Two Photon Absorption - Revision history

Cmditradmin: /* External Links */

2011-04-05T20:05:55Z

External Links

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	==External Links==		==External Links==

	see[[wikipedia: Two-photon_absorption]]		see [[wikipedia: Two-photon_absorption]]

	[[category:third order NLO]]		[[category:third order NLO]]

Cmditradmin: /* External Links */

2011-04-05T20:05:42Z

External Links

← Older revision		Revision as of 13:05, 5 April 2011
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	==External Links==		==External Links==

	see[wikipedia: Two-photon_absorption]		see[[wikipedia: Two-photon_absorption]]

	[[category:third order NLO]]		[[category:third order NLO]]

Cmditradmin at 20:05, 5 April 2011

2011-04-05T20:05:24Z

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	<br clear='all'>		<br clear='all'>
	TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.		TPA can be exploited in many technological applications, including microfabrication, optical limiting, and 3D microscopy.

			==External Links==

			see[wikipedia: Two-photon_absorption]

	[[category:third order NLO]]		[[category:third order NLO]]

Cmditradmin: /* Measuring the Two-Photon Absorption Cross Section */

2011-04-05T19:47:48Z

Measuring the Two-Photon Absorption Cross Section

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	== Measuring the Two-Photon Absorption Cross Section ==		== Measuring the Two-Photon Absorption Cross Section ==
	[[Image:Tpa_measurement.png\|thumb\|400px\|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]		[[Image:Tpa_measurement.png\|thumb\|400px\|Schematic of an optical setup for a two-photon induced fluorescence experiment. The red line represents the excitation beam, the blue line is the two-photon induced fluorescence, the dashed line is a beam splitter and the solid black lines are filters (to control the beam intensity or to block the excitation beam before the PMT detectors).]]
	One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be ~~focussed~~ or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the ~~number~~ number of molecules excited by two-photon absorption per unit time and volume, ''N''<sub>TP</sub>, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:		One of the techniques used to measure the TPA cross section of a material is based on two-photon induced fluorescence. In this measurement, a laser beam is propagated through the sample of interest (the beam can be focused or collimated) and the fluorescence light that is emitted by the excited molecules after two-photon absorption is collected and measured by a detector (for example a photomultiplier tube, PMT). As seen above, the number of molecules excited by two-photon absorption per unit time and volume, ''N''<sub>TP</sub>, is proportional to the TPA cross section of the material at that excitation wavelength, the concentration of molecules and the square of the photon flux. The number of fluorescence photon emitted by these molecules is then:

	:<math>n_{fl} = \eta N_{TP}\,\!</math>,		:<math>n_{fl} = \eta N_{TP}\,\!</math>,

Cmditradmin at 18:43, 9 August 2010

2010-08-09T18:43:45Z

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	<td style="text-align: left; width: 33%">[[Introduction to Third-order Processes and Materials \|Previous Topic]]</td>		<td style="text-align: left; width: 33%">[[Introduction to Third-order Processes and Materials \|Previous Topic]]</td>
	<td style="text-align: center; width: 33%">[[Main_Page#Third-order Processes, Materials & Characterization \|Return to Third-order Processes, Materials & Characterization Menu]]</td>		<td style="text-align: center; width: 33%">[[Main_Page#Third-order Processes, Materials & Characterization \|Return to Third-order Processes, Materials & Characterization Menu]]</td>
	<td style="text-align: right; width: 33%">[[~~Two~~-~~Photon Absorption~~ \| Next Topic]]</td>		<td style="text-align: right; width: 33%">[[Characterization of Third-order Materials\| Next Topic]]</td>
	</tr>		</tr>
	</table>		</table>

Cmditradmin: /* Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules */

2010-07-20T17:57:13Z

Proposed Model to Enhance TPA Cross Sections in Symmetrical Molecules

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	If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seen that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .		If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seen that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .

