## Abstract

Multi-photon absorption and excitation properties of CdSe quantum dots in hexane with different dot-sizes have been investigated. The two- and three-photon absorption (2PA and 3PA) coefficients were measured by using ~160-fs laser pulses at wavelengths of ~775-nm and ~1300-nm, respectively. The dependence of one-, two- and three-photon induced fluorescence spectra as well as their double-exponential decay on the dot-sizes was studied. Based on the fluorescence emission spectra and temporal decay constants for a given sample solution excited by one-, two-and three-photon absorption, it can be concluded that the transition pathways for fluorescence emission and decay under one-, two- and three-photon excitation are nearly identical. The optical power limiting capabilities based on 2PA and 3PA mechanisms are demonstrated separately. In addition, a saturation behavior of 3PA at ~1300 nm was observed.

©2007 Optical Society of America

## 1. Introduction

Recently, semiconductor quantum dots (QDs) have become a very interesting new type of nonlinear material for multi-photon related fundamental studies [1–11] and applications, such as frequency upconversion lasing and amplification [12,13], optical power limiting [14–15], frequency upconversion imaging [16–18] and labeling [19], and optical switching [20]. In comparison with organic multi-photon active materials [21, 22], inorganic semiconductor QDs may provide the advantages of high photo-chemical and photo-physical stabilities, and therefore can withstand a higher laser intensity (and energy) level for power limiting and stabilization purposes. So far, most reported multi-photon studies on QDs were focused on two-photon absorption and excitation processes. In this study, both two- and three-photon excitation properties of CdSe QDs solutions in hexane are presented. Based on our results, we demonstrate that QDs can be promising candidates for optical power limiting application.

## 2. Materials and linear optical property

The nonlinear optical materials utilized for this study are CdSe QDs with five different dot-sizes in their solution phase (in hexane). CdSe QDs were prepared by the hot colloidal synthesis method. Briefly, the reaction was carried out with 1.116 g tetradecylphosphonic acid, using a Cd:Se ratio of 2:4 (2 mmol CdO, 4 mmol Se-TOP), 3 g hexadecylamine and 1 g of dodecylamine at ~260 °C.

Figure 1 shows the TEM images of CdSe with various sizes. These QDs are referred as QD1 to QD5. The average sizes for these five samples were estimated to be 2.0, 2.7, 3.1, 3.4 and 3.9 nm, respectively. The CdSe QDs can be easily dissolved in hexane, with relatively high solubility (from 40 to 80 mg/mL).

Figure 2 shows the linear absorption spectral curves of QDs samples in hexane with a 1-cm path-length and at low concentration levels. For comparison, the linear absorption curve for hexane is also included in the same figure by the dotted-line curve. From Fig. 2, one can see that there is an obvious red-shift of the spectral peak in the long-wavelength tail when the QD-size increases. On the other hand, there is no intrinsic linear absorption for the QDs in the entire spectral range from 600 to 1500 nm. Although there is a small peak around 920 nm and two big peaks around 1200 nm and 1400 nm, they are solely due to the hexane solvent. Based on the spectral feature shown in Fig. 2, ~775 and ~1300 nm wavelength were chosen for effective two- and three-photon excitation of these CdSe QDs.

## 3. One-, two-, and three-photon induced fluorescence spectral property

The two-photon absorption induced fluorescence spectra for five QDs solution samples at low concentration levels were excited by ~775-nm and ~160-fs laser pulses from a Ti:sapphire oscillator/amplifier system (CPA-2010 from Clark-MXR, Inc.) operating at a repetition rate of 1 kHz. The results, recorded by a grating spectrometer (Holo Spec from Kaiser Optical System), are shown in Fig. 3 by the dash-dotted curves. A salient feature in Fig. 3 is that the emission peak wavelength is remarkably red-shifted with increasing the dot-size, i.e. from ~518 nm for QD1 sample shifted to ~570 nm for QD5 sample. By using the second harmonic generation (~388 nm) of the same ~775-nm and ~160-fs laser pulses, we could also measure one-photon induced fluorescence spectra for these sample solution at low concentration levels, the results are also shown in Fig. 3 by the solid-line curves. From Fig. 3 one can see that the one- and two-photon induced spectral curves for each given QDs sample are nearly the same.

