbing light are parallel, the intensity can be described with the
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| [[Beer–Lambert law|Bouguer law]],
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| :<math>(2)~~ ~~ \frac{\mathrm{d} I}{\mathrm{d}z}=-AI</math>
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| where <math>z</math> is coordinate in the direction of propagation.
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| Substitution of (1) into (2) gives the equation
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| :<math>(3)~~ ~~ \frac{\mathrm{d}I}{\mathrm{d}z}=-\frac{\alpha~ I}{1+I/I_0} </math>
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| With the dimensionless variables <math>u=I/I_0</math>, <math>t=\alpha z</math>,
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| equation (3) can be rewritten as
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| :<math>(4)~~ ~~ \frac{\mathrm{d}u}{\mathrm{d}t}=\frac{-u}{1+u} </math>
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| The solution can be expressed in terms of the [[Wright Omega function]] <math>\omega</math>:
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| :<math>(5)~~ ~~ u = \omega(-t) </math>
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| ==Relation with Lambert W function==
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| The solution can be expressed also through the related [[Lambert W function]].
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| Let <math>u=V\big(-\mathrm{e}^t\big)</math>. Then
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| :<math>(6)~~ ~~ -\mathrm{e}^t V'\big(-\mathrm{e}^t\big)= - \frac{V\big(-\mathrm{e}^t\big)}{1+V\big(-\mathrm{e}^t\big)} </math>
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| With new independent variable <math>p=-\mathrm{e}^t</math>,
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| Equation (6) leads to the equation
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| :<math>(7)~~ ~~ V'(p)=\frac{V(p)}{p\cdot (1+V(p))}</math>
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| The formal solution can be written
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| :<math>(8)~~ ~~ V(p)=W(p-p_0)</math>
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| where <math> p_0</math> is constant, but the equation <math>V(p_0)=0</math> may correspond to the non-physical value of intensity
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| (intensity zero) or to the unusual branch of the Lambert W function.
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| ==Saturation fluence==
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| For pulsed operation, in the limiting case of short pulses, absorption can be expressed through the fluence
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| :<math>(9)~~ ~~ F=\int_{0}^t I(t) \mathrm{d}t</math>
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| where time <math>t</math> should be small compared to the relaxation time of the medium; it is assumed that the intensity is zero at <math>t<0 </math>.
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| Then, the saturable absorption can be written as follows:
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| :<math>(10)~~ ~~ A=\frac{\alpha}{1+F/F_0} </math>
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| where saturation fluence <math> F_0</math> is constant.
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| In the intermediate case (neither cw, nor short pulse operation), the rate equations for [[Excited state|excitation]] and [[Relaxation (physics)|relaxation]] in the [[optical medium]] must be considered together.
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| Saturation fluence is one of the factors that determine [[Lasing threshold|threshold]] in the gain media and limits the storage of energy in a pulsed [[disk laser]].<ref name="storage">{{cite journal
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| |author=D.Kouznetsov.
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| |title=Storage of energy in disk-shaped laser materials
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| |journal=Research Letters in Physics
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| |year=2008
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| |pages= 717414
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| |doi=10.1155/2008/717414
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| |bibcode = 2008RLPhy2008E..17K
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| }}
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| </ref>
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| ==Mechanisms and examples of saturable absorption==
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| Absorption saturation, which results in decreased absorption at high incident light intensity, competes with other mechanisms (for example, increase in temperature, formation of [[F-Center|color centers]], etc.), which result in increased absorption.<ref name="stretched">{{cite journal
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| | author= J. Koponen, M.Söderlund, H.J. Hoffman, D. Kliner, J. Koplow, J.L. Archambault, L. Reekie, P.St.J. Russell and D.N. Payne
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| | title=Photodarkening measurements in large mode area fibers
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| | journal=[[Proceedings of SPIE]]
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| | volume=6553
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| | issue= 5
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| | pages=783–9
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| | year=2007
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| | doi=10.1117/12.712545
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| | pmid=17645476
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| }}</ref><ref name="liekki">{{cite journal
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| | author= L. Dong
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| | coauthors= J. L. Archambault, L. Reekie, P. St. J. Russell, and D. N. Payne
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| | title=Photoinduced absorption change in germanosilicate preforms: evidence for the color-center model of photosensitivity
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| | journal=[[Applied Optics]]
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| | volume=34
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| | pages=3436–40
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| | year=1995
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| | doi=10.1364/AO.34.003436
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| | pmid=21052157
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| | issue=18
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| |bibcode = 1995ApOpt..34.3436D }}</ref>
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| In particular, saturable absorption is only one of several mechanisms that produce [[self-pulsation]] in lasers, especially in [[semiconductor laser]]s.<ref name="paoli">
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| {{cite journal
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| |author=Thomas L. Paoli
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| |title=Saturable absorption effects in the self-pulsing (AlGa)As junction laser
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| |journal=Appl. Phys. Lett.
