## Abstract

Based on the simplified five-level rate-equation theory, we investigate the hybrid third-order optical nonlinear processes which combine the instantaneous nonlinearity and the one-photon absorption induced excited-state nonlinearity. We obtain the analytical third-order nonlinear absorption and refraction coefficients originating from the singlet and triplet excited-state effects. We explore the photodynamic process and give the corresponding level diagram in the nanosecond, picosecond, and femtosecond regimes. We develop the pulse-duration-dependent Z-scan theory for characterizing the simultaneous instantaneous nonlinearity and cumulative effect of the excited-state nonlinearity. We also demonstrate the validity of the presented theory to analyze the experimental results.

© 2010 Optical Society of America

## 1. Introduction

Interaction of light with matter, which leads to abundant nonlinear optical phenomena, plays a crucial role in modern optics [1]. Optical nonlinearity may modify either the direction of propagation, phase, or spatiotemporal characteristics of an optical pulse through nonlinear medium. Alternatively, for nonlinear optical materials, the ability of transferring, processing and storing information offers the basis of photonic technology. To exploit the potential applications of the nonlinear optical material, it is imperative to accurately determine the nonlinear parameters and then to understand the photodynamic behaviors. The nonlinear optical processes may involve electronic Kerr effect, multiphoton absorption, and excited-state nonlinearity [2].

There is an extensive interest in understanding the excited-state nonlinearity of organic materials due to the academic interest and the practical applications [3–6]. The excited-state nonlinearity has been investigated in a variety of organic materials such as organic dyes [7, 8], conjugated molecules and polymers [9–13], charge transfer molecules [14], and miscellaneous molecules [15–17]. In addition, the experimental observations have revealed that the optical nonlinearities depend strongly on the pulse duration of the excitation laser [14–19]. The optical nonlinearities measured in nanosecond regime could be two orders of magnitude greater that in picosecond regime [15–19]. To gain insight into the underlying mechanism behind the observed optical nonlinearities, some energy-level diagrams have been presented, such as three-level [20], four-level [3], and five-level models [6,17]. To understand the photodynamic process, multilevel rate-equation analysis, which requires a complex numerical computation, is usually adopted. For the experimentalists, it is highly expected to seek an analytical theory that could adequately explain and reliably predict the observed optical nonlinearities within a reasonable degree of approximation.

In the present article, based on the simplified five-level rate-equation theory, we investigate the hybrid third-order optical nonlinear processes which combine the instantaneous nonlinearity and the cumulative nonlinearity of excited-state nonlinearity induced by one-photon absorption (1PA). We give the high-accuracy approximate analytical results for the effective third-order nonlinear coefficients caused by excited-state absorption and refraction. We explore the nonlinear photodynamics process and give the corresponding level diagrams at different timescales. We develop the pulse-duration-dependent Z-scan theory under the thin sample approximation, with simultaneous instantaneous nonlinearity and cumulative nonlinearity caused by the excited-state effect. We also verify the validity of the present theory by analyzing the published experimental data.

## 2. Theory

As is well known, the photodynamic process for the excited-state enhancement of the third-order nonlinear absorption (NLA) process can be described by the five-level model illustrated in Fig. 1(a). This five-level model involves the excited-state absorption (ESA) from excited singlet and triplet states, as well as two-photon absorption (2PA) between singlet states. The photodynamic process in this system can be described as follows. Electrons in the singlet ground state *S*_{0} can be promoted to the singlet excited states *S*_{1} and *S _{h}*, based on 1PA and 2PA, respectively. The electrons in

*S*

_{1}may relax to

*S*

_{0}by a radiative or nonradiative transition, undergo the intersystem crossing to the lowest triplet state

*T*

_{1}through a spin-flip process, or be promoted to the higher-lying singlet state

*S*by absorbing another photon. For an electron in

_{h}*T*

_{1}, it may relax to

*S*

_{0}by a phosphorescence process or be promoted to

*T*by absorbing another photon. Consequently, the nonlinear absorption process in this system happens the transition processes as follows: (i)

_{h}*S*

_{0}→

*S*

_{1}by 1PA, (ii)

*S*

_{0}→

*S*by instantaneous 2PA, (iii)

_{h}*S*

_{0}→

*S*

_{1}→

*S*by a two-step 2PA, and (iv)

_{h}*T*

_{1}→

*T*by ESA. At the same time, 1PA populates new electronic state, which could be an excited bound state in an organic molecule. Besides the optical Kerr effect caused by the distortion of the electron cloud, the significant population redistribution also produces an additional change in the refractive index, leading to the 1PA-induced excited-state refraction (ESR).

