Saturable absorption dynamics in the triplet system and triplet excitation induced singlet fluorescence of some organic molecules

Chemical Physics 263 (2001) 471±490

www.elsevier.nl/locate/chemphys

Saturable absorption dynamics in the triplet system and triplet excitation induced singlet ¯uorescence of some organic molecules H. Gratz, A. Penzkofer * Institut II ± Experimentelle und Angewandte Physik, Universit at Regensburg, Universit atstraûe 31, D-93053 Regensburg, Germany Received 5 July 2000; in ®nal form 11 October 2000

Abstract The triplet saturable absorption behaviour of the xanthene dyes eosin Y, erythrosin B, and rose bengal and of the fullerene molecule C70 is studied. The molecules are excited to the S1 -state by intense picosecond pulses (wavelength kP ˆ 527 nm). They relax dominantly to the triplet system by intersystem crossing. The triplet±triplet saturable absorption is investigated with time-delayed intense picosecond pulses (wavelength kL ˆ 1054 nm) in the transparency region of the molecules in the singlet ground state. Higher excited-state triplet absorption cross-sections and higher excited-state triplet relaxation times are determined by numerical simulation of the experimental results. Time-resolved ¯uorescence measurements reveal higher excited-state triplet to singlet back-intersystem-crossing and multi-step triplet photoionization. Additionally the two-photon absorption cross-sections at kL ˆ 1054 nm are determined by measurement of the fundamental pulse two-photon induced ¯uorescence relative to the second-harmonic pulse singlephoton induced ¯uorescence. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Triplet saturable absorption; Triplet±singlet intersystem crossing; Triplet±singlet intersystem absorption; Triplet photoionization; Two-photon absorption; Eosin Y; Erythrosin B; Rose bengal; Fullerene molecule C70

1. Introduction Halogenated xanthene dyes like eosin Y, erythrosin B, and rose bengal [1±3] and fullerene molecules like C70 [4] have high quantum yields of triplet formation by S1 ±T1 intersystem crossing (Refs. [5±7] and references therein). Therefore the triplet system of these molecules may be strongly

* Corresponding author. Tel.: +49-941-943-2107; fax: +49941-943-2754. E-mail address: [email protected] (A. Penzkofer).

populated by laser excitation [7±16] and triplet spectroscopy becomes accessible [8,10±19]. The T1 triplet-absorption cross-section spectra have been reported for eosin Y [8,10], erythrosin B [10,20], rose bengal [10,13], and C70 [10,21±24]. Reverse intersystem crossing from excited triplet states to the singlet system has been reported for eosin Y [10,11], erythrosin B [10,11] and rose bengal [10,11,13±16]. The absorption dynamics of molecules in the singlet system is well studied [25±30] and widely applied. The singlet ground-state depopulation of organic molecules is the basis for laser action in dye lasers (population inversion caused by S0 -state

0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 3 5 5 - 4

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absorption and relaxed S1 -state emission) [31,32]. Singlet saturable absorption of organic molecules (absorption bleaching by ground-state depopulation) is widely applied for passive Q-switching and mode locking of lasers [29,33,34]. Alternatively S0 ground-state depopulation of organic molecules may enhance the absorption (reverse saturable absorption, excited-state absorption larger than ground-state absorption) [29,35,36]. Reverse saturable absorbers are used as optical limiters (protection against high-intensity laser radiation) [37,38]. The singlet saturable absorption spectroscopy and reverse saturable absorption spectroscopy give information on the singlet excited-state absorption dynamics and deliver excited-state absorption cross-sections and excited-state lifetimes [30,39]. The absorption dynamics of molecules in the triplet system is less well analysed and is quite complex. Molecules have ®rst to be transferred to the triplet system since the ground state of organic molecules is the singlet S0 state. Therefore two-pulse techniques are applied (triplet-state population and triplet-state excitation) [11±16]. Triplet±triplet absorption may cause photochemical reactions, photoionization and triplet±singlet back-intersystem crossing [11±16,40,41]. Laser action of organic molecules in the triplet system by T2 ±T1 stimulated emission has not been reported so far since the T2 -state lifetime is of the order of 1 ps [5]. On the contrary the triplet± triplet absorption is a limiting factor in the cw singlet±singlet laser operation of organic molecules because of triplet population by S1 ±T1 intersystem crossing and T1 -triplet population accumulation due to a long T1 -state lifetime (weak T1 ±S0 intersystem crossing and weak T1 ±S0 radiative relaxation) [31,32,42]. The combined singlet±triplet reverse saturable absorption (rS0 < rT1 , triplet population by intersystem crossing after singlet system excitation followed by enhanced absorption due to T1 -triplet absorption within the triplet state lifetime) is applied in optical limiters of fast switch-on time (subnanosecond to nanosecond time scale) and long persistence (microsecond to millisecond time scale) [37,38]. Especially fullerene molecules (C60 [36,37,43,44], C70 [43,45]) and porphyrine mole-

cules (weakly absorbing Q-band) [38,46] are used as singlet±triplet optical limiters [38,47]. Triplet±triplet saturable absorption (T1 -state depopulation by intense laser pulse excitation, and T1 -absorption cross-section rT1 less than T2 absorption cross-section rT2 ) and triplet±triplet reverse saturable absorption (T1 -state depopulation, and rT1 < rT2 ) have not been reported so far to the best of our knowledge. Like the singlet saturable absorption spectroscopy for the singlet system, the triplet saturable absorption spectroscopy will give information on the triplet excitedstate absorption dynamics and will deliver triplet excited-state absorption cross-sections and triplet excited-state lifetimes. Here triplet saturable absorption studies on eosin Y, erythrosin B, rose bengal, and C70 are performed at a wavelength of kL ˆ 1054 nm. A ®rst pump pulse at kP ˆ 527 nm populates the triplet state and a second time-delayed pulse at kL ˆ 1054 nm causes the excitation in the triplet system. Higher excited triplet-state absorption cross-sections and higher excited triplet state lifetimes are determined by numerical simulation of experimental intensity dependent transmission data. Higher excited-state triplet to singlet backintersystem crossing is observed by time-resolved ¯uorescence detection. Higher excited-state triplet to singlet intersystem absorption is discussed. The quantum eciency of S1 -state population by higher excited-state triplet±singlet intersystem crossing or higher excited-state triplet±singlet absorption is determined. The back-intersystem crossing or intersystem absorption connects the triplet±triplet excited-state absorption with the singlet±singlet excited-state absorption. Therefore, the singlet excited-state absorption, rS1 ;L , at kL ˆ 1054 nm is involved which is measured separately with shortly delayed probe pulses at kL . An analysis of the intensity dependence of the timeresolved ¯uorescence signals reveals a partial photoionization by multi-step triplet excited-state absorption. The two-photon absorption crosssections of the investigated molecules at kL ˆ 1054 nm are determined by measuring the fundamental pulse …kL † induced two-photon ¯uorescence relative to the second-harmonic pulse …kP † induced single-photon ¯uorescence.

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2. Experimental The structural formulae of the xanthene dyes eosin Y, erythrosin B, and rose bengal are displayed in Fig. 1. They are dissolved in methanol. The fullerene C70 is dissolved in toluene. The dyes and solvents were purchased from Aldrich. They are used without further puri®cation. The m†† and lowest-state ground-state singlet …rS0 …~ m†† absorption cross-section spectra are triplet …rT1 …~ shown in Fig. 2 (redrawn from Ref. [10]). The experiments are carried out with a modelocked Nd:phosphate glass laser system [48]. Second-harmonic pulses (kP ˆ 527 nm) are used for triplet state T1 preparation via singlet S1 -state excitation and intersystem crossing. Time-delayed fundamental pulses (kL ˆ 1054 nm) are used for

Fig. 1. Structural formulae of investigated xanthene dyes.

