Optical Dynamics of Picosecond Pulse Trains in Aluminum and Zinc Tetracarboxy-Phthalocyanines

Abstract

The nonlinear properties and photophysical dynamics of aluminum and zinc tetracarboxy-phthalocyanines (AlPc and ZnPc) were studied using pulse trains of a 532 nm wavelength, which contain 25 subpulses with a 100 ps width and 13 ns spacing. Considering its interaction with long-duration pulses, the energy structure of phthalocyanine could be substituted by a five-level pattern. The nonlinear transmissions of pulse trains in AlPc and ZnPc were simulated by means of equations about the population rate coupled with the paraxial field equation of two-dimensional space. The well-known Crank–Nicholson numerical method was applied to the theoretical simulation. The results demonstrate that both phthalocyanines are efficient as optical limiters. In its low-intensity region, AlPc shows a much better OL effect than ZnPc. But in the region with high intensity, their energy transmittances are nearly the same. The nonlinear transmission of a pulse is susceptible to the state lifetime and cross section of one-photon absorption. Tetracarboxy-phthalocyanines have advantageous photophysical properties for applications in nonlinear optical areas, such as nonlinear optical devices like optical limiters. Adding central metals such as Al and Zn to phthalocyanines could enhance their photodynamic properties, making them potential optical limiters and photosensitizers.

1. Introduction

Macrocycles such as phthalocyanines, porphyrins and their derivatives have been a focus of attention due to their wide applications in nonlinear optics as optical limiters [1,2,3,4]; medical areas like photomedicine, therapy and diagnostics [5,6,7,8,9,10]; chemical sensors [11,12]; molecular switches [13,14,15]; etc. The primary advantage of these materials lies in their large conjugated $\pi$-electron ring structure [16,17], which improves their photophysical parameters such as their absorption region, lifetimes, transition dipole moments, cross sections and quantum yields [18,19,20,21].

At present, phthalocyanines and porphyrins acting as optical limiters have received more and more notice. The most well-known optical limiting (OL) mechanism in organic molecular materials is two-photon absorption (TPA), including coherent one-step and sequential two-step TPA. The most typical two-step TPA process is reverse saturable absorption (RSA), which requires that the absorption cross section of the excited state is far bigger than that of the ground state [22,23,24,25,26,27]. Almost all phthalocyanines and porphyrins satisfy this requirement. Thus, finding or synthesizing new phthalocyanine-like or porphyrin-like materials has become one of the hot topics of research in this field [28,29,30,31,32,33,34]. In addition, binding the materials to proteins or micelles [35,36,37,38,39,40,41,42,43,44,45] and changing the influence factors of their solutions [46,47,48,49] are also effective means of promoting their OL properties.

Recently, tetracarboxyl-substituted aluminum and zinc phthalocyanines (AlPc and ZnPc) were synthesized and their photophysical parameters were studied [50,51]. Both AlPc and ZnPc are typical RSA materials with an emphatically larger absorption cross section in their excited state than their ground state. Picosecond pulse trains were applied as the laser sources in our simulation to study the nonlinear properties and photophysical dynamics of AlPc and ZnPc. The well-known Crank–Nicholson scheme [52,53] has second-order accuracy and stable convergence for any small or big size of time step and was applied in the simulation to solve a paraxial intensity equation coupled with equations about the population rate.

2. Method

The structures of aluminum and zinc tetracarboxy-phthalocyanines (AlPc and ZnPc) are shown in Figure 1. The laser sources we used in this work were long-duration pulses, so the energy-level distribution of AlPc and ZnPc can be substituted by the well-known generalized five-level pattern, including the singlet ground $S_0$, first excited $S_1$, higher excited $S_n$, triplet first excited $T_1$ and higher excited $T_n$ states (Figure 2). During the interactions of long-duration picosecond pulse trains with AlPc and ZnPc, we found that the primary optical absorption paths are two-step TPA processes $(S_0\to S_1)\times (S_1\to S_n)$ or $(S_0\to S_1)\times (T_1\to T_2)$. Since we hypothesized that the incident frequency is near the one-photon resonance absorptions, one-step TPA and one-photon absorption $S_0\to S_n$ are ignored in our simulations.

A series of subpulses make up one pulse train as follows:

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

(1)

Here, n is the subpulse serial number, and the total number $n_{tot}=25$.

We supposed that each subpulse has a rectangle temporal shape intensity [54,55]:

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

(2)

Here, $t_0=[(n_{tot}-1)\Delta +\tau ]/2$ and ${\tau }_e=10\Delta /3$. In the simulations, the spacing between subpulses and the duration of each subpulse are assumed to be $\Delta =13\mathrm{ns}$ and $\tau =100\mathrm{ps}$, respectively, which are same as the values used in the experiment in [50]. $r_0=2\mathrm{mm}$ is the beam width of each initial subpulse.

