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| [[File:Two photon absorption.png|right|thumb|200px|Energy scheme of a two photon excitation upconversion process]]
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| {{Expert-subject|Physics|date=February 2009}}
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| '''Two-photon absorption''' ('''TPA''') is the simultaneous absorption of two [[photon]]s of identical or different frequencies in order to excite a [[molecule]] from one state (usually the [[ground state]]) to a higher energy [[excited state|electronic state]]. The energy difference between the involved lower and upper states of the molecule is equal to the sum of the energies of the two photons. Two-photon absorption is a third-order process several orders of magnitude weaker than [[absorption (electromagnetic radiation)|linear absorption]]. It differs from linear absorption in that the strength of absorption depends on the square of the light [[fluence|intensity]], thus it is a [[nonlinear optics|nonlinear optical]] process.
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| ==Background==
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| The phenomenon was originally predicted by [[Maria Goeppert-Mayer]] in 1931 in her doctoral dissertation.<ref name=Goeppert-Mayer_1931>{{cite journal | author =Goeppert-Mayer M | title = Über Elementarakte mit zwei Quantensprüngen | journal = Annals of Physics | year = 1931 | volume = 9 | issue =3| pages = 273–95 | bibcode=1931AnP...401..273G | doi = 10.1002/andp.19314010303 }}</ref> Thirty years later, the invention of the [[laser]] permitted the first experimental verification of the TPA when [[two-photon-excited fluorescence]] was detected in a [[europium]]-doped crystal<ref>W. Kaiser and C.G.B. Garrett, "Two-photon excitation in CaF<sub>2</sub>:Eu<sup>2+</sup>," ''Physical Review Letters'' '''7''', 229–232 (1961)</ref> and subsequently observed in a cesium vapor.<ref>I.D. Abella, "Optical double-quantum absorption in cesium vapor," ''Physical Review Letters,'' '''9''', 453 (1962)</ref>
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| TPA is a [[nonlinear optics|nonlinear optical]] process. In particular, the imaginary part of the third-order nonlinear [[Electric susceptibility|susceptibility]] is related to the extent of TPA in a given molecule. The [[selection rule]]s for TPA are therefore different from for one-photon absorption (OPA), which is dependent on the first-order susceptibility. For example, in a [[centrosymmetric molecule]], one- and two-photon allowed transitions are mutually exclusive. In [[quantum chemistry|quantum mechanical]] terms, this difference results from the need to conserve [[angular momentum]]. Since photons have spin of ±1, one-photon absorption requires excitation to involve an electron changing its molecular orbital to one with an angular momentum different by ±1. Two-photon absorption requires a change of +2, 0, or −2.
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| The third order can be rationalized by considering that a second order process creates a polarization with the doubled frequency. In the third order, by difference frequency generation the original frequency can be generated again. Depending on the phase between the generated polarization and the original electric field this leads to the [[Kerr effect]] or to the '''two-photon absorption'''. In second harmonic generation this difference in frequency generation is a separated process in a cascade, so that the energy of the fundamental frequency can also be absorbed. In harmonic generation, multiple photons interact simultaneously with a molecule with no absorption events. Because n-photon harmonic generation is essentially a scattering process, the emitted wavelength is exactly 1/n times the incoming fundamental wavelength.<ref>http://www.svi.nl/SecondHarmonicGeneration</ref> This may be better called three photon absorption. In the next paragraph resonant two photon absorption via separate one-photon transitions is mentioned, where the absorption alone is a first order process and any fluorescence from the final state of the second transition will be of second order; this means it will rise as the square of the incoming intensity. The [[Virtual state (physics)|virtual state]] argument is quite orthogonal to the anharmonic oscillator argument. It states for example that in a semiconductor, absorption at high energies is impossible if two photons cannot bridge the band gap. So, many materials can be used for the Kerr effect that do not show any absorption and thus have a high damage threshold.