			There is an optimal amount of charge transfer that is needed to get the largest cross section.

			[[Image:Tpa transdip boa.png\|thumb\|300px\|Transition dipole moment and transition energy ]]

	<br clear='all'>		<br clear='all'>

Mrumi: /* Applications for TPA */

2010-06-11T01:24:59Z

Applications for TPA

Mrumi: /* Design of TPA Chromophores */

2010-06-11T01:14:40Z

Design of TPA Chromophores

Mrumi: /* Examples of Two-Photon Absorbing Materials */

2010-06-11T01:09:49Z

Examples of Two-Photon Absorbing Materials

Mrumi: /* Calculations of the TPA Cross Section in a Donor-Acceptor Molecule */

2010-06-11T00:51:22Z

Calculations of the TPA Cross Section in a Donor-Acceptor Molecule

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	=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===		=== Calculations of the TPA Cross Section in a Donor-Acceptor Molecule===
	[[Image:Tpa_donaracceptor.png\|thumb\|300px\|TPA calculations for stilbene]]		[[Image:Tpa_donaracceptor.png\|thumb\|300px\|TPA calculations for stilbene.]]
	A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of ''δ'' for the transition from the ground to the first excited state (S<sub>0</sub> to S<sub>1</sub>, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).		A simple donor/acceptor stilbene with an amino group and a formyl group on the phenyl rings in para position has been used a model compound for calculations of the TPA cross section. The figure at right shows the molecule in two resonance structures and the calculated TPA cross section as a function of the bond order alternation (BOA; i.e. the difference between the bond order in adjacent CC bonds in the vinylene bridge), which changes going from one resonance structure to the other. The value of ''δ'' for the transition from the ground to the first excited state (S<sub>0</sub> to S<sub>1</sub>, left plot) is small for large negative values of the BOA (corresponding to the resonance structure on the left side), reaches a maximum when the BOA increases, and then it goes to zero when the BOA approaches the cyanine limit (BOA = 0). The same trend as a function of BOA is obtained for the change in dipole moments (see inset).
	The TPA cross section for the transition to the second excited state (S<sub>0</sub> to S<sub>2</sub>, right plot) exhibits a more complicated behavior as a function of BOA and multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S<sub>1</sub> is very close to half of the energy for the transition to S<sub>2</sub>, the detuning term becomes small and ''δ'' increase; this situation is referred to a "double resonance".		The TPA cross section for the transition to the second excited state (S<sub>0</sub> to S<sub>2</sub>, right plot) exhibits a more complicated behavior as a function of BOA and multiple peaks are present, in part because of changes in the detuning term <math>E_{ge} - \hbar \omega\,\!</math>. When the energy for the transition to S<sub>1</sub> is very close to half of the energy for the transition to S<sub>2</sub>, the detuning term becomes small and ''δ'' increase; this situation is referred to a "double resonance".

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	=== Two-Photon Initiated Polymerization and 3D Microfabrication ===		=== Two-Photon Initiated Polymerization and 3D Microfabrication ===
	[[Image:Tpa_crosslinked.png\|thumb\|300px\|TPA can be used to stimulate cross linking in a polymer.]]		[[Image:Tpa_crosslinked.png\|thumb\|300px\|TPA can be used to stimulate cross linking in a polymer.]]
	Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control on the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:		Two-photon absorption can be used to initiate photo-polymerization on a precise microscopic scale in 3D. A beam can be focused at a desired position in a polymer precursor, an initiator in the precursor is excited by two-photon absorption, photoactivating the polymerization or cross-linking in the material. This process is confined to the vicinity of the excitation volume. When the laser beam is moved to a different position, the polymerization or cross-linking is initiated in the new location. Then the non-crosslinked portions (those that have not been exposed to the laser beam) can be washed away by a solvent. This process can be used for microfabrication, that is the fabrication of a polymeric structure at the microscale with good control of the feature sizes in three dimensions, by scanning the laser beam in a prescribed pattern. This process has been used to fabricate structures in various types of resins and high degree of complexity. Selected example can be found in the following publications:


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	[[Image:Tpa initiators.png\|thumb\|300px\|New dyes increase the effective range of power where 3D "writing" can occur.]]		[[Image:Tpa initiators.png\|thumb\|300px\|New dyes increase the effective range of power where 3D "writing" can occur.]]

