Abstract

The range of organic compounds whose degenerate two-photon absorption (2PA) spectrum has been reported has increased rapidly in recent years, in parallel with the growing interest in applications based on the 2PA process. The comparison of results from different techniques is not always straightforward, and experimental conditions employed may vary significantly. We overview the concepts underlying 2PA measurements and the common assumptions and approximations used in the data analysis for various techniques. The importance of selecting appropriate excitation regimes under which measurements should be performed and of avoiding contributions from absorption mechanisms in addition to 2PA will be emphasized.

© 2010 Optical Society of America

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  104. For example, in some cases [101, 102, 103], the cross section is defined to “count” directly the number of molecules excited, instead of the photons absorbed, leading to the relationship nm(2)=δ̃Ngϕ2.(I)    In this case, the propagation equation (1) becomesdϕdz=−2δ̃Ngϕ2.(II)    Obviously, the right-hand sides of the two versions of the propagation equation [Eqs. (1, II)] and of the equation describing the excitation rate due to 2PA [Eqs. (6, I)] differ only by a constant factor and become equivalent if δ=2δ̃.(III)
  105. More precisely, any absorption process that would take place at this stage, via 1PA, 2PA or other nonlinear processes, would reflect properties of the excited state r, not of the ground state g, and thus describe a very different sample, from a spectroscopic point of view. All processes, though, contribute to the attenuation of the propagating beam and Eq. (1) would have to be appropriately modified. This situation will also be discussed in Subsection 3.2. ESA following 2PA will also be considered in Subsection 4.3.
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  107. It is generally assumed that the fluorescence quantum yield is the same after 1PA and 2PA excitation, as for most molecules the system relaxes, after absorption, to the lowest excited state, r, by internal conversion, irrespective of the photon energy and the manner of excitation, as discussed above. However, there are exceptions to this general rule, as some molecules are known that exhibit a wavelength dependent quantum yield or emission spectrum, and fluorescence emission from upper excited states.
  108. Δm(2) is equivalent to the “saturation parameter” α discussed by Xu and Webb [17], pp. 518–520.
  109. It should be kept in mind that Eq. (14) was obtained under a series of assumptions (for example regarding the change in photon flux through the sample). Thus, strictly speaking, Eq. (14) is also an approximate description of the fraction of molecules excited.
  110. The total number of photons absorbed per pulse depends on the excitation volume. For a collimated beam, as the case considered in this section, the volume scales with the sample path length L. However, for a focused beam, only a path length of the order of twice the Rayleigh range of the beam, z0=πw02n∕λ (if this is shorter than L), may have to be considered [29], at least as a first approximation, because of the decrease in photon flux away from the focal plane. See also examples in Subsection 3.1b. We have used here for the Rayleigh range z0 the definition typically introduced for Gaussian beams, for which it corresponds to the distance along z between the waist and the point at which the beam radius is 2w0 [111]. The term confocal parameter is also sometimes used when referring to the quantity z0 (see, for example, [112], p. 117) or 2z0 [see, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966)].
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  114. The 1PA equivalent of parameter Δph(2) is σN0L, the fraction of light absorbed in the limit of thin sample or weak absorption.
  115. In the case of the 2PIF method, an additional limitation on the concentration is imposed by reabsorption of the emitted photons within the material (inner filter effect). This effect is observed to different extents depending on how the fluorescence signal is collected (for example, under backscattering conditions or at 90° with respect to the incident beam).
  116. J. Segal, Z. Kotler, M. Sigalov, A. Ben-Asuly, V. Khodorkovsky, “Two-photon absorption properties of (N-carbazolyl)-stilbenes,” Proc. SPIE 3796, 153–159 (1999).
    [CrossRef]
  117. D. Beljonne, W. Wenseleers, E. Zojer, Z. Shuai, H. Vogel, S. J. K. Pond, J. W. Perry, S. R. Marder, J.-L. Brédas, “Role of dimensionality on the two-photon absorption response of conjugated molecules: the case of octupolar compounds,” Adv. Funct. Mater. 12, 631–641 (2002).
    [CrossRef]
  118. B. Strehmel, A. M. Sarker, H. Detert, “The influence of σ and π acceptors on two-photon absorption and solvatochromism of dipolar and quadrupolar unsaturated organic compounds,” ChemPhysChem 4, 249–259 (2003).
    [CrossRef] [PubMed]
  119. G. P. Bartholomew, M. Rumi, S. J. K. Pond, J. W. Perry, S. Tretiak, G. C. Bazan, “Two-photon absorption in three-dimensional chromophores based on [2.2]-paracyclophane,” J. Am. Chem. Soc. 126, 11529–11542 (2004).
    [CrossRef] [PubMed]
  120. S. K. Lee, W. J. Yang, J. J. Choi, C. H. Kim, S.-J. Jeon, B. R. Cho, “2,6-Bis[4-(p-dihexylaminostyryl)-styryl]anthracene derivatives with large two-photon cross sections,” Org. Lett. 7, 323–326 (2005).
    [CrossRef] [PubMed]
  121. When a photon counting approach is used, signal discrimination procedures can be used to reject noise.
  122. R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975).
    [CrossRef]
  123. We will introduce beams with a Gaussian spatial profile in Subsection 3.2. To explain the results in Fig. 4, it is sufficient to mention that the beam size depends on z as follows: w0(1+(z∕z0)2)1∕2. The total number of excited molecules is obtained by integrating Nm(2)(τ) over the excitation volume. The result for a generic path length L and for a sample with the focal plane at L∕2 (and with photon flux ϕ0 at z=0) is ∫VNm(2)(τ)dV=12δτN0∫−L∕2L∕2dz∫0∞2πrdrϕ2=12δτN0πw02z0ϕ022arctanL2z0=δN0Nph22τ(2nλarctanL2z0).   The asymptotic value for L≫z0 is in this case δN0Nph2πn∕(2τλ).
  124. M. Rumi, J. E. Ehrlich, A. A. Heikal, J. W. Perry, S. Barlow, Z. Hu, D. McCord-Maughon, T. C. Parker, H. Röckel, S. Thayumanavan, S. R. Marder, D. Beljonne, J.-L. Brédas, “Structure-property relationships for two-photon absorbing chromophores: bis-donor diphenylpolyene and bis(styryl)benzene derivatives,” J. Am. Chem. Soc. 122, 9500–9510 (2000).
    [CrossRef]
  125. For example, if the flux is reduced by 5% by 2PA [ϕ(z)∕ϕ(0)=0.95] and each time the photon flux is measured with an uncertainty of 2% (the two measurements before and after the beam are independent), from error propagation, the relative uncertainty in δ would be over 50%.
  126. It should be kept in mind that this common definition of β implicitly assumes that approximation (ii) is valid, as this parameter is proportional to N0. β, usually called the “2PA coefficient,” is a macroscopic equivalent of the molecular parameter δ. In cases for which (ii) is not valid, β is not a constant of the material, but depends on ϕ through Ng, the concentration of molecules in the ground state (Ng≤N0).
  127. V. I. Bredikhin, M. D. Galanin, V. N. Genkin, “Two-photon absorption and spectroscopy,” Sov. Phys. Usp. 16, 299–321 (1974).
    [CrossRef]
  128. In Subsection 4.3 we will discuss how ESA can also limit the applicability of Eq. (29) in NLT measurements, especially for ns pulse durations. At the moment, we are considering instead the case of materials for which 2PA is the only absorption process that can take place at the excitation wavelength λ.
  129. In this context, the first argument for the function ϕ is the coordinate along the propagation direction, z, the origin being the front face of the material; the second argument refers to the radial distance, r, from the axis z. Thus, in this notation ϕ(0,0) corresponds to the peak on-axis flux at the entrance of the material.
  130. The initial pulse energy is now E(0)=τEph∫0∞ϕ(0,0)e−2r2∕w022πrdr=τEphϕ(0,0)πw02∕2.
  131. When the results for the f/60 case in Fig. 7 are compared with those in Fig. 6 for the same pulse energy, it can be seen that the transmittance is larger in the former situation, even when ground state depletion is neglected. This is due to the effect of focusing: even if L<z0 in the f/60 case, the beam size increases slightly away from the focal plane within the sample; instead, the beam was assumed to be perfectly collimated in Fig. 6.
  132. This change in beam size is indeed exploited in closed-aperture Z-scan experiments [30] to measure the nonlinear refractive index of a material (which is related to the real part of the susceptibility χ(3)).
  133. To differentiate between the examples discussed above, where z represented the position within the sample along the propagation direction, and the current example, where z is the position of the front face of the sample with respect to the focal plane, the z dependence of ϕ is now expressed with a subscript.
  134. This is true not only for a Gaussian beam, but also for a beam with a generic beam profile, as long as the z dependence of the flux is known.
  135. For this type of beam the relationship between pulse energy and peak flux isE=Eph∫0∞ϕ0(r=0)e−2r2∕w022πrdr∫−∞∞e−t2∕τ̃2dt=Ephϕ0(r=0)ππw02τ̃2.
  136. The value of the transmittance at a generic point within the sample can be obtained by substituting z′, the distance of the point from the front face, for L.
  137. It should be remembered that we are assuming that 1PA absorption is negligible.
  138. W. Zhao, J. H. Kim, P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” Proc. SPIE 2229, 131–147 (1994).
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  147. In this context, we define the time average of a function Y as follows:⟨Y(t)⟩=f∫TT+1∕fY(t)dt,   where T is a generic time and Y(t) is a periodic function with period 1∕f.
  148. This assumes that τfl≪1∕f, so that effectively all molecules that have been excited during the pulse duration τ have decayed back to the ground state. For other indirect methods 1∕f needs to be large with respect to the time constant of the process to be monitored (thermal time constant, phosphorescence lifetime,…). If this requirement is not satisfied, the sample conditions are different every time a new pulse arrives, and ground state depletion may become relevant after a large number of pulses. In the case of single pulse measurements, the integration time for the signal needs to be long with respect to the time constant of the process monitored, if results for different materials are to be compared under the same excitation conditions.
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  152. For ideal monochromatic light with constant intensity and in the absence of noise, g(2)=1. For a square pulse of duration τ, g(2)=(τf)−1. For a Gaussian pulse with 1∕e width τ, g(2)=(τf(2π)1∕2)−1.
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  167. An expression similar to Eq. (46) could be written in the case of a molecule undergoing intersystem crossing from r to the lowest level in the triplet manifold and then being excited to a higher-lying triplet state. In this case, σex would represent a triplet–triplet ESA cross section.
  168. The quantity ϕτ corresponds to the photon fluence, or the number of photons per unit area of the beam, Nph∕a [Nph from Eq. (17) for the case of a pulse with constant profile in space and time].
  169. Even if ESA occurs from the triplet state, as described in note [167], the fluorescence quantum yield and intersystem crossing rates would be unchanged if Nex is small, and thus ⟨S⟩ would still not be affected by ESA and would not depend on pulse duration.
  170. This numerical example referred to excitation with ns pulses. However, similar conditions for ESA can be reached by using fs pulses from an amplified laser, for example, with ϕ values only slightly larger than those considered in case 3.1.b.
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  173. Unfortunately, there are some typographical errors in the solution of the system of equation reported by Kleinschmidt et al. [162].
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  188. The case of different excitation volumes in the two samples could be handled by including appropriate correction terms in the collection efficiency factor, G. Beside the obvious case of samples with different thicknesses in a collimated beam, the excitation volume is also different if the samples do not have the same refractive index and L≫z0.
  189. Due to the relatively low concentration used in 2PA measurements, it is often assumed that the refractive index of the solution is the same as that of the pure solvent.
  190. For example, the refractive index dependence of G varies with the optical setup configuration on both the excitation and the emission sides. A few cases, often used in the literature, are briefly considered here. If the excitation beam is collimated and the fluorescence signal is imaged at 90° with respect to the excitation propagation direction [124], the situation is similar to that employed in many commercial spectrofluorimeters, for which the signal dependence on nfl is described as nfl−2 [113]. If the excitation beam is focused in the sample, the excitation volume in the material depends on n, and consequently G depends on both n and nfl. In the focused beam configuration described by Beljonne, et al. M. A. Albota, C. Xu, W. W. Webb, “Two-photon fluorescence excitation cross sections of biomolecular probes from 690 to 960 nm,” Appl. Opt. 37, 7352–7356 (1998)].
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  196. If necessary, the attenuation of the beam can be accounted for in the equations to be used to process the data. In some experimental configurations for fluorescence signal collection, reabsorption may also have to be corrected for.
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2009 (2)