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| Two-photon absorption can be measured by several techniques. Two of them are two-photon excited fluorescence (TPEF) and nonlinear transmission (NLT). [[Pulsed laser]]s are most often used because TPA is a third-order nonlinear optical process, and therefore is most efficient at very high [[intensity (physics)|intensities]]. Phenomenologically, this can be thought of as the third term in a conventional [[Anharmonicity|anharmonic oscillator]] model for depicting vibrational behavior of molecules. Another view is to think of light as [[photon]]s. In nonresonant TPA two photons combine to bridge an energy gap larger than the energies of each photon individually. If there were an intermediate state in the gap, this could happen via two separate one-photon transitions in a process described as "resonant TPA", "sequential TPA", or "1+1 absorption". In nonresonant TPA the transition occurs without the presence of the intermediate state. This can be viewed as being due to a "virtual" state created by the interaction of the photons with the molecule.
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| The "nonlinear" in the description of this process means that the strength of the interaction increases faster than linearly with the electric field of the light. In fact, under ideal conditions the rate of TPA is proportional to the square of the field intensity. This dependence can be derived quantum mechanically, but is intuitively obvious when one considers that it requires two photons to coincide in time and space. This requirement for high light intensity means that lasers are required to study TPA phenomena. Further, in order to understand the TPA [[spectrum]], [[Monochrome|monochromatic]] light is also desired in order to measure the TPA cross section at different [[wavelengths]]. Hence, tunable pulsed lasers (such as frequency-doubled Nd:YAG-pumped [[optical parametric oscillator|OPO]]s and [[optical parametric amplifier|OPA]]s) are the choice of excitation.
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| ==Measurements==
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| ===Absorption rate===
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| The [[Beer's law]] for one photon absorption:
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| :<math>I(x) = I_0 e^{-\alpha\,c\,x} \,</math>
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| changes to
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| :<math>I(x) = \frac{I_0}{1 + \beta c x I_0} \,</math>
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| for TPA with light intensity as a function of path length or cross section x as a function of [[concentration]] c and the initial light intensity I<sub>0</sub>. The [[absorption coefficient]] α now becomes the '''TPA coefficient''' β. (Note that there is some confusion over the term β in nonlinear optics, since it is sometimes used to describe the [[hyperpolarizability|second-order polarizability]], and occasionally for the molecular two-photon cross-section. More often however, is it used to describe the bulk 2-photon optical density of a sample. The letter δ or σ is more often used to denote the molecular two-photon cross-section.)
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| ===Units of cross-section===
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| The molecular two-photon cross-section is usually quoted in the units of Goeppert-Mayer ('''GM''') (after its discoverer, Nobel laureate [[Maria Goeppert-Mayer]]), where 1 GM is 10<sup>−50</sup> cm<sup>4</sup> s photon<sup>−1</sup>.<ref>Powerpoint presentation @ chem.ucsb.edu www.chem.ucsb.edu/~ocf/lecture_ford.ppt Link</ref> Considering the reason for these units, one can see that it results from the product of two areas (one for each photon, each in cm<sup>2</sup>) and a time (within which the two photons must arrive to be able to act together). The large scaling factor is introduced in order that 2-photon absorption cross-sections of common dyes will have convenient values.
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| ==Development of the field and potential applications==
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| Until the early 1980s, TPA was used as a [[Spectroscopy|spectroscopic]] tool. Scientists compared the OPA and TPA spectra of different organic molecules and obtained several fundamental structure property relationships. However, in late 1980s, applications were started to be developed. Peter Rentzepis suggested applications in [[3D optical data storage]]. Watt Webb suggested microscopy and imaging. Other applications such as [[3D microfabrication]], optical logic, autocorrelation, pulse reshaping and optical power limiting were also demonstrated <ref> A. Hayat, A. Nevet, P. Ginzburg, M. Orenstein "Applications of two-photon processes in semiconductor photonic devices", Semicond. Sci. Technol. '''26''', 083001 (2011).</ref>
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| ===Microfabrication and lithography===
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| One of the most distinguishing features of TPA is that the rate of absorption of light by a molecule depends on the square of the light's intensity. This is different than OPA, where the rate of absorption is linear with respect to input intensity. As a result of this dependence, if material is cut with a high power [[laser]] beam, the rate of material removal decreases very sharply from the center of the beam to its periphery. Because of this, the "pit" created is sharper and better resolved than if the same size pit were created using normal absorption.
