Tin Phthalocyanine Nanoprobes with Symmetric Macrocyclic Structures: Nonlinear Dynamics of Pulse Trains with Tunable ps/ns Subpulse Widths and Enhanced Optical Limiting for MEMS Microdevices

Abstract

Tin phthalocyanine (SnPc) nanoprobes with strong reverse saturable absorption (RSA) are extremely needed for photoacoustic (PA) molecular imaging. The optical properties and dynamics of SnPc nanoprobes by pulse trains were studied. During the propagating of pulse trains in SnPc, the electronic structure of SnPc is simplified to the five-state energy model. The pulse train contains 25 subpulses with space 13 ns, and the widths of subpulses were set as $3.5$ ps, 35 ps, 350 ps, $3.5$ ns, 10 ns, 20 ns, 35 ns and 100 ns, respectively. In this work, we solved two-dimensional paraxial field coupled with rate equations employing the Crank–Nicholson numerical method. The results reveal the unique optical properties and outstanding optical limiting (OL) effects of SnPc nanoprobes, indicating huge application potential as optical limiters, sensors and switches.

1. Introduction

In recent years, with the development of modern laser technology, progress in high-power laser calls have called for effective protection for experimental devices and human eyes [1,2,3,4,5]. Among various types of organic materials, increasing research attention has been paid to phthalocyanines and porphyrins and their derivatives, attributing to their excellent optical limiting (OL) effects and low synthesis costs. Accordingly, they are widely used as optical limiters, optical sensors or switches [6,7,8,9,10,11,12,13].

Phthalocyanines and porphyrins show specific optical characteristics, thanks to their large conjugated $\pi$-electrons ring structures [14,15]. Therefore, they have typical reverse saturable absorption (RSA) features, namely larger absorption cross sections in excited states than that in ground states [16,17,18,19,20,21]. During the interaction of organic materials with long-duration pulses, two-step two-photon absorption (TPA) primarily occurs, which is a representative RSA process. It is worth noting that symmetric 18 $\pi$-electrons conjugated ring exist in macrocyclic structures of phthalocyanines (Pcs) [22,23,24], prompting nonlinear optical response along with thermal and chemical stability [25,26,27]. Pcs, therefore, have been devised as nanoprobes to be applied to organic solar cells [28,29,30,31], light-emitting diodes [32,33,34,35] as well as photomedicine therapy, diagnostics and photoacoustic imaging [36,37,38,39,40,41].

Recently, tin phthalocyanine (SnPc) nanoprobes were designed and the nonlinear optical absorption was studied with different laser pulse widths in experimental work [42]. Phthalocyanines interacting with pulse trains in experiments were widely studied [43,44,45,46]. In this work, we investigated the dynamical interaction of pulse train with SnPc. The pulse train was composed of 25 subpulses, spaced 13 ns apart. These pulse train parameters, such as subpulses number and spacing between subpulses, were set according to the experiments in works [47,48,49,50,51]. In our previous work, we also discussed how pulse train parameters take effect on OL properties of materials such as naphthalocyanines and phthalocyanines [52]. There were several works about OL properties of materials versus the pulse-width in one single pulse [53,54,55]. However, subpulse widths modification of pulse trains in OL effects were rarely studied. Especially, there was barely research to study the distinction about OL effects by pulse trains with nanosecond and picosecond subpulse widths in one work. In order to study the nonlinear absorption distinction between picosecond and nanosecond subpulse widths in SnPc nanoprobes, the subpulse widths in this work were set as $3.5$ ps, 35 ps, 350 ps, $3.5$ ns, 10 ns, 20 ns, 35 ns and 100 ns, respectively, in accordance with the experiment [42]. Coupled rate equations of population and paraxial intensity equation were solved by the Crank–Nicholson method with second-order accuracy in time and space [56,57]. Our results displayed the novel finding that each subpulse produces cumulative nonlinear response. When the subpulse width increases, OL effects of SnPc occur earlier and stronger. This work provides new ideas for optical limiters, optical sensors or switches applications by using pulse trains with wider subpulse widths. Furthermore, the proposed pulse width regulation strategy provides a theoretical thread for the design of micro-electro-mechanical systems (MEMS)-integrated micro-optical limiting devices, micro-laser protection coatings or optical switches in phthalocyanines [58,59,60] and MEMS optical modules [61,62,63].