Each subpulse propagated in the materials complies with the following paraxial equation, depending on distance z and time t [54]:

$$\left({\displaystyle \frac{\partial }{\partial z}}-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial }{\partial t}}\right)I_n\left(t\right)=-N\sum _{j>i}{\sigma }_{ij}({\rho }_i-{\rho }_j)I_n\left(t\right).$$

(3)

Here, c is the velocity of light in a vacuum. $\rho$ denotes the population of each state, marked by i or j. Both concentrations of AlPc and ZnPc were supposed to be $N=1.0\times {10}^{24}/{\mathrm{m}}^3$. ${\sigma }_{ij}$ is the absorption cross section of the corresponding one-photon process $i\to j$.

The dynamical equations about the population rate of all five states can be described as follows [56]:

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

(4)

Here, $\Gamma$ and $\gamma$ are the population decay and pump rates of the states, respectively. Specifically, ${\gamma }_c$ denotes the intersystem crossing (ISC) rate in the population transition process $S_1\to T_1$. In addition, $\gamma \left(t\right)$, ${\gamma }_S\left(t\right)$ and ${\gamma }_T\left(t\right)$ are pump rates for the corresponding processes $S_0\to S_1$, $S_1\to S_n$ and $T_1\to T_2$. The pump rate $\gamma$ depends on the one-photon absorption cross section $\sigma$ as follows:

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

(5)

Here, ${\sigma }_{ij}$ is decided by the resonant one-photon absorption frequency [56]. $\omega =2\pi c/\lambda$ is the frequency of the incident pulse trains, where wave length $\lambda =532\mathrm{nm}$ is same as the experimental value [50].

The total pulse energy transmitted is expressed as follows:

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

(6)

where the total energy $J\left(z\right)$ is calculated using the following integral formula:

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

(7)

$I(t,r,z)$ is the instantaneous intensity.

3. Results and Discussion

The photophysical parameters of AlPc and ZnPc we used in the simulations (

Table 1. Photophysical parameters of AlPc and ZnPc at $\lambda =532\mathrm{nm}$ [50,51]. ${\gamma }_c=1/{\tau }_{isc}$ and $\Gamma =1/\tau$.

	${\mathit{\tau }}_{{\mathit{S}}_1}$	${\mathit{\tau }}_{{\mathit{T}}_1}$	${\mathit{\tau }}_{\mathit{isc}}$	${\mathit{\sigma }}_{{\mathit{S}}_0{\mathit{S}}_1}\left({\mathbf{cm}}^2\right)$	${\mathit{\sigma }}_{{\mathit{S}}_1{\mathit{S}}_{\mathit{n}}}\left({\mathbf{cm}}^2\right)$	${\mathit{\sigma }}_{{\mathit{T}}_1{\mathit{T}}_2}\left({\mathbf{cm}}^2\right)$
Compounds	(ns)	$(\mathsf{\mu }$s)	(ns)	$\times {\mathbf{10}}^{-\mathbf{17}}$	$\times {\mathbf{10}}^{-\mathbf{17}}$	$\times {\mathbf{10}}^{-\mathbf{17}}$
AlPc	5.6	340	26	0.71	2.0	1.3
ZnPc	2.8	240	4.5	0.22	1.3	1.4

) were extracted from experiments [50,51]. In addition, we set ${\tau }_{S_n}=1\mathrm{ps}$ and ${\tau }_{T_1}=2\mathsf{\mu }\mathrm{s}$ [29].

Figure 3 shows the energy transmittances $\mathcal{T}\left(L\right)$ of AlPc and ZnPc for different peak intensities of the pulse trains at $L=0.5\mathrm{mm}$ and $L=1.0\mathrm{mm}$. Both AlPc and ZnPc show excellent OL effects due to the RSA effect. The trends of the two curves are quite similar at different distances, except that the transmittances of AlPc and ZnPc are lower at long distances $L=1.0\mathrm{mm}$ than at $L=0.5\mathrm{mm}$. Compared to ZnPc, AlPc displays a significantly stronger absorption rate. The convergent values of the transmittances for AlPc and ZnPc could be roughly estimated by the following expression, which is determined by the cross section of one-photon absorption, ${\sigma }_{S_0{S}_1}$ or ${\sigma }_{T_1{T}_2}$ (Table 1):

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

(8)

In the low-intensity region, the transmittance of AlPc is much lower than that of ZnPc due to its larger ${\sigma }_{S_0{S}_1}$, while in the high-intensity region the transmittances of AlPc and ZnPc are quite close to each other due to their almost identical ${\sigma }_{T_1{T}_2}$ (Table 1).

We plotted the transmittances $\mathcal{T}\left(L\right)$ of AlPc and ZnPc depending on the distances in Figure 4, with the peak intensity of incident pulse train $I_0=1.0\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$. This clearly shows that both transmittances decrease as the distance increases due to the absorption process, as more organic molecules take part in it.