_{h}Under the five-level model, the rate equations for populations of states can be described by

*σ*

_{0},

*σ*

_{S1}, and

*σ*

_{T1}are the absorptive cross-sections of the ground state

*S*

_{0}, the first excited singlet state

*S*

_{1}, and the lowest triplet state

*T*

_{1}, respectively.

*β*is the 2PA coefficient from

*S*

_{0}to

*S*.

_{h}*τ*is the lifetime of the state

_{j}*j*(

*j*=

*S*

_{1},

*S*,

_{h}*T*

_{1},

*T*) and

_{h}*τ*is the intersystem crossing time from

_{isc}*S*

_{1}to

*T*

_{1}, in particular, their typical values for the organic materials at room temperature [14–17] are shown in Fig. 1(a).

*N*and

*N*are the total number density and the number density of the state

_{j}*j*, respectively.

*h̄ω*is the incident photon energy.

Under the thin sample approximation, the intensity and phase changes of the optical field within the sample obey the following equations

*z*′ is the propagation length inside the sample,

*γ*is the optical Kerr refraction index,

*η*and

_{S}*η*are the refractive cross-sections of the first excited singlet state and the lowest triplet state, respectively, and

_{T}*k*is the wave vector in free space.

It should be noted that Eqs. (1) and (2) are sufficient for describing the characteristics of the transmitted optical field. However, for the data analysis, the complex numerical calculation is required [8]. In fact, the timescale of the pulse duration for the laser irradiation is usually tens of nanoseconds to hundreds of femtoseconds. In what follows, we neglect the populations on both *S _{h}* and

*T*because both

_{h}*τ*

_{Sh}and

*τ*

_{Th}are much smaller than the used pulse duration. Furthermore, we assume that the

*S*

_{0}state is not sufficiently depleted for the incident optical intensity, i.e.

*N*≪

_{S1}*N*and

*N*≪

_{T1}*N*, and then we can take

*N*≈

_{S0}*N*. Considering the laser pulses with a Gaussian temporal profile, we have

*I*(

*t*) =

*I*

_{0}exp(−

*t*

^{2}

*/τ*

^{2}), where

*τ*is the half-width at

*e*

^{−}

^{1}of the maximum for the pulse duration (

*τ*and

_{F}*τ*are related by ${\tau}_{F}=2\sqrt{\text{ln}2\tau}$, where

*τ*is the full width at half maximum of the pulse). The above-mentioned approximations are adopted because this condition (Z-scan measurements at relatively low irradiance) is easily satisfied. Most importantly, within the reasonable approximation, the obtained theory could adequately interpret and reliably predict the photodynamic process of the materials under the pulsed excitation of different time scales, as we demonstrate below. Under these conditions, the population densities of

_{F}*S*

_{1}and

*T*

_{1}are given by

*τ*(with ${\tau}_{S}^{-1}={\tau}_{{S}_{1}}^{-1}+{\tau}_{\mathit{isc}}^{-1}$) is the effective lifetime of the first singlet state,

_{S}*α*

_{0}=

*σ*

_{0}

*N*is the linear absorption coefficient, and

*φ*=

_{T}*τ*/

_{S}*τ*is the triplet quantum yield.

_{isc}Substituting Eq. (3) into Eq. (2), we obtain

With Eq. (5), the optical field at the exit plane of the sample is determined by

*L*

^{eff}= [1 – exp(−

*α*

_{0}

*L*)]/

*α*

_{0}is the effective length of sample and

*L*is the physical length of sample.

Based on the Huygens-Fresnel diffraction integral method [1], we obtain the optical field with a propagation distance *d* from the exit plane of the sample

*J*

_{0}() is the Bessel function of zero order. If |

*q*(0, 0;0)| < 1, one could yield the analytical expression of

*E*(

_{a}*r*,

_{a}*z; t*) by applying the Gaussian decomposition method [21–23].