M. Pawlicki, H. A. Collins, R. G. Denning, H. L. Anderson, “Two-photon absorption and the design of two-photon dyes,” Angew. Chem. Int. Ed. 48, 3244–3266 (2009).
[CrossRef]

H. M. Kim, B. R. Cho, “Two-photon materials with large two-photon cross sections. Structure-property relationship,” Chem. Commun. (Cambridge)153–164 (2009).

2008 (4)

M. Rumi, S. Barlow, J. Wang, J. W. Perry, S. R. Marder, “Two-photon absorbing materials and two-photon-induced chemistry,” Adv. Polym. Sci. 213, 1–95 (2008).

G. S. He, L.-S. Tan, Q. Zheng, P. N. Prasad, “Multiphoton absorbing materials: molecular designs, characterizations, and applications,” Chem. Rev. 108, 1245–1330 (2008).
[CrossRef] [PubMed]

M. Samoc, A. Samoc, M. G. Humphrey, M. P. Cifuentes, B. Luther-Davies, P. A. Fleitz, “Z-scan studies of dispersion of the complex third-order nonlinearity of nonlinear absorbing chromophores,” Mol. Cryst. Liq. Cryst. 485, 894–902 (2008).
[CrossRef]

N. S. Makarov, M. Drobizhev, A. Rebane, “Two-photon absorption standards in the 550–1600 nm excitation wavelength range,” Opt. Express 16, 4029–4047 (2008).
[CrossRef] [PubMed]

2007 (1)

C. N. LaFratta, J. T. Fourkas, T. Baldacchini, R. A. Farrer, “Multiphoton fabrication,” Angew. Chem. Int. Ed. 46, 6238–6258 (2007).
[CrossRef]

2006 (2)

J. Arnbjerg, M. Johnsen, P. K. Frederiksen, S. E. Braslavsky, P. R. Ogilby, “Two-photon photosensitized production of singlet oxygen: optical and photoacoustic characterization of absolute two-photon absorption cross sections for standard sensitizers in different solvents,” J. Phys. Chem. A 110, 7375–7385 (2006).
[CrossRef] [PubMed]

S.-Y. Tseng, W. Cao, Y.-H. Peng, J. M. Hales, S.-H. Chi, J. W. Perry, S. R. Marder, C. H. Lee, W. N. Herman, J. Goldhar, “Measurement of complex χ(3) using degenerate four-wave mixing with an imaged 2-D phase grating,” Opt. Express 14, 8737–8744 (2006).
[CrossRef] [PubMed]

2005 (3)

R. L. Sutherland, M. C. Brant, J. Heinrichs, J. E. Rogers, J. E. Slagle, D. G. McLean, P. A. Fleitz, “Excited-state characterization and effective three-photon absorption model of two-photon-induced excited-state absorption in organic push-pull charge-transfer chromophores,” J. Opt. Soc. Am. B 22, 1939–1948 (2005).
[CrossRef]

S. K. Lee, W. J. Yang, J. J. Choi, C. H. Kim, S.-J. Jeon, B. R. Cho, “2,6-Bis[4-(p-dihexylaminostyryl)-styryl]anthracene derivatives with large two-photon cross sections,” Org. Lett. 7, 323–326 (2005).
[CrossRef] [PubMed]

S.-J. Chung, M. Rumi, V. Alain, S. Barlow, J. W. Perry, S. R. Marder, “Strong, low-energy two-photon absorption in extended amine-terminated cyano-substituted phenylenevinylene oligomers,” J. Am. Chem. Soc. 127, 10844–10845 (2005).
[CrossRef] [PubMed]

2004 (2)

G. P. Bartholomew, M. Rumi, S. J. K. Pond, J. W. Perry, S. Tretiak, G. C. Bazan, “Two-photon absorption in three-dimensional chromophores based on [2.2]-paracyclophane,” J. Am. Chem. Soc. 126, 11529–11542 (2004).
[CrossRef] [PubMed]

H.-B. Sun, S. Kawata, “Two-photon photopolymerization and 3D lithographic microfabrication,” Adv. Polym. Sci. 170, 169–273 (2004).
[CrossRef]

2003 (5)

K. Kamada, K. Matsunaga, A. Yoshino, K. Ohta, “Two-photon-absorption-induced accumulated thermal effect on femtosecond Z-scan experiments studied with time-resolved thermal-lens spectrometry and its simulation,” J. Opt. Soc. Am. B 20, 529–537 (2003).
[CrossRef]

H. Guo, H. Jiang, L. Luo, C. Wu, H. Guo, X. Wang, H. Yang, Q. Gong, F. Wu, T. Wang, M. Shi, “Two-photon polymerization of gratings by interference of a femtosecond laser pulse,” Chem. Phys. Lett. 374, 381–384 (2003).
[CrossRef]

B. Strehmel, A. M. Sarker, H. Detert, “The influence of σ and π acceptors on two-photon absorption and solvatochromism of dipolar and quadrupolar unsaturated organic compounds,” ChemPhysChem 4, 249–259 (2003).
[CrossRef] [PubMed]

A. Karotki, M. Drobizhev, M. Kruk, C. Spangler, E. Nickel, N. Mamardashvili, A. Rebane, “Enhancement of two-photon absorption in tetrapyrrolic compounds,” J. Opt. Soc. Am. B 20, 321–332 (2003).
[CrossRef]

J. Yoo, S. K. Yang, M.-Y. Jeong, H. C. Ahn, S.-J. Jeon, B. R. Cho, “Bis-1,4-(p-diarylaminostyryl)-2,5-dicyanobenzene derivatives with large two-photon absorption cross-sections,” Org. Lett. 5, 645–648 (2003).
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2002 (5)

R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, E. W. Van Stryland, “Two-photon spectroscopy and analysis with a white-light continuum probe,” Opt. Lett. 27, 270–272 (2002).
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S. J. K. Pond, M. Rumi, M. D. Levin, T. C. Parker, D. Beljonne, M. W. Day, J.-L. Brédas, S. R. Marder, J. W. Perry, “One- and two-photon spectroscopy of donor-acceptor-donor distyrylbenzene derivatives: effect of cyano substitution and distorsion from planarity,” J. Phys. Chem. A 106, 11470–11480 (2002).
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R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” IEEE J. Quantum Electron. 38, 1205–1216 (2002).
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P. Tian, W. S. Warren, “Ultrafast measurement of two-photon absorption by loss modulation,” Opt. Lett. 27, 1634–1636 (2002).
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D. Beljonne, W. Wenseleers, E. Zojer, Z. Shuai, H. Vogel, S. J. K. Pond, J. W. Perry, S. R. Marder, J.-L. Brédas, “Role of dimensionality on the two-photon absorption response of conjugated molecules: the case of octupolar compounds,” Adv. Funct. Mater. 12, 631–641 (2002).
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2001 (2)