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| ===3D photopolymerization===
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| In [[3D microfabrication]], a block of gel containing monomers and a 2-photon active [[Radical initiator|photoinitiator]] is prepared as a raw material. Application of a focused laser to the block results in polymerization only at the focal spot of the laser, where the intensity of the absorbed light is highest. The shape of an object can therefore be traced out by the laser, and then the excess gel can be washed away to leave the traced solid.
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| ===Imaging===
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| The human body is not [[transparency (optics)|transparent]] to [[visible light|visible]] wavelengths. Hence, one photon imaging using [[fluorescent dye]]s is not very efficient. If the same dye had good two-photon absorption, then the corresponding excitation would occur at approximately two times the wavelength at which one-photon excitation would have occurred. As a result, it is possible to use excitation in the [[far infrared#Different regions in the infrared|far infrared]] region where the human body shows good transparency. It is sometimes said, incorrectly, that Rayleigh scattering is relevant to imaging techniques such as two-photon. According to [[Rayleigh scattering|Rayleigh's scattering law]], the amount of scattering is proportional to <math>1/\lambda^4</math>, where <math>\lambda</math> is the wavelength. As a result, if the wavelength is increased by a factor of 2, the Rayleigh scattering is reduced by a factor of 16. However, Rayleigh scattering only takes place when scattering particles are much smaller than the wavelength of light (the sky is blue because air molecules scatter blue light much more than red light). When particles are larger, scattering increases approximately linearly with wavelength: hence clouds are white since they contain water droplets. This form of scatter is known as [[Mie scattering]] and is what occurs in biological tissues. So, although longer wavelengths do scatter less in biological tissues, the difference is not as dramatic as Rayleigh's law would predict.
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| ===Optical power limiting===
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| Another area of research is ''optical power limiting''. In a material with a strong nonlinear effect, the absorption of light increases with intensity such that beyond a certain input intensity the output intensity approaches a constant value. Such a material can be used to limit the amount of optical power entering a system. This can be used to protect expensive or sensitive equipment such as [[sensor]]s, can be used in protective goggles, or can be used to control noise in laser beams.
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| ===Photodynamic therapy===
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| [[Photodynamic therapy]] (PDT) is a method for treating [[cancer]]. In this technique, an organic molecule with a good triplet quantum yield is excited so that the [[triplet state]] of this molecule interacts with [[oxygen]]. The ground state of oxygen has triplet character. This leads to triplet-triplet annihilation, which gives rise to singlet oxygen, which in turn attacks cancerous cells. However, using TPA materials, the window for excitation can be extended into the [[infrared]] region, thereby making the process more viable to be used on the human body.
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| ===Optical data storage===
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| The ability of two-photon excitation to address molecules deep within a sample without affecting other areas makes it possible to store and retrieve information in the volume of a substance rather than only on a surface as is done on the [[DVD]]. Therefore, [[3D optical data storage]] has the possibility to provide media that contain [[terabyte]]-level data capacities on a single disc.
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| ==TPA compounds==
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| To some extent, linear and 2-photon absorption strengths are linked. Therefore, the first compounds to be studied (and many that are still studied and used in e.g. 2-photon microscopy) were standard dyes. In particular, laser dyes were used, since these have good photostability characteristics. However, these dyes tend to have 2-photon cross-sections of the order of 0.1-10 GM, much less than is required to allow simple experiments. | |
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| It was not until the 1990s that rational design principles for the construction of two-photon-absorbing molecules began to be developed, in response to a need from imaging and data storage technologies, and aided by the rapid increases in computer power that allowed quantum calculations to be made. The accurate quantum mechanical analysis of two-photon absorbance is orders of magnitude more computationally intensive than that of one-photon absorbance, requiring highly correlated calculations at very high levels of theory.
