1. Introduction

Blue-emitting polyfluorenes exhibit strong two-photon absorption (TPA) properties in the near infrared (NIR). Determination of the two-photon and single-photon absorption (SPA) cross-sections on both the ground state and the excited states is very useful for better understanding the material-related transition mechanisms and for optimizing the corresponding semiconductor device properties. Stimulated emission depletion spectroscopy [1], two-photon microstructure writing [2] and imaging [3] with high resolutions, and two-photon detection devices [4,5] are potential applications of TPA materials. Two-photon-excitation time-resolved spectroscopy is a powerful tool to study the organic and inorganic semiconductors.

A number of methods have been demonstrated to measure the two-photon absorption cross-sections of organic molecules.[6–11] Measurement of the intensity dependence of the transmission is the most commonly used technique to resolve the TPA cross-section at a specific wavelength [6]. Using spatially dispersed white-light supercontinuum as the light source and measuring the TPA spectrum, the absorption cross-section can be measured over a broad band with the influence from nondegenerate TPA excluded [7]. The fluorescence method [8] can be considered as a background-free technique, where the TPA cross-section is measured by comparing the single- and two-photon excited fluorescence. A pump-probe scheme was demonstrated to measure the nondegenerate TPA cross-sections of organic molecules.[9]

In most cases, influences from the solvent, excited state absorption, and nonlinear optical effects during propagation through the sample need to be eliminated to improve the accuracy of the measurements. Furthermore, the absorption cross-section of the excited state is generally difficult to measure directly. However, if the absorption cross-sections are measured on the basis of mutually correlated transition schemes, the background-free absorption characteristics of multiple channels can be evaluated simultaneously.

In this work, we describe simultaneous determination of the ground state two-photon absorption and excited state single-photon absorption cross-sections at 1.55 eV in the prototypical member of polyfluorene: poly(9,9-dioctylfluorene-co-benzothiadiazole) (F8), making use of the correlation between the single- and two-photon processes, i.e. the single-photon pumping and two-photon probing spectroscopy. In the transient absorption (TA) measurements, the femtosecond pulses at 3.1 eV were used as the pump and those at 1.55 eV as the probe. The probe was delayed about 4 ps from the pump in the absorption cross-section measurements and had a relatively high intensity varying from 199 μJ/cm² to 1.99 mJ/cm² in order to obtain sufficiently strong TPA. The probe-intensity dependence of the TA enables simultaneous determination of the ground state TPA and the singlet exciton absorption cross-sections at the probe frequency.

2. Theory

2.1 Basic principles

The scheme of transitions with single-photon pumping and two-photon probing spectroscopy is depicted in Fig. 1. Single-photon excitation at 2 ω produced population on the odd-parity state of 1B_u and depopulation on the ground state (1A_g ) with an absorption coefficient α ₁ The time-delayed probe pulse at stakes part in two absorption processes: two-photon absorption from the ground state to the even-parity state of mA_g with an absorption coefficient of β ₂ and single-photon absorption from the lowest singlet exciton state of 1B_u to higher-lying state of kA_g with an absorption coefficient of β₁ The population densities on 1A_g , 1B_u , and mA_g states are N₁ , N₂ , and N₃ , respectively. Because of the depopulation of the ground state by the preceding pump and the long lifetime (a few hundred picoseconds [12][13]) of the singlet excited state of 1B_u , the two-photon absorption of the probe will be reduced or partially bleached. In principle, the reduction of the TPA would appear as a positive signal in the TA. However, because the absorption by the excited state is far larger than the bleaching effect of the TPA, this modulation will display as a reduction in the TA amplitude.

 

Fig. 1. Transition mechanisms related to the transient absorption in single-photon pumping (at 2ω) and two-photon probing (at 2ω) spectroscopy.

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Because the two-photon absorption increases with increasing pulse intensity, the bleaching effect becomes more pronounced with increasing probe intensity. Consequently, higher probe intensities give rise to smaller amplitude of TA signals. In this way, we can evaluate quantitatively the absorption properties of both the ground state and the excited state using the probe-intensity dependence of the TA at the two-photon frequency.