## 3. Discussion

To characterize the optical nonlinearities, the time-averaging technique has been extensively exploited in the Z-scan measurement due to its high sensitivity, experimental simplicity, and simultaneous determination of signs and magnitudes of the nonlinear (absorption and refraction) coefficients [22]. For the incident pulses with a Gaussian spatiotemporal profile, we obtain the normalized energy transmittance of the open-aperture (OA) Z-scan by

*p*(

*z, t*) = [

*β*+

*α*(

_{S}*t*) +

*α*(

_{T}*t*)]

*I*(

*z*)

*L*

^{eff}and $I(z)={I}_{00}/(1+{z}^{2}/{z}_{0}^{2})$. Here

*I*

_{00}and

*z*

_{0}are the on-axis peak intensity and the Rayleigh length of the Gaussian beam, respectively. For the closed-aperture (CA) Z-scan measurements, we have

*ω*

_{0}is the waist radius of the Gaussian beam. The linear transmittance of the aperture in the far field is $S=1-\text{exp}(-2{R}_{a}^{2}/{\omega}_{a}^{2})$, where

*R*is the aperture radius and

_{a}*ω*is the beam radius at the aperture in the linear regime.

_{a}Generally speaking, the determination of both the photophysical parameters (*β*, *γ*, *σ _{S}*,

*σ*,

_{T}*η*, and

_{S}*η*) and the response times (

_{T}*τ*and

_{S}*τ*) requires the complex calculations for the best fitting of the experimental data (both OA and CA Z-scans) with Eqs. (10) and (11). In what follows, we simplify the procedure to estimate the nonlinear parameters. Using the time-average approximation as we presented previously [24], we obtain the effective nonlinear coefficients arising from the cumulative effect of 1PA-induced excited-state nonlinearity as

_{T}*α*and

_{S}*α*are the effective third-order NLA coefficients originating from the singlet and triplet ESAs, whereas

_{T}*n*and

_{S}*n*are the effective third-order nonlinear refraction (NLR) indices from the singlet and triplet excited-state refractions (ESRs), respectively. To determine the effective parameters (

_{T}*α*,

_{S}*α*,

_{T}*n*, and

_{S}*n*), one should perform numerical integration. Actually, the typical values of

_{T}*τ*and

_{S}*τ*are about picosecond and sub-microsecond orders, respectively [14–17]. In addition, the pulse duration of the laser excitation is generally tens of nanoseconds to hundreds of femtoseconds, which satisfies the condition of

_{T}*τ*≪

*τ*. Accordingly, we find the analytical expressions for

_{T}*α*and

_{S}*α*at three conditions, as summarized in Table 1. Similarly, the results for

_{T}*n*and

_{S}*n*are listed in Table 2. When

_{T}*τ*≪

*τ*≪

_{S}*τ*, the 1PA-induced excited-state nonlinearities caused by the excited singlet and triplet states depend on the pulse duration. In this instance, the nonlinearities originate from the cumulative effect and exhibit the fluence-dependence instead of the intensity-dependence. For

_{T}*τ*≪

_{S}*τ*≪

*τ*, however, the nonlinearities originating from the singlet and triplet states become independent and dependent of the pulse duration, respectively. In this case, the nonlinearities induced by the singlet and triplet states are treated as the instantaneous and cumulative effects, respectively.

_{T}Taking the typical photophysical parameters of organic materials [14–17] as listed in Table 3, we explore the effective nonlinear coefficients from the singlet and triplet states, by using numerical simulation with Eq. (12) at different pulse durations, as displayed by the symbols in Fig. 2. For comparison, the corresponding analytical results (given in Tables 1 and 2) are also shown in Fig. 2 by the solid lines. Obviously, the analytical results are in good agreement with the numerical simulations. It can also be found that the dependence of the NLA coefficient on the pulse duration is similar to that of the NLR index, as shown in Fig. 2.

In the subpicosecond regime, as illustrated in Fig. 2, the 1PA-induced excited-state nonlinearity becomes negligible. In this case, the instantaneous nonlinearity is the dominant mechanism in the whole nonlinear process. The NLA process can be described by 2PA in Fig. 1(b). Accordingly, the measured NLA and NLR coefficients with the subpicosecond pulses are closer to the intrinsic values. In a few to hundreds of picosecond timescales, the 1PA-induced singlet excited-state nonlinearity mainly contributes to the observed nonlinear effect as shown in Fig. 2. In this instance, the photodynamic process can be described as Fig. 1(c). The laser pulse promotes an electron from *S*_{0} to *S*_{1} by absorbing one photon. Subsequently, the electron can be excited to *S _{h}* by absorbing another photon, resulting in 1PA-induced ESA. Thus, Fig. 1(c) can adequately explain the NLA process in the picosecond regime. Under the pulsed excitation of subnanosecond to few nanoseconds, as shown in Fig. 2, the contributions of the singlet and the triplet state nonlinearities are comparable. The photodynamic process can be illustrated as Fig. 1(a) except for 2PA. The observed nonlinear effect mainly originates from the 1PA-induced excited state nonlinearities because the instantaneous nonlinear process is negligible and can be justifiably omitted in the analysis. For tens-of-nanoseconds and longer pulsed irradiance, the resultant nonlinear effect basically arises from the triplet nonlinear process, as illustrated in Fig. 2. This result is consistent with the experimental observation [14]. In the instance, NLA can be interpreted by four-level diagram as Fig. 1(d). For an electron in