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triplet saturable absorption, multi-step triplet photoionization back-intersystem crossing, triplet± singlet absorption (delay time td ˆ 20 ns; intense pulses), and S1 singlet excited-state absorption (delay time td ˆ 25 ps; attenuated pulses). Twophoton ¯uorescence measurements at kL ˆ 1054 nm are carried out whereby the fundamental pulse two-photon ¯uorescence signals are compared with the second-harmonic pulse single-photon ¯uorescence signals (time delay td P 1 min). The experimental setup is shown in Fig. 3. A mode-locked laser oscillator generates a train of picosecond laser pulses. A single pulse is selected with a laser-triggered Kerr shutter [49] and increased in energy with a laser ampli®er (wavelength kL ˆ 1054 nm, duration DtL ˆ 6 ps, energy WL  3 mJ). Part of the fundamental laser energy is frequency doubled in a CDA crystal [50] (kP ˆ 527 nm, WP  1 mJ). The second-harmonic light is separated from the fundamental light by a harmonic beam splitter, HBS. The input peak intensity of the second-harmonic pulses at the sample position is determined by energy transmission measurement through the sample SA1 of rhodamine 6G in ethanol (slow saturable absorber) [51] with the photodetectors PD1 and PD2. The fundamental light pulses are time delayed relative to the second-harmonic pulses by an optical delay line, DL. The spatial overlap of the second-harmonic (®rst pulse) and the fundamental (second pulse) pump beams in the sample S is controlled by a camera system (objective, Ob, and diode-array detector, DA). The singlet excited-state absorption cross-sections, rS1 ;L , of the samples are determined by strongly attenuating the fundamental pulses (no absorption saturation) and by using a delay time of td ˆ 25 ps (negligible triplet population at this short delay time). The probe pulse polarization is orientated to an angle of 54.7° (magic angle) relative to the second-harmonic pump pulse polarization in order to avoid anisotropy e€ects. The polarization was rotated with a 1054 nm half-wave plate. For the triplet saturable absorption studies a time delay of td ˆ 20 ns is used. The input fundamental pulse peak intensity is determined by energy transmission measurement through the

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Fig. 2. Singlet±singlet absorption cross-section spectra, rS0 (solid curves, lower abscissa) and triplet±triplet absorption cross-section spectra, rT1 (dashed curves, upper abscissa) of investigated dyes. Origins of upper abscissa are shifted to S0 ±T1 transition wave number m~T1 of lower abscissa in order to visualize the same energetic level position. The excitation wave number positions m~T1 ‡ m~L and m~T1 ‡ m~P are indicated.

Fig. 3. Experimental setup. M.L.Laser, mode-locked Nd:phosphate glass laser: SW, Kerr shutter; Ampli®er, Nd:phosphate glass laser ampli®er; SHG, CDA crystal for second-harmonic generation; HBS, harmonic beam splitter; DL, optical delay line; L1±L5, lenses; S, sample; PD1±PD6, photo-diodes; SP, grating spectrometer; MCP, micro-channelplate photomultiplier: Ob, camera objective; DA, diode-array detector.

sample SA2 of the Kodak dye A9860 in 1,2dichloroethane (fast saturable absorber) [52] with the photodetectors PD3 and PD4. The energy transmission through the samples, S, is measured with the photodetectors PD4 and PD5. For the ¯uorescence studies a time-resolved ¯uorescence detection is carried out with a fast micro-channelplate photomultiplier, MCP (Hamamatsu type R1564 U-01). The sideward ¯uorescence is collimated by lens L3 and focused to the spectrometer SP in front of the detector MCP by lens L4. The ¯uorescence signals are also collected with lens L5 and measured with the photodetector PD6. In the case of weak second pulse ¯uorescence the micro-channelplate photomultiplier is overexposed by the ®rst pulse ¯uorescence and the ®rst pulse ¯uorescence is therefore recorded with photodetector PD6. For the two-photon absorption cross-section [53] determination at kL ˆ 1054 nm the samples are excited separately by fundamental laser pulses (two-photon ¯uorescence) and second-harmonic laser pulses (single-photon ¯uorescence) and the ¯uorescence signals are compared.

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3. Results The S1 excited-state absorption at kL ˆ 1054 nm is measured by S1 -state population with intense picosecond second-harmonic pump pulses at kP ˆ 527 nm and probing the transmission at kL ˆ 1054 nm shortly after excitation (delay time td ˆ 25 ps) with strongly attenuated fundamental laser pulses. The experimental parameters of initial small-signal transmission, T0P , at kP ˆ 527 nm, the second-harmonic pump laser peak intensity, I0P , and the fundamental probe pulse energy transmission, TEL , are listed in Table 1. From the experimental data the singlet excited-state absorption cross-sections, rS1 ;L are extracted below by numerical simulation of the absorption dynamics [46]. The triplet±triplet saturable absorption is studied by intense picosecond pump pulse (kP ˆ 527 nm) S1 -state population (®rst pump pulse), excitation transfer to the triplet system by intersystem crossing, and T1 -state depletion by time-delayed intense picosecond fundamental pulse excitation (kL ˆ 1054 nm, second pulse). The neat energy transmissions TEL , of the timedelayed excitation pulses versus their input peak intensities, I0L , are displayed in Fig. 4 (eosin Y in methanol), Fig. 5 (erythrosin B in methanol), Fig. 6 (rose bengal in methanol), and Fig. 7 (C70 in toluene). The neat energy transmissions of the dye solutions are measured relative to the solvents, i.e. TEL …I0L † ˆ TEL …I0L , solution†=TEL …I0L; solvent†. The small signal transmissions, T0P , at kP ˆ 527 nm and the applied ®rst excitation pulse intensities, I0P , are listed in the ®gure captions. The sample length is ` ˆ 1 mm in all cases. The transmission through the solvent methanol is 100% at kL independent of the applied excitation intensity, I0L . The transmission through toluene is shown in Fig.

Fig. 4. Energy transmission of time-delayed fundamental pump laser pulses at kL ˆ 1054 nm through eosin Y in methanol. Sample length ` ˆ 1 mm. Time delay td ˆ 20 ns. Small-signal transmission at second-harmonic wavelength, kP ˆ 527 nm is T0P ˆ 0.0015. Second-harmonic pump pulse intensity is I0P ˆ 1:0  1010 W cmÿ2 . Ratio of laser beam radii is rL =rP ˆ 0.72. Circles are experimental data. Curves are calculated neglecting back-intersystem transfer and using parameters of Table 2 together with (1) rex;T ˆ 0; (2) rex;T ˆ 1  10ÿ16 cm2 , rT;I ˆ 0, sex;T ˆ 1  10ÿ12 s; (3) rex;T ˆ 2  10ÿ16 cm2 , rT;I ˆ 0, sex;T ˆ 1  10ÿ12 s; (4) rex;T ˆ 1  10ÿ16 cm2 , rT;I ˆ 0, sex;T ˆ 1  10ÿ13 s.

7b. There is a reduction in transmission above I0L  2  1010 W cmÿ2 probably due to optical damage (dielectric breakdown) [54,55]. For the xanthene dyes the energy transmission, TEL , ®rst slightly decreases with rising I0L (reverse saturable absorption) and then strongly increases. The initial decrease indicates that the triplet excited-state absorption cross-section, r2T , is larger than the triplet T1 absorption cross-section, rT1

Table 1 Experimental parameters and results of singlet excited-state absorption, rS1 ;L at kL ˆ 1054 nma

a

Dye

Solvent

T0P

I0P (109 W cmÿ2 )

TEL

rS1 ;L …10ÿ18 cm2 †

Eosin Y Erythrosin B Rose bengal C70

Methanol Methanol Methanol Toluene

0.0188 0.013 0.011 0.082

4.4 5.5 7 7

0.968 0.918 0.805 0.96

3.7 6.8 6.2 6.4

Singlet S1 -state population with pump pulse at kP ˆ 527 nm. Ratio of laser beam radii rL =rP ˆ 0:72.

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10

ÿ2

Fig. 5. Energy transmission, TEL …I0P ˆ 1:0  10 W cm , td ˆ 20 ns†, versus I0L for erythrosin B in methanol. ` ˆ 1 mm, T0P ˆ 0:0039. rL =rP ˆ 0:72. Circles are measured. Curves are calculated using parameters of Table 2 except stated below. (a) No inclusion of intersystem crossing, intersystem absorption, and triplet photo-ionization. The curves belong to (1) rex;T ˆ 0; (2) rex;T ˆ 1  10ÿ16 cm2 , sex;T ˆ 1 ps; (3) rex;T ˆ 1  10ÿ16 cm2 , sex;T ˆ 0:5 ps; (4) rex;T ˆ 1  10ÿ16 cm2 , sex;T ˆ 0:1 ps. Dotted curve: rex;T ˆ 2  10ÿ16 cm2 , sex;T ˆ 0:5 ps. (b) Inclusion of intersystem crossing …/2T;S ˆ 0:00125, /ex;T;S ˆ 0:02† and triplet photo-ionization …rT;I ˆ 2  10ÿ17 cm2 †. The solid curves belong to (1) sex;T ˆ 0:5 ps, rsol ˆ 0; (2) sex;T ˆ 0:5 ps, rsol ˆ 5  10ÿ18 cm2 ; (3) sex;T ˆ 0:5 ps, rsol ˆ 1  10ÿ17 cm2 . Dotted curve: sex;T ˆ 0:3 ps, rsol ˆ 5  10ÿ18 cm2 . (c) Inclusion of intersystem absorption …fT1 ;S ˆ 0:00175, fex;T;S ˆ 0:015† and triplet photo-ionization …rT;I ˆ 4  10ÿ16 cm2 †. The varied parameters are: (1) sex;T ˆ 0:5 ps, rsol ˆ 0; (2) sex;T ˆ 0:5 ps and rsol ˆ 5  10ÿ18 cm2 ; (3) sex;T ˆ 0:3 ps, rsol ˆ 5  10ÿ18 cm2 ; (4) sex;T ˆ 0:2 ps, rsol ˆ 5  10ÿ18 cm2 ; and (5) sex;T ˆ 0:1 ps, rsol ˆ 5  10ÿ18 cm2 .