2.2 Single-photon excitation

The rate equation describing the population on the 1B_u state due to the single-photon excitation is given by:

where τ ₂ is the lifetime of 1B_u , N ₁=N ₀- N ₂ is the population on the ground state with N ₀ equal the density of states on the ground state before excitation, P₁ is the single-photon pumping rate defined as$P_1\left(t\right)=\frac{{\sigma }_{2\omega }^{\left(1\right)}{I}_{2\omega }\left(t\right)}{2\mathit{ħ}\mathit{\omega }}$ is the intensity of the pump pulses, and ${\mathrm{\sigma }}_{2\omega }^{\left(1\right)}$ is the ground state absorption cross-section at 2ω. For simplicity, we assume a rectangular pulse shape so that I _2ω(t)≈I _2ω and P₁ (t) is a constant of $P_1=\frac{{\sigma }_{2\omega }^{\left(1\right)}{I}_{2\omega }}{2\mathit{ħ}\mathit{\omega }}$for 0<t<τ_p where τ_p is the pulse length. Therefore, an analytical solution to equation (1) can be obtained as:

and N₁ =N ₀-N₂ .

In the measurements (Section 3) we used the solution sample of F8 that has a thickness of 1 mm, therefore, we have to take into account the pump depletion during propagation in the sample. Therefore, I _2ω actually decays with propagating along the z-axis that is perpendicular to the sample surface, which can be described approximately as I _2ω(z) = I _2ω0 exp(- ${\sigma }_{2\omega }^{\left(1\right)}$ N ₁ z). For extremely dilute solutions, this depletion should be small.

2.3 Two-photon probing

In single-photon pumping and two-photon probing spectroscopy, the probe arrives with a sufficiently large delay (t ₀) after the pump so that there is no temporal overlap between them. Thus, the possibly complicated interaction between them is avoided. On the other hand, this delay should be small enough with respect to the lifetime of the excited state so that the decay of the 1B_u excitons is negligibly small. In the section4, we have performed the measurement at a time-delay of 4 ps (t ₀≈4 ps) using a pulse length of about 250 fs. In this way, the probe pulses will be absorbed by the ground-state molecules through two-photon excitation and by the 1B_u excitons with a presumably constant density. The corresponding rate equations can be written as:

Where $P_2=\frac{{\sigma }_{\omega }^{\left(2\right)}{I}_{\omega }^2}{2\mathit{ħ}\mathit{\omega }}$ is the two-photon excitation rate and $P_3=\frac{{\sigma }_{\omega }^{\left(2\right)}{I}_{\omega }}{\mathit{ħ}\mathit{\omega }}$ evaluates the absorption rate of the probe pulses by 1B_u excitons. I_ω is the intensity of the probe pulses at the two-photon frequency ω. ${\sigma }_{\omega }^{\left(2\right)}$ and ${\sigma }_{\omega }^{\left(1\right)}$ are the two-photon absorption and excited state absorption cross-sections, respectively. The rate equations in (3)–(5) apply to both cases when the pump is switched on and switched off. However, the initial conditions (arrival of the probe pulse at t ₀) are different. When the pump is switched on:

$N_2\left(t\right)=N_0\frac{P_1{\tau }_2}{1+P_1{\tau }_2}\left[1-e^{-\left(\frac{1}{{\tau }_2}+P_1\right){\tau }_P}\right]·e^{\frac{-\left(t_0-{\tau }_P\right)}{{\tau }_2}}$ and N ₁|_t=t0 = N ₀ - N ₂|_t=t0and when the pump is switched off: N ₂|_t=t0=0 and N ₁|_t=t0 =N ₀.

Obviously, the excited state absorption on 1B_u at the probe has been included in both cases. Using extremely dilute solution sample (0.1 mg/ml) in the experimental work, we intend to avoid charge generation and charge absorption processes. Considering quick relaxation from mA_g to 1B_u (~100 fs)[14][15], we did not take into account further excitations from mA_g to higher-lying excited states. Additionally, we did not include the possible transitions to the triplet states in the above model.

2.4 Relating TA to the absorption cross-sections

The propagation equation describing the absorption mechanisms of the probe pulses at ω with an intensity of I_ω reads:

The transient absorption measurement is an evaluation of:

where $I_{\omega }^{\mathit{\text{on}}}$ and $I_{\omega }^{\mathit{\text{off}}}$ are the transmitted pulse intensities of the probe with the pump switched on and switched off, respectively, and both can be obtained by solving (6) in combination with the rate equations.

Because β ₁ is related to the exciton absorption cross-section ${\sigma }_{\omega }^{\left(1\right)}$ at ω through β ₁ = ${\sigma }_{\omega }^{\left(1\right)}$ N ₂, β ₂ is related to the ground state two-photon absorption cross-section ${\sigma }_{\omega }^{\left(2\right)}$ through β ₂ = ${\sigma }_{\omega }^{\left(2\right)}$ N ₁, and the population densities N ₁ and N ₂ are both dependent on ${\sigma }_{2\omega }^{\left(1\right)}$ and I _2ω, a series of measurements of ΔT/T at different probe intensities I_ω will give a simultaneously determination of ${\sigma }_{\omega }^{\left(1\right)}$ and ${\sigma }_{\omega }^{\left(2\right)}$.

