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Saturable absorption

For saturable transport, see facilitated diffusion.

Saturable absorption is a property of materials where the absorption of light decreases with increasing light 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, the ground state of a saturable absorber material is excited into an upper energy state at such a rate that there is insufficient time for it to decay back to the ground state before the ground state becomes depleted, causing the absorption to saturate. The key parameters for a saturable absorber are its wavelength range (where in the electromagnetic spectrum it absorbs), its dynamic response (how fast it recovers), and its saturation intensity and fluence (at what intensity or pulse energy it saturates).

Saturable absorber materials are useful in laser cavities. For instance, they are commonly used for passive Q-switching.

Contents

Within the simple model of saturated absorption, the relaxation rate of excitations does not depend on the intensity. Then, for the continuous-wave (cw) operation, the absorption rate (or simply absorption) A {\displaystyle A} is determined by intensity I {\displaystyle I} :

( 1 ) A = α 1 + I / I 0 {\displaystyle (1)~~~~A={\frac {\alpha }{1+I/I_{0}}}}

where α {\displaystyle \alpha } is linear absorption, and I 0 {\displaystyle I_{0}} is saturation intensity. These parameters are related with the concentration N {\displaystyle N} of the active centers in the medium, the effective cross-sections σ {\displaystyle \sigma } and the lifetime τ {\displaystyle \tau } of the excitations.

In the simplest geometry, when the rays of the absorbing light are parallel, the intensity can be described with the Beer–Lambert law,

( 2 ) d I d z = A I {\displaystyle (2)~~~~{\frac {\mathrm {d} I}{\mathrm {d} z}}=-AI}

where z {\displaystyle z} is coordinate in the direction of propagation. Substitution of (1) into (2) gives the equation

( 3 ) d I d z = α I 1 + I / I 0 {\displaystyle (3)~~~~{\frac {\mathrm {d} I}{\mathrm {d} z}}=-{\frac {\alpha ~I}{1+I/I_{0}}}}

With the dimensionless variables u = I / I 0 {\displaystyle u=I/I_{0}} , t = α z {\displaystyle t=\alpha z} , equation (3) can be rewritten as

( 4 ) d u d t = u 1 + u {\displaystyle (4)~~~~{\frac {\mathrm {d} u}{\mathrm {d} t}}={\frac {-u}{1+u}}}

The solution can be expressed in terms of the Wright Omega function ω {\displaystyle \omega } :

( 5 ) u = ω ( t ) {\displaystyle (5)~~~~u=\omega (-t)}

The solution can be expressed also through the related Lambert W function. Let u = V ( e t ) {\displaystyle u=V{\big (}-\mathrm {e} ^{t}{\big )}} . Then

( 6 ) e t V ( e t ) = V ( e t ) 1 + V ( e t ) {\displaystyle (6)~~~~-\mathrm {e} ^{t}V'{\big (}-\mathrm {e} ^{t}{\big )}=-{\frac {V{\big (}-\mathrm {e} ^{t}{\big )}}{1+V{\big (}-\mathrm {e} ^{t}{\big )}}}}

With new independent variable p = e t {\displaystyle p=-\mathrm {e} ^{t}} , Equation (6) leads to the equation

( 7 ) V ( p ) = V ( p ) p ( 1 + V ( p ) ) {\displaystyle (7)~~~~V'(p)={\frac {V(p)}{p\cdot (1+V(p))}}}

The formal solution can be written

( 8 ) V ( p ) = W ( p p 0 ) {\displaystyle (8)~~~~V(p)=W(p-p_{0})}

where p 0 {\displaystyle p_{0}} is constant, but the equation V ( p 0 ) = 0 {\displaystyle V(p_{0})=0} may correspond to the non-physical value of intensity (intensity zero) or to the unusual branch of the Lambert W function.

For pulsed operation, in the limiting case of short pulses, absorption can be expressed through the fluence

( 9 ) F = 0 t I ( t ) d t {\displaystyle (9)~~~~F=\int _{0}^{t}I(t)\mathrm {d} t}

where time t {\displaystyle t} should be small compared to the relaxation time of the medium; it is assumed that the intensity is zero at t < 0 {\displaystyle t<0} . Then, the saturable absorption can be written as follows:

( 10 ) A = α 1 + F / F 0 {\displaystyle (10)~~~~A={\frac {\alpha }{1+F/F_{0}}}}

where saturation fluence F 0 {\displaystyle F_{0}} is constant.

In the intermediate case (neither cw, nor short pulse operation), the rate equations for excitation and relaxation in the optical medium must be considered together.

