Two-Step Two-Photon Absorption Dynamics in π-π Conjugated Carbazole-Phthalocyanine/Graphene Quantum Dot Hybrids Under Picosecond Pulse Excitation

Abstract

In carbazole-substituted phthalocyanine complexes 2,3,9,10,16,17,23,24-octakis-(3,6-dibromo-9Hcarbazol) phthalocyaninato zinc(II) (Pc 2) and 2,3,9,10,16,17,23,24-Octakis-(9H-carbazol-9-yl) phthalocyaninato zinc(II) (Pc 4) and their conjugated complexes to graphene quantum dots (GQDs), we studied the nonlinear absorption and propagating of picosecond pulse trains. Each pulse train contains 25 subpulses with width 100 ps seperated by space 13 ns. During the interaction with pulse trains, the structures of Pcs can be simplified to the five-state energy model. In our calculations, the coupled rate equations and two-dimensional paraxial field were solved using the Crank–Nicholson numerical method. The effects of substituted carbazoles and conjugated GQDs were investigated. Pcs and their conjugated complexes with GQDs exhibit optical limiting (OL) properties, and GQDs could decrease the OL of Pcs. One-photon absorption cross section ${\sigma }_{S_0{S}_1}$ or ${\sigma }_{T_1{T}_2}$ is the critical factor to determine the limiting value of energy transmittance in weak- or strong-intensity regions, respectively. The two-step two-photon absorption (TPA) tunnel $(S_0\to S_1)\times (T_1\to T_2)$ is the main absorption mechanism; therefore, the effective population transfer time ${\tau }_{ST}$ from $S_0$ to $T_1$ is another critical factor that is determined by one-photon absorption cross section ${\sigma }_{S_0{S}_1}$ and intersystem crossing time ${\tau }_{isc}$. Through further exploration it is found that a high incident intensity will lead to an asymmetric shape of output intensity due to different absorption mechanisms in the front and latter subpulses of the pulse train.

1. Introduction

In the last half century, lasers have been applied in many fields. But the advancement of high power laser brings about great danger to optical devices and human eyes [1]. As a result, organic materials have attracted great attention due to their nonlinear optical susceptibilities, low synthesis costs and great practical significances [2,3,4,5]. Those materials that meet the above requirements have the well-known optical limiting (OL) effects and they are widely used as optical limiters [6,7,8], sensors [9,10], and switches [11,12,13]. New synthetic OL organic materials attract great research interest due to their specific performances in transmitting low intensities and absorbing destructive high intensities of lasers.

Among various types of OL molecules, porphyrins, naphthalocyanines, and phthalocyanines are the most competitive materials. Owing to large conjugated $\pi$-electron ring structures [14,15], their absorption cross sections of excited states are bigger than those of ground states, which is known as the reverse saturable absorption (RSA) properties [16,17,18,19,20]. RSA is typical two-step sequential two-photon absorption (TPA), which often occurs in the process of long duration pulses with low intensity in OL molecules. In contrast, one-step coherent TPA often occurs in the cases of short duration pulses with high intensity. Compared to porphyrins and naphthalocyanines, phthalocyanines (Pcs) have most notably 18 $\pi$-electros planar macrocyclic structures, which leads to distinctive optical and electronic properties as well as outstanding thermal and chemical stability [21,22,23]. Consequently, Pcs are studied further for medical applications in photomedicine, therapy and diagnostics [24,25] and for potential compounds in organic solar cells [26,27,28,29] and light-emitting diodes [30,31,32,33].

The nonlinear optical behaviours of graphene quantum dots (GQDs) linked to Pcs were studied [34,35] because GQDs can be functionalized and synthesized easily and are soluble in aqueous or nonaqueous solvents [36]. Delocalized $\pi$-electrons in GQDs can promote $\pi$-$\pi$ interactions with other $\pi$-electrons systems such as Pcs. However, the OL effects of Pcs linked with GQDs by $\pi$-$\pi$ interactions have rarely been studied.

