Two-photon absorption
Encyclopedia
Two-photon absorption 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 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 linear absorption
. It differs from linear absorption in that the strength of absorption depends on the square of the light intensity
, thus it is a nonlinear optical
process.
permitted the first experimental verification of the TPA when two-photon-excited fluorescence was detected in a europium
-doped crystal., and then observed in a vapor (cesium) in 1962 by Isaac Abella
TPA is a nonlinear optical
process. In particular, the imaginary part of the third-order nonlinear susceptibility
is related to the extent of TPA in a given molecule. The selection rule
s for TPA are therefore different than 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 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.
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. 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
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.
Two-photon absorption can be measured by several techniques. Two of them are two-photon excited fluorescence (TPEF) and nonlinear transmission (NLT). Pulsed lasers are most often used because TPA is a third-order nonlinear optical process, and therefore is most efficient at very high intensities
. Phenomenologically, this can be thought of as the third term in a conventional 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.
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
, 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 OPO
s and OPA
s) are the choice of excitation.
changes to
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 I0. The absorption coefficient α now becomes the TPA cross section β. (Note that there is some confusion over the term β in nonlinear optics, since it is sometimes used to describe the 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.)
.
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
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.
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.
, the amount of scattering is proportional to
, where 
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.
s, can be used in protective goggles, or can be used to control noise in laser beams.
(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.
. Therefore, 3D optical data storage
has the possibility to provide media that contain terabyte
-level data capacities on a single disc.
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.
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. 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.
Compounds with interesting TPA properties also include various porphyrin
derivatives, conjugated polymers and even dendrimers. In one study 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.
so that
Where
is the two-photon absorption coefficient, 
is the absorption coefficient 
is the transition rate for TPA, 
is the irradiance
,
is the photon frequency and the thickness of the slice is 
.
The SI units of the coefficient are:
Where N is the number density of molecules per cm3, E is the photon energy (J), σ2 is the two-photon absorption cross section (cm4s/molecule).
If β (m/W) is multiplied by 10−9 it can be converted to the CGS system (cal/cm s/erg).
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
picosecond to subpicosecond durations, noticeably reduced TPA coefficient have been obtained.
Water absorbs UV radiation near 125 nm exiting the 3a1 orbital
leading to dissociation
into OH⁻ and H⁺. Through TPA this dissociation can be achieved by two photons near 266 nm. 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.
A study from Jan 2002 used a femtosecond laser tuned to 0.22 Picoseconds found the coefficient of D2O to be 42±5 10−11(cm/W) whereas H2O was 49±5 10−11(cm/W)
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s of identical or different frequencies in order to excite a molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
from one state (usually the ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
) to a higher energy electronic state
Excited state
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited 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 linear absorption
Absorption (electromagnetic radiation)
In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...
. It differs from linear absorption in that the strength of absorption depends on the square of the light intensity
Fluence
In physics, fluence is the flux integrated over time. For particles, it is defined as the total number of particles that intersect a unit area in a specific time interval of interest, and has units of m–2...
, thus it is a nonlinear optical
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
process.
Background
The phenomenon was originally predicted by Maria Goeppert-Mayer in 1931 in her doctoral dissertation. Thirty years later, the invention of the laserLaser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
permitted the first experimental verification of the TPA when two-photon-excited fluorescence was detected in a europium
Europium
Europium is a chemical element with the symbol Eu and atomic number 63. It is named after the continent of Europe. It is a moderately hard silvery metal which readily oxidizes in air and water...
-doped crystal., and then observed in a vapor (cesium) in 1962 by Isaac Abella
Isaac Abella
Isaac David Abella is Professor of Physics at The University of Chicago. He specializes in laser physics, quantum optics, and spectroscopy. Isaac is the cousin of Irving Abella....
TPA is a nonlinear optical
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
process. In particular, the imaginary part of the third-order nonlinear susceptibility
Electric susceptibility
In electromagnetism, the electric susceptibility \chi_e is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field...
is related to the extent of TPA in a given molecule. The selection rule
Selection rule
In physics and chemistry a selection rule, or transition rule, formally constrains the possible transitions of a system from one state to another. Selection rules have been derived for electronic, vibrational, and rotational transitions...
s for TPA are therefore different than 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 mechanical
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
terms, this difference results from the need to conserve angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
. 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.
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
Kerr effect
The Kerr effect, also called the quadratic electro-optic effect , is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric...