	Regular photo-initiators are not excellent two photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).		Regular photo-initiators are not excellent two-photon absorbers. A femtosecond laser can supply a beam with very large power and thus it may be possible, in some cases, to use them as initiators under two-photon excitation conditions. However, at this laser power damage to the material could occur. At the same time, if the laser power is too low, the microscopic structure obtained by the photopolymerization may not have sufficient mechanical strength after removal of the unexposed resin, or the process may not be reliable enough. To achieve good reliability and fidelity in the microfabrication process it is desirable to utilize photoinitiators that are efficient and that have a wide dynamic range in writing power. For conventional initiators used under two-photon excitation conditions, the writing power range (that is the ratio between the power at which damage starts occurring and the minimum power that gives well-formed structures) is usually small, for example it is between 1 and 2.5 for the first three initiators in the table at right. Newly developed two-photon absorbing dyes expand this range, up to a value of 50 for the bottom two examples in the table. This increase results mainly from the fact that these dyes have a much larger TPA cross section than conventional initiators. This lets you write more accurately and faster because the beam does not have remain in the same place as long. The figures are SEM images of the same nominal structure fabricated: below the writing threshold, resulting in an incomplete structure after removal of the unexposed resin (top image), within the writing power range (middle image), and above the damage threshold of the material (bottom image).
	<br clear='all'>		<br clear='all'>

	=== Fluorescent and Refractive Bit Optical Data Storage ===		=== Fluorescent and Refractive Bit Optical Data Storage ===
	[[Image:Tpa_optical_storage.png\|thumb\|300px\|Fluorescent and Refractive Bit Optical Data Storage]]		[[Image:Tpa_optical_storage.png\|thumb\|300px\|Fluorescent and Refractive Bit Optical Data Storage.]]

	This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.		This two-photon induced polymerization technique can be used in optical data storage applications. For example the compound shown at right is non-fluorescent, but it becomes fluorescent when the pendant acrylate groups are incorporated in an acrylate polymer. When this compound is mixed with other acrylate monomers and exposed to laser light at an appropriate wavelength and intensity, TPA-induced polymerization takes place and the resulting polymer is fluorescent. The resin in the unexposed areas remains non-fluorescent. The image at the bottom left shows two rows of fluorescent bits, obtained by two-photon induced polymerization, on the dark background of the unexposed resin.
	After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.		After laser exposure, the polymer also has higher density, due to cross-linking, than the unexposed portions of the resin. If the density goes up and the polarizability of the material stays the same, the susceptibility goes up and the refractive index goes up. Thus, it is also possible to "read" the bits based on the contrast in refractive index. Peter Rentzepis at the UC Urvine is using this method to create a 3D optical memory.
	Because the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of ~~bit~~ in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.		Because of the three-dimensional confinement of the excitation volume that is characteristic of the TPA process, it is possible to write and read sets of bits in multiple layers within the material, each layer independently from the one above and the one below. The image on the right side shows two such layers: the "1" bits (dark spots) can easily be seen in each of the layers, as well as one "0" bit per layer.
	The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm<sup>3</sup> are achievable).		The ability to write on hundreds of different planes increases the amount of information that can be stored in a given volume of material (gigabits or terabits of data per cm<sup>3</sup> are achievable).
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	=== Photochemistry Generated via an Intramolecular Electron Transfer ===		=== Photochemistry Generated via an Intramolecular Electron Transfer ===
	[[Image:Tpa_photochemistry.png\|thumb\|300px\|]]		[[Image:Tpa_photochemistry.png\|thumb\|300px\|]]
	Another method to initiate this process with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two photon absorption.		Another method to initiate this process is with electron transfer. A two-photon absorbing dye connected to a photoactive group will absorb the two photons and cause photoinduced electron transfer (PET) producing a radical anion and radical cation. This group can cleave to give rise to photoproducts. There is a history of doing this kind of chemistry not necessarily with dyes connected to each other and not with two-photon absorption.
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	If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area.<br clear='all'>		If a particular organelle or cell structure can be labeled with fluorescent TPA dye, by scanning a laser beam at the appropriate wavelength and recording the fluorescence emitted by the dye, it is possible to obtain a 3D mapping of the distribution of the dye with submicron resolution and thus a model of the structure to which the fluorescent dye is attached. Laser scanning fluorescence microscopy using TPA dyes is nowadays extensively used for imaging in biology or other area.<br clear='all'>
	A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large δ values) and are highly fluorescent.		A key factor in this technology is the availability of suitable labeling molecules that are affective two-photon absorbers (i.e. have large ''δ'' values) and are highly fluorescent.

