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, 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 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.
Phenomenology of saturable absorption
Within the simple model of saturated absorption, the relaxation rate of excitations does not depend on the intensity. Then, for the continuous-wave operation, the absorption rate (or simply absorption) is determined by intensity :
where is linear absorption, and is saturation intensity. These parameters are related with the concentration of the active centers in the medium, the effective cross-sections and the lifetime of the excitations.
Relation with 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,
where is coordinate in the direction of propagation. Substitution of (1) into (2) gives the equation
With the dimensionless variables , , equation (3) can be rewritten as
The solution can be expressed in terms of the Wright Omega function :
Relation with Lambert W function
The solution can be expressed also through the related Lambert W function. Let . Then
With new independent variable , Equation (6) leads to the equation
The formal solution can be written
where is constant, but the equation 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
where time should be small compared to the relaxation time of the medium; it is assumed that the intensity is zero at . Then, the saturable absorption can be written as follows:
where saturation fluence is constant.
Mechanisms and examples of saturable absorption
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.
Microwave and Terahertz saturable absorption
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 
Saturable X-ray absorption
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 nanometres (5.3×10−7 in)). 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.
- 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: 717414. Bibcode:2008RLPhy2008E..17K. doi:10.1155/2008/717414.
- Koponen J, Söderlund M, Hoffman HF, Kliner D, Koplow J, Archambault JL, Reekie L, Russell P.St.J., Payne DN (2007). "Photodarkening measurements in large mode area fibers". Proceedings of SPIE. Fiber Lasers IV: Technology, Systems, and Applications. 6553 (5): 783–9. Bibcode:2007SPIE.6453E..40K. 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 2010-05-21.
- 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.
- 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". Nature Nanotechnology. 3 (12): 738–742. Bibcode:2008NatNa...3..738W. doi:10.1038/nnano.2008.312.
- 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): 20, 23201. 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". Nature Physics. 5 (9): 693–696. Bibcode:2009NatPh...5..693B. doi:10.1038/nphys1341.