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. 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
Virtual state (physics)
The term virtual state is commonly used to refer to two different types of states in physical systems. It may refer to a very short-lived, unobservable quantum state or a real, but unstable, 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.
Two-photon absorption can be measured by several techniques. Two of them are two-photon excited fluorescence (TPEF) and nonlinear transmission (NLT). Pulsed lasers are most often used because TPA is a third-order nonlinear optical process, and therefore is most efficient at very high intensities
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
. Phenomenologically, this can be thought of as the third term in a conventional anharmonic oscillator
Anharmonicity
In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in simple harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be...
model for depicting vibrational behavior of molecules. Another view is to think of light as photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
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.
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
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
, monochromatic
Monochrome
Monochrome describes paintings, drawings, design, or photographs in one color or shades of one color. A monochromatic object or image has colors in shades of limited colors or hues. Images using only shades of grey are called grayscale or black-and-white...
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 OPO
Optical parametric oscillator
An optical parametric oscillator is a parametric oscillator which oscillates at optical frequencies. It converts an input laser wave into two output waves of lower frequency by means of second order nonlinear optical interaction. The sum of the output waves frequencies is equal to the input wave...
s and OPA
Optical parametric amplifier
An optical parametric amplifier, abbreviated OPA, is a laser light source that emits light of variable wavelengths by an optical parametric amplification process.-Optical parametric generation :...
s) are the choice of excitation.
Absorption rate
The Beer's law for OPA:changes to
for TPA with light intensity as a function of path length or cross section x as a function of concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...
c and the initial light intensity I0. The absorption coefficient α now becomes the TPA cross section β. (Note that there is some confusion over the term β in nonlinear optics, since it is sometimes used to describe the second-order polarizability
Hyperpolarizability
The hyperpolarizability, a nonlinear-optical property of a molecule, is the second-order electric susceptibility per unit volume. See nonlinear optics....
, 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.)
Units of cross-section
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 10−50 cm4 s photon−1. Considering the reason for these units, one can see that it results from the product of two areas (one for each photon, each in cm2) 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.Two-photon emission
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. TPE is important for applications in astrophysics, contributing to the continuum radiation from planetary nebulae (theoretically predicted for them in ref[] and observed in ref[]). TPE in condensed matter and specifically in semiconductors was only recently observed, with emission rates nearly 5 orders of magnitude weaker than OPE, with potential applications in quantum informationQuantum information
In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system...
.
Development of the field and potential applications
Until the early 1980s, TPA was used as a spectroscopicSpectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...
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
3D optical data storage
3D optical data storage is the term given to any form of optical data storage in which information can be recorded and/or read with three dimensional resolution ....
. Watt Webb suggested microscopy and imaging. Other applications such as 3D microfabrication, optical logic, autocorrelation, pulse reshaping and optical power limiting were also demonstrated
Microfabrication and lithography
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 laserLaser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
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.
3D photopolymerization
In 3D microfabrication, a block of gel containing monomers and a 2-photon active photoinitiatorInitiator
An initiator can refer to:* A person that takes an initiative in making something happen.* Modulated neutron initiator, a neutron source used in some nuclear weapons...
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.
Imaging
The human body is not transparent to visible wavelengths. Hence, one photon imaging using fluorescent dyes 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 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's scattering lawRayleigh scattering
Rayleigh scattering, named after the British physicist Lord Rayleigh, is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through...
, the amount of scattering is proportional to
, where 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.Optical power limiting
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 sensorSensor
A sensor is a device that measures a physical quantity and converts it into a signal which can be read by an observer or by an instrument. For example, a mercury-in-glass thermometer converts the measured temperature into expansion and contraction of a liquid which can be read on a calibrated...
s, can be used in protective goggles, or can be used to control noise in laser beams.
Photodynamic therapy
Photodynamic therapyPhotodynamic therapy
Photodynamic therapy is used clinically to treat a wide range of medical conditions, including malignant cancers, and is recognised as a treatment strategy which is both minimally invasive and minimally toxic...
(PDT) is a method for treating cancer
Cancer
Cancer , known medically as a malignant neoplasm, is a large group of different diseases, all involving unregulated cell growth. In cancer, cells divide and grow uncontrollably, forming malignant tumors, and invade nearby parts of the body. The cancer may also spread to more distant parts of the...