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| The most important features of strongly TPA molecules were found to be a long conjugation system (analogous to a large antenna) and substitution by strong donor and acceptor groups (which can be thought of as inducing nonlinearity in the system and increasing the potential for charge-transfer). Therefore, many [[push-pull olefin]]s exhibit high TPA transitions, up to several thousand GM.<ref>''Mechanisms for enhancement of two-photon absorption in donor–acceptor conjugated chromophores'' T. Kogej, D. Beljonne, F. Meyers, J.W. Perry, S.R. Marder, J.L. Bre ́das [[Chem Phys Letters]] '''1998''', 298, 1-6</ref> It is also found that compounds with a real intermediate energy level close to the "virtual" energy level can have large 2-photon cross-sections as a result of resonance enhancement.
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| Compounds with interesting TPA properties also include various [[porphyrin]] derivatives, conjugated [[polymers]] and even [[dendrimers]]. In one study <ref>''Strong Two-Photon Absorption of Singlet Diradical Hydrocarbons'' Kenji Kamada, Koji Ohta, TakashiKubo,Akihiro Shimizu, Yasushi Morita, Kazuhiro Nakasuji, Ryohei Kishi, Suguru Ohta, Shin-ichi Furukawa, Hideaki Takahashi, and Masayoshi Nakano [[Angew. Chem. Int. Ed.]] '''2007''', 46, 3544 –3546 {{DOI|10.1002/anie.200605061}}</ref> a [[diradical]] [[resonance contribution]] for the compound depicted below was also linked to efficient TPA. The TPA wavelength for this compound is 1425 nanometer with observed TPA cross section of 424 GM.
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|
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| :[[File:DiradicalApplicationinTPA.png|300px|Diradical Application in TPA]]
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| ==TPA Coefficients==
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| The two photon absorption coefficient is defined by the relation <ref>
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| {{cite book
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| | last = Bass
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| | first = Michael
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| | title = HANDBOOK OF OPTICS Volume I
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| | publisher = McGraw-Hill Professional; 2 edition (September 1, 1994)
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| | year = 1994
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| | at=9 .32
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| | isbn = 0-07-047740-X
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| }}
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| </ref>
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| <math>-\frac{dI}{dz}=\alpha I+\beta I^{2} </math>
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| so that
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| <math>\beta (\omega)=\frac{2 \hbar \omega}{I^{2}} W_T^{(2)}(\omega)=\frac{N}{E}\sigma^{(2)}</math>
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| Where <math>\beta</math> is the two-photon absorption coefficient, <math>\alpha</math> is the absorption coefficient, <math>W_T^{(2)}(\omega)</math> is the transition rate for TPA per unit volume, <math>I</math> is the [[irradiance]], <math>\omega</math> is the photon frequency and the thickness of the slice is <math>dz</math>. N is the number density of molecules per cm<sup>3</sup>, E is the photon energy (J), σ<sup>(2)</sup> is the two-photon absorption cross section (cm<sup>4</sup>s/molecule).