In recent experimental works [37,38], carbazole-substituted phthalocyanine complexes 2,3,9,10,16,17,23,24-octakis-(3,6-dibromo-9Hcarbazol) phthalocyaninato zinc(II) (Pc 2) and 2,3,9,10,16,17,23,24-Octakis-(9H-carbazol-9-yl) phthalocyaninato zinc(II) (Pc 4) and their conjugated complexes to graphene quantum dots (GQDs) via $\pi$-$\pi$ interactions were synthesized. Their photophysical properties and nonlinear optical parameters in solvent spectrophotometric-grade dimethyl sulfoxide (DMSO) were also reported. Pulse trains were widely used to studied the novel nonlinear absorption and dynamical transmissions in phthalocyanines [39,40] or porphyrins [41,42,43]. Therefore we are interested in exploring the nonlinear dynamics of picosecond pulse trains transmittion in Pc 2, Pc 4 and their nanoconjugates in solvent DMSO. The parameters of pulse trains were set according to those experiments [39,40,41,42,43]. In order to solve the population rate equations coupled with the paraxial intensity equation, we applied the notable Crank–Nicholson method, which has a second-order accurate and stable convergence for any size of time-step [44,45].

2. Method

The structures of phthalocyanines 2,3,9,10,16,17,23,24-octakis-(3,6-dibromo-9Hcarbazol) phthalocyaninato zinc(II) (Pc 2) and 2,3,9,10,16,17,23,24-Octakis-(9H-carbazol-9-yl) phthalocyaninato zinc(II) (Pc 4) and graphene quantum dots (GQDs) are presented in Figure 1. A generalized five-state energy model (Figure 2) can substitute for phthalocyanines when they are interacting with long duration pulses such as picosecond pulse trains [46,47]. Thermal effects are obvious when considering vibrational energy states. In this work we considered the electronic energy states without vibrational energy states, so we will not discuss thermal effects. During the propagating of pulse trains in phthalocyanines, one-photon absorption $S_0\to S_n$ and the one-step TPA were ignored since the frequency of pulses is close to the one-photon resonance absorptions. Therefore the only possible absorption channels are two-step TPA $(S_0\to S_1)\times (S_1\to S_n)$ or $(S_0\to S_1)\times (T_1\to T_2)$.

In our simulations, the pulse train comprises several subpulses,

$$I\left(t\right)=\sum _{n=0}{I}_n\left(t\right),n=0,1,\dots ,n_{tot}-1.$$

(1)

Here n marks the subpulse serial number. The total number is $n_{tot}=25$ in this work. Each subpulse intensity has temporal rectangle shape [48,49],

$$I_n\left(r\right)=I_{0}exp\left[-{\left({\displaystyle \frac{n\Delta -t_0}{{\tau }_e}}\right)}^{2}ln2\right]exp\left[-{\left({\displaystyle \frac{r}{r_0}}\right)}^{2}ln2\right].$$

(2)

Here $t_0=[(n_{tot}-1)\Delta +\tau ]/2$, ${\tau }_e=10\Delta /3$. The spacing between adjacent subpulses is $\Delta =13$ ns, and each subpulse has duration $\tau =100$ ps and beam width $r_0=2$ mm initially. The evolution equation on time t and distance z for each subpulse is described by the paraxial equation [48],

$$\left({\displaystyle \frac{\partial }{\partial z}}-{\displaystyle \frac{n_0}{c}}{\displaystyle \frac{\partial }{\partial t}}\right)I_n\left(t\right)=-N\sum _{j>i}{\sigma }_{ij}({\rho }_i-{\rho }_j)I_n\left(t\right).$$

(3)

Here c is light speed in vacuum and $N=1.0\times {10}^{24}/m^3$ is material concentration. $n_0$ is the refraction index. According to our calculation, the refraction index has minimal impact on OL behvious of Pcs, so here we set $n_0=1$. Subscripts i and j denote different energy states of studied material, and ${\sigma }_{ij}$ is a cross section of one-photon absorption from state i to j. The populations $\rho$ of all five states follow the dynamical rate equations as below [50]

