How Micelles Influence the Optical Limiting Properties of Zinc Porphyrins and J-Aggregates for Picosecond Pulse Trains

Abstract

In this work, we studied nonlinear dynamics and optical limiting (OL) effects of pulse trains in zinc porphyrins meso-tetrakis methylpyridiniumyl (Zn${}^{2+}$TMPyP) and meso-tetrakis sulfonatophenyl (Zn${}^{2+}$TPPS) and porphyrin J-aggregates. The environments of zinc porphyrins were selected as aqueous solutions and micelles of sodium dodecyl sulfate (SDS) and cetyltrimethyl ammonium bromide (CTAB). Our numerical results show that both Zn${}^{2+}$TMPyP and Zn${}^{2+}$TPPS are good optical limiters in all solutions. Zn${}^{2+}$TPPS in aqueous solutions shows the best OL effect. Micelles of SDS and CTAB produced less OL effects than the aqueous solutions. The main reason lies in the first excited singlet state and intersystem crossing depending on the electronic structures in different environments.

1. Introduction

Porphyrins have advantages as optical limiters and switching devices [1,2,3,4,5,6,7] basing on their extremely extended $\pi$-conjugated structures [8,9]. In addition, their competitive features such as intense transition dipole moments, broad absorption region in a visible spectrum, and large quantum yield for triplet state [10,11,12,13] made them be widely used in therapy and diagnostics [14,15,16,17,18,19,20].

Owing to the advantages of porphyrins’ structure, many researchers tried to make some modifications to increase their molecular properties by adding different central metal ions or peripherally substitutes [11,21,22,23,24,25], putting them in different solvents [26,27,28,29], binding to some proteins [30,31,32,33,34] et al. Recently, zinc(II) meso-tetrakis methylpyridiniumyl (Zn${}^{2+}$TMPyP), zinc(II) meso-tetrakis sulfonatophenyl (Zn${}^{2+}$TPPS), their J-aggregates, and interactions with different micelles were the focus in a number of different studies [35,36,37,38,39,40]. Works about the interaction of micelles such as cetyltrimethylammonium bromide (CTAB) with porhyrins were studied [41,42,43,44,45,46,47]. Their results indicated that CTAB is beneficial to pharmaceutical practice and pharmacological system of supramolecular porphyrin-based assembles.

In this work, we studied nonlinear dynamics and optical limiting (OL) of Zn${}^{2+}$TMPyP, Zn${}^{2+}$TPPS and their J-aggregates in the absence and in the presence of sodium dodecyl sulfate (SDS) and CTAB micelles. Porphyrins are classic reverse saturable absorption (RSA) molecules with larger excited state absorption cross sections than the ground state one. We considered the studied compounds as a five-state scheme. During propagation of long duration pulse trains, singlet–singlet or singlet–triplet absorption is the fundamental nonlinear absorption mechanism. Our numerical results reveal that Zn${}^{2+}$TPPS has better OL effects than Zn${}^{2+}$TMPyP, and the presence of micelles diminished the OL properties of porphyrins J-aggregates.

2. Method

The molecular structures of zinc(II) meso-tetrakis methylpyridiniumyl (Zn${}^{2+}$TMPyP) and zinc(II) meso-tetrakis sulfonatophenyl (Zn${}^{2+}$TPPS) were shown in Figure 1. A five-state scheme in Figure 2 was used instead of the structures. We considered the interactions of molecules with pulse trains, hence the two-step two-photon absorption can be $(S_0\to S_1)\times (S_1\to S_n)$ or $(S_0\to S_1)\times (T_1\to T_2)$.

The pulse train consists of a series of subpulses,

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

(1)

n and $n_{tot}=20$ are the serial and total numbers of subpulses. Each subpulse was supposed to have the temporal shape of a rectangle [48,49],

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

(2)

In this work, we considered picosecond pulse trains. Taking into account the data in the experiment [50], subpulse spacing is set as $\Delta$ = 13.2 ns, and the duration for each subpulse is $\tau$ = 70 ps. In addition, $r_0$ = 2 mm is the beam width of each initial subpulse. $t_0=[(n_{tot}-1)\Delta +\tau ]/2$, ${\tau }_e=10\Delta /3$.

