1. Introduction

The proliferation of laser systems with ultrafast pulse durations gives the opportunity to investigate high-order nonlinear optical effects. Apart from fundamental academic interest, such investigations are motivated by potential applications for ultrafast all-optical devices and ultrafast information processing. One of the investigations is multiphoton absorption (MPA), which is of direct interest to fluorescence imaging [1], frequency-up-conversion lasing [2], optical microfabrication [3], optical power limiting [4], optical data storage [5], and photodynamic therapy [6]. Recently, studies of higher-order MPA related processes, in particular, three- and four-photon absorption (3PA and 4PA), have received considerable attention [7–11]. From the viewpoint of both fundamental research and technological applications, unambiguous characterization is crucial to full understanding of MPA processes.

As well known, MPA conventionally refers to the simultaneous absorption of n identical photons, promoting an electron from the ground state of a system, S ₀, to an excited state, S ₁, by virtual intermediate states [12]. This process is called a one-step n-photon absorption. For instance, Fig. 1(a) shows the one-step 4PA process. However, many of the strongest MPA molecules are hybrid chromophores where theMPA is accompanied by excited-state absorption (ESA). Under the excitation of high-power ultrafast-pulsed laser, the system simultaneously absorbs n identical photons, promoting an electron from S ₀ to S ₁ by virtual states. Subsequently, the electron is excited to higher-lying state S_h by absorbing another single photon, resulting in MPA-induced ESA. We name this process as a two-step (n+1)-photon absorption. As an example, Fig. 1(b) illustrates an effective 4PA process of 3PA-induced ESA process. Many investigations have revealed that the MPA-induced ESA response of a given sample is different in the magnitude — depending on the laser characteristics, in particular, on the laser pulse duration [12–16]. Two-photon-induced ESA has been experimentally and theoretically investigated [14–19]. For a 3PA-induced ESA process, observations have been experimentally reported for CdS [13] and SrTiO₃ [20]. However, there is no report in the literature on the unambiguous characterization of 3PA-induced ESA.

In the work presented here, we theoretically investigate a three-level two-step 4PA process of 3PA-induced ESA. The effective 4PA coefficient, which depends on the 3PA cross-section, excited-state photophysical properties, and laser pulse duration, is analytically presented. The analytical expression allows ones to unambiguously yet conveniently separate the contribution of 3PA from the effective 4PA in the case that both processes occur simultaneously.

 

Fig. 1. Schematic diagram of (a) one-step four-photon absorption and (b) two-step four-photon absorption.

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2. Theory

Under the excitation of ultrafast laser pulses, a simplified three-level model can adequately explain nonlinear absorption processes in polyatomic molecules because long-lived states such as triplet-triplet state can be justifiably neglected [15]. Figure 1(b) illustrates the proposed mechanism for three-level two-step 4PA process. This model is similar to the model of two-photon-induced ESA, in which 2PA is replaced by 3PA from the ground state [18]. The photodynamics of the system is described as follows. Initial absorption of the laser pulse promotes electrons from S ₀ to S ₁, by simultaneous absorption of three photons. From S ₁, the electrons relax to S ₀ or are promoted to S_h, by absorbing single photons. The propagation equation of the pulse intensity is given by

and the time derivatives of populations of states S ₀, S ₁, and S_h are governed by

where z′ is the propagation length inside the sample; σ ₃ and σ _e are the molecular 3PA and ESA cross-sections, respectively; τ _e and τ _h are the lifetimes of S ₁ and S_h, respectively; and h̄ω is the incident photon energy.

Usually the relaxation time τ _h from S_h to S ₁ is very short, so the electrons pumped onto S_h jump back promptly, and the population on S_h is close to zero. In this analysis, we assume that for the laser intensities used, the ground state S ₀ is not sufficiently depleted, therefore $N_{S_1}\ll N$ and $N_{S_h}\ll N$ , we set $N_{S_0}\simeq N$ . Under these conditions, Eq. (2) becomes

Solving this equation, the population density of S ₁ is given by

The above expression can be written as

where

To characterize the MPA properties of a material, time averaging techniques have been extensively exploited in both Z-scan and nonlinear transmission experiments [15, 21]. It should be noted that integrating Eq. (1) over time yields an equation that describes the attenuation of the pulse fluence. The last term on the right-hand side will contain an integral, ∫^+∞ _-∞ F(t)I ⁴(t)dt. The equation is unchanged if we replace this integral with 〈F〉 ∫^+∞ _-∞ I ⁴(t)dt, where

is the temporal average of F(t) weighted by intensity distribution I ⁴(t).