2. Method

The structure of tin phthalocyanine (SnPc) is presented in Figure 1. When interacting with long-duration pulse such as pulse trains [42], the electronic energy structure of SnPc can be simplified to the generalized five-state model (Figure 2). In these cases, together with the incident pulse frequency close to the resonant one-photon absorptions, one-step TPA and nonresonant one-photon absorption $S_0\to S_n$ can be neglected during the transmission of pulse trains in SnPc. Accordingly, the most preferential channels of population absorption are two-step TPA $(S_0\to S_1)\times (S_1\to S_n)$ or $(S_0\to S_1)\times (T_1\to T_2)$.

A pulse train contains a sequence of subpulses marked with serial number n as follows:

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

(1)

where $n_{tot}=25$ is set as the total subpulses number. Considering the radial distribution of the pulse trains, the intensity of each subpulse with temporal rectangle shape is described as follows [64,65]:

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

(2)

where $t_0=[(n_{tot}-1)\Delta +\tau ]/2$, ${\tau }_e=10\Delta /3$, and $\Delta =13$ ns is set as the subpulses spacing. Radial parameter $r_0=2$ mm is initial beam width. In the light of the experiment [42], the duration of each subpulse $\tau$ is set as $3.5$ ps, 35 ps, 350 ps, $3.5$ ns, 10 ns, 20 ns, 35 ns and 100 ns, respectively, in this work.

The following paraxial equation describes the evolution of each subpulse depending on time t and distance z [64]:

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

(3)

where $N=1.0\times {10}^{24}/m^3$ is set as the concentration and c is the light speed in vacuum. Since refraction index has almost no impact on OL behaviour according to the calculation, here, we set refraction index $n_0=1$. Different states are denoted by i, and j. ${\sigma }_{ij}$ is the absorption cross section of one-photon transition from state i to j. The following set of equations are the rate equations of populations $\rho$ [66],

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

(4)

where $\Gamma$ and $\gamma$ are the decay and pump rates separately. Specifically, $\gamma \left(t\right)$, ${\gamma }_S\left(t\right)$, ${\gamma }_T\left(t\right)$ denote the pump rate of processes $S_0\to S_1$, $S_1\to S_n$ and $T_1\to T_2$, respectively, and ${\gamma }_c$ marks the transition rate of intersystem crossing (ISC) $S_1\to T_1$. Pump rate $\gamma$ can be calculated by the following expression depending on resonant one-photon absorption cross section $\sigma$ [66],

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

(5)

where $\omega =2\pi c/\lambda$ is the incident frequency of pulse trains, and wave length $\lambda =532$ nm is set based on the experiment [42].

Total energy transmittance is calculated to investigate the OL effects,

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

(6)

where total pulse train energy $J\left(z\right)$ of integral over instantaneous intensity $I(t,r,z)$ is calculated by

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

(7)

3. Results and Discussion

Photophysical parameters in

Table 1. Photophysical parameters of SnPc at $\lambda =532$ nm [42]. ${\gamma }_c=1/{\tau }_{isc}$, $\Gamma =1/\tau$.

	${\mathit{\tau }}_{{\mathit{S}}_1}$	${\mathit{\tau }}_{\mathit{isc}}$	${\mathit{\sigma }}_{{\mathit{S}}_0{\mathit{S}}_1}$ (cm2)	${\mathit{\sigma }}_{{\mathit{S}}_1{\mathit{S}}_{\mathit{n}}}$ (cm2)	${\mathit{\sigma }}_{{\mathit{T}}_1{\mathit{T}}_2}$ (cm2)
Compounds	(ns)	(ns)	$\mathbf{\times }{\mathbf{10}}^{-\mathbf{18}}$	$\mathbf{\times }{\mathbf{10}}^{-\mathbf{17}}$	$\mathbf{\times }{\mathbf{10}}^{-\mathbf{17}}$
2	$4.44$	$3.36$	$2.1$	$2.3$	$6.7$

of SnPc were extracted from the experiment [42]. In addition, we set the relaxation time of state $T_1$ as ${\tau }_{T_1}=100$ $\mathrm{\mu }$s [67].