The paraxial equation of time t and distance z for each subpulse is expressed as [48]

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

(3)

$N=2.9\times {10}^{23}/m^3$ is the concentration of molecules used in the experiment [50]. c is the speed of light in vacuum. Subscripts i and j were used to distinguish different states; then, ${\sigma }_{ij}$ represents one-photon absorption cross section from low to high state $i\to j$. All state populations $\rho$ can be calculated by the dynamical rate equations [51],

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

(4)

$\mathsf{\Gamma }$ denotes the decay rate of the excited state’s population and $\gamma$ marks the opposite pump rate from low to high state. ${\gamma }_c$ is the population transition rate of intersystem crossing (ISC) $S_1\to T_1$, and $\gamma \left(t\right)$, ${\gamma }_S\left(t\right)$, ${\gamma }_T\left(t\right)$ are for the transitions $S_0\to S_1$, $S_1\to S_n$, $T_1\to T_2$, respectively. Population pump rate $\gamma$ is well known to relate to the corresponding cross section $\sigma$,

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

(5)

$\omega$ is incident frequency of pulse trains and wave length is set as $\lambda =2\pi c/\omega$ =532 nm in our calculations.

The transmittance of total energy for pulse trains was also calculated to estimate the OL effect,

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

(6)

The entrance and propagation distance of pulse trains is set as $z_0=0$ and L. Total energy $J\left(z\right)$ is the integral of instantaneous intensity $I(t,r,z)$ as follows:

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

(7)

3. Results and Discussion

In our calculations, useful photophysical parameters of ZnTMPyP and ZnTPPS in aqueous solutions and their porphyrin J-aggregates with micelles of SDS and CTAB were collected in

Table 1. Photophysical parameters of ZnTMPyP and ZnTPPS in aqueous solutions and their porphyrin J-aggregates with micelles of SDS and CTAB at $\lambda$ = 532 nm [50]. ${\gamma }_c=1/{\tau }_{isc}$, $\mathsf{\Gamma }=1/\tau$.

	${\mathit{\tau }}_{{\mathit{S}}_1}$	${\mathit{\tau }}_{\mathbf{isc}}$	${\mathit{\sigma }}_{{\mathit{S}}_0{\mathit{S}}_1}\left({\mathit{m}}^2\right)$	${\mathit{\sigma }}_{{\mathit{S}}_1{\mathit{S}}_{\mathit{n}}}\left({\mathit{m}}^2\right)$	${\mathit{\sigma }}_{{\mathit{T}}_1{\mathit{T}}_2}\left({\mathit{m}}^2\right)$
Compounds	$\left(\mathit{ns}\right)$	$\left(\mathit{ns}\right)$	$\times {10}^{-21}$	$\times {10}^{-21}$	$\times {10}^{-21}$
ZnTMPyP	$1.3$	$2.0$	$1.5$	$6.1$	$4.9$
ZnTMPyP + SDS	$1.4$	$1.5$	$1.4$	$4.0$	$2.4$
ZnTPPS	$1.8$	$2.2$	$1.7$	$4.3$	$7.0$
ZnTPPS + CTAB	$1.7$	$3.2$	$1.5$	$4.0$	$2.2$

from the experiment [50]. ${\tau }_{S_n}$ is set as 100 fs for ZnTMPyP and ZnTMPyP+SDS, and $1.3$ ps for ZnTPPS and ZnTPPS+CTAB, respectively [50]. In addition, concentrations for all compounds are set as $2.9\times {10}^{23}$/m${}^3$ in consistent with the experiment [50]. Unfortunately, it did not give the lifetime of state $T_1$ in experiment [50]. We set ${\tau }_{T_1}$ for all compounds as $1.0\times {10}^{-3}$ s according to another two experiments [26,30].

Firstly, we studied the energy transmittance $\mathcal{T}\left(L\right)$ depending on peak intensity of incident pulse trains in Figure 3 at distances L = 0.5 mm and L = 1.0 mm for all compounds. ZnTMPyP and ZnTPPS in aqueous solutions show lower transmittances than their J-aggregates with SDS and CTAB. For weak intensities, this difference is small. While the peak intensity increases higher than $I_0=1.0\times {10}^{12}$ W/m${}^2$, the gap becomes obvious. In addition, we can notice that ZnTPPS in aqueous solutions has better OL behaviors than ZnTMPyP. At longer distance L = 1.0 mm, the overall trend is almost the same, but energy transmittances of all compounds decrease more rapidly, which is caused by more molecules taking part in the absorption.