D. A. Oulianov, I. V. Tomov, A. S. Dvornikov, P. M. Rentzepis, “Observations on the measurement of the two-photon absorption cross-section,” Opt. Commun. 191, 235–243 (2001).
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P. K. Frederiksen, M. Jørgensen, P. R. Ogilby, “Two-photon photosensitized production of singlet oxygen,” J. Am. Chem. Soc. 123, 1215–1221 (2001).
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2000 (4)

T. J. Bunning, S. M. Kirkpatrick, L. V. Natarajan, V. P. Tondiglia, D. W. Tomlin, “Electronically switchable gratings formed using ultrafast holographic two-photon-induced photopolymerization,” Chem. Mater. 12, 2842–2844 (2000).
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P. T. C. So, C. Y. Dong, B. R. Masters, K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2, 399–429 (2000).
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M. Rumi, J. E. Ehrlich, A. A. Heikal, J. W. Perry, S. Barlow, Z. Hu, D. McCord-Maughon, T. C. Parker, H. Röckel, S. Thayumanavan, S. R. Marder, D. Beljonne, J.-L. Brédas, “Structure-property relationships for two-photon absorbing chromophores: bis-donor diphenylpolyene and bis(styryl)benzene derivatives,” J. Am. Chem. Soc. 122, 9500–9510 (2000).
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O.-K. Kim, K.-S. Lee, H. Y. Woo, K.-S. Kim, G. S. He, J. Swiatkiewicz, P. N. Prasad, “New class of two-photon-absorbing chromophores based on dithienothiophene,” Chem. Mater. 12, 284–286 (2000).
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1999 (7)

P. Kaatz, D. P. Shelton, “Two-photon fluorescence cross-section measurements calibrated with hyper-Rayleigh scattering,” J. Opt. Soc. Am. B 16, 998–1006 (1999).
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J. W. Perry, S. Barlow, J. E. Ehrlich, A. A. Heikal, Z.-Y. Hu, I.-Y. S. Lee, K. Mansour, S. R. Marder, H. Röckel, M. Rumi, S. Thayumanavan, X. L. Wu, “Two-photon and higher-order absorptions and optical limiting properties of bis-donor substituted conjugated organic chromophores,” Nonlinear Opt. 21, 225–243 (1999).

P. Chen, D. A. Oulianov, I. V. Tomov, P. M. Rentzepis, “Two-dimensional Z scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043–7050 (1999).
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K. A. Drenser, R. J. Larsen, F. P. Strohkendl, L. R. Dalton, “Femtosecond, frequency-agile, phase-sensitive-detected, multi-wave-mixing nonlinear optical spectroscopy applied to π-electron photonic materials,” J. Phys. Chem. A 103, 2290–2301 (1999).
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B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, J. Qin, H. Röckel, M. Rumi, X.-L. Wu, S. R. Marder, J. W. Perry, “Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication,” Nature 398, 51–54 (1999).
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S. M. Kirkpatrick, J. W. Baur, C. M. Clark, L. R. Denny, D. W. Tomlin, B. R. Reinhardt, R. Kannan, M. O. Stone, “Holographic recording using two-photon-induced photopolymerization,” Appl. Phys. A 69, 461–464 (1999).
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J. Segal, Z. Kotler, M. Sigalov, A. Ben-Asuly, V. Khodorkovsky, “Two-photon absorption properties of (N-carbazolyl)-stilbenes,” Proc. SPIE 3796, 153–159 (1999).
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1998 (7)

P. A. Fleitz, M. C. Brant, R. L. Sutherland, F. P. Strohkendl, R. J. Larsen, L. R. Dalton, “Nonlinear measurements on AF-50,” Proc. SPIE 3472, 91–97 (1998).
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J. Swiatkiewicz, P. N. Prasad, B. A. Reinhardt, “Probing two-photon excitation dynamics using ultrafast laser pulses,” Opt. Commun. 157, 135–138 (1998).
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M. Albota, D. Beljonne, J.-L. Brédas, J. E. Ehrlich, J.-Y. Fu, A. A. Heikal, S. T. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X.-L. Wu, C. Xu, “Design of organic molecules with large two-photon absorption cross sections,” Science 281, 1653–1656 (1998).
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G. Witzgall, R. Vrijen, E. Yablonovitch, V. Doan, B. J. Schwartz, “Single-shot two-photon exposure of commercial photoresist for the production of three-dimensional structures,” Opt. Lett. 23, 1745–1747 (1998).
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For example, the refractive index dependence of G varies with the optical setup configuration on both the excitation and the emission sides. A few cases, often used in the literature, are briefly considered here. If the excitation beam is collimated and the fluorescence signal is imaged at 90° with respect to the excitation propagation direction [124], the situation is similar to that employed in many commercial spectrofluorimeters, for which the signal dependence on nfl is described as nfl−2 [113]. If the excitation beam is focused in the sample, the excitation volume in the material depends on n, and consequently G depends on both n and nfl. In the focused beam configuration described by Beljonne, et al. M. A. Albota, C. Xu, W. W. Webb, “Two-photon fluorescence excitation cross sections of biomolecular probes from 690 to 960 nm,” Appl. Opt. 37, 7352–7356 (1998)].
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B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, P. N. Prasad, “Highly active two-photon dyes: design, sythesis, and characterization toward application,” Chem. Mater. 10, 1863–1874 (1998).
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W. G. Fisher, E. A. Wachter, F. E. Lytle, M. Armas, C. Seaton, “Source-corrected two-photon excited fluorescence measurements between 700 and 880 nm,” Appl. Spectrosc. 52, 536–545 (1998).
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1997 (12)

P. A. Fleitz, R. L. Sutherland, “Investigating the nonlinear optical properties of molten organic materials,” Proc. SPIE 3146, 24–30 (1997).
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J. E. Ehrlich, X. L. Wu, I.-Y. S. Lee, Z.-Y. Hu, H. Röckel, S. R. Marder, J. W. Perry, “Two-photon absorption and broadband optical limiting with bis-donor stilbenes,” Opt. Lett. 22, 1843–1845 (1997).
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R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
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J. A. Hermann, “Nonlinear optical absorption in thick media,” J. Opt. Soc. Am. B 14, 814–823 (1997).
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G. S. He, L. Yuan, N. Cheng, J. D. Bhawalkar, P. N. Prasad, L. Brott, S. J. Clarson, B. A. Reinhardt, “Nonlinear optical properties of a new chromophore,” J. Opt. Soc. Am. B 14, 1079–1087 (1997).
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S. Delysse, P. Raimond, J.-M. Nunzi, “Two-photon absorption in non-centrosymmetric dyes,” Chem. Phys. 219, 341–351 (1997).
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W. G. Fisher, E. A. Wachter, M. Armas, C. Seaton, “Titanium:sapphire laser as an excitation source in two-photon spectroscopy,” Appl. Spectrosc. 51, 218–226 (1997).
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S. Hughes, J. M. Burzler, “Theory of Z-scan measurements using Gaussian–Bessel beams,” Phys. Rev. A 56, R1103–R1106 (1997).
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P. R. Callis, “Two-photon-induced fluorescence,” Annu. Rev. Phys. Chem. 48, 271–297 (1997).
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P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251–293 (1997).
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A. S. Dvornikov, P. M. Rentzepis, “Novel organic ROM materials for optical 3D memory devices,” Opt. Commun. 136, 1–6 (1997).
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S. Maruo, O. Nakamura, S. Kawata, “Three-dimensional microfabrication with two-photon-absorbed photopolymerization,” Opt. Lett. 22, 132–134 (1997).
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1996 (3)

P. B. Chapple, P. J. Wilson, “Z-scan with near-Gaussian laser beams,” J. Nonlinear Opt. Phys. Mater. 5, 419–436 (1996).
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J. M. Burzler, S. Hughes, B. S. Wherrett, “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
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C. Xu, W. W. Webb, “Measurement of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm,” J. Opt. Soc. Am. B 13, 481–491 (1996).
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1995 (4)

C. Xu, J. Guild, W. W. Webb, W. Denk, “Determination of absolute two-photon excitation cross sections by in situ second-order autocorrelation,” Opt. Lett. 20, 2372–2374 (1995).
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S. Hughes, J. M. Burzler, G. Spruce, B. S. Wherrett, “Fast Fourier transform techniques for efficient simulation of Z-scan measurements,” J. Opt. Soc. Am. B 12, 1888–1893 (1995).
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S. M. Mian, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Guassian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
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P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, C. Cremer, “Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,” Appl. Phys. Lett. 66, 1698–1700 (1995).
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1994 (4)

S. W. Hell, S. Lindek, E. H. K. Stelzer, “Enhancing the axial resolution in far-field light microscopy: two-photon 4Pi confocal fluorescence microscopy,” J. Mod. Opt. 41, 675–681 (1994).
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W. Zhao, J. H. Kim, P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” Proc. SPIE 2229, 131–147 (1994).
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K. McEwan, R. Hollins, “Two-photon-induced excited-state absorption in liquid crystal media,” Proc. SPIE 2229, 122–130 (1994).
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A. A. Said, C. Wamsley, D. J. Hagan, E. W. Van Stryland, B. A. Reinhardt, P. Roderer, A. G. Dillard, “Third- and fifth-order optical nonlinearities in organic materials,” Chem. Phys. Lett. 228, 646–650 (1994).
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1993 (5)