	== Summary ==		== Summary ==

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	[[Image:Tpa_trans_dipole.png\|thumb\|400px\|]]		[[Image:Tpa_trans_dipole.png\|thumb\|400px\|]]

	We summarize here the expression for δ as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:		We summarize here the expression for ''δ'' as a function of molecular parameters for a centrosymmetric molecule within the three-level model approximation and discuss how this can be used to derive guidelines for the design of chromophores with large TPA cross section:


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	This suggests that to obtain large δ values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:		This suggests that to obtain large ''δ'' values, one of the following strategies can be used, at least in the case of centrosymmetric molecules, to increase the magnitude of one or both transition dipole moments:

	*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.		*Increase the distance between the donors, so as to increase the distance over which the charge is transferred.
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	The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for δ in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The ~~trendd~~ in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.		The diagram reports the TPA cross section (numbers under the molecular structures, in GM units) for a series of molecules with phenylene-vinylene conjugated backbones, to show the effect of donors (shown in blue) and acceptors (shown in red), as well as longer molecular chain lengths. The trend observed for ''δ'' in these compounds is consistent with the expectations based on the three-level model. The combination of these design strategies allows to achieve cross section on the order of thousands of GM. The trend in cross section when the molecular structure is changed are further illustrated in the following two sub-sections.


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	=== Chain-Length Dependence ===		=== Chain-Length Dependence ===
	[[Image:Tpa_chainlength.png\|thumb\|300px\|TPA spectra for molecules with different conjugation lengths.]]		[[Image:Tpa_chainlength.png\|thumb\|300px\|TPA spectra for molecules with different conjugation lengths.]]
	The graph shows that as the length of the conjugated bridge increases, the energy of the two photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.		The graph shows that as the length of the conjugated bridge increases, the energy of the two-photon absorption band decreases (the maximum in TPA band shifts to longer wavelength) and the magnitude of the TPA cross section increases, due to the increase of the transition dipole moment <math>M_{ge}\,\!</math>.
	<br clear='all'>		<br clear='all'>

	=== Effect of D/A Substitution ===		=== Effect of D/A Substitution ===
	[[Image:Tpa_donaracceptor_substitute.png\|thumb\|300px\|Trends for δ, detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]		[[Image:Tpa_donaracceptor_substitute.png\|thumb\|300px\|Trends for ''δ'', detuning energy, and transition dipole moments of a series of quadrupolar molecules.]]