Furthermore, shown in Fig. 4 are the measured three-photon absorption induced fluorescence spectra of QDs solution samples at high concentration levels, excited by ~1300-nm laser pulses that were obtained from an optical parametric generator (OPG) pumped by the ~775-nm and ~160-fs laser pulses. The propagation path-length of the measured fluorescence signals inside the solutions was kept at ~2 mm. Comparing Fig. 4 to Fig. 3 one can see a very obvious red-shift of the peak emission wavelength position for each given QDs sample. This apparent red-shift is due to reabsorption of the fluorescence signal propagating in a high concentration solution sample, which can be ignored in a low concentration sample.

To identify the reabsorption effects in high concentration samples, the linear transmission spectra of three QD4 solution samples with different concentrations and path-lengths were measured. The obtained transmission spectral curves are shown in Fig. 5, in which one can see that the edge of transparent spectral range is shifted towards the longer wavelength direction following the increase of either the concentration or the optical path-length. On the other hand, also shown in Fig. 5 are the one-photon induced emission spectral curve measured in a low concentration sample and the three-photon induced emission spectral curve measured in a high concentration sample. It can be seen in the latter case that most portions of the emission spectral components fall into the strong linear absorption range; therefore the measured emission spectral peak position is actually determined by the edge of transmission curve for a given path-length.

Finally, to verify the two- and three-photon excitation identities, we measured the fluorescence intensity as a function of intensities of the input ~775-nm and ~1300-nm laser pulses, respectively. The experimental data are shown in Fig. 6 in logarithmic scales, in which the data excited at ~775 nm can be well fitted by a straight line with a slope of 2.05, while those data excited at ~1300 nm can be well fitted by another straight line with a slope of 3.2. These results basically indicated the two- and three-photon excitation features at these two wavelengths.

## 4. One-, two-, and three-photon excited fluorescence decay property

The one-, two-, and three-photon induced fluorescence decay curves were measured using a high-speed streak camera system (C-5680-22 from Hamamatsu) with excitation wavelengths of ~388 nm, ~775 nm and ~1300 nm, respectively. Shown in Fig. 7 are the measured normalized decay curves for five (from QD1 to QD5) solution samples of high concentrations, excited by ~775-nm and ~160-fs laser pulses. All these normalized decay curves can be well fitted by the same double-exponential formula,

though for different samples the best-fitting constants are different. Similarly, Fig. 8 shows the measured fluorescence decays of two sample solutions (QD2 and QD3), excited by ~1300-nm and ~160-fs laser pulses.

Furthermore, the one-photon induced fluorescence decay behaviors of the five sample solutions were also measured by using the 388-nm laser pulses. In all these cases, the decay curves can be basically fitted by using the same Eq. (1). Finally, for comparison purposes, the best-fitting decay constants for these measured QDs solution samples under one-, two- and three-photon excitation conditions are summarized in Table 1. It is noted that the similar double exponential decay behaviors of one-photon excited fluorescence emission from other semiconductor QDs (GaAs, InAs, and InAs/GaAs) were previously reported [23–25].

Based on the values listed in Table 1, the following conclusions can be drawn: (i) there is a fast decay process with a decay constant in several nanoseconds, and a slow decay process with a decay constant in several tens of nanoseconds; (ii) for a given sample solution, the double-exponential decay constants remain the same (within our experimental uncertainty) under one-, two- and three-photon excitation conditions; (iii) the decay constants are getting shorter as the dot-size increases.

## 5. Two-photon absorption property and optical limiting performance

It is well known that two- or three-photon absorption (2PA or 3PA) is one of the best approaches for optical power limiting [21, 22]. In comparison with other methods (such as reverse saturable absorption, induced refraction or scattering) for this application, the multi-photon absorption techniques exhibit the advantages of negligible linear attenuation (nearly 100% transmission for very weak input signals), and almost instantaneous response for the intensity change of the input laser signals.