Saturation fluence is one of the factors that determine threshold in the gain media and limits the storage of energy in a pulsed disk laser.

Absorption saturation, which results in decreased absorption at high incident light intensity, competes with other mechanisms (for example, increase in temperature, formation of color centers, etc.), which result in increased absorption. In particular, saturable absorption is only one of several mechanisms that produce self-pulsation in lasers, especially in semiconductor lasers.

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. The saturable absorption response of graphene is wavelength independent from UV to IR, mid-IR and even to THz frequencies. In rolled-up graphene sheets (carbon nanotubes), saturable absorption is dependent on diameter and chirality.

Saturable absorption can even take place at the microwave and terahertz band (corresponding 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.[non-primary source needed]

Saturable absorption has been demonstrated for X-rays. In one study, a thin 50 nanometres (2.0×10−6 in) foil of aluminium was irradiated with soft X-ray laser radiation (wavelength 13.5 nm). The short laser pulse knocked out core L-shell electrons without breaking the crystalline structure of the metal, making it transparent to soft X-rays of the same wavelength for about 40 femtoseconds.[non-primary source needed]

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  4. L. Dong; J. L. Archambault; L. Reekie; P. St. J. Russell; D. N. Payne (1995). "Photoinduced absorption change in germanosilicate preforms: evidence for the color-center model of photosensitivity". Applied Optics. 34 (18): 3436–40. Bibcode:1995ApOpt..34.3436D. doi:10.1364/AO.34.003436. PMID 21052157.
  5. Thomas L. Paoli (1979). "Saturable absorption effects in the self-pulsing (AlGa)As junction laser". Appl. Phys. Lett. 34 (10): 652. Bibcode:1979ApPhL..34..652P. doi:10.1063/1.90625.
  6. Kuzmenko, A. B.; van Heumen, E.; Carbone, F.; van der Marel, D. (2008). "Universal infrared conductance of graphite". Phys Rev Lett. 100 (11): 117401. arXiv:0712.0835. Bibcode:2008PhRvL.100k7401K. doi:10.1103/PhysRevLett.100.117401. PMID 18517825.
  7. Zhang, Han; Tang, Dingyuan; Knize, R. J.; Zhao, Luming; Bao, Qiaoliang; Loh, Kian Ping (2010). "Graphene mode locked, wavelength-tunable, dissipative soliton fiber laser"(PDF). Applied Physics Letters. 96 (11): 111112. arXiv:1003.0154. Bibcode:2010ApPhL..96k1112Z. doi:10.1063/1.3367743. Archived from the original(PDF) on 2010-11-15.
  8. Z. Sun; T. Hasan; F. Torrisi; D. Popa; G. Privitera; F. Wang; F. Bonaccorso; D. M. Basko; A. C. Ferrari (2010). "Graphene Mode-Locked Ultrafast Laser". ACS Nano. 4 (2): 803–810. arXiv:0909.0457. doi:10.1021/nn901703e. PMID 20099874.
  9. F. Bonaccorso; Z. Sun; T. Hasan; A. C. Ferrari (2010). "Graphene photonics and optoelectronics". Nature Photonics. 4 (9): 611–622. arXiv:1006.4854. Bibcode:2010NaPho...4..611B. doi:10.1038/NPHOTON.2010.186.
  10. F. Wang; A. G. Rozhin; V. Scardaci; Z. Sun; F. Hennrich; I. H. White; W. I. Milne; A. C. Ferrari (2008). "Wideband-tuneable, nanotube mode-locked, fibre laser"(PDF). Nature Nanotechnology. 3 (12): 738–742. Bibcode:2008NatNa...3..738W. doi:10.1038/nnano.2008.312. PMID 19057594.
  11. T. Hasan; Z. Sun; F. Wang; F. Bonaccorso; P. H. Tan; A. G. Rozhin; A. C. Ferrari (2009). "Nanotube–Polymer Composites for Ultrafast Photonics". Advanced Materials. 21 (38–39): 3874–3899. doi:10.1002/adma.200901122.
  12. Zheng; et al. (2012). "Microwave and optical saturable absorption in graphene". Optics Express. 20 (21): 23201–14. Bibcode:2012OExpr..2023201Z. doi:10.1364/OE.20.023201. PMID 23188285..
  13. "Transparent Aluminum Is 'New State Of Matter'". sciencedaily.com. July 27, 2009. Retrieved29 July 2009.
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Saturable absorption