(see Fig. 14). Finally higher lying triplet states are depleted leading to a rise of transmission (see below). For the fullerene dye C70 in toluene the transmission is nearly constant (slightly decreasing up to I0L ˆ 2  1010 W cmÿ2 ) over the whole pump pulse intensity range. The temporal ¯uorescence emission caused by the ®rst frequency-doubled pump pulse …kP † and

Fig. 6. Energy transmission, TEL …I0P ˆ 1:0  1010 W cmÿ2 , td ˆ 20 ns†, versus I0L for rose bengal in methanol. ` ˆ 1 mm. T0P ˆ 0:023. rL =rP ˆ 0:72. Circles are measured. Curves are calculated for parameters of Table 2, except stated below. (a) No inclusion of intersystem crossing, intersystem absorption, and photo-ionization. The solid curves belong to (1) rex;T ˆ 0; (2) rex;T ˆ 1  10ÿ16 cm2 , sex;T ˆ 1 ps; (3) rex;T ˆ 2  10ÿ16 cm2 , sex;T ˆ 1 ps; and (4) rex;T ˆ 2  10ÿ16 cm2 , sex;T ˆ 0:1 ps. Dotted curve: rex;T ˆ 1  10ÿ16 cm2 , sex;T ˆ 1 ps, and s2T ˆ 1 ps. (b) Inclusion of intersystem crossing …/2T;S ˆ 0:0115, /ex;T;S ˆ 0:03, rT;I ˆ 5  10ÿ18 cm2 †. Curve 1, rsol ˆ 0: curve 2, rsol ˆ 5  10ÿ18 cm2 . (c) Inclusion of intersystem absorption …fT1 ;S ˆ 0:012, fex;T;S ˆ 0:02, rT;I ˆ 4  10ÿ16 cm2 †. Curve 1, rsol ˆ 0; curve 2, rsol ˆ 5  10ÿ18 cm2 .

the second time-delayed fundamental pump pulse …kL † is observed perpendicular to the excitation direction (Fig. 3). In Fig. 8 experimental ¯uorescence traces are shown for eosin Y in methanol (a), erythrosin B in methanol (b), rose bengal in methanol (c), and C70 in toluene (d). The ®rst peaks are the direct S1 ±S0 ¯uorescence signals due to S0 ±S1 excitation by the ®rst excitation pulse (kP ˆ 527 nm). The weak second peaks observed for erythrosin B, rose bengal, and C70 coincide with the time-delayed fundamental pump pulse excitation (kL ˆ 1054 nm). For eosin Y no secondary ¯uorescence peak can be resolved in Fig.

H. Gratz, A. Penzkofer / Chemical Physics 263 (2001) 471±490

Fig. 7. (a) Energy transmission, TEL …I0P ˆ 1:57  1010 W cmÿ2 , td ˆ 20 ns†, versus I0L for C70 in toluene: ` ˆ 1 mm; T0P ˆ 0:031: rL =rP ˆ 0:85. Circles are measured. Curves are calculated for parameters of Table 2, except rT;I ˆ 0, and (1) rex;T ˆ 0; (2) rex;T ˆ 2  10ÿ17 cm2 , sex;T ˆ 1 ps; (3) rex;T ˆ 2  10ÿ17 cm2 , sex;T ˆ 0:1 ps; (4) rex;T ˆ 2  10ÿ17 cm2 , sex;T ˆ 50 fs. Dotted curve: rT;I ˆ 0; rex;T ˆ 1:5  10ÿ17 cm2 , and sex;T ˆ 50 fs. (b) Energy transmission, TEL , through solvent toluene.

8a (back-intersystem transfer is too weak to show up in Fig. 8a; ®rst signal on oscilloscope has to be overexposed to see a tiny second signal peak). In Fig. 9, for all investigated dyes, two-photon ¯uorescence signals are shown which were obtained by fundamental pulse excitation. For erythrosin B and rose bengal the two-photon ¯uorescence signals are negligibly small compared to the second pulse ¯uorescence signals of Fig. 8. For eosin Y in methanol and C70 in toluene the two-photon ¯uorescence signals are comparable to the ¯uorescence signals obtained by the 20 ns delayed fundamental pulses in the double pulse excitation experiments (see Figs. 8, 10 and 13). 00 0 =SF;max , The ¯uorescence signal ratios, SF;max versus I0L for a ®xed I0P value are displayed by circles in Fig. 10 (eosin Y in methanol), Fig. 11 (erythrosin B in methanol), Fig. 12 (rose bengal in 00 is the methanol) and Fig. 13 (C70 in toluene). SF;max

477

Fig. 8. Fluorescence signal traces due to ®rst second-harmonic pulse (kP ) and time-delayed fundamental pulse …kL † excitation. Sample length ` ˆ 1 mm. (a) Eosin Y in methanol: T0P ˆ 0:0021; I0P ˆ 2:8  1010 W cmÿ2 ; I0L ˆ 1:1  1010 W cmÿ2 . (b) Erythrosin B in methanol: T0P ˆ 0:0011; I0P ˆ 4:1  1010 W cmÿ2 ; I0L ˆ 1:3  1010 W cmÿ2 . (c) Rose bengal in methanol: T0P ˆ 0:039. I0P ˆ 2:7  1010 W cmÿ2 ; I0L ˆ 1:3  1010 W cmÿ2 . (d) C70 in toluene: T0P ˆ 0:044; I0P ˆ 2:8  1010 W cmÿ2 ; I0L ˆ 1:7  1010 W cmÿ2 . 0 second ¯uorescence peak and SF;max is the ®rst ¯uorescence peak. In the measurements the fundamental laser beam diameter was a factor of 3.3 times larger than the second-harmonic laser beam diameter. For eosin Y in methanol the second fundamental pulse induced ¯uorescence signal is weak (circles in Fig. 10). It could be detected only 00 = for I0L > 5  109 W cmÿ2 . A ratio of SF;max 0 SF;max  0:0045 was obtained for I0L ˆ 2:6  1010 W cmÿ2 . For erythrosin B in methanol (circles in Fig. 11) a back-intersystem-transfer signal could be detected down to I0L ˆ 3  108 W cmÿ2 . The 00 0 =SF;max increased up to a value of signal ratio SF;max 0.05 at I0L  8  109 W cmÿ2 and then remained approximately constant. The highest back-intersystem-crossing eciency was observed for rose bengal in methanol (circles in Fig. 12). A fundamental laser induced ¯uorescence signal could be detected down to I0L  108 W cmÿ2 . The sig00 0 =SF;max , increased up to 0.08 at nal ratio, SF;max

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Fig. 9. Two-photon ¯uorescence signals of fundamental pulse …kL † excitation. Sample length ` ˆ 1 mm. (a) Eosin Y in methanol: T0P ˆ 0:021; I0L ˆ 1:1  1010 W cmÿ2 . (b) Erythrosin B in methanol: T0P ˆ 0:0011; I0L ˆ 9:2  109 W cmÿ2 . (c) Rose bengal in methanol: T0P ˆ 0:039; I0L ˆ 9:1  109 W cmÿ2 . (d) C70 in toluene: T0P ˆ 0:044; I0P ˆ 1:9  1010 W cmÿ2 .