*S*

_{1}, it may relax to

*S*

_{0}or undergo a spin-flip process to

*T*

_{1}. From

*T*

_{1}, the electron may relax to

*S*

_{0}by a phosphorescence process or be promoted to

*T*by absorbing another photon. It should be noted that the above-mentioned conclusions are applicable for organic materials in general.

_{h}As we discussed previously [23], one can easily obtain the OA Z-scan trace by

As illustrated in Fig. 3, the Z-scan signal strongly depends on the pulse duration for a given on-axis peak intensity. The valley depth of the OA Z-scan Δ*T _{V}* (= 1 –

*T*) and the normalized peak-valley transmittance difference in the CA Z-scan Δ

_{V}*T*(=

_{PV}*T*–

_{P}*T*, where

_{V}*T*and

_{P}*T*are the normalized transmittance at the peak and the valley, respectively) as a function of the pulse duration

_{V}*τ*are shown in Figs. 4(a) and 4(b), respectively. Both Δ

*T*and Δ

_{V}*T*are monotonously nonlinear increasing functions of

_{PV}*τ*.

Based on the above-mentioned discussions, the excited-state photophysical parameters can be exploited fully by performing both the OA and CA Z-scan measurements [22] under the laser excitation with the fixed photon energy in the femtosecond, picosecond, and nanosecond regimes. Under the excitation of hundreds of femtoseconds laser pulses, the instantaneous nonlinear coefficients (both *β* and *γ*) could be extracted by using the femtosecond-pulsed Z-scan theory [23]. In tens of picosecond timescales, the singlet excited-state photophysical parameters (*σ _{S}* and

*η*) could be evaluated unambiguously by the presented Z-scan theory by ignoring the triplet state contribution. For the nanosecond pulsed excitation, the best fittings between the experimental data and the presented Z-scan theory with the known

_{S}*σ*and

_{S}*η*give the triplet state parameters of both

_{S}*σ*and

_{T}*η*. The singlet and triplet excited-state lifetimes,

_{T}*τ*

_{S}and

*τ*

_{T}, could be estimated by additional experiments such as femtosecond and nanosecond transient difference absorption [16] or pump-probe measurements [19]. In addition, the intensity-dependent Z-scan should be carried out to confirm that the observed nonlinearities are indeed a third-order process.

To check the validity of the presented theory, we have analyzed the published experimental results. Figure 5 shows the OA Z-scan traces for [(C_{2}H_{5})_{4}N]_{2}[Cu(C_{3}S_{5})_{2}] (DCu1) in acetone solution with *τ _{F}* = 40 ps (squares) and 18 ns (circles), respectively [25]. Fan

*et al*. [25] qualitatively analyzed the experimental results based on a five-level model. In this analysis, we take the typical lifetimes of

*τ*≃ 100 ps,

_{S}*τ*≃ 2 ns, and

_{isc}*τ*≃ 0.1 ms for organic molecules and fit the experimental data by Eq. (13) with the known parameters of

_{T}*L*= 2 mm,

*I*

_{00}= 1.48 GW/cm

^{2},

*h̄ω*≃ 1.18 eV, and

*α*

_{0}= 0.253 cm

^{−1}. The best fitting shown in Fig. 5 by the solid lines gives

*β*= 0.1 cm/GW,

*σ*= 2.6 × 10

_{S}^{−}

^{18}cm

^{2}, and

*σ*= 5.4 × 10

_{T}^{−}

^{18}cm

^{2}. These values are comparable with the findings for other organic molecules [15–17], which indicates that the presented theory could adequately interpret and give an insight into the physical mechanism for the observed optical nonlinearities in the nanosecond and picosecond regimes.