Actually, the three absorption cross-sections of ${\sigma }_{\omega }^{\left(2\right)}$, ${\sigma }_{\omega }^{\left(1\right)}$, and ${\sigma }_{2\omega }^{\left(1\right)}$ can be determined simultaneously using TA dynamics data measured at different probe intensities. However, we found that the solution to the equations would then become sensitive to the accuracy of the TA measurements. As a result, noise in the TA data introduced relatively large errors in the calculation. Considering this sensitivity and the fact that the ground state single-photon absorption cross-section ${\sigma }_{2\omega }^{\left(1\right)}$ is relatively easy to measure, we used the measured value of ${\sigma }_{2\omega }^{\left(1\right)}$ as a given parameter to determine ${\sigma }_{\omega }^{\left(2\right)}$ and ${\sigma }_{\omega }^{\left(1\right)}$, ensuring the reliability of this method. Furthermore, moderately lower pump intensities can improve the accuracy, avoiding the saturation of the excitation.

3. Experimental method

In the experiment, we first measured ${\sigma }_{\omega }^{\left(1\right)}$ using the common absorption-spectrum method with a HP UV/VIS absorption spectrometer. The sample was a F8/p-Xylene solution with a concentration of 0.1 mg/ml and a thickness of 1 mm. Figure 2 demonstrates the absorption spectrum of the solution with the chemical structure of F8 shown in the inset. The peak absorption of F8 is around 3.2 eV, implying strong two-photon absorption around 1.6 eV. In our experiment, the pump pulse is centered around 400 nm (3.1 eV) and the probe around 800 nm (1.55eV). The absorption is approximately 0.96 OD at 3.1 eV for our dilute solution sample. To eliminate the background, we used 1-mm-thick p-Xylene solvent as the blank. ${\sigma }_{2\omega }^{\left(1\right)}$ was measured to be 1.38×10^-16 cm² at 400 nm (3.1 eV), where we have used the repeat unit of the F8 molecule chain as the chromophore. In the transient absorption measurement, the pump and the probe powers were measured using a 3sigma single-channel Laser Power/Energy meter, and their beam sizes at the sample were measured using the knife-edge method. The pump pulses at 3.1 eV were produced by doubling the ~100-fs fundamental pulses from a Ti:sapphire amplifier operating at 1 kHz. A small fraction of the fundamental pulses at 1.55 eV was used as the probe. ΔT/T was measured as a function of the delay between the pump and the probe pulses at different intensities of both the pump and probe pulses. To rule out the possibility of the influence from the nonlinearity of the detection system, we fixed both the reference and the signal levels before the monochromator using continuously adjustable attenuators with large dynamic ranges.

 

Fig. 2. Absorption spectrum of the F8/p-Xylene solution with a concentration of 0.1 mg/ml. Inset: the chemical structure of F8.

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4. Measurement results and discussion

4.1 Probe intensity dependence

Figure 3 shows a group of measurements of the TA dynamics with a pump intensity of about 96 μJ/cm², while the probe intensity varied from 600 μJ/cm² (open triangles), 990 μJ/cm² (open squares), to 1.99 mJ/cm² (open circles). To show the decay dynamics of the 1B_u state, TA measurements with relatively large delay ranges (-5 ~ 110 ps) are presented in Fig. 3. We attribute the reduction of the transient absorption amplitude with increasing the probe intensity to the bleaching of TPA. Using the ΔT/T values in Fig. 3 at a 4-ps delay (marked by the dash-dotted vertical line) from the pump pulses and the measured pump and probe intensities, we calculated ${\sigma }_{\omega }^{\left(2\right)}$ and ${\sigma }_{\omega }^{\left(1\right)}$ using the theory in section 2. In determining the measurement of TA at a given delay, we used an average value of four adjacent data (a delay window of 400 fs) centered around the specified data point. For a 4-ps delay, we can assume no decay of the population on 1B_u . The correct values of ${\sigma }_{\omega }^{\left(2\right)}$ and ${\sigma }_{\omega }^{\left(1\right)}$ should satisfy the three TA data at a 4-ps delay simultaneously. A TPA cross-section of 4.2×10^-21 cm⁴/GW was obtained for ${\sigma }_{\omega }^{\left(2\right)}$ and a value of 3.3×10^-19 cm² was calculated for the excited state absorption cross-section ${\sigma }_{\omega }^{\left(1\right)}$. Here we have also used the repeat unit of the F8 molecule chain as the chromophore. These absorption cross-sections are large when compared with other organic molecules [11] and their values vary within a range of about ±15% when using different groups of TA measurements with the probe intensity ranging from 199 μJ/cm² to 1.99 mJ/cm². The decay rate of the TA dynamics was found almost independent of the probe intensity in the long-term range, implying that the decay dynamics is mainly dependent on the long-lived 1B_u population excited by the single-photon pumping at 3.1 eV. This can be confirmed by comparing with the TA dynamics at 2 eV, which is shown by the filled circles in Fig. 3 and has been normalized to the dynamics at 1.55 eV with a probe intensity of 1.99 mJ/cm² (open circles).