D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
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R. Trebino, D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
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A. A. Rehms, P. R. Callis, “Two-photon fluorescence excitation spectra of aromatic amino acids,” Chem. Phys. Lett. 208, 276–282 (1993).
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L. W. Tutt, T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
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R. L. Sutherland, E. Rea, L. V. Natarajan, T. Pottenger, P. A. Fleitz, “Two-photon absorption and second hyperpolarizability measurements in diphenylbutadiene by degenerate four-wave mixing,” J. Chem. Phys. 98, 2593–2603 (1993).
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1992 (4)

M. E. Orczyk, M. Samoc, J. Swiatkiewicz, N. Manickam, M. Tomoaia-Cotisel, P. N. Prasad, “Optical heterodyning of the phase-tuned femtosecond optical Kerr gate signal for the determination of complex third-order susceptibilities,” Appl. Phys. Lett. 60, 2837–2839 (1992).
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S. E. Braslavsky, G. E. Heibel, “Time-resolved photothermal and photoacoustic methods applied to photoinduced processes in solution,” Chem. Rev. 92, 1381–1410 (1992).
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S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93, 277–282 (1992).
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E. S. Wu, J. H. Strickler, W. R. Harrell, W. W. Webb, “Two-photon lithography for microelectronic application,” Proc. SPIE 1674, 776–782 (1992).
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1991 (3)

M. Zhao, Y. Cui, M. Samoc, P. N. Prasad, M. R. Unroe, B. A. Reinhardt, “Influence of two-photon absorption on third-order nonlinear optical processes as studied by degenerate four-wave mixing: the study of soluble didecyloxy substituted polyphenyls,” J. Chem. Phys. 95, 3991–4001 (1991).
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J. H. Strickler, W. W. Webb, “Three-dimensional optical data storage in refractive media by two-photon point excitation,” Opt. Lett. 16, 1780–1782 (1991).
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N. Pfeffer, F. Charra, J. M. Nunzi, “Phase and frequency resolution of picosecond optical Kerr nonlinearities,” Opt. Lett. 16, 1987–1989 (1991).
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1990 (3)

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
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M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
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J. H. Strickler, W. W. Webb, “Two-photon excitation in laser scanning fluorescence microscopy,” Proc. SPIE 1398, 107–118 (1990).
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1989 (1)

D. A. Parthenopoulos, P. M. Rentzepis, “Three-dimensional optical storage memory,” Science 245, 843–845 (1989).
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1988 (2)

R. D. Jones, P. R. Callis, “A power-squared sensor for two-photon spectroscopy and dispersion of second-order coherence,” J. Appl. Phys. 64, 4301–4305 (1988).
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R. D. Jones, P. R. Callis, “Two-photon spectra of inductively perturbed naphthalenes,” Chem. Phys. Lett. 144, 158–164 (1988).
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1987 (1)

J. A. Wilder, G. L. Findley, “Construction of a two-photon photoacoustic spectrometer,” Rev. Sci. Instrum. 58, 968–974 (1987).
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1986 (3)

J. K. Rice, R. W. Anderson, “Two-photon, thermal lensing spectroscopy of monosubstituted benzenes in the B2u1(Lb1)←A1g1(A1) and B1u1(La1)←A1g1(A1) transition regions,” J. Phys. Chem. 90, 6793–6800 (1986).
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P. Sperber, A. Penzkofer, “S0–Sn two-photon absorption dynamics of rhodamine dyes,” Opt. Quantum Electron. 18, 381–401 (1986).
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S. M. Kennedy, F. E. Lytle, “p-Bis(o-methylstyryl)benzene as a power-squared sensor for two-photon absorption measurements between 537 and 694 nm,” Anal. Chem. 58, 2643–2647 (1986).
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1985 (1)

1984 (1)

1983 (1)

H. L. Fang, T. L. Gustafson, R. L. Swofford, “Two-photon absorption photothermal spectroscopy using a synchronously pumped picosecond dye laser. Thermal lensing spectra of naphthalene and diphenylbutadiene,” J. Chem. Phys. 78, 1663–1669 (1983).
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1982 (2)

B. Dick, G. Hohlneicher, “Importance of initial and final states as intermediate states in two-photon spectroscopy of polar molecules,” J. Chem. Phys. 76, 5755–5760 (1982).
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S. Li, C. Y. She, “Two-photon absorption cross-section measurements in common laser dyes at 1.06 μm,” Opt. Acta 29, 281–287 (1982).
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1981 (2)

I. M. Catalano, A. Cingolani, “Absolute two-photon fluorescence with low-power cw lasers,” Appl. Phys. Lett. 38, 745–747 (1981).
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O. S. Mortensen, E. N. Svendsen, “Initial and final molecular states as “virtual states” in two-photon processes,” J. Chem. Phys. 74, 3185–3189 (1981).
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1980 (4)

D. M. Friedrich, W. M. McClain, “Two-photon molecular electronic spectroscopy,” Annu. Rev. Phys. Chem. 31, 559–577 (1980).
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D. S. Kliger, “Thermal lensing: a new spectroscopic tool,” Acc. Chem. Res. 13, 129–134 (1980).
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P. M. Johnson, “Molecular multiphoton ionization spectroscopy,” Acc. Chem. Res. 13, 20–26 (1980).
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M. C. Johnson, F. E. Lytle, “Studies of the single-mode and multimode cw dye lasers as sources for obtaining power-squared corrected two-photon spectra,” J. Appl. Phys. 51, 2445–2449 (1980).
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1979 (5)

A. C. Tam, C. K. N. Patel, “Two-photon absorption spectra and cross-section measurements in liquids,” Nature 280, 304–306 (1979).
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R. J. M. Anderson, G. R. Holtom, W. M. McClain, “Two-photon absorptivities of the all trans α,ω-diphenylpolyenes from stilbene to diphenyloctatetraene via three wave mixing,” J. Chem. Phys. 70, 4310–4315 (1979).
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H. P. Trommsdorff, R. M. Hochstrasser, G. R. Meredith, “Raman and two-photon resonances in the three wave mixing in organic crystals: benzene, naphthalene and biphenyl,” J. Lumin. 18/19, 687–692 (1979).
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M. Bass, E. W. Van Stryland, A. F. Stewart, “Laser calorimetric measurement of two-photon absorption,” Appl. Phys. Lett. 34, 142–144 (1979).
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R. R. Birge, B. M. Pierce, “A theoretical analysis of the two-photon properties of linear polyenes and the visual chromophores,” J. Chem. Phys. 70, 165–178 (1979).
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1978 (1)

J. H. Brannon, D. Madge, “Absolute quantum yield determination by thermal blooming. Fluorescein,” J. Phys. Chem. 82, 705–709 (1978).
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1977 (5)

M. Mardelli, J. Olmsted, “Calorimetric determination of the 9,10-diphenyl-anthracene fluorescence quantum yield,” J. Photochem. 7, 277–285 (1977).
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R. J. M. Anderson, G. R. Holtom, W. M. McClain, “Absolute two-photon absorptivity of trans-stilbene near the two-photon absorption maximum via three wave mixing,” J. Chem. Phys. 66, 3832–3833 (1977).
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A. M. Bonch-Bruevich, T. K. Razumova, I. O. Starobogatov, “Single- and two-photon spectroscopy of liquid media using the pulsed acousto-optical effect,” Opt. Spectrosc. 42, 45–48 (1977).

A. J. Twarowski, D. S. Kliger, “Multiphoton absorption spectra using thermal blooming. I. Theory,” Chem. Phys. 20, 253–258 (1977).
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L. Singer, Z. Baram, A. Ron, S. Kimel, “The two-photon phosphorescence excitation spectrum of triphenylene,” Chem. Phys. Lett. 47, 372–376 (1977).
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1975 (2)

P. Esherick, P. Zinsli, M. A. El-Sayed, “The low energy two-photon spectrum of pyrazine using the phosphorescence photoexcitation method,” Chem. Phys. 10, 415–432 (1975).
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R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975).
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1974 (6)

V. I. Bredikhin, M. D. Galanin, V. N. Genkin, “Two-photon absorption and spectroscopy,” Sov. Phys. Usp. 16, 299–321 (1974).
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J. Krasiński, S. Chudzyński, W. Majewski, M. Głódź, “Experimental dependence of two-photon absorption efficiency on statistical properties of laser light,” Opt. Commun. 12, 304–306 (1974).
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J. Kleinschmidt, S. Rentsch, W. Tottleben, B. Wilhelmi, “Measurement of strong nonlinear absorption in stilbene-chloroform solutions, explained by the superposition of the two-photon absorption and one-photon absorption from the excited state,” Chem. Phys. Lett. 24, 133–135 (1974).
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D. J. Bradley, G. H. C. New, “Ultrashort pulse measurements,” Proc. IEEE 62, 313–345 (1974).
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W. M. McClain, “Two-photon molecular spectroscopy,” Acc. Chem. Res. 7, 129–135 (1974).
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1973 (1)

R. M. Hochstrasser, H.-N. Sung, J. E. Wessel, “High resolution two-photon excitation spectra,” J. Chem. Phys. 58, 4694–4695 (1973).
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1972 (4)