Recently, He *et al*. [26] have reported the giant third-order optical nonlinearities of the thin films containing anionic tetracarboxylic copper phthalocyanine [CuPc(COONa)_{4}] and cationic polydiallydimethylammonium chloride (PDDA) under the excitation of picosecond and nanosecond pulses. Figures 6(a) and 6(b) display the Z-scan traces of the 26-bilayer CuPc(COONa)_{4}/PDDA film at *τ _{F}* = 21 ps and

*τ*= 4 ns, respectively [26]. The valley-to-peak configurations shown by symbols in Fig. 6 are the results of dividing the CA Z-scan by the OA Z-scan. The experimental data indicates that the NLA coefficient and the NLR index of the film measured at

_{F}*τ*= 4 ns are two orders larger than those at

_{F}*τ*= 21 ps [26]. In what follows, we analyze the experimental data using our simplified theory and gain insight into the physical mechanism of the observed nonlinearities. We take the typical lifetimes of

_{F}*τ*≃ 100 ps,

_{S}*τ*≃ 2 ns, and

_{isc}*τ*≃ 0.1 ms for organic molecules and the known parameters of

_{T}*L*= 285 nm,

*α*

_{0}≃ 2.0 × 10

^{4}cm

^{−}

^{1},

*h̄ω*= 2.34 eV,

*S*= 0.13,

*I*

_{00}= 1.77 × 10

^{−}

^{2}GW/cm

^{2}at

*τ*= 4 ns, and

_{F}*I*

_{00}= 3.65 GW/cm

^{2}at

*τ*= 21 ps. The best fittings shown in Fig. 6 indicate that

_{F}*β*≃ 0,

*γ*≃ 0,

*σ*= 2.5 × 10

_{S}^{−}

^{18}cm

^{2},

*σ*= 9.4 × 10

_{T}^{−}

^{18}cm

^{2},

*η*= 0.41 × 10

_{S}^{−}

^{22}cm

^{3}, and

*η*= 5.1 × 10

_{T}^{−}

^{22}cm

^{3}. The difference between the fitting curves and the CA Z-scan experimental data is understood as follows. In Ref. 26, the relatively rough data processing method was adopted for determining the contribution of NLR, which is the CA Z-scan trace divided by the corresponding the OA one. Consequently, the interval between the valley and the peak of the obtained nominal CA Z-scan trace becomes larger than that of the CA Z-scan trace for the pure NLR [27]. The evaluated excited-state parameters are comparable with the typical values listed in Table 3. The resultant NLA coefficients of 2.1 × 10

^{4}cm/GW at

*τ*= 21 ps and 8.7 × 10

_{F}^{5}cm/GW at

*τ*= 4 ns are twice larger than those reported by He

_{F}*et al*. [26]. The total NLR indices (3.4 × 10

^{−}

^{2}cm

^{2}/GW at

*τ*= 21 ps and 4.3 cm

_{F}^{2}/GW at

*τ*= 4 ns) are closer to the reported ones (3.7 × 10

_{F}^{−}

^{2}cm

^{2}/GW at

*τ*= 21 ps and 4.1 cm

_{F}^{2}/GW at

*τ*= 4 ns). The observed optical nonlinearities can be understood as follows. The electronic structure of the film can be simplified as a five-level model, as illustrated in Fig. 1(a). In tens of picoseconds timescale, the singlet excited-state nonlinearity dominates the contribution to the observed nonlinear effect, whereas the triplet excited-state optical nonlinearity is the dominant mechanism under the pulsed excitation of few nanoseconds.

_{F}## 4. Conclusion

In summary, we have investigated the enhancement of third-order optical nonlinearities by the 1PA-assisted excited state process under reasonable approximations. We have obtained the analytical results for the effective third-order nonlinearities caused by the 1PA-assisted excited-state absorption and refraction. We have studied the dominant photodynamics process of the system and discussed the corresponding level diagram in the nanosecond, picosecond, and femtosecond regimes. We have developed the pulse-duration-dependent Z-scan theory for characterizing both the instantaneous nonlinearity and the cumulative nonlinearity of excited-state nonlinear effect. We have also demonstrated that the presented theory can well describe the experimental results.

## Acknowledgments

This work was supported in part by the National Science Foundation of China under 10704042 and 10934003. J. Chen also acknowledges the support from the Program for New Century Excellent Talents in University.

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