	The table shows that by making these molecules quadrupolar, that is attaching electron-~~rich~~ and/or electron-~~poor~~ functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).		The table shows that by making these molecules quadrupolar, that is attaching electron-donating and/or electron-withdrawing functional groups on the conjugated backbone in a centrosymmetric arrangement, the TPA cross section increases significantly with respect to molecules without substituents. This is mainly due to the increase in the transition dipole moment <math>M_{ee^\prime}\,\!</math>: in fact this parameter is much larger in the molecule with donors at the termini of the molecule (second row in the table), than the one without donors (first row); the transition moment becomes even larger when the molecules contains both donor and acceptor groups (last three rows).
	<br clear='all'>		<br clear='all'>

← Older revision		Revision as of 18:09, 10 June 2010
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	[[Image:Tpa_spectra.png\|thumb\|400px\|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]		[[Image:Tpa_spectra.png\|thumb\|400px\|Two-photon and linear absorption spectra of the molecule shown (in toluene solutions).]]

	Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S<sub>1</sub> (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S<sub>2</sub> (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S<sub>1</sub> and then fluorescence emission from S<sub>1</sub>. There is no fluorescence emission from S<sub>2</sub> because, in most cases, the relaxation from S<sub>2</sub> to S<sub>1</sub> is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S<sub>2</sub> to S<sub>0</sub> is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the ~~grounds~~ and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S<sub>2</sub> (or S<sub>n</sub>). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S<sub>1</sub>). Most molecules behave according to Kasha's rule, but a few exceptions are ~~know~~, such as azulene.		Here is a centrosymmetric molecule with a conjugated backbone and donor groups on both ends. The energy level diagram on the left side of the figure, similar to that discussed at the beginning of the section, shows the allowed transitions for this molecules. There can be one-photon excitation into S<sub>1</sub> (this transition is not two-photon allowed because the molecule is centrosymmetric) and two-photon excitation into S<sub>2</sub> (again for symmetry reason this transition is not one-photon allowed). After excitation, rapid relaxation can occur by internal conversion back to S<sub>1</sub> and then fluorescence emission from S<sub>1</sub>. There is no fluorescence emission from S<sub>2</sub> because, in most cases, the relaxation from S<sub>2</sub> to S<sub>1</sub> is much faster than the fluorescence lifetime. In centrosymmetric molecules this can be easily understood, because the transition from S<sub>2</sub> to S<sub>0</sub> is symmetry forbidden for one photon, therefore the transition dipole moment for this transition is close to zero and the coupling between the ground and the excited state is very small, resulting in a long radiative lifetime of the excited state. However even if the molecule was not centrosymmetric, the internal conversion relaxation from a higher-lying excited state is generally so fast that there still would not be fluorescence from S<sub>2</sub> (or S<sub>n</sub>). This is known as "Kasha's rule", which was described by Michael Kasha and which states that, irrespective of the electronic state of the molecule reached by excitation, fluorescence will only occur from the lowest lying excited state (S<sub>1</sub>). Most molecules behave according to Kasha's rule, but a few exceptions are known, such as azulene.

	The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).		The figure also shows the one-photon (blue line) and two-photon absorption (red line) spectra of the molecule and the fluorescence emission spectrum (green line).
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	[[Image:Tpa_laserdyes.png\|thumb\|300px\|]]		[[Image:Tpa_laserdyes.png\|thumb\|300px\|]]

	Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in ~~goppert~~-mayer (GM) units: 1 GM = 1 × 10<sup>-50</sup> cm<sup>4</sup> s molecules<sup>−1</sup> photon<sup>−1</sup>. The unit ~~are names~~ in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.		Xu and Webb measured the TPA cross section for various laser dyes and other commercially available compounds. The values at the peak of the two-photon absorption band for a selection of these compounds are reported at right. The TPA cross section is given here in goeppert-mayer (GM) units: 1 GM = 1 × 10<sup>-50</sup> cm<sup>4</sup> s molecules<sup>−1</sup> photon<sup>−1</sup>. The unit is named in honor of Maria Goeppert Mayer, the German physicist that predicted the process of two-photon absorption in 1931. TPA was not actually observed experimentally until the early 60s, when lasers were developed that had sufficient intensity to lead to measurable effects in materials.
	One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.		One dye shown here has a cross section of about 300 GM, the other are in the range 10-100 GM.