. In this technique, an organic molecule with a good triplet quantum yield is excited so that the triplet state
Triplet state
A spin triplet is a set of three quantum states of a system, each with total spin S = 1 . The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one.In physics, spin is the angular momentum...
of this molecule interacts with oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...
. 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
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...
region, thereby making the process more viable to be used on the human body.
Optical data storage
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 DVDDVD
A DVD is an optical disc storage media format, invented and developed by Philips, Sony, Toshiba, and Panasonic in 1995. DVDs offer higher storage capacity than Compact Discs while having the same dimensions....
. Therefore, 3D optical data storage
3D optical data storage
3D optical data storage is the term given to any form of optical data storage in which information can be recorded and/or read with three dimensional resolution ....
has the possibility to provide media that contain terabyte
Terabyte
The terabyte is a multiple of the unit byte for digital information. The prefix tera means 1012 in the International System of Units , and therefore 1 terabyte is , or 1 trillion bytes, or 1000 gigabytes. 1 terabyte in binary prefixes is 0.9095 tebibytes, or 931.32 gibibytes...
-level data capacities on a single disc.
TPA compounds
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.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.
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
Push-pull olefin
A push-pull olefin is a type of olefin characterized by an electron-withdrawing substituent on one side of the double bond and an electron-donating substituent on the other side. This makes the pi bond very polarized...
s exhibit high TPA transitions, up to several thousand GM. 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.
Compounds with interesting TPA properties also include various porphyrin
Porphyrin
Porphyrins are a group of organic compounds, many naturally occurring. One of the best-known porphyrins is heme, the pigment in red blood cells; heme is a cofactor of the protein hemoglobin. Porphyrins are heterocyclic macrocycles composed of four modified pyrrole subunits interconnected at...
derivatives, conjugated polymers and even dendrimers. In one study a diradical
Diradical
A diradical in organic chemistry is a molecular species with two electrons occupying two degenerate molecular orbitals . They are known by their higher reactivities and shorter lifetimes. In a broader definition diradicals are even-electron molecules that have one bond less than the number...
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.
TPA Coefficients
The two photon absorption coefficient is defined by the relationso that
Where
is the two-photon absorption coefficient, is the absorption coefficient is the transition rate for TPA, is the irradianceIrradiance
Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
,
is the photon frequency and the thickness of the slice is .The SI units of the coefficient are:
Where N is the number density of molecules per cm3, E is the photon energy (J), σ2 is the two-photon absorption cross section (cm4s/molecule).
If β (m/W) is multiplied by 10−9 it can be converted to the CGS system (cal/cm s/erg).
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
picosecond to subpicosecond durations, noticeably reduced TPA coefficient have been obtained.
TPA in Water
laser induced TPA in water was discovered in 1980.Water absorbs UV radiation near 125 nm exiting the 3a1 orbital
Orbital
Orbital may refer to:In chemistry and physics:* Atomic orbital* Molecular orbitalIn astronomy and space flight:* Orbit* Orbital resonance* Orbital period* Orbital plane * Orbital elements* Orbital speed...
leading to dissociation
Dissociation
Dissociation is an altered state of consciousness characterized by partial or complete disruption of the normal integration of a person’s normal conscious or psychological functioning. Dissociation is most commonly experienced as a subjective perception of one's consciousness being detached from...
into OH⁻ and H⁺. Through TPA this dissociation can be achieved by two photons near 266 nm. 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.
A study from Jan 2002 used a femtosecond laser tuned to 0.22 Picoseconds found the coefficient of D2O to be 42±5 10−11(cm/W) whereas H2O was 49±5 10−11(cm/W)
| λ (nm) | pulse duration τ (ps) |  (cm/W) |
|---|---|---|
| 315 | 29 | 4 |
| 300 | 29 | 4.5 |
| 289 | 29 | 6 |
| 282 | 29 | 7 |
| 282 | 0.18 | 19 |
| 266 | 29 | 10 |
| 264 | 0.22 | 49±5 |
| 216 | 15 | 20 |
| 213 | 26 | 32 |
See also
- Virtual particleVirtual particleIn physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...
s are in virtual state where the probability amplitudeProbability amplitudeIn quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
is not conserved. - Two-photon excitation microscopyTwo-photon excitation microscopyTwo-photon excitation microscopy is a fluorescence imaging technique that allows imaging of living tissue up to a very high depth, that is up to about one millimeter. Being a special variant of the multiphoton fluorescence microscope, it uses red-shifted excitation light which can also excite...