In Figure 3, we plotted the energy transmittances $\mathcal{T}\left(L\right)$ of SnPc depending on the incident intensities of pulse trains with different subpulse widths based on Equation (6). One can see clearly that the transmittance evidently decreases when incident intensities increase. SnPc shows obvious OL effects interacting with pulse trains of different subpulse widths. One can also notice that in weak and strong intensity regions, the tendencies of transmittances with different subpulse widths conform to the convergent transmittance as follows:

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

(8)

Regardless of the subpulse widths, the limit values of transmittances are decided on the absorption cross sections of one-photon resonance processes ${\sigma }_{S_0{S}_1}$ and ${\sigma }_{T_1{T}_2}$ (Table 1) for weak and strong intensity regions, respectively. Considering the subpulse widths, one can notice that the transmittances reach the limit value at different intensities. When the incident intensity increases to $I_0=1.0\times {10}^{11}$ W/m², the transmittance of pulse train with subpulse width 100 ns reaches the limit value. However, for the pulse train with shorter subpulse width $3.5$ ps, the transmittance reaches the limit value at much stronger intensity $I_0=4.0\times {10}^{15}$ W/m² approximately. Consequently, the pulse train with wider subpulse width reaches the limit value of energy transmittance at lower incident intensity.

The dependences of energy transmittances $\mathcal{T}\left(L\right)$ of SnPc on propagation distances L for pulse trains with different subpulse widths at $I_0=1.0\times {10}^{11}$ W/m² were plotted in Figure 4. It shows that SnPc is excellent optical limiting material. When distance increases, the transmittance clearly decreases. For pulse trains with different subpulse widths, the descending speeds of transmittances are different. The wider the subpulse width, the shorter the distance required to reach the limit value of transmittance. For the pulse train with subpulse width 100 ns, the transmittance reaches the limit value at a rather short distance, approximately $L=2$ mm. However, for the pulse train with shorter subpulse width $3.5$ ps, the distance exceeds more than $L=10$ mm. The reason of these trends is that wide subpulse width carrying more energy would promote more intense absorption in the second step in the two-step TPA. Therefore, the transmittance of SnPc will reach the limit value much earlier than the cases with shorter subpulse widths.

In Figure 5, we plotted the dynamical populations on states $S_0$ and $T_1$ for pulse trains with subpulse widths $3.5$ ps, $3.5$ ns and 35 ns at $I_0=1.0\times {10}^{11}$ W/m². Since populations are scarcely distributed in states $S_1$, $S_n$ and $T_2$, so ${\rho }_{S_1}$, ${\rho }_{S_n}$ and ${\rho }_{T_2}$ were not drew. One can see clearly that each subpulse produces cumulative nonlinear response. The more subpulses participate in the nonlinear absorption, the more obvious the population inversion. In Figure 5a with short subpulse widths $3.5$ ps, one can see that almost all populations ${\rho }_{S_1}$ stay still in state $S_0$ and very few populations are in state $T_1$, which leads to unachievable second step of the two-step TPA. So the OL effect for pulse train with subpulse width $3.5$ ps is weak. When the subpulse width increases to $3.5$ ns in Figure 5b, one can see that approximately 56% of populations ${\rho }_{S_0}$ were transformed to ${\rho }_{T_1}$. For the case with subpulse width 35 ns in Figure 5c, almost all populations ${\rho }_{S_1}$ are distributed in state $T_1$. Large populations ${\rho }_{T_1}$ would be conducive to the second-step absorption $T_1\to T_2$ of the two-step TPA, which promotes the OL effect greatly.