J. P. Hermann, J. Ducuing, “Dispersion of the two-photon cross section in rhodamine dyes,” Opt. Commun. 6, 101–105 (1972).
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J. P. Hermann, J. Ducuing, “Absolute measurement of two-photon cross sections,” Phys. Rev. A 5, 2557–2568 (1972).
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D. J. Bradley, M. H. R. Hutchinson, H. Koetser, T. Morrow, G. H. C. New, M. S. Petty, “Interactions of picosecond laser pulses with organic molecules. I. Two-photon fluorescence quenching and singlet states excitation in Rhodamine dyes,” Proc. R. Soc. London Ser. A 328, 97–121 (1972).
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D. J. Bradley, M. H. R. Hutchinson, H. Koetser, “Interactions of picosecond laser pulses with organic molecules. II. Two-photon absorption cross-sections,” Proc. R. Soc. London Ser. A 329, 105–119 (1972).
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1971 (3)

H. P. Weber, “Two-photon-absorption laws for coherent and incoherent radiation,” IEEE J. Quantum Electron. QE-7, 189–195 (1971).
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W. M. McClain, “Excited state symmetry assignment through polarized two-photon absorption studies of fluids,” J. Chem. Phys. 55, 2789–2796 (1971).
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J. N. Demas, G. A. Crosby, “The measurement of photoluminescence quantum yields. A review,” J. Phys. Chem. 75, 991–1024 (1971).
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1970 (2)

P. R. Monson, W. M. McClain, “Polarization dependence of the two-photon absorption of tumbling molecules with application to liquid 1-chloronaphthalene and benzene,” J. Chem. Phys. 53, 29–37 (1970).
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H. E. Rowe, T. Li, “Theory of two-photon measurement of laser output,” IEEE J. Quantum Electron. QE-6, 49–67 (1970).
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1969 (5)

M. D. Galanin, B. P. Kirsanov, Z. A. Chizhikova, “Luminescence quenching of complex molecules in a strong laser field,” JETP Lett. 9, 304–306 (1969).

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E. I. Blount, J. R. Klauder, “Recovery of laser intensity from correlation data,” J. Appl. Phys. 40, 2874–2875 (1969).
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P. G. Seybold, M. Gouterman, J. Callis, “Calorimetric, photometric, and lifetime determinations of fluorescence yields of fluorescein dyes,” Photochem. Photobiol. 9, 229–242 (1969).
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1968 (2)

H. P. Weber, R. Dändliker, “Method for measurement the shape asymmetry of picosecond light pulses,” Phys. Lett. A 28, 77–78 (1968).
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H. P. Weber, R. Dändliker, “Intensity interferometry by two-photon excitation of fluorescence,” IEEE J. Quantum Electron. QE-4, 1009–1013 (1968).
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1967 (4)

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, “Two-photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216–218 (1967).
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R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Science (3)

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

D. A. Parthenopoulos, P. M. Rentzepis, “Three-dimensional optical storage memory,” Science 245, 843–845 (1989).
[CrossRef] [PubMed]

M. Albota, D. Beljonne, J.-L. Brédas, J. E. Ehrlich, J.-Y. Fu, A. A. Heikal, S. T. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X.-L. Wu, C. Xu, “Design of organic molecules with large two-photon absorption cross sections,” Science 281, 1653–1656 (1998).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. I. Bredikhin, M. D. Galanin, V. N. Genkin, “Two-photon absorption and spectroscopy,” Sov. Phys. Usp. 16, 299–321 (1974).
[CrossRef]

Other (60)

In Subsection 4.3 we will discuss how ESA can also limit the applicability of Eq. (29) in NLT measurements, especially for ns pulse durations. At the moment, we are considering instead the case of materials for which 2PA is the only absorption process that can take place at the excitation wavelength λ.

In this context, the first argument for the function ϕ is the coordinate along the propagation direction, z, the origin being the front face of the material; the second argument refers to the radial distance, r, from the axis z. Thus, in this notation ϕ(0,0) corresponds to the peak on-axis flux at the entrance of the material.

The initial pulse energy is now E(0)=τEph∫0∞ϕ(0,0)e−2r2∕w022πrdr=τEphϕ(0,0)πw02∕2.

When the results for the f/60 case in Fig. 7 are compared with those in Fig. 6 for the same pulse energy, it can be seen that the transmittance is larger in the former situation, even when ground state depletion is neglected. This is due to the effect of focusing: even if L<z0 in the f/60 case, the beam size increases slightly away from the focal plane within the sample; instead, the beam was assumed to be perfectly collimated in Fig. 6.

This change in beam size is indeed exploited in closed-aperture Z-scan experiments [30] to measure the nonlinear refractive index of a material (which is related to the real part of the susceptibility χ(3)).

To differentiate between the examples discussed above, where z represented the position within the sample along the propagation direction, and the current example, where z is the position of the front face of the sample with respect to the focal plane, the z dependence of ϕ is now expressed with a subscript.

This is true not only for a Gaussian beam, but also for a beam with a generic beam profile, as long as the z dependence of the flux is known.

For this type of beam the relationship between pulse energy and peak flux isE=Eph∫0∞ϕ0(r=0)e−2r2∕w022πrdr∫−∞∞e−t2∕τ̃2dt=Ephϕ0(r=0)ππw02τ̃2.

The value of the transmittance at a generic point within the sample can be obtained by substituting z′, the distance of the point from the front face, for L.

It should be remembered that we are assuming that 1PA absorption is negligible.

For example, if the flux is reduced by 5% by 2PA [ϕ(z)∕ϕ(0)=0.95] and each time the photon flux is measured with an uncertainty of 2% (the two measurements before and after the beam are independent), from error propagation, the relative uncertainty in δ would be over 50%.

It should be kept in mind that this common definition of β implicitly assumes that approximation (ii) is valid, as this parameter is proportional to N0. β, usually called the “2PA coefficient,” is a macroscopic equivalent of the molecular parameter δ. In cases for which (ii) is not valid, β is not a constant of the material, but depends on ϕ through Ng, the concentration of molecules in the ground state (Ng≤N0).

The 1PA equivalent of parameter Δph(2) is σN0L, the fraction of light absorbed in the limit of thin sample or weak absorption.

In the case of the 2PIF method, an additional limitation on the concentration is imposed by reabsorption of the emitted photons within the material (inner filter effect). This effect is observed to different extents depending on how the fluorescence signal is collected (for example, under backscattering conditions or at 90° with respect to the incident beam).

We will introduce beams with a Gaussian spatial profile in Subsection 3.2. To explain the results in Fig. 4, it is sufficient to mention that the beam size depends on z as follows: w0(1+(z∕z0)2)1∕2. The total number of excited molecules is obtained by integrating Nm(2)(τ) over the excitation volume. The result for a generic path length L and for a sample with the focal plane at L∕2 (and with photon flux ϕ0 at z=0) is ∫VNm(2)(τ)dV=12δτN0∫−L∕2L∕2dz∫0∞2πrdrϕ2=12δτN0πw02z0ϕ022arctanL2z0=δN0Nph22τ(2nλarctanL2z0).   The asymptotic value for L≫z0 is in this case δN0Nph2πn∕(2τλ).

E. W. Van Stryland, M. Sheik-Bahae, “Z-scan,” in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, M. G. Kuzyk and C. W. Dirk, eds. (Marcel Dekker, 1998), pp. 655–692.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 1996).

P. N. Prasad, D. J. Williams, Introduction to Nonlinear Optical Effects in Molecules and Polymers (Wiley, 1991).

S. Kershaw, “Two-photon absorption,” in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, M. G. Kuzyk and C. W. Dirk, eds. (Marcel Dekker, 1998), pp. 515–654.

R. R. Birge, “One-photon and two-photon excitation spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, ed. (Academic, 1983), pp. 109–174.

H. J. Eichler, P. Günther, D. W. Pohl, Laser-Induced Gratings (Springer-Verlag, 1986).
[CrossRef]

S. M. Kuebler, “Studies of the third-order nonlinear optical properties of some materials by degenerate four-wave mixing,” D. Phil. thesis (Univ. of Oxford, 1997).

R. A. Norwood, “Four wave mixing,” in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, M. G. Kuzyk and C. W. Dirk, eds. (Marcel Dekker, 1998), pp. 693–765.

In this context, we define the time average of a function Y as follows:⟨Y(t)⟩=f∫TT+1∕fY(t)dt,   where T is a generic time and Y(t) is a periodic function with period 1∕f.

This assumes that τfl≪1∕f, so that effectively all molecules that have been excited during the pulse duration τ have decayed back to the ground state. For other indirect methods 1∕f needs to be large with respect to the time constant of the process to be monitored (thermal time constant, phosphorescence lifetime,…). If this requirement is not satisfied, the sample conditions are different every time a new pulse arrives, and ground state depletion may become relevant after a large number of pulses. In the case of single pulse measurements, the integration time for the signal needs to be long with respect to the time constant of the process monitored, if results for different materials are to be compared under the same excitation conditions.

For ideal monochromatic light with constant intensity and in the absence of noise, g(2)=1. For a square pulse of duration τ, g(2)=(τf)−1. For a Gaussian pulse with 1∕e width τ, g(2)=(τf(2π)1∕2)−1.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2000).