I0 ˆ 1010 W cmÿ2 and remained approximately constant at higher fundamental pulse intensities. For C70 in toluene the ¯uorescence quantum eciency is low [7]. The ¯uorescence ratio in a limited intensity range is presented by circle in Fig. 13a. 00 0 =SF;max , of fundamental pulse The ratios, SF;max 00 ; to two-photon ¯uorescence signal height, SF;max second-harmonic pulse single-photon ¯uorescence 0 , versus I0L for a ®xed I0P value signal height, SF;max are indicated by triangles in Figs. 10±13. The pulse focusing conditions and the I0P values are the same as for the 20 ns delayed two-pulse ¯uorescence measurements. For eosin Y in methanol and C70 in toluene the two-photon ¯uorescence signals are remarkable compared to the 20 ns time-delayed second pulse ¯uorescence signals. The dots in Figs. 10 and 13 present the ¯uorescence contribution from triplet±singlet back transfer (20 ns delayed second pulse ¯uorescence minus 1 min delayed

00 Fig. 10. Ratio of ¯uorescence signal, SF;max , induced by time0 delayed fundamental laser pulse to ¯uorescence signal, SF;max , caused by initial second-harmonic excitation pulse. Sample, eosin Y in methanol. Sample length ` ˆ 1 mm. Small signal transmission, T0P ˆ 0:0015. Second-harmonic pump pulse peak intensity I0P ˆ 4  1010 W cmÿ2 . Ratio of beam radii, rL =rP ˆ 00 0 =SF;max for td ˆ 20 ns. 3:3. Circles are experimental ratios SF;max Triangles are experimental ratios for td P 1 min (second pulse induced two-photon ¯uorescence). Dots represent second pulse ¯uorescence at td ˆ 20 ns minus two-photon ¯uorescence contribution. Curves are calculated using parameters of Table 2. (a) Simulation of triplet±singlet intersystem crossing. Solid curve, /2T;S ˆ 1  10ÿ3 ; dashed curve, /ex;T;S ˆ 1  10ÿ3 ; dash-dotted curve, /2T;S ˆ 1  10ÿ3 and /ex;T;S ˆ 1  10ÿ3 ; dotted curve, …2† two-photon absorption contribution with rL ˆ 5:8  10ÿ50 cm4 s. (b) Simulation of triplet±singlet absorption. Solid curve, fT1 ;S ˆ 5  10ÿ4 ; dashed curve, fex;T;S ˆ 5  10ÿ4 ; dash-dotted curve, fT1 ;S ˆ 1:4  10ÿ3 and fex;T;S ˆ 0.

fundamental-pulse ¯uorescence). For erythrosin B and rose bengal the two-photon ¯uorescence contributions are negligible compared to the triplet±singlet back-transfer contributions. 4. Theory The absorption and emission experiments are analysed by using the energy level system of Fig. 14. The singlet energy level system is determined

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479

00 0 Fig. 11. Fluorescence ratio, SF;max =SF;max , for erythrosin B in methanol: ` ˆ 1 mm; I0P ˆ 4:2  1010 W cmÿ2 . T0P ˆ 0:001; rL =rP ˆ 3:3. Circles are measured for td ˆ 20 ns. Curves are calculated using parameters of Table 2. (a) In¯uence of triplet± singlet intersystem crossing and of two-photon absorption. Solid curves, T2 ±S intersystem crossing with (1) /2T;S ˆ 0:02, (2) /2T;S ˆ 0:01, and (3) /2T;S ˆ 0:005. Dashed curves, Tn ±S intersystem crossing with …10 † /ex;T;S ˆ 0:04, …20 † /ex;T;S ˆ 0:02, and 00 0 …30 † /ex;T;S ˆ 0:01. Triangle, SF;max =SF;max ratio measured for td P 1 min (two-photon ¯uorescence). Dotted curve, two-pho…2† ton absorption contribution with rL ˆ 8:5  10ÿ50 cm4 s. (b) Combined e€ects of triplet±singlet intersystem crossing and triplet photo-ionization. Curves are calculated for /2T;S ˆ 0:00125, /ex;T;S ˆ 0:02, and (1) rT;I ˆ 0, (2) rT;I ˆ 5  10ÿ18 cm2 , (3) rT;I ˆ 1  10ÿ17 cm2 , and (4) rT;I ˆ 2  10ÿ17 cm2 . (c) In¯uence of triplet±singlet intersystem absorption. Solid curves, T1 ±S absorption with rT1 ;S ˆ fT1 ;S rT1 where (1) fT1 ;S ˆ 0:02, (2) fT1 ;S ˆ 0:01, and (3) fT1 ;S ˆ 0:005. Dashed curves, Tn ±S absorption with rex;T;S ˆ fex;T;S r2T where …10 † fex;T;S ˆ 0:04, …20 † fex;T;S ˆ 0:02, and …30 † fex;T;S ˆ 0:01. (d) Combined e€ects of triplet±singlet absorption and triplet photo-ionization. Curves are calculated for fT1 ;S ˆ 0:00175, fex;T;S ˆ 0.015, and rT;I ˆ 0 (1), 2  10ÿ17 cm2 , (2), 1  10ÿ16 cm2 (3), 2  10ÿ16 cm2 (4), and 4  10ÿ16 cm2 (5).

00 0 Fig. 12. Fluorescence ratio, SF;max =SF;max , for rose bengal in methanol. ` ˆ 1 mm. I0P ˆ 3:9  1010 W cmÿ2 . T0P ˆ 0:023, rL =rP ˆ 3.3. Circles are measured for td ˆ 20 ns. Curves are calculated using parameters of Table 2. (a) In¯uence of triplet± singlet intersystem crossing and two-photon absorption. Solid curves, T2 ±S intersystem crossing with (1) /2T;S ˆ 0:04, (2) /2T;S ˆ 0:02, and (3) /2T;S ˆ 0:01. Dashed curves, Tn ±S intersystem crossing with …10 † /ex;T;S ˆ 0:04, …20 † /ex;T;S ˆ 0:02, 00 0 =SF;max ratio measured for …30 † /ex;T;S ˆ 0:01. Triangle, SF;max td P 1 min (two-photon ¯uorescence). Dotted curve, two-pho…2† ton absorption contribution with rL ˆ 3:75  10ÿ50 cm4 s. (b) Combined e€ects of triplet±singlet intersystem crossing and triplet photo-ionization. Curves are calculated for /2T;S ˆ 0:0115, /ex;T;S ˆ 0:03, and (1) rT;I ˆ 0, (2) rT;I ˆ 5  10ÿ18 cm2 , (3) rT;I ˆ 1  10ÿ17 cm2 , (4) rT;I ˆ 2  10ÿ17 cm2 . (c) In¯uence of triplet±singlet intersystem absorption. Solid curves, T1 ±S absorption with rT1 ;S ˆ fT1 ;S rT1 where (1) fT1 ;S ˆ 0:04, (2) fT1 ;S ˆ 0:02, (3) fT1 ;S ˆ 0:01. Dashed curves, Tn ±S absorption with rex;T;S ˆ fex;T;S rex;T where …10 † fex;T;S ˆ 0:04, …20 † fex;T;S ˆ 0:02, …30 † fex;T;S ˆ 0:01. (d) Combined e€ects of triplet±singlet absorption and triplet photo-ionization. Curves are calculated for fT1 ;S ˆ 0:012, fex;T;S ˆ 0:02 and (1) rT;I ˆ 0, (2) rT;I ˆ 1  10ÿ17 cm2 , (3) rT;I ˆ 2  10ÿ17 cm2 , (4) rT;I ˆ 5  10ÿ17 cm2 , (5) rT;I ˆ 1  10ÿ16 cm2 , (6) rT;I ˆ 2  10ÿ16 cm2 , and (7) rT;I ˆ 4  10ÿ16 cm2 .

by the S0 -absorption cross-section spectrum, rS0 …~ m†, of the molecules (solid curves in Fig. 2). The triplet energy level system is determined by the m†, of T1 -absorption cross-section spectrum, rT1 …~ the molecules (dashed curves in Fig. 2). The energy level positions, m~T1 , of the triplet ground-states are given by the short-wavelength positions of the

phosphorescence spectra of the dyes (values are given in Table 2) [4,56±59]. The gas-phase ionization energy positions may be determined by gasphase photoelectron spectroscopy [60] or may be approximately calculated by quantum chemical calculations [61]. The ionization energy positions, m~I , in solutions are approximately 1.5±2 eV

480

H. Gratz, A. Penzkofer / Chemical Physics 263 (2001) 471±490

00 0 Fig. 13. Fluorescence ratio, SF;max =SF;max , for C70 in toluene. ` ˆ 1 mm. I0P ˆ 1  1010 W cmÿ2 . T0P ˆ 0:044, rL =rP ˆ 3.3. Circle, second pulse delay td ˆ 20 ns. Dots, second pulse ¯uorescence at td ˆ 20 ns minus two-photon ¯uorescence contribution. Curves are calculated using parameters of Table 2. (a) Simulation of triplet±singlet intersystem crossing and twophoton absorption. Solid curve, /2T;S ˆ 0:01. Dashed curve, /ex;T;S ˆ 0:01. Dash-dotted curve, /2T;S ˆ 0:008 and /ex;T;S ˆ 0. Triangles, second pulse delay td P 1 min. Dotted curve, ¯uo…2† rescence due to two-photon absorption with rL ˆ 2:5  10ÿ49 4 cm s. (b) Simulation of triplet±singlet absorption. Solid curve, fT1 ;S ˆ 0:01. Dashed curve, fex;T;S ˆ 0:01. Dash-dotted curve, fT1 ;S ˆ 0:007 and fex;T;S ˆ 0.

