Developing a time-domain method for simulating statistical behavior of many-emitter systems in the presence of electromagnetic field

In this paper, one of the major shortcomings of the conventional numerical approaches is alleviated by introducing the probabilistic nature of molecular transitions into the framework of classical computational electrodynamics. The main aim is to develop a numerical method which is capable of capturing the statistical attributes caused by the interactions between a group of spontaneous as well as stimulated emitters and the surrounding electromagnetic field. The electromagnetic field is governed by classical Maxwell's equations, while energy is absorbed from and emitted to the (surrounding) field according to the transitions occurring for the emitters, which are governed by time-dependent probability functions. These probabilities are principally consistent with quantum mechanics. In order to validate the proposed method, it is applied to three different test cases: directionality of fluorescent emission in a corrugated single-hole gold nanodisk, spatial and temporal coherence of fluorescent emission in a hybrid photonic-plasmonic crystal, and stimulated emission of a core-shell SPASER (surface plasmon amplification by stimulated emission of radiation). The results are shown to be closely comparable to the experimental results reported in the literature.

Figure 1

Jablonski diagram showing electronic transitions of a molecule with radiative and nonradiative transitions from or to singlet ($S$) and triplet ($T$) states. The simplified model for a fluorescent molecule is shown in the inset.

Figure 2

Amplitude of $\mathbf{p}$ as a function of $\omega$.

Figure 3

The modified FDTD algorithm for implementing the procedure required for simulating emitters-EM field interactions. All emitters are at the ground state at the beginning of simulation and updated during the main time-marching loop. The state-updating steps are elaborated in Fig. 4.

Figure 4

State-updating procedure for emitters. In this flowchart, $E{F}_i$ is the emission flag of $i\mathrm{th}$ emitter, which is on while the wave packet is emitting. ${\mathbf{E}}_i\left(t\right)$ is the electric field at the position of the $i\mathrm{th}$ emitter, ${\mathbf{F}}_i\left({\tilde{t}}_i\right)$ is the emitted electric wave function, and $t_i$ is the time elapsed since the $i\mathrm{th}$ emitter has been transited to its current state. In each time step, if the emission flag of an emitter (e.g., the $i\mathrm{th}$ one) is on, the emission continues. If the flag is off, the state of the emitter should be checked. In case the emitter is at the ground state, the absorption probability $P_{02i}\left(t\right)$ is compared to a randomly generated number, $r_i$, and consequently, either the transition to level 2 takes place or the dipole moment of the emitter is updated. In other cases (i.e., the emitter is at level 1 or 2) the same procedure is followed with the corresponding probability function.

Figure 5

Schematic of the plasmonic nanodisk with the fluorescent molecules positioned at the central hole of it as proposed by Aouani et al. [42]. The three-dimensional view is shown on top, and the cross-sectional view is presented at the bottom. The geometrical parameters are set according to [42] as $a=220$ nm, $b=140$ nm, $c=440$ nm, $d=200$ nm, $h_1=190$ nm, and $h_2=65$ nm.

Figure 6

Polar distribution of the long-time and ensemble averaged intensity for emissions detected from the plasmonic nanodisk. Red (light gray) and green (dark gray) curves correspond to the results obtained for Alexa Fluor 647 (${\lambda }_{em}=670$ nm) and rhodamine 6G (${\lambda }_{em}=560$ nm), respectively. Numerical results (solid lines) are compared with the results reported by Aouani et al. [42] (dashed lines).

Figure 7

Schematic of the HPPS proposed by Shi et al. [13]. The $y–z$ cross-sectional view of the structure is shown at the bottom, in which the fluorescent-doped PVA filling is also marked.

Figure 8

The time evolution of the electric field produced by a pulse train in the photonic-crystal part of the HPPS proposed by Shi et al. [13]. The field distribution on a surface passing through the center of the polystyrene spheres in the $x–y$ plane (see the structure in Fig. 7) is illustrated at selective time steps. The $y$-directed incident pump pulse enters the domain of the structure from the center of $x–z$ plane ($y=0$). Here the first four pulses of the train are depicted. The fluorescent emission of the molecules is observed as small spots disturbing the pump field.

Figure 9

Wavelength spectrum that is obtained using the proposed numerical method. The peak corresponding to the reflected portion of the excitation wave is observed at $\lambda =532$ nm, and the emission peak occurs at $\lambda =594$ nm with the bandwidth of $\left|\mathrm{\Delta }\lambda \right|=7.1$ nm. The inset shows the emission spectrum as reported by Shi et al. [13].

Figure 10

Young's double-slit fringe visibility as a function of the separation distance of the double slits calculated using the proposed numerical method. The main plot shows the absolute value of visibility, and in the inset, the maxima and minima of visibility are plotted as red (light gray) bars. The results reported in Fig. 4(c) of Ref. [13] are also plotted as blue (dark gray) bars for comparison.

Figure 11

Schematic of the core-shell-type SPASER.

Figure 12

Emission spectrum of SPASER. The Lorentzian functions and their summation are matched to the data and shown in red (dark gray), purple (light gray), and black, respectively. The results reported by Noginov et al. [48] are shown in the inset.