The nonlinear transmission of a one-photon transparent but two-photon absorbing medium excited by a focused laser beam with a hat-top pulse shape can be expressed as following [26],

where *I*
_{0} is the peak intensity of the input laser beam inside the medium, *l* is the thickness of the sample, and *β* is the 2PA coefficient of the medium. From Eq. (2) one can see that for a given nonlinear medium, if the nonlinear transmission is measured as a function of the input intensity (*I*
_{0}), the *β* value can be readily determined. In our experiment for 2PA measurements, the beam size and divergence angle of the input 775-nm laser beam were ~4 mm and ~0.25 mrad, which was focused by an *f* =20-cm lens in the center of a 1-cm path-length quartz cuvette filled with a given solution sample. The measured 2PA coefficient (*β*) values for five QDs solution sample of high concentrations are summarized in Table 2. Furthermore, we assume a close-packed monolayer of the mixed surfactant on the surface of QD of diameter *d _{p}*. In this case the total weight of a nanoparticle plus the monolayer is (1/6)

*πd*

_{p}^{3}

*ρ*+ (

*πd*

_{p}^{2}/

*a*)(

*M*/

*N*), where

_{A}*d*is the diameter of the particle,

_{p}*ρ*the density of the QD (5.81 g/cm

^{3}),

*a*the head area per molecule of the surfactant,

*M*the average of the molecular weights of three surfactants (235.27), and

*N*is Avogadro’s number, and the mass fraction of the surfactants adsorbed on the QD surface can be calculated. Here, it is being assumed that the three surfactants dodecyl amine, hexadecylamine and tetradecylphosphonic acid are adsorbed to the QD surface to an equal extent, and their average head parking area is 21 Å

_{A}^{2}as usually calculated for similar amhiphilic molecules [27]. Thus, one can finally determine the 2PA cross-section values per chemical unit or per nanoparticle for five measured QDs samples. Here, the relation between the 2PA coefficient (

*β*) and 2PA cross-section (σ

_{2}or σ′

_{2}) is

where *N*
_{0} is the CdSe molecular (chemical unit) number per unit volume and *N*′_{0} is the CdSe QDs’ number per unit volume. Such determined cross-section values are also given in Table 2 for comparison.

The 2PA cross-section values calculated per chemical unit and per dot as a function of dot’s diameters are shown in Figs. 9(a) and 9(b) separately. In Fig. 9(a) one can see that the cross-section (per unit) is an increasing function of the dot’s size, which reflects a volume effect; accordingly, in Fig. 9(b) the measured cross-section (per dot) values can be well fitted by a solid-line curve of *y*=*ax*
^{4}, revealing the same volume effect. The similar size-dependent 2PA cross-section (per dot) behavior of CdSe and CdTe QDs has been recently reported by *Pu et al*. [28].

We will now try to understand the observed size dependence of the 2PA cross section using a model for the third-order nonlinear susceptibility of a semi-conductor QD [29]:

Here *V* is the dot volume, and the *A* and *B* coefficients are related to one-photon and two-photon transition matrix elements, respectively. Also, *ω*
_{1} = *E*
_{1} - *ω* and *ω*
_{2σ} = *E*
_{2σ} - 2*ω*, where *ω* is the photon energy, *E _{1}* and

*E*are the energies of one-pair electron-hole state and two-pair electron-hole (bi-exciton) state, respectively, and

_{2σ}*γ*is the phenomenological damping constant. To quantitatively analyze nonlinear susceptibility, we used the following material parameters of CdSe semi-conductor: bulk bandgap energy - 1.865 eV, effective electron mass - 0.12 of free-electron mass, effective hole mass - 0.44 of free electron mass, bulk exciton binding energy - 16 meV, and background dielectric constant - 9.53. We consider two different regimes: the strong quantum confinement regime, when the dot radius is smaller than the hole Bohr radius of CdSe (1.1 nm), and the moderate confinement regime when the dot radius is smaller than the electron Bohr radius (4.2 nm) but larger than the hole Bohr radius. In the regime of strong confinement, the Coulomb energy is negligible compared to the confinement energy, and the bi-exciton energy is approximately the sum of two Wannier exciton energies (

*E*

_{2σ}= 2

*E*

_{1}). We confirmed this numerically and found that both energies (

*E*and

_{1}*E*) are much larger than corresponding one-photon and two-photon energies. In this case the leading term in the denominators of Eq. (4) is the confinement energy, which is inversely proportional to the square of the dot radius (