In Figure 5, we plotted the dynamical populations ${\rho }_{S_0}$ and ${\rho }_{T_1}$ when $I_0=1.0\times {10}^{13}{\mathrm{W}/\mathrm{m}}^2$, which occupy almost all populations. The populations in other states are too small to be ignored. During the pulse train’s interaction with organic molecules, the majority of populations undergo the transferring process $S_0\to T_1$, and this process is earlier and more complete for populations in AlPc than those in ZnPc. A large population in state $T_1$ together with a large one-photon absorption cross section ${\sigma }_{T_1{T}_2}$ could apparently benefit to two-step TPA ($(S_0\to S_1)\times (T_1\to T_2)$) and thus lead to strong OL behaviors. Therefore, AlPc shows a lower energy transmittance than ZnPc in Figure 3 and Figure 4. This fast and effective process in AlPc is mainly determined by the longer lifetime ${\tau }_{S_1}=5.6\mathrm{ns}$ of state $S_1$ and larger cross section of its one-photon absorption ${\sigma }_{S_0{S}_1}=0.71\times {10}^{-17}{\mathrm{cm}}^2$ compared to those of ZnPc (Table 1). The results indicate that the lighter metal Al could accelerate the population accumulation in state $S_1$. However, the heavier metal Zn could enhance spin–orbit coupling in the population transfer from a singlet to triplet state with a shorter ${\tau }_{isc}$. Generally, the central metal Al, when linked to phthalocyanines, has more obvious OL effects than Zn.

Based on the above discussion, one could see that the two-step TPA path $(S_0\to S_1)\times (S_1\to S_n)$ is neglected due to it barely populating $S_1$. A large population ${\rho }_{T_1}$ favors a strong second-step absorption $T_1\to T_2$ of TPA and thus leads to a strong OL effect. Therefore, the main TPA path is $(S_0\to S_1)\times (T_1\to T_2)$, where the population transferring process $S_0\to T_1$ is quite an important part of the whole TPA process. In Figure 6, we plotted the effective times ${\tau }_{ST}$ needed to study the population transferring process $S_0\to T_1$ [54]. In the two sides with low intensities, the population transferring time ${\tau }_{ST}$ is long. On the contrary, ${\tau }_{ST}$ is quite fast in the central region with a high intensity. Also, one can notice that AlPc needs a shorter transferring time than ZnPc, which is determined by the combined effect of ${\tau }_{S_1}$, ${\tau }_{T_1}$ and ${\tau }_{isc}$. AlPc has longer lifetimes, ${\tau }_{S_1}$ and ${\tau }_{T_1}$, while ZnPc has a shorter ${\tau }_{isc}$. Their advantages lead to them having quite a fast and high-intensity population transferring process $S_0\to T_1$. But in terms of their overall effect, AlPc shows a slightly shorter ${\tau }_{ST}$ in the central high-intensity region. This is another important piece of evidence that explains that AlPc shows better OL behaviors than ZnPc in Figure 3 and Figure 4.

In Figure 7, we studied the emitted pulse train intensities of AlPc at $L=1\mathrm{mm}$ and $L=10\mathrm{mm}$. In Figure 7a, we plotted the output intensities at the beam center $r=0$. The peak intensities at $L=1\mathrm{mm}$ and $L=10\mathrm{mm}$ are $3.65\times {10}^{12}{\mathrm{W}/\mathrm{m}}^2$ and $4.83\times {10}^9{\mathrm{W}/\mathrm{m}}^2$, respectively, which clearly shows a three-orders-of-magnitude decrease at $L=10\mathrm{mm}$ compared to that at $L=1\mathrm{mm}$. Compared to Figure 7b, one can clearly see that the shape of the output intensity becomes asymmetrical in Figure 7c, and the peak of the output intensity occurs earlier, at $140\mathrm{ns}$ for $L=10\mathrm{mm}$ rather than $149\mathrm{ns}$ for $L=1\mathrm{mm}$. In the front part of the output intensity, a two-step TPA $(S_0\to S_1)\times (T_1\to T_2)$ dominates the absorption process. After the population accumulating in state $T_1$, at the latter part of the intensity mainly one-photon absorption occurs $T_1\to T_2$. These two different absorption mechanisms make the shape of the output intensity asymmetrical.

4. Conclusions

We theoretically simulated the nonlinear absorption and propagation of picosecond pulse trains in tetracarboxyl-substituted aluminum and zinc phthalocyanines (AlPc and ZnPc). Both AlPc and ZnPc show interesting nonlinear absorption processes and excellent optical limiting effects. In its low-intensity region, AlPc shows a much better OL effect than ZnPc. But in the region with high intensity, their energy transmittances are nearly the same. Considering their interaction with long-duration pulses, there is mainly a two-step TPA involving singlet and triplet states $(S_0\to S_1)\times (T_1\to T_2)$. The nonlinear transmission of the pulse train is sensitive to photophysical parameters, such as the state lifetime and one-photon absorption cross section. The results indicate that tetracarboxy-phthalocyanines have advantageous photophysical properties for applications in nonlinear optical areas, such as nonlinear optical devices like optical limiters. Adding central metals such as Al and Zn to phthalocyanines could enhance their photodynamic properties, making them potential optical limiters and photosensitizers. In addition, increasing the thickness or concentration of phthalocyanines in their interaction with laser pulses would effectively increase the efficiency of their OL effects.