(12000±16000 cmÿ1 ) lower than the gas-phase ionization energies [3,62±64]. Some approximate values are given in Table 2 [65,66]. The ®rst pulse singlet absorption and emission dynamics is described by singlet±singlet absorption, singlet-system relaxation, singlet-system photodegradation, and triplet population by S1 ±T intersystem crossing. The 20 ns delayed second pulse absorption and emission dynamics is described by triplet±triplet absorption, triplet±singlet back-intersystem crossing, triplet to singlet intersystem absorption, triplet multi-step photoionization, and singlet ground-state two-photon absorption. The initial second-harmonic pump pulse (frequency mP ) excites molecules from the ground-

state S0 (level 1S) to a vibronic Franck±Condon state 2S0 in the S1 band. From there the molecules relax to a temporal thermal equilibrium state 2S in the S1 -band with a relaxation time sFC . Starting from the S1 -band there occurs excited-state absorption (cross-section, rS1 ;P ) to a higher excited singlet state (level 3S). Higher excited molecules in the singlet system relax back to the S1 -band with a rate constant …1 ÿ /D;Sn †sÿ1 ex;S and they photodegrade with a rate constant of /D;Sn sÿ1 ex;S . /D;Sn is the quantum yield of photodegradation of the Sn state. The S1 -state lifetime is sS1 . The molecules in the S1 -band relax to the ground state (radiative relaxation with ¯uorescence quantum yield /F , and internal conversion) and to the T1 -triplet state (intersystem crossing, quantum eciency /S1 ;T ). The time-delayed fundamental pump pulse (frequency m~L , delay time td ˆ 20 ns  sF ) excites molecules from the lowest triplet-state T1 to a vibronic Franck±Condon state 2T0 . From there the molecules relax to a temporal equilibrium state 2T in the T2 -band with a relaxation time sFC . The T2 band relaxes with a time constant s2T mainly to the T1 -band. From the levels 2T0 and 2T there occurs successive excitation to higher lying levels 3T and 4T with the absorption cross-sections r2T;L and r3T;L , which we approximate by rex;T . Light absorption from level 4T leads to photoionization with solvated ion and electron formation (level I, absorption cross-section rT;I ). The higher excited triplet states relax to lower lying triplet states with time constants s3T and s4T , which we approximate by a single time constant sex;T . From the excited triplet states 2T, 3T, and 4T there occurs backintersystem crossing to the singlet system with the quantum eciencies /2T;S , /3T;S , and /4T;S . For simplicity the eciencies /3T;S , and /4T;S are approximated by a single constant /ex;T;S . The ionized molecules are thought to absorb some laser light due to solvated electron absorption at the fundamental laser wavelength, kL (absorption cross-section, rsol;L ) [67±69], and do not recombine measurably within the fundamental pump pulse duration. Their recombination and subsequent relaxation in the singlet and triplet systems is expected to occur on a microsecond to millisecond time scale [3].

H. Gratz, A. Penzkofer / Chemical Physics 263 (2001) 471±490

481

Fig. 14. Energy level scheme applied in numerical simulations.

Besides triplet±singlet back-intersystem crossing there may occur some back transfer of triplet molecules to the singlet system by triplet±singlet absorption. Spin±orbit coupling causes that the triplet-states have a singlet admixture and the singlet-states have a triplet admixture [3,42]. This admixture causes triplet±singlet absorption and emission and vice versa. The triplet±singlet absorption processes are indicated in Fig. 14 by transitions with the cross-sections rT1;S , r2T;S , r3T;S . In simulations we approximate r2T;S and r3T;S by rex;T;S . The molecules transferred from the triplet system to the singlet system by either back-intersystem crossing or triplet±singlet absorption cause S1 excited-state absorption for the fundamental pump laser (absorption cross-section, rS1 ;L ). The molecules remaining in the triplet system after time-delayed fundamental pulse excitation accumulate quickly in the T1 state and return to the singlet ground state S0 by T1 ±S0 intersystem crossing and T1 ±S0 emission (phosphorescence) with a time constant sph (phosphorescence lifetime) on a microsecond to millisecond time scale. Two-

photon absorption of the fundamental laser pulse causes singlet ground-state excitation with a two…2† photon absorption cross-section rL . The dynamics of the level population number densities, Ni …r; z0 ; t0 ; h†, i ˆ 1S, 2S, 2S0 , 3S, by initial second-harmonic pump pulse, IP …r; z0 ; t0 †, absorption is described by the following equation system [5]. 0 oN1S 3 rS0 ;P cos2 …h† 0 0 ˆÿ …N1S ÿ N2S 0 †IP 0 ot h mP 0 ÿ
N 0 ‡ N2S 0 ‡ 1 ÿ /S1 ;T 2S sS1 0 N 0 ÿ N 1S N1T ÿ 1S ‡ ; sor sph

…1†

0 oN2S 3 rS0 ;P cos2 …h† 0 0 0 ˆ …N1S ÿ N2S 0 †IP ot0 h mP 0 0 0 N2S ÿ N2S;th N2S0 3 rS1 ;P cos2 …h† ÿ ÿ ÿ sFC sS1 ! h mP 0 0 0 N2S0 N2S 0 ÿ N 2S0 0 0  N2S0 ÿ 0 N3S IP ÿ ; 0 N2S ‡ N2S sor 0

…2†

482

H. Gratz, A. Penzkofer / Chemical Physics 263 (2001) 471±490

Table 2 Spectroscopic parameters of dyes needed for simulations and obtained results (all parameters belong to room temperature) Eosin Y (methanol)

Erythrosin B (methanol)

Rose bengal (methanol)

C70 (toluene)

Comments

3:2  10ÿ16 2:9  10ÿ17 3:7  10ÿ18 4:65  10ÿ17 2.0 [5] 0.5 60 330 [80] 1 [5] 15 600 [56,57] 56 000b 0.56 [5] 0.0029 …1  0:2†  10ÿ16 1 …5:8  1†  10ÿ50

3:6  10ÿ16 3:0  10ÿ17 6:8  10ÿ18 4:6  10ÿ17 0.53 [6] 0.5 60 200 [81] 1.5 [9] 15 800 [58] 52 000b 0.90 [10] 0.0048 …1  0:2†  10ÿ16 0.5 …8:5  1:4†  10ÿ50

1:16  10ÿ16 2:9  10ÿ17 6:2  10ÿ18 7:8  10ÿ17 0.59 [6] 0.5 60 150 [81] 2.2 [9] 14 500 [59] 55 000b 0.9 [10] 0.0039 …1  0:2†  10ÿ16 1 …3:75  0:5†  10ÿ50

3:5  10ÿ17 5:0  10ÿ17 6:4  10ÿ18 9  10ÿ18 0.65 [7] 0.5 60 23 [10] 1 [7] 12 600 [4] 49 300 [66] 0.99915 [7] 0.0 …1:8  0:2†  10ÿ17 0.05 …2:5  0:4†  10ÿ49

Fig. 2 Ref. [10] Table 1 Ref. [10]

Intersystem crossing ®t 10ÿ3 /2T;S /ex;T;S 10ÿ3 rT;I (cm2 ) rsol (cm2†

1:25  10ÿ3 0.02 2  10ÿ17 3  10ÿ18

0.0115 0.03 5  10ÿ18 3  10ÿ18

0.008 0

This This This This

Intersystem absorption ®t 1.4 10ÿ3 fT1 ;S 0 fex;T;S rT;I (cm2 )

1:75  10ÿ3 0.015 4  10ÿ16

0.012 0.02 4  10ÿ16

0.007 0

This work This work This work

rS0 ;P (cm2 ) rS1 ;P (cm2 ) rS1 ;L (cm2 ) rT1 (cm2 ) sF (ns) sFC (ps) sex;S (fs) sor (ps) s2T (ps) m~T1 (cmÿ1 ) m~I (cmÿ1 )a /S1 ;T /D;Sn c rex;T (cm2 ) sex;T (ps) …2† rL (cm4 s)

Assumed [79] Assumed [76]

This work This work This work work work work work

a

Gas-phase values or calculated semi-empirical values are reduced by 12 000 cmÿ1 to take care of solvent ionization potential reduction [62±65]. b Gas-phase ionization calculated using a semi-empirical PM3 program package [65]. c Values calculated here (Eqs. (1)±(10)) with experimental data of Ref. [10].