In solving Eq. (1), the transmitted intensity is numerically calculated and then it is integrated over time and space to give the transmitted energy. We assume that a reasonable approximation to the transmitted energy can be yielded by replacing F(t) with its corresponding time average 〈F〉 in Eq. (7), solving Eq. (1) to determine the transmitted intensity, and then integrating temporally and spatially to obtain the transmitted energy. We also assume a temporal Gaussian profile for the incident pulses. With these approximations, Eq. (1) can be revised as

where

Here α ₃=σ ₃ N is the 3PA coefficient [3PA process is usually quantified by 3PA cross-section σ _3PA through the conversion formula σ _3PA=α ₃(h̄ω)²/N=σ ₃(h̄ω)²], α ₄ is the effective 4PA coefficient, and τ is the half-width at e ^-1 of the maximum for the pulse duration. Obviously, Eq. (9) predicts that the effective 4PA coefficient depends on 3PA coefficient, ESA cross-section, lifetime of the first excited state, and laser pulse duration. In the limit of τ _e≪τ, Eq. (9) becomes α ₄=σ _e α ₃ τ _e/3h̄ω. Under the extreme condition of τ _e≫τ, Eq. (9) gives α ₄=√πσ _e α ₃ t/(3√3h̄ω). In particular, we find an empirical analytic expression as follows:

Figure 2 shows the effective 4PA coefficient α ₄ as a function of the pulse-duration τ for τ _e=1 ps obtained by numerical simulation (circles) and our analytic solution (solid line), respectively. Clearly, our analytical solution agrees well with the numerical simulation. For the 3PA-induced ESA process, the ESA lifetimes and ESA cross-section are of a few picoseconds and usually close to 10^-17 cm², respectively [10, 13, 20]. It is interesting to note that the difference of effective 3PA coefficients measured in the nanosecond regime and in the femtosecond regime for the same sample could be too large in order of magnitude [12, 13]. This fact implies that, as illustrated in Fig. 2, in the nanosecond regime, the 3PA-induced ESA could play a more important role than the pure 3PA process. With the excitation of femtosecond laser pulses, however, 3PA is the dominant mechanism of whole nonlinear absorption process.

 

Fig. 2. Pulse-width dependence of effective 4PA coefficient for τ _e=1 ps. The circles are numerical simulations, while the solid line is the analytical result. The effective 4PA coefficient is normalized to σ _e α ₃ τ _e/3h̄ω.

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At low intensity, 3PA process is predominant because the $N_{S_1}$ population is not sufficient and no electron could be excited from S ₁ to S_h. At excitation intensity exceeding a critical value, the competition between 3PA and ESA occurs. That is, Eqs. (8) and Eq. (12) are valid only above a critical intensity. Using the steady-state condition in Eq. (3), we estimate the critical intensity

where $N_{S_1}^c$ is the critical population of S ₁. Under the excitation of low intensities (I<I_c), the sample only exhibits 3PA effect. When the intensity is larger than the critical intensity, however, the simultaneous appearance of 3PA and 3PA-induced-ESA effects is observable.

The popular yet effective techniques to characterizeMPA related studies are Z-scan and nonlinear transmittance measurements [12, 21]. In solving Eq. (8), we assume that the spatiotemporal profiles of incident pulses are Gaussian. To simulate the z-dependent nonlinear transmission, the transmitted intensity is numerically calculated and then it is integrated over time and space to give the transmitted energy. We determine the same Z-scan traces using Eqs. (1) and (3), and (8) and (9), respectively. It indicates that the effective 4PA theory of 3PA-induced ESA we presented correctly describes the Z-scan measurements.

For a thin sample with simultaneous 3PA and effective 4PA, we obtain the normalized energy transmittance T(z,p,h), in polynomial series form, as follows

Here p=p ₀/(1+z ²/z ² ₀) and h=h ₀/(1+z ²/z ² ₀), where p ₀=(2α ₃ I ² ₀₀ L)^1/2 and h ₀=(3α ₄ I ³ ₀₀ L)^1/3 are the on-axis peak phase shifts caused by the 3PA and 4PA processes, respectively; I ₀₀ is the on-axis peak intensity at the beam waist; z ₀ is the Rayleigh length of the Gaussian beam; and L is the sample physical length. The coefficient amn in Eq. (12) is listed in Table 1. The above empirical expression has an utmost error of 0.3 % with the exact numerical results when 0≤p ₀≤π and/or 0≤h ₀≤π. For the pure 3PA (α ₄=0) and 4PA (α ₃=0), Eq. (12) degenerates into the following forms, respectively:

Table 1. Coefficient a_mn when 0≤p₀≤π and 0≤h₀≤π.

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In Eq. (12), there are two free parameters, namely α ₃ and α ₄. Nevertheless, the α ₃ value is unambiguously obtained by Eq. (13) from the measurements at low intensities where ESA is negligible. With the measured α ₃, the value of α ₄ can be extracted from the z-dependent nonlinear transmission measurements at high intensity by the use of Eq. (12).

It should be emphasized that Eq. (12) is applicable to the following three cases: (i) simultaneous occurrence of 3PA and 3PA-induced ESA processes in polyatomic molecules, (ii) appearance of both 3PA and 3PA-generated free-carrier absorption in semiconductors such as in CdS [13], and (iii) interplay between 3PA and intrinsic 4PA, a situation that exists, for example, in a conjugated polymer at 1700 nm [7].

3. Conclusion

In summary, we have proposed a three-level two-step 4PA model of 3PA-induced ESA. We have derived an analytical expression for the effective 4PA coefficient that depends on the 3PA cross-section, excited-state photophysical properties, and laser pulse duration. We have also presented the analytical theory of the z-dependent nonlinear transmission for straightforwardly yet unambiguously characterizing 3PA-induced ESA.