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| The SI units of the beta coefficient are m/W. If β (m/W) is multiplied by 10<sup>−9</sup> it can be converted to the CGS system (cal/cm s/erg).<ref>
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| {{cite book
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| | last = Marvin
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| | first = Weber
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| | title = HANDBOOK OF OPTICAL MATERIALS
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| | publisher = The CRC Press
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| | series = Laser and Optical Science and Technology Series
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| | year = 2003
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| | at=APPENDIX V
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| | isbn = 978-0-8493-3512-9
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| }}
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| </ref>
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| Due to different laser pulses the TPA coefficients reported has differed as much as a factor 3. With the transition towards shorter laser pulses, from
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| picosecond to subpicosecond durations, noticeably reduced TPA coefficient have been obtained.<ref name="femto">''Femtosecond Measurements of Two-Photon Absorption Coefficients at λ = 264 nm in Glasses, Crystals, and Liquids''Adrian Dragonmir, John G. McInerney, and David N. Nikogosyan Applied Optics, Vol. 41, Issue 21, pp. 4365-4376 (2002) {{doi|10.1364/AO.41.004365}}</ref>
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| ===TPA in Water===
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| Laser induced TPA in water was discovered in 1980.<ref>D. N. Nikogosyan and D. A. Angelov, ''Formation of free radicals in water under high-power laser UV irradiation,'' Chem. Phys. Lett. 77, 208 –210 ͑1981͒.</ref>
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| Water absorbs UV radiation near 125 nm exiting the 3a1 [[Atomic orbital|orbital]] leading to [[dissociation (chemistry)|dissociation]] into OH⁻ and H⁺. Through TPA this dissociation can be achieved by two photons near 266 nm.<ref>''Two photon photodissociation of H2O via the B state'' J. Underwood and C. Wittig, Chem.Phys. Lett. 386 (2004) 190-195.</ref> Since water and heavy water have different vibration frequencies and inertia they also need different photon energies to achieve dissociation and have different absorption coefficients for a given photon wavelength.
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| A study from Jan 2002 used a femtosecond laser tuned to 0.22 Picoseconds found the coefficient of D2O to be 42±5 10<sup>−11</sup>(cm/W) whereas H2O was 49±5 10<sup>−11</sup>(cm/W) <ref name="femto"/>
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| {| class="wikitable sortable"
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| |+ TPA Coefficients for Water <ref name="femto"/>
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| |-
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| ! λ (nm) !! pulse duration τ (ps) !!<math>\beta\times 10^{11}</math> (cm/W)
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| |-
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| | 315 || 29 || 4
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| |-
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| | 300 || 29 || 4.5
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| |-
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| | 289 || 29 || 6
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| |-
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| | 282 || 29 || 7
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| |-
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| | 282 || 0.18 || 19
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| |-
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| | 266 || 29 || 10
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| |-
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| | 264 || 0.22 || 49±5
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| |-
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| | 216 || 15 || 20
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| |-
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| | 213 || 26 || 32
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| |}
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| ==Two-photon emission==
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| The opposite process of TPA is two-photon emission (TPE), which is a single electron transition accompanied by the emission of a photon pair. The energy of each individual photon of the pair is not determined, while the pair as a whole conserves the transition energy. The spectrum of TPE is therefore very broad and continuous.<ref>J. Chluba and R. A. Sunyaev, ''Induced two-photon decay of the 2s level and the rate of cosmological hydrogen recombination'', Astron.Astrophys. '''446''', 39 (2006).</ref> TPE is important for applications in astrophysics, contributing to the continuum radiation from planetary [[nebulae]] (theoretically predicted for them in <ref>L. Spitzer and J. Greenstein, ''Continuous emission from planetary nebulae'', Astrophys.J. '''114''', 407 (1951)</ref> and observed in <ref>G. A. Gurzadyan, ''Two-photon emission in planetary nebula IC 2149'', Publ.Astr.Soc.Pac. '''88''', 526 (1976)</ref>). TPE in condensed matter and specifically in semiconductors was only recently observed,<ref>A. Hayat, P. Ginzburg, M. Orenstein, [http://www.nature.com/nphoton/journal/v2/n4/abs/nphoton.2008.28.html ''Observation of Two-Photon Emission from Semiconductors''], Nature Photon. '''2''', 238 (2008)</ref> with emission rates nearly 5 orders of magnitude weaker than one-photon spontaneous emission, with potential applications in [[quantum information]].
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| ==See also==
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| *[[Virtual particle]]s are in virtual state where the [[probability amplitude]] is not conserved.
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| *[[Two-photon excitation microscopy]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.calctool.org/CALC/chem/photochemistry/2pa Web-based calculator for the rate of 2-photon absorption]
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| {{DEFAULTSORT:Two-Photon Absorption}}
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| [[Category:Nonlinear optics]]
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