_{2σ}*1/R*). This is because the photon energy of 1.6 eV (corresponding to the wavelength of 775 nm) almost matches the bulk bandgap energy of CdSe. The imaginary part of third-order susceptibility [see Eq. (4)] is thus proportional to the fifth power of the dot radius. Accordingly, the 2PA cross-section per dot

^{2}scales as the eighth power of the dot radius. In this simple analysis we assumed that the damping constant, *γ*, is independent of the dot’s size. Let us now analyze the moderate confinement regime. In this case excitonic and bi-excitonic energies become i) comparable, because of the large bi-exciton binding energy, and ii) close to the bulk bandgap energy, because the confinement energy is much smaller compared to that under strong confinement regime. Let us assume for a moment that resonant conditions, *ω* - *E*
_{1} << *γ*, 2*ω* - *E*
_{2σ} << *γ*, are fulfilled.

In this case the 2PA cross section should not change as the size of the dots grows. Indeed, the third-order susceptibility [see Eq. (4)] becomes inversely proportional to the cube of the dot radius and, accordingly, the 2PA cross-section per dot [see Eq. (5)] does not depend on the dot size. This is in disagreement with the results of the measurements, which show the growth of the 2PA cross section [see Fig. 9(b)]. Apparently, the reason for such a discrepancy is our initial assumption of size-independent damping constant. However, it is a known fact that electron scattering on surface defects becomes increasingly important as the size of semiconductor particles decreases [30], which results in size-dependent dephasing rate of electron coherence. Therefore, in order to correctly describe nonlinear susceptibility we have to introduce the size-dependent damping constant [31, 32]:

where *γ*
_{0} quantifies various electron-hole decay mechanisms and *C* is a material parameter [31]. For metallic particles the parameter *C* is proportional to the Fermi velocity of the electrons. Since in semi-conductor nanoparticles the excited electrons usually occupy the metallic cation orbitals [30], we assumed the parameter *C* is proportional to the Fermi velocity in cadmium which equals 1.62×10^{6}m/sec. Now, in the moderate confinement regime the imaginary part of the nonlinear susceptibility does not depend on the dot size and, hence, the 2PA cross-section per dot is proportional to the cube of the dot radius (*R ^{3}*) if the resonant conditions,

*ω*-

*E*

_{1}<<

*γ*, 2

*ω*-

*E*

_{2σ}<<

*γ*, are fulfilled. In the off-resonant case, corresponding to the strong confinement regime, the imaginary part of the third-order susceptibility becomes proportional to the fourth power of the dot radius (

*R*) and the 2PA cross-section is then proportional to the seventh power of the dot radius (

^{4}*R*). The solid-line curve in Fig. 9(b) is the fourth power of the dot size fitting, which seems consistent with our simple analysis. Similar size dependence of nonlinear optical response was observed [33] and theoretically studied [34] in nano-sized metal particles.

^{7}It is worthwhile to note that the fluorescence life-times observed in our time resolved measurements become longer as the size of QDs decreases (Table 1). One could argue that the coupling of the excited electrons to the surface is rather weak since it is known that modification of the surface can lead to longer life-times [35]. One reason for this is that the electrons are no longer trapped on the surface and non-radiative decay by, for example, dipole-dipole interaction with ligands is strongly reduced. The fluorescence life-time is then mainly determined by radiative exciton recombination. The other reason for the observed increase of the life-times could be the lower concentration ratio of solvent molecules to dots for smaller QDs. In this case collisional quenching which depends on concentration of solvent molecules (quenchers) is suppressed, leading to longer life-times.

As an example of optical limiting, Fig. 10(a) shows the measured nonlinear transmission of QD5 sample solution as a function of the focused intensity of the input 775-nm laser pulses. The measured data can be well fitted by using Eq. (2) with a best fitting parameter of β=0.0063 cm/GW. Figure 10(b) shows the output pulse energy versus the input laser pulse energy. Both curves shown in Figs. 10(a) and 10(b) indicated a superior optical limiting performance, the nonlinear transmission value decreased from the initial near 100% to the final value of ~34% at the input energy level of ~5.7 μJ.