 N0 0 0 /S1 ;T  0 o N1T N1T ÿ N 1T 0 1T ˆ N ÿ ; ‡ N ÿ 0 2S 2S o t0 sS1 sph sor

0 0 N2S oN2S 3 rS1 ;P cos2 …h† 0 ˆ ÿ o t0 sFC h mP



0 N2S

! 0 N2S N0 0 N3S IP ÿ 2S ÿ 0 0 N2S ‡ N2S0 sS1

ÿ
‡ 1 ÿ /D;Sn 0 o N3S o t0

ˆ

0

0 N3S

sex;S

ÿ

0 N2S

ÿ sor

0 N 2S

;

…5† /D;Sn 0 o ND0 ˆ N ; 0 ot sex;S 3S …3† oIP ˆ ÿ3 rS0 ;P IP o z0

2

3 rS1 ;P cos …h† 0 0 0 …N2S ‡ N2S 0 ÿ N †IP 3S h mP ÿ

0 N 3S 0

0 N3S N0 ÿ ÿ 3S sex;S sor

;

…6†

Z

p=2 0

Z ÿ 3 rS1 ;P IP …4†

p=2 0

0 0 2 …N1S ÿ N2S 0 † cos …h† sin …h† dh

0 0 0 …N2S ‡ N2S 0 ÿ N † 3S

 cos2 …h† sin …h† dh ÿ a…2† IP2 ;

…7†

H. Gratz, A. Penzkofer / Chemical Physics 263 (2001) 471±490 0

Ni ˆ

Z 0

p=2

Z

i ˆ 1S; 2S; 2S0 ; 3S; 1T; D;  0 N2S 0 ;th

ˆ

TL ˆ

Ni0 …h† sin …h† dh;

0 …N2S

‡

0 N2S 0 † exp

 h m2S;2S0 ÿ : kB #

…8†

1

0

…9†

The singlet excited-state absorption cross-section, rS1 ;L , is probed with an attenuated magicangle polarized fundamental laser pulse shortly after ®rst second-harmonic pump pulse excitation (delay time td ˆ 25 ps). At the moment of probing the orientation-averaged and length-averaged level populations, N i …r; td †, are Z 1 ` 0 N i …r; td † ˆ N …r; z; td † dz; …11† ` 0 i where ` is the sample length. The probe pulse transmission is given by

r exp… ÿ r2 =rL2 † 

 expf ÿ ‰rS1 ;L N 2S …r; td † ‡ rT1 N 1T …r; td †Š`gdr  Z

The moving frame transformation t0 ˆ t ÿ nz=c0 and z0 ˆ z is used, where t is the time, z is the propagation coordinate, n is the refractive index, and c0 is the vacuum light velocity. h is the angle between the molecular transition dipole moment and the polarization direction of the pump laser [70]. sor is the reorientation time of the transition 0 dipole moment. N i is the orientation averaged 0 population number density of level i. N 2S0 ;th is the thermal population number density of level 2S0 . hm2S;2S0 ˆ hc0 m~2S;2S0 is the energy di€erence between level 2S0 and 2S. kB is the Boltzmann constant, and …2† # is the temperature. The term ÿaP IP2 in Eq. (7) takes care of two-photon absorption of the solvent …2† at wavelength kP . aP is the two-photon absorption …2† coecient (a ˆ 2  10ÿ10 cm Wÿ1 for toluene [7], …2† aP  0 for methanol). The initial conditions are Ni0 …r; t0 ˆ ÿ1; z0 ; 0 …r; t0 ˆ ÿ1; z0 ; h† ˆ N0 , and IP …r; h† ˆ 0 except N1S 0 0 t ; z ˆ 0† ˆ I0P exp …ÿr2 =rP2 † exp…ÿt02 =tP2 †, where r is the radial coordinate, rP is the 1=e-intensity pump beam radius, and tP is half the 1=e-intensity pump pulse duration. The ®rst pulse energy transmission, TEP , is R1 R1 r dr ÿ1 IP …r; `; t0 † dt0 0 R R : …10† TEP ˆ 1 1 r dr ÿ1 IP …r; 0; t0 † dt0 0

483

0

1

 r exp… ÿ r2 =rL2 † dr :

…12†

The saturable absorption dynamics in the triplet system caused by intense fundamental pulse excitation at a time delay long compared to the S1 state lifetime, sS1 , (td ˆ 20 ns) is described in the following. The involved initial level populations are 00 …r; z0 ; t0 ˆ ÿ1; h† N1S 0

0

0

ˆ N 1S …r; z0 ; te † ‡ ‰N 2S …r; z0 ; te † ‡ N 2S0 …r; z0 ; te † 0 …r; z0 ; te †…1 ÿ /D;Sn †Š…1 ÿ /S1 ;T † ‡ N3S

 f1 ÿ exp ‰ÿ…td ÿ te †=sS1 Šg;

…13†

00 …r; z0 ; t0 ˆ ÿ1; h† N2S 0

0

0 0 0 ˆ ‰N 2S …r; z0 ; te † ‡ N2S 0 …r; z ; te † ‡ N 3S …r; z ; te †

 …1 ÿ /D;Sn †Š…1 ÿ /S1 ;T † exp ‰ÿ…td ÿ te †=sS1 Š; …14† 00 …r; z0 ; t0 ˆ ÿ1; h† N1T 0

0

0 0 ˆ N 1T …r; z0 ; te † ‡ ‰N 2S …r; z0 ; te † ‡ N2S 0 …r; z ; te †

‡ N3S …r; z0 ; te †…1 ÿ /D;Sn †Š/S1 ;T  f1 ÿ exp ‰ÿ…td ÿ te †=sS1 Šg; Ni00 …r; z0 ; t0 ˆ ÿ1; h† ˆ 0;

…15†

i ˆ 2T; 2T0 ; 3T; 4T; I; …16†

where te is the time position at the end of the initial pulse (te ˆ 4tP is used in calculations). Including triplet±triplet intersystem crossing, triplet±singlet intersystem absorption, triplet photoionization, and singlet two-photon absorption, the di€erential equation system for the timedelayed fundamental laser pulse absorption reads:

484

H. Gratz, A. Penzkofer / Chemical Physics 263 (2001) 471±490 00

00 00 00 00 oN1S N2S N1T …N1S ÿ N 1S † ˆ …1 ÿ / † ‡ ÿ S ;T 1 0 ot sS1 sph sor …2†

5rL cos4 …h†

ÿ

2

2…hmL †

00 2 N1S IL ;

…17†

00 oN2S 3rS1 ;L cos2 …h† 00 00 ˆ ÿ …N2S ÿ N4S †IL ot0 hmL …2†

‡

5rL cos4 …h† 2…hmL †

‡ /2T;S

00 2 N1S IL ÿ

2

ÿ

00

00 N2S N 00 ‡ 4S sS1 sex;S

00 N2S ÿ N 2S ; sor

…18†

00 oN4S 3rS1 ;L cos2 …h† 00 00 ˆ …N2S ÿ N4S †IL ot0 hmL

00

00

…19†

00 00 00 oN1T N2S N1T 3 cos2 …h† ˆ / ÿ ÿ S ;T 1 hmL ot0 sS1 sph 00 00 00  ‰rT1 …N1T ÿ N2T 0 † ‡ rT ;S N 1T ŠIL 1 00

00 N 00 0 ‡ N2T N 00 ÿ N 1T ‡ …1 ÿ /2T;S † 2T ÿ 1T ; …20† s2T sor " 3 cos2 …h† 00 00 ˆ r1T;L …N1T ÿ N2T 0† hmL

00 ÿ rex;T N2T 0 ÿ

!

00 00 N2T 0 ‡ N2T

# 00 ÿ rex;T;S N2T 0 IL ÿ



1 sFC

‡

00 N3T

 1 00 N2T 0 s2T

00

ÿ

00 N2T 0 ÿ N 2T0 ; sor

…23†

00 oN4T 3 cos2 …h† 00 00 00 ˆ ‰rex;T …N3T ÿ N4T † ÿ rT;I N4T ŠIL 0 ot hmL

ÿ

00 N4S N 00 ÿ N 4S ÿ 4S ; sex;S sor

00 N2T 0

00 N3T N 00 ‡ …1 ÿ /ex;T;S † 4T sex;T sex;T 00 00 N3T ÿ N3T ÿ ; sor

ÿ 00

00 oN2T 0 ot0

…22†

00 oN3T 3 cos2 …h† h 00 00 00 ˆ rex;T …N2T ‡ N2T 0 ÿ N 3T † ot0 hmL i 00 00 00 ÿ N4T † ÿ rex;T;S N3T IL ÿ rex;T …N3T

3 cos2 …h† 00 ‡ ‰rT1 ;S N1T ‡ rex;T;S hmL

ÿ

00 N2T N 00 ‡ …1 ÿ /ex;T;S † 3T s2T sex;T

N 00 ÿ N 2T ÿ 2T ; sor

00 00 00 N2T ‡ N2T N 00 ‡ N4T 0 ‡ /ex;T;S 3T s2T sex;T

00 00  …N2T ‡ N3T †ŠIL ÿ

" 00 00 N2T oN2T 3 cos2 …h† 0 00 ˆ ÿ rex;T N2T hmL ot0 sFC ! # 00 N2T 00 00 N3T ‡ rex;T;S N2T IL ÿ 00 00 N2T0 ‡ N2T