For example, in some cases [101, 102, 103], the cross section is defined to “count” directly the number of molecules excited, instead of the photons absorbed, leading to the relationship nm(2)=δ̃Ngϕ2.(I)    In this case, the propagation equation (1) becomesdϕdz=−2δ̃Ngϕ2.(II)    Obviously, the right-hand sides of the two versions of the propagation equation [Eqs. (1, II)] and of the equation describing the excitation rate due to 2PA [Eqs. (6, I)] differ only by a constant factor and become equivalent if δ=2δ̃.(III)

More precisely, any absorption process that would take place at this stage, via 1PA, 2PA or other nonlinear processes, would reflect properties of the excited state r, not of the ground state g, and thus describe a very different sample, from a spectroscopic point of view. All processes, though, contribute to the attenuation of the propagating beam and Eq. (1) would have to be appropriately modified. This situation will also be discussed in Subsection 3.2. ESA following 2PA will also be considered in Subsection 4.3.

S. M. Kirkpatrick, C. Clark, R. L. Sutherland, “Single state absorption spectra of novel nonlinear optical materials,” in Thin Films for Optical Waveguide Devices and Materials for Optical Limiting, K. Nashimoto, R. Pachter, B. W. Wessels, A. K.-Y. Jen, K. Lewis, R. Sutherland, and J. W. Perry, eds., Vol. 597 of Materials Research Society Symposium Proceedings (Materials Research Society, 2000), pp. 333–338.

It is generally assumed that the fluorescence quantum yield is the same after 1PA and 2PA excitation, as for most molecules the system relaxes, after absorption, to the lowest excited state, r, by internal conversion, irrespective of the photon energy and the manner of excitation, as discussed above. However, there are exceptions to this general rule, as some molecules are known that exhibit a wavelength dependent quantum yield or emission spectrum, and fluorescence emission from upper excited states.

Δm(2) is equivalent to the “saturation parameter” α discussed by Xu and Webb [17], pp. 518–520.

It should be kept in mind that Eq. (14) was obtained under a series of assumptions (for example regarding the change in photon flux through the sample). Thus, strictly speaking, Eq. (14) is also an approximate description of the fraction of molecules excited.

D. M. Hercules, “Theory of luminescence processes,” in Fluorescence and Phosphorescence Analysis. Principles and Applications, D. M. Hercules, ed. (Interscience, 1966), pp. 1–40.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 2nd ed. (Kluver Academic/Plenum, 1999).
[CrossRef]

The 2PA cross section δ is proportional to the square modulus of the two-photon tensor mentioned earlier in the section (see, for example, [29, 40, 41]).

The symbols σ2 and σ(2) are also commonly used in the literature to represent the 2PA cross section.

Typical units for δ are m4s/(molecule photon) in the SI system and cm4s/(molecule photon) in the cgs system, or m4s and cm4s, respectively, if omitting dimensionless parameters. The derived unit GM (Göppert-Mayer) is also frequently used and is defined as follows: 1 GM≡1×10−50 cm4s∕(molecule photon)=1×10−58 m4s∕(molecule photon).

Typical SI units for δ* are cm4/(molecule W) or cm4/(molecule GW). Sometimes “molecule” is omitted (because it is dimensionless), and the units are simply reported as cm4∕W or cm4∕GW.

W. Denk, D. W. Piston, W. W. Webb, “Two-photon molecular excitation in laser-scanning microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum Press, 1995), pp. 445–458.
[CrossRef]

If ΔE represents the uncertainty in the energy of the system after the interaction of the molecule with the first photon, it follows from the uncertainty principle that ΔEτv≥ℏ [17, 40]. For example, if E1=E2=1.55 eV (photon wavelength=800 nm), ΔE∼E1, and then τv≥4.3×10−16 s.

W. M. McClain, R. A. Harris, “Two-photon molecular spectroscopy in liquids and gases,” in Excited States, E. C. Lim, ed. (Academic, 1977), Vol. 3, pp. 1–56.
[CrossRef]

B. Strehmel, V. Strehmel, “Two-photon physical, organic and polymer chemistry: theory, techniques, chromophore design, and applications,” in Advances in Photochemistry, D. C. Neckers, W. S. Jenks, and T. Wolff, eds. (Wiley, 2007), Vol. 29, pp. 111–354.

C. Xu, W. W. Webb, “Multiphoton excitation of molecular fluorophores and nonlinear laser microscopy,” in Nonlinear and Two-Photon-Induced Fluorescence, Vol. 5 of Topics in Fluorescence Spectroscopy, J. Lakowicz, ed. (Plenum, 1997), pp. 471–540.

S. M. Kuebler, M. Rumi, “Nonlinear optics—applications: three-dimensional microfabrication,” in Encyclopedia of Modern Optics, B. D. Guenther, D. G. Steel, and L. Bayvel, eds. (Elsevier, 2004), Vol. 3, pp. 189–206.

E. W. Van Stryland, M. A. Woodall, “Photoacoustic measurement of nonlinear absorption in solids,” in Laser Induced Damage in Optical Materials, 1980, National Bureau of Standards Special Publication620 (National Bureau of Standards, 1981), pp. 50–57.
[CrossRef]

When a photon counting approach is used, signal discrimination procedures can be used to reject noise.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, 1971).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).

See, for example, P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics (Cambridge Univ. Press, 1990).

B. Dick, R. M. Hochstrasser, H. P. Trommsdorff, “Resonant molecular optics,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, 1987), Vol. 2, pp. 159–212.
[CrossRef]

The case of different excitation volumes in the two samples could be handled by including appropriate correction terms in the collection efficiency factor, G. Beside the obvious case of samples with different thicknesses in a collimated beam, the excitation volume is also different if the samples do not have the same refractive index and L≫z0.

Due to the relatively low concentration used in 2PA measurements, it is often assumed that the refractive index of the solution is the same as that of the pure solvent.

P. A. Fleitz, R. L. Sutherland, T. J. Bunning, “Z-scan measurements on molten diphenylbutadiene in the isotropic liquid state,” in Materials for Optical Limiting, R. Crane, K. Lewis, E. Van Stryland, and M. Khoshnevisan, eds., Vol. 374 of Materials Research Society Symposium Proceedings (Materials Research Society, 1995), pp. 211–216.

Unfortunately, there are some typographical errors in the solution of the system of equation reported by Kleinschmidt et al. [162].

An expression similar to Eq. (46) could be written in the case of a molecule undergoing intersystem crossing from r to the lowest level in the triplet manifold and then being excited to a higher-lying triplet state. In this case, σex would represent a triplet–triplet ESA cross section.

The quantity ϕτ corresponds to the photon fluence, or the number of photons per unit area of the beam, Nph∕a [Nph from Eq. (17) for the case of a pulse with constant profile in space and time].

Even if ESA occurs from the triplet state, as described in note [167], the fluorescence quantum yield and intersystem crossing rates would be unchanged if Nex is small, and thus ⟨S⟩ would still not be affected by ESA and would not depend on pulse duration.

This numerical example referred to excitation with ns pulses. However, similar conditions for ESA can be reached by using fs pulses from an amplified laser, for example, with ϕ values only slightly larger than those considered in case 3.1.b.

If necessary, the attenuation of the beam can be accounted for in the equations to be used to process the data. In some experimental configurations for fluorescence signal collection, reabsorption may also have to be corrected for.

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Figures (21)

Fig. 1
Fig. 1

Schematic energy level diagram showing the excitation of a molecule from the ground state, g, to an excited state, f, located at energy E f above the g state by the absorption of two photons (vertical solid arrows). The photons can have the same energy, E 1 (degenerate case, E f = 2 E 1 ), or different energies, E 1 and E 2 (nondegenerate case, E f = E 1 + E 2 ). After excitation, the system relaxes quickly to state r, the lowest vibronic level of the lowest-energy excited state, by internal conversion or vibrational relaxation (dashed arrow). The system finally returns to the ground state by radiative or nonradiative pathways (bold dashed arrow).

Fig. 2
Fig. 2

Schematic representation of the attenuation of a beam incident on a slab of material that exhibits two-photon absorption. Because of the nonlinear nature of the phenomenon, the transmittance of the material depends on the intensity of the incident beam (here represented by the width of the arrows): a weak beam (top) is absorbed to a smaller degree than an intense beam (bottom).

Fig. 3
Fig. 3

Magnitude of thermal lensing signal induced by 2PA as a function of the laser intensity of the excitation beam. The dotted line is the best fit of the data points to signal = A I 2 , where A is a constant and I is the laser intensity. Sample, toluene; excitation wavelength λ = 497.5 nm . Reproduced with permission from Rice and Anderson [78]. Copyright 1986 American Chemical Society.

Fig. 4
Fig. 4

Number of molecules excited over the whole excitation volume, N m ( 2 ) ( τ ) × V , as a function of path length in the sample. For L < 2 z 0 , V is assumed to be π w 0 2 L , and the system response is given by Eq. (24) (straight line through the origin). For L > 2 z 0 , V = 2 π w 0 2 z 0 , and the system response is governed by Eq. (25) (horizontal line). The dashed portion of these lines represents the regions where Eqs. (24, 25) are to be considered approximate estimates. For comparison, the total number of molecules excited by a beam with a Gaussian spatial profile (obtained by integrating the appropriate value of N m ( 2 ) ( τ ) over the excitation volume, V N m ( 2 ) ( τ ) d V ) is shown for comparison (dotted curve). N ph is the same in all cases.

Fig. 5
Fig. 5

Fraction of photons per pulse absorbed Δ ph ( 2 ) (left-hand axis) and of molecules excited Δ m ( 2 ) (right-hand axis) by 2PA for 5 ns , 10 Hz pulses at 700 nm (see legend for list of other parameters) as a function of pulse energy; the solid curve is from the approximate Eq. (15), the dashed curve from Eq. (14).