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| |volume=34
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| |page=652
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| |year=1979
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| |doi=10.1063/1.90625
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| |bibcode = 1979ApPhL..34..652P
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| |issue=10
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| }}
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| </ref>
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| One atom thick layer of carbon, [[graphene]], can be seen with the naked eye because it absorbs approximately 2.3% of white light, which is ''π'' times [[fine-structure constant]].<ref>{{Cite journal |title=Universal infrared conductance of graphite |first1=A. B.|last1=Kuzmenko |first2=E. |last2=van Heumen |first3=F. |last3=Carbone |first4=D. |last4=van der Marel |journal=Phys Rev Lett|volume=100 |pages=117401| doi=10.1103/PhysRevLett.100.117401 |year=2008 |pmid=18517825 |issue=11 |bibcode=2008PhRvL.100k7401K|arxiv = 0712.0835 }}</ref> The saturable absorption response of graphene is wavelength independent from UV to IR, mid-IR and even to THz frequencies.<ref>{{cite journal|title=Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser|journal=Applied Physics Letters|volume=96|page=111112|year=2010|url=http://www.sciencenet.cn/upload/blog/file/2010/3/20103191224576536.pdf|doi = 10.1063/1.3367743 |bibcode = 2010ApPhL..96k1112Z|id=|arxiv=1003.0154|issue=11}}</ref><ref>{{cite journal
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| |author=Z. Sun
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| |coauthors=T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari
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| |title=Graphene Mode-Locked Ultrafast Laser
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| |journal=[[ACS Nano]]
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| |volume=4
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| |pages=803
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| |year=2010
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| |doi=10.1021/nn901703e
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| }}
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| </ref><ref>{{cite journal
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| |author=F. Bonaccorso
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| |coauthors=Z. Sun, T. Hasan, and A. C. Ferrari
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| |title=Graphene photonics and optoelectronics
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| |journal=[[Nature Photonics]]
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| |volume=4
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| |pages=611
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| |year=2010
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| |doi=10.1038/NPHOTON.2010.186
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| |bibcode = 2010NaPho...4..611B |arxiv = 1006.4854 }}
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| </ref> In rolled-up graphene sheets ([[carbon nanotubes]]), saturable absorption is dependent on diameter and chirality.<ref>{{cite journal
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| |author= F. Wang
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| |coauthors= A. G. Rozhin, V. Scardaci, Z. Sun, F. Hennrich, I. H. White, W. I. Milne, and A. C. Ferrari
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| |title= Wideband-tuneable, nanotube mode-locked, fibre laser
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| |journal=[[Nature Nanotechnology]]
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| |volume=3
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| |pages= 738
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| |year=2008
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| |doi= 10.1038/nnano.2008.312
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| |bibcode = 2008NatNa...3..738W }}
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| </ref><ref>{{cite journal
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| |author= T. Hasan
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| |coauthors= Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari
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| |title= Nanotube–Polymer Composites for Ultrafast Photonics
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| |journal=Advanced Materials
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| |volume=21
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| |pages= 3874
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| |year=2009
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| |doi= 10.1002/adma.200901122
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| }}
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| </ref>
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| ==Microwave and Terahertz saturable absorption==
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| Saturable absorption can even take place at the Microwave and Terahertz band (correpsonding to a wavelength from 30 μm to 300 μm). Some materials, for example [[graphene]], with very weak energy band gap (several meV), could absorb photons at Microwave and Terahertz band due to its interband absorption.In one report, microwave absorbance of graphene always decreases with increasing the power and reaches at a constant level for power larger than a threshold value. The microwave saturable absorption in graphene is almost independent of the incident frequency, which demonstrates that graphene may have important applications in graphene microwave photonics devices such as: microwave saturable absorber, modulator, polarizer,microwave signal processing, broad-band wireless access networks, sensor networks, radar, satellite communications, and so on <ref name=Zheng>{{Cite journal|author=Zheng ''et al.''|title=Microwave and optical saturable absorption in graphene|journal=OPTICS EXPRESS
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| |year=2012
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| |volume=20
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| |pages= 20,23201|url=http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-21-23201}}.</ref>
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| .
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| ==Saturable X-ray absorption==
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| Saturable absorption has been demonstrated for X-rays. In one study, a thin {{convert|50|nm|in}} foil of [[aluminium]] was irradiated with soft [[X-ray]] [[laser]] radiation ([[wavelength]] {{convert|13.5|nm|in}}). The short laser pulse knocked out core [[Electron shell|L-shell]] electrons without breaking the [[crystal]]line structure of the metal, making it transparent to soft X-rays of the same wavelength for about 40 [[femtoseconds]].<ref name=scid>{{cite web |title=Transparent Aluminum Is ‘New State Of Matter’ |url=http://www.sciencedaily.com/releases/2009/07/090727130814.htm |date=July 27, 2009 |publisher=sciencedaily.com |accessdate=29 July 2009}}</ref><ref name=Nagler>{{Cite journal|last1=Nagler|first1=Bob|last2=Zastrau|first2=Ulf|last3=Fustlin|first3=Roland R.|last4=Vinko|first4=Sam M.|last5=Whitcher|first5=Thomas|last6=Nelson|first6=A. J.|last7=Sobierajski|first7=Ryszard|last8=Krzywinski|first8=Jacek|last9=Chalupsky|first9=Jaromir|title=Turning solid aluminium transparent by intense soft X-ray photoionization|journal=Nature Physics|volume=5|pages=693|year=2009|doi=10.1038/nphys1341|bibcode = 2009NatPh...5..693B|issue=9}}</ref>
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| ==See also==
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| *[[Two-photon absorption]]
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| ==References==
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| {{reflist|35em}}
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| [[Category:Nonlinear optics]]
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