…21†

00 N4T N 00 ÿ N 4T ÿ 4T ; sex;T sor

…24† 00

oNI00 3 cos2 …h† N 00 ÿ N I 00 ˆ rT;I N4T IL ÿ I ; 0 ot sor hmL Z p=2 oIL 00 00 00 ˆ ÿ3 I ‰rS1 ;L …N2S ÿ N4S † ‡ r1T …N1T L o z0 0 00 00 00 00 ÿ N2T 0 † ‡ rex;T …N 2T ‡ N2T0 ÿ N4T †

…25†

00 00 00 00 ‡ rex;T;S …N2T ‡ N2T ‡ r1T;S N1T 0 ‡ N 3T †

00 ‡ rsol NI00 Š cos2 …h† sin …h† dh ‡ rT;I N4T Z p=2 …2† r 00 N1S cos4 …h† sin …h† dh: ÿ 5 L IL2 hmL 0

The time-delayed fundamental transmission is R1 R1 r dr ÿ1 IL …r; `; t0 † dt0 0 R1 : TEL ˆ R 1 r dr ÿ1 IL …r; 0; t0 † dt0 0

pulse

…26†

energy …27†

The S0 ±S1 two-photon absorption terms in Eqs. (17), (18) and (26) are discussed in Refs. [71,72]. The ®rst pulse induced ¯uorescence signal 0 , is proportional to0 the S1 -state maximum, SF;max level population number density, N S1 …te †, at the

H. Gratz, A. Penzkofer / Chemical Physics 263 (2001) 471±490

end of the ®rst pulse excitation. This population number density is given by Z 1 Z ` 0 0 N S1 …te † ˆ 2pr dr ‰N 2S …r; z; te † 0

‡

0 0 N 2S0 …r; z; te †

0

‡ N 3S …r; z; te †

 …1 ÿ /D;Sn †Š dz:

…28†

The time-delayed pulse induced ¯uorescence signal 00 , is proportional 00to the S1 -state maximum, SF;max level population number density, N S1 …te † at the end of second excitation. It is given by Z 1 Z ` 00 00 N S1 …te † ˆ 2pr dr ‰N 2S …r; z; te † 0

0

00

‡ N 4S …r; z; te †Š dz:

…29†

The ¯uorescence signal ratio is 00

00 N S …te † SF;max : ˆ 01 0 SF;max N S1 …te †

…30†

The length-averaged number density of photoionized molecules, N001 …r; te †, at the end of the second pump pulse is given by N001 …r; te † ˆ

1 `

Z

` 0

NI00 …r; z; te † dz;

…31†

and the fraction of photoionized molecules, gI …r; te † is given by gI …r; te † ˆ

N00I …r; te † ; N0

…32†

where N0 is the total molecule number density. 5. Data analysis 5.1. Singlet±singlet excited-state absorption crosssection rS1 ;L The S1 excited-state absorption cross-section, rS1 ;L , at kL ˆ 1054 nm is obtained by ®tting Eq. (12) to the experimental transmission data in Table 1. The T1 excited-state absorption cross-section rT1 ;L at kL ˆ 1054 nm which enters Eq. (12) has been determined previously [10] and is listed in

485

Table 2. The best ®tting rS1 ;L values are given in Table 1. They are needed in the analysis of the triplet saturable absorption dynamics which involves the singlet system by intersystem crossing and intersystem absorption. 5.2. Triplet excited-state absorption and excitedstate relaxation The saturable absorption behaviour of the timedelayed fundamental laser pulses (kL ˆ 1054 nm) is analysed to gain information on the triplet excited-state absorption, rex;T , and the higher excited-state triplet relaxation time, sex;T . The triplet absorption cross-section, rT1 , has been determined previously by picosecond pump and time-delayed picosecond light continuum probing [10] (values are given in Table 2). The T1 state absorption recovery time, s2T , of eosin Y in methanol has been determined earlier by picosecond double pulse transient absorption measurements [5]. A value of s2T  1 ps was obtained. For erythrosin B a value of s2T  1:5 ps and for rose bengal a value of s2T  2:2 ps have been estimated in ref. [9] where the higher excited-state triplet± singlet intersystem crossing was studied. In Ref. [16] a lifetime of s2T ˆ 5:8 ps has been reported for rose bengal in phosphate bu€ered saline. The present analysis is not quite sensitive to s2T since rex;T is slightly larger than rT1 and the depletion of the T1 population does not show up in strong transmission changes. In a ®rst ®tting of the experimental TEL -data only rex;T and sex;T are varied while triplet±singlet intersystem transfer is neglected. The solid curves 1 in Figs. 4±7a show the expected saturable triplet absorption behaviour in the absence of triplet excited-state absorption, rex;T ˆ 0. The other solid curves are calculated for di€erent rex;T and sex;T values. The best ®tting rex;T values are determined by adjusting to the transmission behaviour in the low to medium intensity region (107 ±109 W cmÿ2 ), and the best ®tting sex;T values are determined by adjusting to the transmission behaviour in the high intensity region (109 ±1011 W cmÿ2 ). The shorter sex;T the more dicult it is to bleach the dyes. In a second step transmissions have been calculated using the best ®tting rex;T and sex;T values

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determined above together with the intersystem crossing and photoionization parameters determined in the next section. For eosin Y and C70 the in¯uence of the determined intersystem transfer and photoionization is negligible (curve 2 in Fig. 4 and curve 3 in Fig. 7 are practically not in¯uenced). For erythrosin B and rose bengal the best ®tting back-intersystem crossing parameters are used in Figs. 5b and 6b, respectively. The solvated electron (and possible solvated ion) absorption cross-section, rsol , is varied. A reasonable ®t is obtained using rsol ˆ …3  2†  10ÿ18 cm2 for both dyes in methanol. This absorption cross-section is reasonable for solvated electrons in methanol [68]. In Figs. 5c and 6c the curves are calculated for erythrosin B and rose bengal, respectively, using the best ®tting intersystem absorption cross-sections. For erythrosin B (Fig. 5c) the solvated electron absorption cross-section is set to rsol ˆ 5  10ÿ18 cm2 and the higher excited triplet state relaxation time is varied between sex;T ˆ 0:5 ps (curve 1) and 0.1 ps (curve 4). None of the calculated curves ®ts well to the experimental data. For rose bengal (Fig. 6c) the solvated electron absorption cross-section is varied (curve 1: rsol ˆ 0, curve 2: rsol ˆ 5  10ÿ18 cm2 ). The experimental data agree reasonably well to curve 2 …rsol ˆ 5  10ÿ18 cm2 †. The e€ects of back-intersystem crossing (Fig. 6b) and intersystem absorption (Fig. 6c) on the nonlinear energy transmission are too small to allow a distinction between intersystem crossing and intersystem absorption for rose bengal. The in¯uence of the T1 -absorption recovery time, s2T , on the saturable absorption is tested in Fig. 6a for rose bengal in methanol. The dotted curve belongs to the same parameter as the solid curve 2 except s2T ˆ 1 ps for the dotted curve and s2T ˆ 2.2 ps for the solid curve 2. The similarity of the curves shows that the transmission behaviour is not quite sensitive to s2T . 5.3. Triplet±singlet intersystem crossing, triplet± singlet intersystem absorption, and triplet photoionization The experimental intensity dependence of the 00 0 =SF;max , of Figs. 10± ¯uorescence signal ratio, SF;max

13 (eosin Y, erythrosin B, rose bengal, C70 ) is compared with numerical calculations (Eq. (30)) to gain information on the intersystem crossing parameters, /2T;S and /ex;T;S , the intersystem absorption parameters, rT1 ;S and rex;T;S , and the absorption cross-section rT;I leading to triplet ionization. The solid curves in Figs. 10a±13a are calculated for various quantum yields, /2T;S , of excited-state singlet population by T2 ±S intersystem crossing (/ex;T;S ˆ 0). The dashed curves in Figs. 10a±13a belong to various quantum yields, /ex;T;S , of excited-state singlet population by Tn ±S intersystem crossing …n > 2; /2T;S ˆ 0†. The curves show that /2T;S alone and /ex;T;S alone cannot ®t the experimental data. The curves in Figs. 11b, and 12b are calculated for the best ®tting combination of /2T;S and /ex;T;S . The ®t was made in the low to medium intensity range (I0L K 2  109 W cmÿ2 ). For the various curves the absorption cross-section for photoionization from level 4T to I is varied. The best ®tting /2T;S , /ex;T;S , and rT;I parameters are listed in Table 2. For eosin Y in methanol (dash-dotted curve in Fig. 10a) and for C70 in toluene (dash-dotted curve in Fig. 13a) only one curve /2T;S and /ex;T;S combination for rT;I ˆ 0 is shown. The second pulse ¯uorescence signals are too weak and inaccurate and the two-photon absorption contributions are too strong at high excitation intensities, I0L , to allow an accurate parameter determination for eosin Y and C70 . In Figs. 10b (eosin Y), 11c (erythrosin B), 12c (rose bengal), and 13b (C70 ) ¯uorescence signal 00 0 =SF;max , are calculated for T1 ±S interratios, SF;max system absorption (r1T;S ˆ fT1 ;S rT1 , solid curves), and Tn ±S intersystem absorption (r2T;S ˆ r3T;S ˆ rex;T;S ˆ fex;T;S rex;T , dashed curves). Again fT1 ;S and fex;T;S alone cannot ®t the experimental data. The curves in Fig. 11d (erythrosin B) and 12d (rose bengal) are calculated for the best ®tting combination of fT1 ;S and fex;T;S together with varying ionization cross-sections, rT;I . The fT1 ;S and fex;T;S ®t is made in the low and medium intensity range (I0L K 2  109 W cmÿ2 ). High ionization absorption cross-sections, rT;I , are needed to ®t the theoretical curves to the experimental data points at high pump laser intensities. The best