	See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481 <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352 <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536 <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>		See Xu and Webb, J. Opt. Soc. Am. 1996, vol. 13, p. 481; <ref>C. Xu, JOSA B, 1996;</ref> Albota et al., Appl. Opt. 1998, vol. 37, p. 7352; <ref>M. Albota, Appl. Opt., 1998;</ref> Fisher et al., Appl. Spectrosc. 1998, vol. 52, p. 536. <ref>W. G. Fisher, Appl. Spectr., 1998.</ref>
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	[[Image:Tpa_af50.png\|thumb\|500px\|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]		[[Image:Tpa_af50.png\|thumb\|500px\|Values of δ for compound AF-50 (structure shown) from various measurements reported in the literature. τ is the pulse duration used.]]

	The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity δ is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the problems ~~in measurement~~ in the NLO field in general, as ~~this~~ are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a δ value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S<sub>1</sub> to a higher state Sn, if the population of S<sub>1</sub> becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross ~~sections~~ that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that ~~depend~~ on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").		The chart shows various measurements of the TPA cross section for the molecule AF-50. It can be seen that the values vary over many orders of magnitude. What is the reason for these differences? Is the problem intrinsic in the material or due to the way in which the parameter was measured? The measurements were indeed made using various techniques and conditions. The quantity ''δ'' is a molecular characteristic and it should not depend on the experimental conditions and optical set-up used. The variation in value in the chart is in part indicative of the measurement problems in the NLO field in general, as these are affected by relatively large uncertainties under the best of circumstances. In the case of TPA absorption, though, there can be additional problems, because some experimental techniques may not be selective to measure only TPA and for certain intensity ranges and pulse duration other effect may contribute to the observed signal. For example, the experiment that gave a ''δ'' value of 11560 GM was based on the nonlinear transmission (NLT) technique and conducted using pulses with duration of a few nanoseconds. It is now recognized that for pulse duration this long, many materials exhibit other nonlinear absorption processes in addition to TPA, in particular there can be excited state absorption (ESA) from S<sub>1</sub> to a higher state S<sub>n</sub>, if the population of S<sub>1</sub> becomes large enough. During the NLT measurement the combined effect of TPA and ESA is seen and results in an apparent TPA cross section that is very large. This is, however, not the "intrinsic" cross section of the material, but an "effective" cross section that depends on the excitation conditions used in the experiment. While this is detrimental for the measurement of the "intrinsic" cross section, the large magnitude of the "effective" cross sections in some materials could be useful, for example, in making coatings for safety glasses that could exclude high intensity laser light (i.e. to achieve "optical limiting").

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	*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)		*Strong nonlinear transmission was observed (that is the transmittance of the material changes as a function of the intensity of the excitation laser beam)

	For this compound, the maximum TPA cross section (at 600 nm), δ was measured to be 210 x 10<sup>-50</sup> cm<sup>4</sup> s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10<sup>-50</sup> cm<sup>4</sup> s/photon.		For this compound, the maximum TPA cross section (at 600 nm), ''δ'' was measured to be 210 x 10<sup>-50</sup> cm<sup>4</sup> s/photon, while for stilbene (i.e. if the two donor groups are removed) δ = 12 x 10<sup>-50</sup> cm<sup>4</sup> s/photon.
	Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. δ for BDAS is very large and it is useful to understand why the donors have this effect.		Thus, the TPA cross section of BDAS is about 20 times that for the molecule without the electron donor groups. ''δ'' for BDAS is very large and it is useful to understand why the donors have this effect.

	'''Interesting features for two-photon applications'''		'''Interesting features for two-photon applications'''
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	Theoretical calculations can help to explain the properties of the BDAS molecule.		Theoretical calculations can help to explain the properties of the BDAS molecule.

	Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to ~~central~~ vinyl group in the middle (becoming more negative) when the molecule is excited from S<sub>0</sub> (''g'') to S<sub>1</sub> (''e'') and to S<sub>2</sub> (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S<sub>1</sub> and S<sub>2</sub> (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is ~~instead~~ much smaller in the case of stilbene.		Calculations show that BDAS has large and symmetrical charge transfer from nitrogens (becoming more positive) to the vinyl group in the middle (becoming more negative) when the molecule is excited from S<sub>0</sub> (''g'') to S<sub>1</sub> (''e'') and to S<sub>2</sub> (''e''') and this charge transfer is reflected primarily in very a large transition dipole moment between S<sub>1</sub> and S<sub>2</sub> (<math>M_{ee^{\prime}}\,\!</math>). The value of <math>M_{ee^{\prime}}\,\!</math> is much smaller in the case of stilbene.

	These results suggest that a large change in quadrupole moment between S<sub>0</sub> and S<sub>2</sub> can lead to large values of δ.		These results suggest that a large change in quadrupole moment between S<sub>0</sub> and S<sub>2</sub> can lead to large values of ''δ''.

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	[[Image:Tpn_bisdonorstilbene_subst.png\|thumb\|300px\| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]		[[Image:Tpn_bisdonorstilbene_subst.png\|thumb\|300px\| Transition energies and transition dipole moments for stilbene and a bid-donor stilbene obtained from quantum-chemical calculations]]
	The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.		The observation above suggests certain design strategies for increasing the dipole or quandrupole moment: if you want to make the dipole (or quadrapole) moment larger you can increase the distance over which charge is separated and you can increase the charge that is separated by that distance, or do both. The first would correspond to increasing the length of the molecule, the second to introducing functional groups that can facilitate the charge transfer. This is the role that the dibutylamino groups play in BDAS with respect to stilbene.
	This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).		This interpretation is supported by results of quantum chemical calculations performed on these molecules (in the calculations, dimethyl- instead of dibutyl-amino groups were considered).
	The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.		The transition dipole moment between the ground and first excited state is 7.2 D for stilbene and 8.9 D for BDAS. The biggest change, though, is observed for <math>M_{ee^{\prime}}\,\!</math>, as the transition dipole goes from 3.1 D in stilbene to 7.4 D for BDAS.

	A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for γ (keeping in mind that δ is proportional to Im γ):		A simplified expression for the maximum TPA cross section for the transition to ''e''' in centrosymmetric molecules can be obtained from the three-level equation given earlier for ''γ'' (keeping in mind that δ is proportional to Im ''γ''):

	:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>		:<math>\delta_{g \rightarrow e^{\prime}} \propto \frac {M^2_{ge} M^2_{ee^{\prime}}} {(E_{ge} - \hbar \omega )^2 \Gamma_{ge^{\prime}}} \,\!</math>
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	'''Transition Dipole Moments'''		'''Transition Dipole Moments'''
	[[Image:Tpa_transdip_densities.png\|thumb\|300px\| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]		[[Image:Tpa_transdip_densities.png\|thumb\|300px\| Change in atomic charges for the transition from ''g'' to ''e'' for stilbene and a bis-donor stilbene.]]
	If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be ~~seems~~ that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .		If the components that contribute to transition dipole moments are located farther from the middle of the molecule the contribution to the transition dipole moment is correspondingly larger. This is illustrated at right, where the change in charge distribution is plotted for the various atoms in the molecules. It can be seen that in BDAS one of the largest changes occurs on the nitrogens, which are located at the opposite ends of the molecule. This change, coupled with the extended distance from the molecular origin results in a large transition dipole moment for this molecule. In stilbene, the charge distribution changes mostly in the central part of the molecule, resulting in a smaller transition dipole moment <math>M_{ee^{\prime}}\,\!</math> .