 

Fig. 3. Dependence of the TA dynamics on the probe (1.55 eV) intensities at a pump (3.1 eV) intensity of 96 μJ/cm². The filled circles are TA dynamics measurements at 2 eV for the lifetime comparison. The calculation of the absorption cross-sections have been performed for a delay of 4 ps, as marked by the dash-dotted vertical line.

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4.2 Pump intensity dependence

Figure 4(a) and (b) demonstrate the pump intensity dependence of the TA dynamics measurements for probe fluences of 1.99 mJ/cm² and 199 μJ/cm², respectively, where the pump fluence varied from 22 μJ/cm² to 176 μJ/cm². Figure 4(c) shows the measured ΔT/T at a 4-ps delay as a function of the pump fluence for a probe fluence of 1.99 mJ/cm² (filled circles) and 199 μJ/cm² (open circles), respectively, and the solid curves are the corresponding simulation results. The calculated values of ${\sigma }_{\omega }^{\left(2\right)}$ and ${\sigma }_{\omega }^{\left(1\right)}$ given in section 4.1 have been used in both simulations. Basically, the simulations fit well to the measurement data, implying a physically reasonable model and reliable evaluations of the absorption cross-sections. At a pump fluence of 176 μJ/cm², larger discrepancy can be observed between the measurement and the simulation. The reason is that the intense pump has begun to saturate the excitation [16], so that the TA data tended to become independent of the pump intensity.

The absolute value of ΔT/T increases faster at a lower probe intensity of 199 μJ/cm² than at a higher probe intensity of 1.99 mJ/cm² with increasing the pump intensity, as can be seen by the different slopes of the two curves in Fig. 4(c). The mechanism can be understood by considering that the TPA is proportional to the square of the probe intensity. That means, the bleaching effect depends nonlinearly on the probe intensity.

Figure 5 gives the contour lines of the absolute value of the calculated ΔT/T as a function of both the pump and the probe fluences, showing that the absolute amplitude of the TA (|ΔT/T|) increases slower at higher probe intensity than at lower probe intensity with increasing the pump intensity, just as have been observed in Fig. 4(c), while it decreases faster at higher pump intensity than at lower pump intensity with increasing the probe intensity. Therefore, we can conclude that the same pump bleaches the TPA more strongly for a higher than for a lower probe intensity, and that this kind of bleaching effect will be enhanced with increasing the pump intensity.

 

Fig. 4. Pump intensity dependence of the TA dynamics at the two-photon frequency. The pump fluence at 3.1 eV was changed from 22 μJ/cm² to 176 μJ/cm² with the probe fluence at 1.55 eV fixed at (a) 199 μJ/cm² and (b) 1.99 mJ/cm², respectively. (c) Comparison between the measurements in (a) and (b) with the simulations using the measured absorption cross-sections.

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Fig. 5. Simulated contour lines showing the absolute values of the TA (|ΔT/T|) as a function of the pump and the probe fluences.

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5. Conclusions

We have investigated the bleaching of two-photon absorption by single-photon excitation in F8 and demonstrated a nearly “pure-molecule” method to determine simultaneously the two-photon absorption and singlet exciton absorption cross-sections. By comparison between the measured TA data and the theoretical simulation, a two-photon absorption cross-section of 4.2×10^-21 cm4/GW and a excited state absorption cross-section of 3.3×10^-19 cm² at 1.55 eV have been resolved for F8/p-Xylene solution. This method makes use of the mutual correlation between different transition channels and can be considered as a background-free technology, giving reliable evaluation on the absorption characteristics. In principle, the two-photon frequency does not necessarily equal half the single-photon frequency for excitation, therefore, this method can be extended for the measurement of the TPA cross-sections at different wavelengths using the same pump. Large two-photon absorption cross-section and large exciton absorption cross-section for the sequential excitation to higher lying states imply potentially efficient charge generation in this kind of conjugated polymers, which is important for photodetection devices.