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®tting fT1 ;S , fex;T;S , and rT;I parameters are listed in Table 2. For eosin Y (dash-dotted curve in Fig. 10b) and for C70 (dash-dotted curve in Fig. 13b) only one fT1 ;S and fex;T;S combination for rT;I ˆ 0 is shown. The back-transfer ¯uorescence signals are too weak to allow a rT;I determination. In Fig. 15 the eciency of photoionization in erythrosin B (Fig. 15a) and rose bengal (Fig. 15b) is illustrated. The length-averaged fraction of ionized molecules, gI …0; te †, (Eq. (32)) at r ˆ 0 and t ˆ te at a ®xed I0P is plotted versus I0L for the best ®tting molecule parameters. The solid curves are calculated for triplet±singlet intersystem crossing while the dashed curves are calculated for triplet± singlet intersystem absorption. Assuming neat

487

triplet±singlet absorption for the triplet±singlet transfer process (dashed curves) results in unlikely high photoionization because of the required extremely high absorption cross-sections of photoionization. 5.4. S0 ±S1 two-photon absorption The fundamental pulse two-photon absorption is studied by exciting separately the samples with second-harmonic pulses (kP , single-photon absorption) and fundamental pulses (kL , two-photon absorption, delay time td P 1 min, complete depopulation of triplet system). The resulting ¯u00 0 =SF;max , are shown by trianorescence ratios, SF;max gles in Figs. 10a±13a for eosin Y, erythrosin B, rose bengal, and C70 , respectively. Numerical cal00 0 =SF;max (Eq. (30)) have been culations of SF;max …2† carried out to determine rL (dotted curves). The obtained two-photon absorption cross-section ®t parameters are listed in Table 2. For eosin Y in ethanol a value of r…2†  7:8  10ÿ50 cm4 s at kL ˆ 1060 nm has been reported in Ref. [73]. Our result for eosin Y in methanol at kL ˆ 1054 nm is slightly smaller. For erythrosin B and rose bengal no two-photon absorption cross-sections have been found in the literature. Two-photon absorption cross-sections at 1064 nm are reported for C70 thin ®lms in Refs. [74,75]. The value derived from Ref. [75] (r…2†  5:8  1049 cm4 s) is a factor of two larger than our value. In Ref. [74] a factor of eight larger value is reported. 6. Discussion

Fig. 15. Calculated fraction of ionized molecules, gI , at r ˆ 0 after second pulse excitation at td ˆ 20 ns as a function of the second pulse intensity I0L . Parameters of Table 2 apply. (a) Erythrosin B in methanol. I0P ˆ 1  1010 W cmÿ2 . Solid curve, triplet±singlet intersystem crossing with /2T;S ˆ 0:00125. /ex;T;S ˆ 0:02, rT;I ˆ 2  10ÿ17 cm2 , and rsol ˆ 3  10ÿ18 cm2 . Dashed curve, triplet±singlet absorption with fT1 ;S ˆ 0:00175, fex;T;S ˆ 0:015, rT;I ˆ 4  10ÿ16 cm2 , and rsol ˆ 3  10ÿ18 cm2 . (b) Rose bengal in methanol. I0P ˆ 1  1010 W cmÿ2 . Solid curve, triplet±singlet intersystem crossing with /2T;S ˆ 0:0115, /ex;T;S ˆ 0:03, rT;I ˆ 5  10ÿ18 cm2 . Dashed curve, triplet± singlet absorption with fT1 ;S ˆ 0:012, fex;T;S ˆ 0:02, rT;I ˆ 4  10ÿ16 cm2 , rsol ˆ 5  10ÿ18 cm2 .

The triplet saturable absorption studies reveal a strong triplet absorption bleaching at high excitation intensities for the investigated xanthene dyes. The numerical simulations reveal that the T2 excited-state absorption is larger than the T1 excited-state absorption (reverse saturable absorption situation). But the higher triplet relaxation time is relatively long (sex;T  1 ps for eosin Y and rose bengal, sex;T  0:5 ps for erythrosin B) leading to the population of higher excited states. Finally a weak absorption of a highly excited-state (4T) or

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of ionized molecules is required to bleach the fundamental pump pulse absorption. The higher excited-state triplet relaxation time observed here is longer than the higher excited-state singlet relaxation time generally observed in the singlet system [76]. For the fullerene molecules C70 in toluene no triplet absorption bleaching is observed. The numerical simulations give a larger T2 excited-state absorption than T1 excited-state absorption (reverse saturable absorption) and a short higher excited-state triplet relaxation time (sex;T  50 fs). The fact that rex;T > rT1 hinders light ampli®cation by T2 ±T1 emission in the case of population 00 00 > N1T †. Therefore no ampli®ed inversion …N2T spontaneous emission [77] or laser action is possible by T2 ±T1 emission for the investigated dyes even in the case of femtosecond pulse excitation (excitation pulse duration shorter than T2 relaxation time) [78]. The time-resolved ¯uorescence signals induced by ®rst frequency-doubled and second time-delayed fundamental pulse excitation clearly show the occurrence of triplet to singlet back transfer for the investigated molecules. The e€ect is especially strong for erythrosin B and rose bengal. The back-transfer was analysed assuming T2 ±S and Tn ±S back-intersystem crossing or T1 ±S and Tn ±S triplet±singlet intersystem absorption. A backintersystem crossing or intersystem absorption ef®ciency in the per thousand (eosin Y, C70 ) to the per cent (erythrosin B and rose bengal) region may explain the observed ¯uorescence behaviour. At high excitation intensities the ¯uorescence signal ratio becomes higher than the yield of back-intersystem transfer because of the multiple higher triplet state population within the laser pulse duration (relaxation time shorter than pulse duration). The saturation of the second pulse induced ¯uorescence signal at high excitation intensities, I0L , requires a removal of molecules from the triplet reservoir by photoionization. The triplet± singlet intersystem crossing process requires reasonably low photoionization absorption crosssections, rT;I , and allows a good ®tting of the energy transmission data. A ®t of the ¯uorescence signals to intersystem absorption requires unlikely high absorption

cross-sections, rT;I , and allows no good ®tting for the energy transmission data (erythrosin B). We think that triplet±singlet absorption may be present, but that back-intersystem crossing dominates. In this context we would like to note that for the porphyrine molecule meso-tetraphenyl-porphine we could verify spectroscopically a 3 Q±1 B absorption (triplet Q-band to singlet Soret band transition) [46]. For eosin Y and C70 the S0 ±S1 two-photon absorption induced ¯uorescence was comparable to the triplet±singlet back-transfer induced ¯uorescence. The two-photon absorption cross-sections have been determined by comparing the fundamental laser pulse induced two-photon ¯uorescence with the second-harmonic laser pulse induced single-photon ¯uorescence. For the xanthene dyes the eciency of triplet± singlet transfer increases form eosin Y to erythrosin B to rose bengal. The heavy atom e€ect of iodide in erythrosin B and rose bengal dominates over the heavy atom e€ect of bromide (lighter atom). 7. Conclusions The reverse saturable absorption and saturable absorption of three xanthene dyes and one fullerene dye in the triplet system have been studied by intense time-delayed double colour picosecond laser pulse excitation. The ®rst short-wavelength pulse populates the triplet system by singlet-state absorption and subsequent intersystem crossing. For the time-delayed long-wavelength triplet excitation pulse there occurs no single-photon singlet ground state absorption. The xanthene dyes have a rather slow Tn -state relaxation time (n > 2) of around one picosecond leading to successive higher triplet level population up to photoionization and resulting in triplet absorption bleaching at high intensities. A laser pulse induced transfer of molecules from the triplet system to the singlet system was observed by time-resolved S1 ±S0 ¯uorescence detection. The excited-state triplet to singlet molecule transfer seems to be dominated by triplet±singlet back-intersystem crossing, but some contributions

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