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{\partial }{\partial t}}{\rho }_{S_0}=-\gamma \left(t\right)({\rho }_{S_0}-{\rho }_{S_1})+{\mathsf{\Gamma }}_{S_1}{\rho }_{S_1}+{\mathsf{\Gamma }}_{T_1}{\rho }_{T_1},\hfill \\ & & ({\displaystyle \frac{\partial }{\partial t}}+{\mathsf{\Gamma }}_{S_1}+{\gamma }_c){\rho }_{S_1}={\mathsf{\Gamma }}_{S_n}{\rho }_{S_n}-{\gamma }_S\left(t\right)({\rho }_{S_1}-{\rho }_{S_n})+\gamma \left(t\right)({\rho }_{S_0}-{\rho }_{S_1}),\hfill \\ & & ({\displaystyle \frac{\partial }{\partial t}}+{\mathsf{\Gamma }}_{S_n}){\rho }_{S_n}={\gamma }_S\left(t\right)({\rho }_{S_1}-{\rho }_{S_n}),\hfill \\ & & ({\displaystyle \frac{\partial }{\partial t}}+{\mathsf{\Gamma }}_{T_2}){\rho }_{T_2}={\gamma }_T\left(t\right)({\rho }_{T_1}-{\rho }_{T_2}),\sum _k{\rho }_k=1.\hfill \end{array}$$

(4)

$\mathsf{\Gamma }$ is decay rate of population, and $\gamma$ is pump rate from low to high state. The population transition rate of intersystem crossing (ISC) via $S_1\to T_1$ and transitions $S_0\to S_1$, $S_1\to S_n$ and $T_1\to T_2$ are denoted by ${\gamma }_c$ and $\gamma \left(t\right)$, ${\gamma }_S\left(t\right)$, ${\gamma }_T\left(t\right)$ separately.

Pump rate $\gamma$ depends on cross section $\sigma$ of resonant one-photon absorption as follows [50]

$${\gamma }_{ij}\left(t\right)={\displaystyle \frac{{\sigma }_{ij}I\left(t\right)}{\hslash \omega }},$$

(5)

The frequency of incident pulse trains is $\omega =2\pi c/\lambda$, where wave length $\lambda =532$ nm was extracted from the experiment [37].

Total energy of pulse trains $J\left(z\right)$ during transmission is instantaneous intensity $I(t,r,z)$ integral

$$J\left(z\right)=2\pi \underset{0}{\overset{R}{\int }}\underset{0}{\overset{\infty }{\int }}I(t,r,z)rdrdt,$$

(6)

Then one can obtain the total energy transmittance

$$\mathcal{T}\left(L\right)={\displaystyle \frac{J(z_0+L)}{J\left(z_0\right)}}.$$

(7)

3. Results and Discussion

The photophysical parameters, as shown in

Table 1. Photophysical parameters of Pc 2, Pc 4, and their corresponding nanoconjugates with GQDs in solvent DMSO at $\lambda =532$ nm [37]. ${\gamma }_c=1/{\tau }_{isc}$, $\mathsf{\Gamma }=1/\tau$.

	${\mathit{\tau }}_{{\mathit{S}}_1}$	${\mathit{\tau }}_{{\mathit{T}}_1}$	${\mathit{\tau }}_{\mathbf{isc}}$	${\mathit{\sigma }}_{{\mathit{S}}_0{\mathit{S}}_1}\left({\mathbf{cm}}^2\right)$	${\mathit{\sigma }}_{{\mathit{T}}_1{\mathit{T}}_2}\left({\mathbf{cm}}^2\right)$
Compounds	(ns)	(μs)	(ns)	$\times {\mathbf{10}}^{-\mathbf{18}}$	$\times {\mathbf{10}}^{-\mathbf{18}}$
2	$2.59$	118	$4.11$	$0.137$	$7.2$
4	$2.41$	252	$4.46$	$0.13$	$7.9$
2-GQDs	$2.86$	110	$4.14$	$0.069$	$2.4$
4-GQDs	$2.64$	184	$4.32$	$0.07$	$5.5$

, of all studied complexes in DMSO were extracted from [37]. Moreover, we set ${\tau }_{S_n}=1$ ps and ${\sigma }_{S_1{S}_n}=1.0\times {10}^{-18}{\mathrm{cm}}^2$ [51].