Fig. 6
Fig. 6

Dependence of the transmittance of a 2PA material when excited by a beam with rectangular [from Eq. (28)] or Gaussian spatial profiles [from Eq. (33)]. (a) Comparison of beams with the same peak photon flux (see also Sutherland [26], p. 504). (b) Comparison of beams with the same pulse energy. Other parameters are N 0 = 6.0 × 10 18 cm 3 , δ = 1000 GM , L = 1.0 cm , τ = 5 ns (rectangular temporal profile in all cases), w 0 = 50 μ m (for the Gaussian case this is the 1 e 2 radius), λ = 700 nm (the scale on the top horizontal axis in (a) is in units of δ N 0 L ϕ ( 0 , 0 ) , and is independent of any specific choice of parameters; this corresponds to the parameter q 0 that will be introduced in Subsection 3.2d).

Fig. 7
Fig. 7

Transmittance of a 2PA material in two different optical configurations. Solid curve, f/10 (focused beam, L = 0.5 cm z 0 ); dashed and dotted curves, f/60 (approximately collimated beam, L = 1 cm < z 0 ). The effect of ground state depletion is included in an approximate way [using Eq. (7)] in the first two cases, but not in the last (dotted curve). In all cases λ = 700 nm , δ = 1000 GM , N 0 = 6.0 × 10 18 cm 3 , τ = 5 ns ; the focal plane is located at z = L 2 .

Fig. 8
Fig. 8

Z-scan transmittance trace for a material exhibiting 2PA (solid curve) as a function of normalized sample position, z z 0 , for q 0 = 0.3 . The corresponding values of q z [from Eq. (37)] as a function of z are also displayed. The transmittance was calculated from Eq. (41), which is equivalent to Eq. (39) for all the values of q z in this example.

Fig. 9
Fig. 9

Change in transmittance, 1 T , observed in a Z-scan experiment at z = 0 as a function of q 0 (solid curve). The effect of truncation of the series in Eq. (41), that is, the inclusion of only terms with m m max , is shown for a few values of m max .

Fig. 10
Fig. 10

(a) Change in transmittance, Δ T = 1 T , at z = 0 in a Z-scan trace for beams with non-Gaussian spatial profiles; the parameter M 2 is a measure of the deviation of the actual profile from the ideal Gaussian shape ( M 2 = 1 ) . Δ T Gaussian is the corresponding change in transmittance for a Gaussian beam with the same waist and pulse energy. Each triangle corresponds to a different beam profile. The slope of the best-fit straight line is 1.77. (b) Beam cross section at z = 0 for one of the beam profiles included in plot (a) and with M 2 = 1.08 . (c) Z-scan traces for the beam with the profile in (b) (solid curve) and for a Gaussian beam with the same beam waist and pulse energy (dashed curve). Reproduced with permission from Chapple and Wilson [143]. Copyright 1996 World Scientific Publishing Company.

Fig. 11
Fig. 11

Dependence of the time-averaged 2PIF signal on the excitation pulse width ( λ = 770 nm , 10 4 M solution of chromophore Indo-1 in water). The line in the log-log plot is the best fit of the experimental data (dots) and has slope 1 . This dependence of S ( t ) [Eq. (45)] on τ 1 through g ( 2 ) indicates that δ is constant over this pulse duration range. Reproduced with permission from Xu and Webb [20]. Copyright 1996 Optical Society of America.

Fig. 12
Fig. 12

2PA spectra of two compounds (molecular structures included in figures) obtained by relative 2PIF measurements using excitation sources with different pulse durations: (a) ns and ps; (b) ns and fs. Adapted from (a) Rumi et al. [124] and (b) Chung et al. [158].

Fig. 13
Fig. 13

Modification of the energy level diagram of Fig. 1 (degenerate case) to include ESA pathways. After the simultaneous absorption of two photons of energy E 1 (solid arrows) and relaxation by internal conversion from state f to r (short dashed arrow), the material can absorb additional photons at the same energy, building up population in excited state s. For very short pulses, direct absorption from the state f, before relaxation, is also possible, leading to population of state t. From any of the excited states the system then relaxes to r by internal conversion (other dashed arrows) and then to the ground state by radiative or nonradiative de-excitation processes (bold dashed arrow).

Fig. 14
Fig. 14

Dependence of the inverse of the transmittance, 1 T , on laser photon flux incident on a trans-stilbene sample. In the original work, q ( 0 ) was the symbol used for the incident flux and q ( L ) that for the transmitted flux after a sample of length L. In our formalism, these correspond to ϕ ( 0 ) and ϕ ( L ) , respectively. Thus the ordinate is the inverse of the sample transmittance (using the nomenclature of the original paper: q ( 0 ) q ( L ) = 1 T ). The scale on the abscissa spans from 0 to 5 × 10 26 photons  cm 2 s 1 . Sample, trans-stilbene in chloroform; λ = 693 nm , τ = 20 ns , L = 20 cm . The open circles are experimental data points, the solid curve a theoretical prediction (see text), and the dashed lines the limiting behaviors for low and high flux. Reproduced from Kleinschmidt et al. [162]. Copyright 1974, with permission from Elsevier.

Fig. 15
Fig. 15

Dependence of the 2PA coefficient or cross section on incident flux from Z-scan measurements. (a) Results for 2,5-bis(benzothiazolyl)-3,4-didecyloxythiophene at λ = 532 nm , using 32 ps pulses. In our formalism of Subsection 3.2d, the abscissa corresponds to ϕ 0 ( r = 0 ; t = 0 ) × E ph , the ordinate to δ eff N 0 E ph . Reproduced from Said et al. [164], Copyright 1994, with permission from Elsevier. (b) Results for compounds AF-50 and AF-250 at λ = 796 nm , using 150 fs pulses. In our formalism, the abscissa corresponds to ϕ 0 ( r = 0 ; t = 0 ) × E ph , the ordinate to δ eff E ph . Reproduced from Swiatkiewicz et al. [165], Copyright 1998, with permission from Elsevier. The molecular structures are displayed to the right of the corresponding plots. The solid lines are the best linear fits to the experimental data [Eq. (49)].

Fig. 16
Fig. 16

Parameter q 0 [defined in Eq. (38)] as a function of pulse energy, E, as obtained from fitting of Z-scan traces for 4-N,N-dimethylamino- 4 -nitrostilbene in chloroform ( λ = 639 nm , τ = 130 fs ). The direct proportionality of q 0 with E indicates that δ is constant and that ESA does not contribute to the beam attenuation. Reproduced with permission from Kamada et al. [33]. Copyright 2003 Optical Society of America.

Fig. 17
Fig. 17

Comparison of the 2PA spectrum for the compound shown at the right in toluene as obtained by Arnbjerg et al. [24] (filled squares) using an absolute method (see text) and as obtained by Pond et al. [160] (open circles) using a relative 2PIF measurement (excitation with fs pulses; reference material, fluorescein, based on [20]). The ordinate axis is the intensity of the signal for the measurement as conducted in [24], and the other spectrum is normalized to the same intensity at the maximum. The inset is the background signal assigned to toluene for the measurement in [24]. Graph reproduced with permission from Arnbjerg et al. [24]. Copyright 2006 American Chemical Society.

Fig. 18
Fig. 18

2PA spectrum obtained by an absolute fluorescence-based method for p-bis(o-methylstyryl)benzene in cyclohexane. The shape of the spectrum is based on the tabulated values reported by Kennedy and Lytle [191]; the 2PA cross section is scaled to match the value at 585 nm , δ = 69 GM , reported by Fisher et al. [192].

Fig. 19
Fig. 19

2PA spectra of selected compounds obtained by absolute fluorescence-based methods: (a) Coumarin 307, (b) fluorescein, (c) Rhodamine B, and (d) p-bis(o-methylstyryl)benzene (the solvent is indicated in the legend) obtained by Xu and Webb [20]. In the case of Coumarin 307, the ordinate displays the quantity η δ . Different symbols refer to different mirror sets used to tune the laser. Reproduced with permission from Xu and Webb [20]. Copyright 1996 Optical Society of America.

Fig. 20
Fig. 20

2PA spectra (blue circles) of seven different compounds obtained by an absolute fluorescence-based method, as reported by Makarov et al. [25]. The compound name and solvent are indicated in the title of each plot. The pink solid curves represent the linear absorption spectrum of each compound. For some compounds results from other sources in the literature are also included (symbols of various shapes and colors; the numbers in square brackets in the legends refer to references cited by Makarov et al.). Reproduced with permission from Makarov et al. [25]. Copyright 2008 Optical Society of America. Continues in Fig. 21.

Fig. 21
Fig. 21

Continued from Fig. 20: 2PA spectra (blue circles) of eight different compounds obtained by an absolute fluorescence-based method, as reported by Makarov et al. [25]; colors and symbols are the same as in Fig. 20. Reproduced with permission from Makarov et al. [25]. Copyright 2008 Optical Society of America.