## 6. Three-photon absorption property and optical limiting performance

Similarly, for 3PA process, the nonlinear transmission can be expressed as [36]

where η is the 3PA coefficient, *I*
_{0} is the input local laser intensity, and *l* is the path-length of the sample. Equation (7) is obtained under the assumption the laser pulses have a rectangular distribution in both temporal and transverse beam profiles. From Eq. (7) we know that by simply measuring the 3PA-induced nonlinear transmission as a function of the input laser intensity, the 3PA coefficient (η) value can be readily determined for a given sample. The measured η values for the five QDs solution samples are summarized in Table 3, based on which one can notice that among the measured five samples, the η value is increasing from QD1 to QD5 sample. Furthermore, one can finally determine the 3PA cross-section values per chemical unit or per nanoparticle for five measured QDs samples. Here, the relation between the 3PA coefficient (η) and 3PA cross-section is

where σ_{3} or σ^{′}
_{3} is the 3PA cross-section value per chemical unit or per nanoparticle, and the definitions of *N* and *N*′_{0} are the same as given for Eq. (3). Such determined cross-section values are given in Table 3 for comparison.

The 3PA cross-section values calculated for per chemical unit and for per dot as a function of dot’s diameters are shown in Figs. 11(a) and 11(b) separately. In Fig. 11(a) one can see that the cross-section (per unit) is a slightly increasing function of dot’s sizes, which reflects a smaller volume effect (comparing to Fig. 9(a) for 2PA); accordingly, in Fig. 11(b) the measured cross-section (per dot) values can be well fitted by a solid-line curve of *y*=*bx*
^{3.3}, revealing the smaller volume effect for our 3PA case.

Figure 12(a) shows the measured 3PA-induced nonlinear transmission of the QD5 sample as a function of the input ~1300-nm and ~160-fs laser beam, which was of ~0.6 mrad divergence and focused by an *f*=10-cm lens into the center of the 1-cm path-length sample cuvette. When the input intensity range is less than ~100 GW/cm^{2}, the experimental data could be well fitted by using Eq. (7) with η=6.9×10^{-5} cm^{3}/GW^{2}. However, when the input levels are higher than 100 GW/cm^{2}, the measured transmission values were deviated from the fitting curve given by Eq. (7). One possible mechanism that leads to this deviation is the 3PA saturation effect [37]. Considering this effect, Eq. (7) can be modified as [37]

where *I _{s,3pa}* is 3PA saturation intensity parameter, and η

_{0}is unsaturated 3PA coefficient when the input intensity is much lower than

*I*. In Fig. 12(a), the dotted-line curve is the best fitting curve given by Eq. (9) with the same η

_{s,3pa}_{0}value and

*I*=270 GW/cm

_{s,3pa}^{2}. This fitting is pretty good within the experimental uncertainty. Finally, shown in Fig. 12(b) are the measured output laser energy data versus the input laser energy change under the same experimental condition, which also exhibits a fairly good optical limiting capability based on 3PA mechanism, operating at ~1300-nm wavelength that is one of the most useful wavelengths for optical telecommunications.

## 7. Conclusions

Two- and three-photon absorption and excitation properties of CdSe quantum dots with different sizes in hexane were studied by using ~160-fs laser pulses at wavelengths of ~775 nm and ~1300 nm, respectively. The measurements of one-, two- and three-photon excited fluorescence emission spectra and decay curves showed that for a given dot’s size sample, the fluorescence under three different excitation conditions was from the same emitting state with a double exponential decay feature: a fast decay process (in several nanoseconds range) and a slow decay process (in 15 - 40 ns range). When the dot’s size increased from ~2 to ~3.9 nm, the peak fluorescence emission wavelength was shifted towards longer wavelength direction, while the fluorescence decay became faster. By using nonlinear transmission method, the 2PA and 3PA coefficient and cross-section values of five sample solutions were measured. Finally, it was experimentally shown that semiconductor nanoparticles could be employed for optical power limiting applications, based on 2PA and/or 3PA mechanisms in near-IR spectral range.

## Acknowledgments

This work was supported by the directorate of chemistry and life science of U. S. Air Force Office of Scientific Research, Washington D.C. The authors are grateful to Dr. T. J. Bunning for providing the photos shown in Fig. 1, and to Prof. A. N. Cartwright for critically reading the manuscript.

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