In Figure 4, we plotted the dependences of the energy transmittance $\mathcal{T}\left(L\right)$ on distance L for all compounds with peak intensity $I_0=1.0\times {10}^{13}$ W/m${}^2$. ZnTMPyP and ZnTPPS with SDS and CTAB have the very approximate curves, while ZnTMPyP and ZnTPPS in aqueous solutions could reach rather low values $\mathcal{T}\left(L\right)=0.19$ and $\mathcal{T}\left(L\right)=0.12$ at L = 2 mm, respectively.

Dynamical populations on five states were calculated, and we found two states $S_0$ and $T_1$ occupied almost all populations. In Figure 5, we plotted dynamical populations ${\rho }_{S_0}$ and ${\rho }_{T_1}$ at $L=0$ with $I_0=1.0\times {10}^{12}$ W/m${}^2$. For all compounds, the populations on state $S_0$ were transferred to $T_1$ mostly. We marked the $100\%$ population transfer times, and ZnTPPS in aqueous solutions has the earliest time $105.6 ns$ among all four compounds. ZnTMPyP with SDS has an earlier time than ZnTMPyP in aqueous solutions. Generally, ${\tau }_{S_1}$ and ${\tau }_{isc}$ are decisive parameters during population transferring in Table 1. Populations pumping on $T_1$ early ended up being instrumental in the one-photon absorption $T_1\to T_2$, resulting in strong two-photon absorption.

Effective population transfer times ${\tau }_{ST}$ during absorption process $S_0\to T_1$ [48] were studied and plotted in Figure 6. We could notice that ${\tau }_{ST}$ for all compounds are quite similar with weak intensities in two wing parts. In the middle part with high intensity, they are different apparently and ZnTPPS has the fastest transfer time ${\tau }_{ST}$. Fast ${\tau }_{ST}$ would promote the residing of populations on state $T_1$, which is mainly determined by the overall effect of ${\tau }_{S_1}$ and ${\tau }_{isc}$ in Table 1. Consequently, ZnTPPS in aqueous solutions has early and fast population transferring time and exhibits the best OL behaviors in Figure 3 and Figure 4.

Taking ZnTPPS as an example, we studied the output intensities after OL absorption at L = 1.0 mm for different concentrations $N=2.9\times {10}^{23}$/m${}^3$, $N=2.9\times {10}^{24}$/m${}^3$ in Figure 7. At $r=0$ in Figure 7a, we could see that the output intensity decreases 2∼3 orders of magnitude with big concentration $N=2.9\times {10}^{24}$/m${}^3$, and the peak intensity occurs earlier than the cause with small concentration. Comparing the two-dimensional pictures Figure 7b,c, we could notice that the shape of output intensity becomes asymmetric with big concentration. This is because the front part is mainly the two-step absorption $(S_0\to S_1)\times (T_1\to T_2)$, while the latter part is dominated by the one-photon absorption $T_1\to T_2$ depending on population ${\rho }_{T_1}$. The different absorption mechanisms lead to the asymmetric phenomenon. Big concentration means more molecules taking part in the nonlinear absorption and therefore the asymmetry becomes more explicit.

4. Conclusions

Dynamical absorption processes of picosecond pulse trains in ZnTMPyP and ZnTPPS in aqueous solutions and their porphyrin J-aggregates with micelles of SDS and CTAB were studied in this work. Interacting with picosecond pulse trains, the studied materials were simplified as a five-level model. We used the Crank–Nicholson numerical method to solve rate equations and the two-dimensional paraxial field. The OL process is mainly two-step absorption $(S_0\to S_1)\times (T_1\to T_2)$. For weak intensities, there is mainly the linear absorption. However, for high intensities, the nonlinear two-step absorption is dominant, and the output intensities would be extremely asymmetric. All compounds are good optical limiters, and ZnTPPS in aqueous solutions shows the best OL properties mainly due to the combined effect of photophysical parameters ${\tau }_{S_1}$ and ${\tau }_{isc}$. In general, micelles of SDS and CTAB would weaken the OL effects of ZnTMPyP and ZnTPPS. Increasing concentration is one effective method to enhance the OL effect.