Figure 6 displayed the populations effective transfer times ${\tau }_{ST}$ of the process $S_0\to T_1$ for pulse trains with subpulse widths $3.5$ ps, $3.5$ ns and 35 ns at $I_0=1.0\times {10}^{11}$ W/m². The values of ${\tau }_{ST}$ at the central part with high intensities are quite small, which means that the transfer process is fast and it is beneficial to the whole two-step TPA. ${\tau }_{ST}$ at two wings with low intensities are large so the OL effects in these regions are weak. Depending on intensity, ${\tau }_{ST}$ is mainly attributed to the absorption cross-section of the one-photon resonance process ${\sigma }_{S_0{S}_1}$ and intersystem crossing time ${\tau }_{isc}$ (Table 1). In addition, all the curves are approximately symmetrical, and the pulse train with subpulse width 35 ns shows the lowest ${\tau }_{ST}$ compared to other two cases with shorter subpulse widths. It is interesting to notice that there is a time delay of the minimal value for the case 35 ns compared to other two cases, $3.5$ ps and $3.5$ ns. The reason for this is that a pulse train with subpulse width 35 ns carrying more energy would lead to longer continuous populations accumulation in state $T_1$.

In order to show the dynamical absorptions intuitively, in Figure 7, we plotted the two-dimensional output intensities of pulse trains with subpulse widths $3.5$ ps, $3.5$ ns and 35 ns when $I_0=1.0\times {10}^{11}$ W/m². At the short propagation distance $L=1$ mm in Figure 7a–d, one can see there is an obvious deformation of the output intensities when the subpulse width increases to a wider value and the output intensity decreases by one order of magnitude. When the distance increases to $L=1$ cm in Figure 7e–h, the deformation of the output intensities becomes even more obvious and the output intensity decreases by two orders of magnitude from subpulse width $3.5$ ps to 35 ns. The deformation phenomenon is caused by different absorption courses during pulse trains propagation in SnPc. In front parts with weak intensities, there are mainly linear absorption process $(S_0\to S_1)$, while in latter parts with high intensities, a nonlinear strong TPA absorption process $(S_0\to S_1)\times (T_1\to T_2)$ dominates due to populations accumulation on state $T_1$ by the populations transferring after front parts. Furthermore, one can notice that the peak values of output intensities occur earlier when the subpulse width increases, which could be seen more clearly in Figure 7a,e. The pulse train with wider subpulse width carrying more energy would promote faster population accumulation on state $T_1$, and subsequently would be conducive to the two-step TPA, which results in better OL effects.

4. Conclusions

Considering the interactions of pulse trains with SnPc, we explored the dynamical absorption of pulse trains with different subpulse widths. SnPc shows excellent OL behaviour with different pulse trains. Absorption cross-sections ${\sigma }_{S_0{S}_1}$ and ${\sigma }_{T_1{T}_2}$ of one-photon resonance processes determine the limit values of transmittances for weak- and high-intensity regions, respectively. The OL effect of SnPc becomes better when the subpulse width increases. With weak intensity or short subpulse width, linear absorption $(S_0\to S_1)$ occurs mainly in SnPc. However with high intensity or wide subpulse width, nonlinear TPA $(S_0\to S_1)\times (T_1\to T_2)$ occurs due to large and fast population accumulation in state $T_1$. In practical applications of SnPc as an optical limiter or in another role, changing the pulse duration is an effective method to improve the OL behaviour and optical response characteristics. The findings suppose important prospects which could offer effective methods to strengthen the optical features of organic compounds via employing pulse trains with wider subpulse widths. Furthermore, the pulse width regulation strategy proposed in this work provides a theoretical reference for the design of MEMS-integrated micro-optical limiting devices, micro-laser protection coatings or optical switches in MEMS optical modules, and future research may focus on the compatibility of SnPc nanoprobe fabrication processes with MEMS technologies.