Saturable absorption Language Watch Edit 160 160 Redirected from Saturable absorber For saturable transport see facilitated diffusion Saturable absorption is a property of materials where the absorption of light decreases with increasing light 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 the ground state of a saturable absorber material is excited into an upper energy state at such a rate that there is insufficient time for it to decay back to the ground state before the ground state becomes depleted causing the absorption to saturate The key parameters for a saturable absorber are its wavelength range where in the electromagnetic spectrum it absorbs its dynamic response how fast it recovers and its saturation intensity and fluence at what intensity or pulse energy it saturates Saturable absorber materials are useful in laser cavities For instance they are commonly used for passive Q switching Contents 1 Phenomenology of saturable absorption 2 Relation with Wright Omega function 3 Relation with Lambert W function 4 Saturation fluence 5 Mechanisms and examples of saturable absorption 6 Microwave and terahertz saturable absorption 7 Saturable X ray absorption 8 See also 9 ReferencesPhenomenology of saturable absorption EditWithin the simple model of saturated absorption the relaxation rate of excitations does not depend on the intensity Then for the continuous wave cw operation the absorption rate or simply absorption A displaystyle A is determined by intensity I displaystyle I 1 A a 1 I I 0 displaystyle 1 A frac alpha 1 I I 0 where a displaystyle alpha is linear absorption and I 0 displaystyle I 0 is saturation intensity These parameters are related with the concentration N displaystyle N of the active centers in the medium the effective cross sections s displaystyle sigma and the lifetime t displaystyle tau of the excitations 1 Relation with Wright Omega function Edit Wright Omega function In the simplest geometry when the rays of the absorbing light are parallel the intensity can be described with the Beer Lambert law 2 d I d z A I displaystyle 2 frac mathrm d I mathrm d z AI where z displaystyle z is coordinate in the direction of propagation Substitution of 1 into 2 gives the equation 3 d I d z a I 1 I I 0 displaystyle 3 frac mathrm d I mathrm d z frac alpha I 1 I I 0 With the dimensionless variables u I I 0 displaystyle u I I 0 t a z displaystyle t alpha z equation 3 can be rewritten as 4 d u d t u 1 u displaystyle 4 frac mathrm d u mathrm d t frac u 1 u The solution can be expressed in terms of the Wright Omega function w displaystyle omega 5 u w t displaystyle 5 u omega t Relation with Lambert W function EditThe solution can be expressed also through the related Lambert W function Let u V e t displaystyle u V big mathrm e t big Then 6 e t V e t V e t 1 V e t displaystyle 6 mathrm e t V big mathrm e t big frac V big mathrm e t big 1 V big mathrm e t big With new independent variable p e t displaystyle p mathrm e t Equation 6 leads to the equation 7 V p V p p 1 V p displaystyle 7 V p frac V p p cdot 1 V p The formal solution can be written 8 V p W p p 0 displaystyle 8 V p W p p 0 where p 0 displaystyle p 0 is constant but the equation V p 0 0 displaystyle V p 0 0 may correspond to the non physical value of intensity intensity zero or to the unusual branch of the Lambert W function Saturation fluence EditFor pulsed operation in the limiting case of short pulses absorption can be expressed through the fluence 9 F 0 t I t d t displaystyle 9 F int 0 t I t mathrm d t where time t displaystyle t should be small compared to the relaxation time of the medium it is assumed that the intensity is zero at t lt 0 displaystyle t lt 0 Then the saturable absorption can be written as follows 10 A a 1 F F 0 displaystyle 10 A frac alpha 1 F F 0 where saturation fluence F 0 displaystyle F 0 is constant In the intermediate case neither cw nor short pulse operation the rate equations for excitation and relaxation in the optical medium must be considered together Saturation fluence is one of the factors that determine threshold in the gain media and limits the storage of energy in a pulsed disk laser 2 Mechanisms and examples of saturable absorption EditAbsorption saturation which results in decreased absorption at high incident light intensity competes with other mechanisms for example increase in temperature formation of color centers etc which result in increased absorption 3 4 In particular saturable absorption is only one of several mechanisms that produce self pulsation in lasers especially in semiconductor lasers 5 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 