Tables (1)

Tables Icon

Table 1 2PA Cross Section of AF-50 and Bis(dibutylamino)stilbene Obtained by Various Experimental Methods for Excitation Wavelength λ and Pulse Duration τ

Equations (72)

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( e 1 M g i ) ( M i f e 2 ) E i E 1 , ( e 2 M g i ) ( M i f e 1 ) E i E 2 ,
d ϕ d z = δ N g ϕ 2 ,
d ϕ d z = σ N g ϕ δ N g ϕ 2
ϕ = I E ph .
d I d z = σ N g I δ E ph N g I 2 .
n ph ( 2 ) = d ϕ d z = δ N g ϕ 2 .
n m ( 2 ) = 1 2 n ph ( 2 ) = 1 2 δ N g ϕ 2 .
N g ( t ) = N 0 0 t n m ( 2 ) ( t ) d t ,
n m ( 2 ) = d N g d t .
d N g d t = 1 2 δ N g ϕ 2 ,
N g ( τ ) = N 0 e ( 1 2 ) δ ϕ 2 τ .
N m ( 2 ) ( τ ) = N 0 N g ( τ ) = N 0 ( 1 e ( 1 2 ) δ ϕ 2 τ ) .
N m ( 2 ) ( τ ) 1 2 δ ϕ 2 τ N 0 .
S η G N m ( 2 ) ( τ ) 1 2 η G δ ϕ 2 τ N 0 ,
Δ m ( 2 ) = N m ( 2 ) ( τ ) N 0 = ( 1 e ( 1 2 ) δ ϕ 2 τ ) ,
Δ m ( 2 ) 1 2 δ ϕ 2 τ .
N ph = E E ph = E λ h c ,
ϕ = N ph a τ = 1 π w 0 2 τ E E ph = λ E π w 0 2 τ h c .
N ph ( 2 ) ( τ ) = 2 N m ( 2 ) ( τ ) V δ ϕ 2 τ N 0 π w 0 2 L .
Δ ph ( 2 ) = N ph ( 2 ) ( τ ) N ph δ ϕ N 0 L .
Δ m ( 2 ) δ N ph 2 2 τ ( π w 0 2 ) 2 ,
Δ ph ( 2 ) δ N ph N 0 L π w 0 2 τ .
approximation (i): Δ ph ( 2 ) 1 or δ ϕ N 0 L 1 ,
approximation (ii): Δ m ( 2 ) 1 or 1 2 δ ϕ 2 τ 1.
Approximation (i): Δ ph ( 2 ) 1 or δ ϕ N 0 L 1 ,
Approximation (ii): Δ m ( 2 ) 1 or 1 2 δ ϕ 2 τ 1.
N ph = 5.0 × 10 9 photons pulse,
ϕ = 6.4 × 10 26 photons  cm 2 s 1 ,
Δ m ( 2 ) = 2.1 × 10 7 ,
Δ ph ( 2 ) = 3.9 × 10 4 .
N ph = 2.0 × 10 11 photons pulse,
ϕ = 1.8 × 10 29 photons  cm 2 s 1 ,
Δ m ( 2 ) = 2.5 × 10 2 ,
Δ ph ( 2 ) = 5.4 × 10 4 .
N m ( 2 ) ( τ ) V = 1 2 δ ϕ 2 τ N 0 ( π w 0 2 L ) = δ N 0 N ph 2 2 τ L π w 0 2 = δ N 0 N ph 2 2 τ n L λ z 0 ( for L < 2 z 0 ) .
N m ( 2 ) ( τ ) V 1 2 δ ϕ 2 τ N 0 ( π w 0 2 ( 2 π w 0 2 n λ ) ) = δ N 0 N ph 2 2 τ 2 n λ ( for L > 2 z 0 ) .
d ϕ ϕ 2 = δ N 0 d z ,
d ϕ ϕ 2 = δ N 0 d z ,
1 ϕ + 1 ϕ ( 0 ) = δ N 0 z .
ϕ ( z ) = ϕ ( 0 ) 1 + δ N 0 z ϕ ( 0 ) .
T = ϕ ( L ) ϕ ( 0 ) = 1 1 + δ N 0 L ϕ ( 0 ) ,
Δ ph ( 2 ) = 1 T = δ N 0 L ϕ ( 0 ) 1 + δ N 0 L ϕ ( 0 ) .
1 T = 1 + β I L ,
ϕ ( 0 , r ) = ϕ ( 0 , 0 ) e 2 r 2 w 0 2 ,
E ( L ) = τ E ph 0 ϕ ( L , r ) 2 π r d r = τ E ph 0 ϕ ( 0 , r ) 1 + δ N 0 L ϕ ( 0 , r ) 2 π r d r = τ E ph 0 ϕ ( 0 , 0 ) e 2 r 2 w 0 2 1 + δ N 0 L ϕ ( 0 , 0 ) e 2 r 2 w 0 2 2 π r d r = τ E ph π w 0 2 2 δ N 0 L ln ( 1 + δ N 0 L ϕ ( 0 , 0 ) ) .
T = E ( L ) E ( 0 ) = τ E ph π w 0 2 2 δ N 0 L ln ( 1 + δ N 0 L ϕ ( 0 , 0 ) ) τ E ph π w 0 2 2 ϕ ( 0 , 0 ) = ln ( 1 + δ N 0 L ϕ ( 0 , 0 ) ) δ N 0 L ϕ ( 0 , 0 ) .
ϕ z ( r ; t ) = ϕ 0 ( r = 0 ; t = 0 ) w 0 2 w z 2 e 2 r 2 w z 2 e t 2 τ ̃ 2 = ϕ z ( r = 0 ; t = 0 ) e 2 r 2 w z 2 e t 2 τ ̃ 2 ,
ϕ 0 ( r = 0 ; t = 0 ) = 2 π π w 0 2 τ ̃ E E ph .
E z ( L ) = E ph + d t 0 2 π r d r ϕ z ( r ; t ) 1 + δ N 0 L ϕ z ( r ; t ) = E ph + d t 0 2 π r d r [ ϕ z ( r = 0 ; t = 0 ) e t 2 τ ̃ 2 ] e 2 r 2 w z 2 1 + δ N 0 L [ ϕ z ( r = 0 ; t = 0 ) e t 2 τ ̃ 2 ] e 2 r 2 w z 2 = E ph π w z 2 2 δ N 0 L + d t ln ( 1 + δ N 0 L [ ϕ z ( r = 0 ; t = 0 ) e t 2 τ ̃ 2 ] ) = E ph π w z 2 2 δ N 0 L + d t ln ( 1 + q z e t 2 τ ̃ 2 ) .
q z δ N 0 L ϕ z ( r = 0 ; t = 0 ) = δ N 0 L ϕ 0 ( r = 0 ; t = 0 ) ( 1 + ( z z 0 ) 2 ) = q 0 ( 1 + ( z z 0 ) 2 ) ,
q 0 δ N 0 L ϕ 0 ( r = 0 ; t = 0 ) .
T z = E z ( L ) E = E ph π w z 2 2 δ N 0 L + ln ( 1 + q z e t 2 τ ̃ 2 ) d t E ph π π w z 2 τ ̃ 2 ϕ z ( r = 0 ; t = 0 ) = + ln ( 1 + q z e t 2 τ ̃ 2 ) d t π τ ̃ q z .
T z = + d t 1 1 m ( q z e t 2 τ ̃ 2 ) m ( 1 ) m + 1 π τ ̃ q z = 1 1 m ( q z ) m ( 1 ) m + 1 + e m t 2 τ ̃ 2 d t π τ ̃ q z = 1 ( q z ) m 1 ( 1 ) m + 1 m m
T z = 0 ( q z ) m ( m ) 3 2 ( valid for 0 q z 1 ) .
T T + 1 f S ( t ) d t = T T + 1 f 1 2 η G δ N 0 ϕ 2 ( t ) d t = 1 2 η G δ N 0 f 1 ϕ 2 ( t )
S ( t ) = 1 2 η G δ N 0 ϕ 2 ( t ) .
g ( 2 ) = ϕ 2 ( t ) ϕ ( t ) 2 ,
S ( t ) = 1 2 η G δ N 0 g ( 2 ) ϕ ( t ) 2 .
d ϕ d z = δ N g ϕ 2 σ ex N ex ϕ ,
( d ϕ ϕ ) 2 PA = δ N g ϕ d z ,
( d ϕ ϕ ) ESA = σ ex N ex d z .
( d ϕ ϕ ) 2 PA = 3.8 × 10 4 cm 1 d z ,
( d ϕ ϕ ) ESA = 6.0 × 10 3 cm 1 d z .
( d ϕ ϕ ) ESA ( d ϕ ϕ ) 2 PA σ ex ( 1 2 δ ϕ 2 τ N 0 ) δ N 0 ϕ = 1 2 σ ex ϕ τ .
δ eff ( ϕ 0 ) δ + u σ ex ϕ 0 ,
δ = δ η N 0 G η N 0 G S ( t ) S ( t ) ,
n m ( 2 ) = δ ̃ N g ϕ 2 .
d ϕ d z = 2 δ ̃ N g ϕ 2 .
δ = 2 δ ̃ .
V N m ( 2 ) ( τ ) d V = 1 2 δ τ N 0 L 2 L 2 d z 0 2 π r d r ϕ 2 = 1 2 δ τ N 0 π w 0 2 z 0 ϕ 0 2 2 arctan L 2 z 0 = δ N 0 N ph 2 2 τ ( 2 n λ arctan L 2 z 0 ) .
E = E ph 0 ϕ 0 ( r = 0 ) e 2 r 2 w 0 2 2 π r d r e t 2 τ ̃ 2 d t = E ph ϕ 0 ( r = 0 ) π π w 0 2 τ ̃ 2 .
Y ( t ) = f T T + 1 f Y ( t ) d t ,

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