Based on Equation (7), we drew energy transmittances $\mathcal{T}\left(L\right)$ of Pc 2, Pc 4 and their GQD nanoconjugates in DMSO in Figure 3. As the peak intensity of incident pulse train increases, all complexes show an evident OL effect; energy transmittances decrease sharply, especially when intensity exceeds $I_0=1.0\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$. With a longer propagation distance $L=2$ mm, one can notice that transmittances could reach rather low values. Among all complexes, Pc 2 and Pc 4 in DMSO exhibit better OL behaviours compared to their GQD nanoconjugates. In the weak-intensity region, which is lower than $I_0=1.0\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$, their transmittances are quite similar and have little changes. In the strong-intensity region, which is higher than $I_0=1.0\times {10}^{15}{\mathrm{W}/\mathrm{m}}^2$, their transmittances significantly decrease and the transmittances of Pc 2 and Pc 4 could drop close to $0.3$ at $L=2$ mm. The tendencies of all curves obey following convergent transmittance

$$\mathcal{T}\left(L\right)=\left\{\begin{array}{c}exp(-N{\sigma }_{S_0{S}_1}L)I_0\to 0,\hfill \\ exp(-N{\sigma }_{T_1{T}_2}L)I_0\to \infty .\hfill \end{array}\right.$$

(8)

So, their limiting values of energy transmittances are determined by their one-photon absorption cross sections ${\sigma }_{S_0{S}_1}$ or ${\sigma }_{T_1{T}_2}$ (Table 1) in weak- or strong-intensity regions, respectively. In the cases with extremely strong laser energy, ${\sigma }_{T_1{T}_2}$ is the critical parameter to select suitable materials as optical limiter. Moreover, one can also raise the performance of the optical limiter by increasing the concentration or thickness.

In Figure 4 we studied the correlations between the energy transmittances $\mathcal{T}\left(L\right)$ of all complexes and propagation distances L with different intensity. It is predictable that transmittance would decrease when distance increases due to more molecules getting involved in nonlinear absorptions. One can see that Pc 2 and Pc 4 show lower transmittances than their GQD nanoconjugates in DMSO. With higher incident peak intensity $I_0=1.0\times {10}^{14}{\mathrm{W}/\mathrm{m}}^2$ compared to $I_0=1.0\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$, the transmittances of all complexes decrease to rather low values. The principal reason for this is that high intensity could strengthen the second step of two-step sequential TPA, which leads to strong nonlinear OL effect.

In order to understand the dynamical absorption process, we investigated population transformation for Pc 4. Since populations on states $S_1$, $S_n$ nad $T_2$ are negligible, so we only plotted ${\rho }_{S_0}$ and ${\rho }_{T_1}$ in Figure 5. With intensity $I_0=1.0\times {10}^{14}{\mathrm{W}/\mathrm{m}}^2$, nearly 60% of populations on state $S_0$ are transferred to $T_1$. But when intensity increases to $I_0=1.0\times {10}^{15}{\mathrm{W}/\mathrm{m}}^2$, almost all populations are pumped from $S_0$ to $T_1$. In this instance, stronger absorption $T_1\to T_2$ will happen. So, we can draw the conclusion that intenser two-step TPA $(S_0\to S_1)\times (T_1\to T_2)$ rather than $(S_0\to S_1)\times (S_1\to S_n)$ leads to rather low energy transmittance (Figure 3).