p times fine structure constant 6 The saturable absorption response of graphene is wavelength independent from UV to IR mid IR and even to THz frequencies 7 8 9 In rolled up graphene sheets carbon nanotubes saturable absorption is dependent on diameter and chirality 10 11 Microwave and terahertz saturable absorption EditSaturable absorption can even take place at the microwave and terahertz band corresponding to a wavelength from 30 mm to 300 mm 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 12 non primary source needed Saturable X ray absorption EditSaturable absorption has been demonstrated for X rays In one study a thin 50 nanometres 2 0 10 6 in foil of aluminium was irradiated with soft X ray laser radiation wavelength 13 5 nm The short laser pulse knocked out core L shell electrons without breaking the crystalline structure of the metal making it transparent to soft X rays of the same wavelength for about 40 femtoseconds 13 14 non primary source needed See also EditTwo photon absorptionReferences Edit Colin S Contesse E Boudec PL Stephan G Sanchez F 1996 Evidence of a saturable absorption effect in heavily erbium doped fibers Optics Letters 21 24 1987 1989 Bibcode 1996OptL 21 1987C doi 10 1364 OL 21 001987 PMID 19881868 D Kouznetsov 2008 Storage of energy in disk shaped laser materials Research Letters in Physics 2008 1 5 Bibcode 2008RLPhy2008E 17K doi 10 1155 2008 717414 Koponen J Soderlund M Hoffman HF Kliner D Koplow J Archambault JL Reekie L Russell P St J Payne DN 2007 Harter DJ Tunnermann A Broeng J Headley Iii C eds Photodarkening measurements in large mode area fibers Proceedings of SPIE Fiber Lasers IV Technology Systems and Applications 6553 5 783 9 Bibcode 2007SPIE 6453E 1EK doi 10 1117 12 712545 L Dong J L Archambault L Reekie P St J Russell D N Payne 1995 Photoinduced absorption change in germanosilicate preforms evidence for the color center model of photosensitivity Applied Optics 34 18 3436 40 Bibcode 1995ApOpt 34 3436D doi 10 1364 AO 34 003436 PMID 21052157 Thomas L Paoli 1979 Saturable absorption effects in the self pulsing AlGa As junction laser Appl Phys Lett 34 10 652 Bibcode 1979ApPhL 34 652P doi 10 1063 1 90625 Kuzmenko A B van Heumen E Carbone F van der Marel D 2008 Universal infrared conductance of graphite Phys Rev Lett 100 11 117401 arXiv 0712 0835 Bibcode 2008PhRvL 100k7401K doi 10 1103 PhysRevLett 100 117401 PMID 18517825 Zhang Han Tang Dingyuan Knize R J Zhao Luming Bao Qiaoliang Loh Kian Ping 2010 Graphene mode locked wavelength tunable dissipative soliton fiber laser PDF Applied Physics Letters 96 11 111112 arXiv 1003 0154 Bibcode 2010ApPhL 96k1112Z doi 10 1063 1 3367743 Archived from the original PDF on 2010 11 15 Z Sun T Hasan F Torrisi D Popa G Privitera F Wang F Bonaccorso D M Basko A C Ferrari 2010 Graphene Mode Locked Ultrafast Laser ACS Nano 4 2 803 810 arXiv 0909 0457 doi 10 1021 nn901703e PMID 20099874 F Bonaccorso Z Sun T Hasan A C Ferrari 2010 Graphene photonics and optoelectronics Nature Photonics 4 9 611 622 arXiv 1006 4854 Bibcode 2010NaPho 4 611B doi 10 1038 NPHOTON 2010 186 F Wang A G Rozhin V Scardaci Z Sun F Hennrich I H White W I Milne A C Ferrari 2008 Wideband tuneable nanotube mode locked fibre laser PDF Nature Nanotechnology 3 12 738 742 Bibcode 2008NatNa 3 738W doi 10 1038 nnano 2008 312 PMID 19057594 T Hasan Z Sun F Wang F Bonaccorso P H Tan A G Rozhin A C Ferrari 2009 Nanotube Polymer Composites for Ultrafast Photonics Advanced Materials 21 38 39 3874 3899 doi 10 1002 adma 200901122 Zheng et al 2012 Microwave and optical saturable absorption in graphene Optics Express 20 21 23201 14 Bibcode 2012OExpr 2023201Z doi 10 1364 OE 20 023201 PMID 23188285 Transparent Aluminum Is New State Of Matter sciencedaily com July 27 2009 Retrieved 29 July 2009 Nagler Bob Zastrau Ulf Fustlin Roland R Vinko Sam M Whitcher Thomas Nelson A J Sobierajski Ryszard Krzywinski Jacek et al 2009 Turning solid aluminium transparent by intense soft X ray photoionization PDF Nature Physics 5 9 693 696 Bibcode 2009NatPh 5 693B doi 10 1038 nphys1341 Retrieved from https en wikipedia org w index php title Saturable absorption amp oldid 1024277202, wikipedia, wiki, book,

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