In Figure 6, we show the effective population transfer times ${\tau }_{ST}$ from state $S_0$ to $T_1$ of all complexes [48], which is an important variable to make sense of the whole RSA process $(S_0\to S_1)\times (T_1\to T_2)$. The curves are almost symmetric, and this trend is mainly dependent on intensity of pulse train. On both wings with low intensities, ${\tau }_{ST}$ needs longer time. Conversely, in the central section with high intensity, ${\tau }_{ST}$ reaches relatively low values. For each intensity, both Pc 2 and Pc 4 show lower values compared to their GQD nanoconjugates. The dependence of ${\tau }_{ST}$ on intensity [48] is mainly determined by the cross section of one-photon absorption ${\sigma }_{S_0{S}_1}$ and intersystem crossing time ${\tau }_{isc}$. From Table 1 we can see that conjugated GQDs would reduce ${\sigma }_{S_0{S}_1}$ and ${\tau }_{isc}$ as well as ${\sigma }_{T_1{T}_2}$ of Pcs via $\pi$-$\pi$ interactions. Short ${\tau }_{ST}$ would lead to fast population accumulation in state $T_1$, which is beneficial for OL results. Therefore conjugated GQDs weaken the OL behaviours of Pcs, as shown in Figure 3 and Figure 4.

We show 2D maps of output intensities with different incident peak intensity $I_0$ of Pc 4 in Figure 7. One can notice that the peak output intensity becomes earlier and higher when $I_0$ increases, which is shown clearly in Figure 7a. With $I_0=1.0\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$ in Figure 7b, the shape is near symmetric. In the cases of high intensities, $I_0=1.0\times {10}^{14}{\mathrm{W}/\mathrm{m}}^2$ and $I_0=1.0\times {10}^{15}{\mathrm{W}/\mathrm{m}}^2$ in Figure 7c,d, their shapes become increasingly asymmetric. This shape deformation is caused by two different absorption mechanisms during pulse train transmission. For the front subpulses of pulse train, the dominant absorption path is linear absorption $(S_0\to S_1)$ in the studied complexes. Then for the latter subpulses of the pulse train, the nonlinear two-step TPA $(S_0\to S_1)\times (T_1\to T_2)$ occurs due to the saturated accumulation of population on state $T_1$ (Figure 5). These two absorption mechanisms become differentiated when $I_0$ increases to a rather high value such as $I_0=1.0\times {10}^{15}{\mathrm{W}/\mathrm{m}}^2$. Compared the output intensities, one can notice that the growth trend becomes slower. When incident peak intensity $I_0$ increases by an order of magnitude from $I_0=1.0\times {10}^{14}{\mathrm{W}/\mathrm{m}}^2$ to $I_0=1.0\times {10}^{15}{\mathrm{W}/\mathrm{m}}^2$, the output peak intensity increases just two times approximatively from $I=2.770\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$ to $I=5.320\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$. This slow growth trend is caused by the significantly greater TPA in the strong-intensity region.

4. Conclusions

We studied the dynamical propagation of picosecond pulse trains in carbazole-substituted phthalocyanine complexes Pc 2 and Pc 4 and their conjugated complexes to graphene quantum dots (GQDs) in solvent DMSO. Pc 2 and Pc 4 have better OL behaviours than their GQD nanoconjugates. One-photon absorption cross section ${\sigma }_{S_0{S}_1}$ or ${\sigma }_{T_1{T}_2}$ is the critical factor to determine the limiting value of energy transmittance in weak- or strong-intensity regions, respectively. During the interaction of pulse trains with studied complexes, almost all populations are pumped from state $S_0$ to $T_1$. The population accumulation on $T_1$ is quite important for the second step of two-step TPA $(S_0\to S_1)\times (T_1\to T_2)$, which will lead to strong RSA absorption. So, the effective population transfer time ${\tau }_{ST}$ from $S_0$ to $T_1$ is another critical factor which is determined by one-photon absorption cross section ${\sigma }_{S_0{S}_1}$ and intersystem crossing time ${\tau }_{isc}$. In addition, high incident intensity will lead to an asymmetric shape of output intensity due to different absorption mechanisms in the front and latter subpulses of pulse train. Accordingly, carbazole-substituted phthalocyanine and their conjugated complexes showed competitive OL parameters. This kind of phthalocyanine can be a strong potential candidate in many applications, such as optical limiters, molecular sensors, or switches.