Effects of a solid matrix on the dye kinetic parameters for Rh800 were experimentally studied. Saturation intensity dependencies were measured with a seeding pulse amplification method using a picosecond and a femtosecond white light supercontinuum source. The kinetic parameters were obtained by fitting experimental dependencies with Yee’s finite-difference time-domain model coupled to the rate equations of the 4-level Rh800-system. The comparison of these parameters (Rh800-solid host) with liquid host parameters revealed a slight change of the radiative lifetime and a strong change of the non-radiative decay rate. This experimentally determined model enables predictive simulations of time-domain responses of active metamaterials.
©2011 Optical Society of America
Organic dye based gain media, which have been widely used for lasing, have recently found new applications. One application implies the compensation of losses in metal-dielectric composites. Here, the dyes involved are in the form of solid films, for example embedded in epoxy. The kinetic parameters of dyes are environment sensitive and have not been well studied in the form of solid films. Having accurate models of the optical gain materials is imperative for achieving the complete compensation of losses in plasmonic elements of optical metamaterials. In the visible range, a loss-free and active negative-index material with Rhodamine800 (Rh800)/epoxy film as a gain medium has been recently experimentally demonstrated , supported by numerical simulations based on the finite element method (FEM) in the frequency domain (FD). Frequency domain, finite element analysis of metamaterials with gain media has been studied extensively [2–4]. Time domain (TD) analysis of metamaterials response is often required. Here we determine kinetic parameters of Rh800 in the form of a solid film modeled by a 4-level system. A seeding pulse amplification method (pump-probe) enables saturation dependences measured with a picosecond (ps) and a femtosecond (fs) white light continuum. The dye parameters are then obtained by fitting the experimental data with the time domain simulations along with quantum yield measurements.
In contrast to FD, time domain analysis provides a time-resolved description of the system kinetics. Yee’s classical Finite-Difference Time-Domain (FDTD) method [5, 6] coupled to a multi-level atomic system through auxiliary differential equations (ADE) has been already used for various studies [7, 8] in homogeneous hosts, and metamaterials [9, 10].
Addressing the need for accurate time domain models of gain media, we provide a detailed study of a dye solid film with pump-probe experiments and a numerical model of the 4-level gain system. The numerical model is matched in order to retrieve the kinetic parameters of the Rh800-epoxy film. Finally, the kinetic parameters of Rh800 dye in epoxy are compared with corresponding parameters of a submonolayer molecular film  and Rh800 in different solvents [12–14]. We employ the ADE-FDTD approach for our analysis.
First, the reference transmission data are obtained from a pump-probe experiment with a uniform slab of Rh800 dye embedded in epoxy and deposited on an ITO-coated glass substrate; the quantum yield for the dye in solid film is measured as well. Then, the parameters of the ADE model are tuned to match these experimental data. The final model with best-fitted parameters adequately represents the core dynamics of the system, providing a well defined physical background for modeling of complex nanostructured active metamaterials. Deviations from experiments are of the order of experimental error.
2. Model Description and Experimental Setup
Modeling of an atomic system usually involves a set of rate equations (SRE). Our consideration is limited to the transitions shown in Fig. 1(a). Thus, the model is valid only for a specific subset of nonlinear media, i.e. dye media. The SRE can be readily obtained [10, 15]:Fig. 2), we found the P ijΔωij/2-terms to only have a minor contribution to the simulated transmission. The following polarization terms Eq. (1)], where Ni are the corresponding occupation densities, P ij are the transition polarizations, and are the transition line-widths, and κij = 6πɛ 0 c 3 γr,ij/(nωij 2) are the coupling coefficients. Further the total lifetime of each level is given by , being the inverse of the total energy decay rate .
Because the total population is conserved, ΣN′i = N′Σ = 0, we eliminate the zero-level equation, obtaining N 0 via . The above formalism is further simplified utilizing a matrix notation (we assume E and P ij being column vectors) and normalized functions , p = (ɛ 0 Ep)−1 [P 30|P 21]T, e = E/Ep, with Ep being the magnitude of the pumping electric field:
ADE [Eqs. (3) and (4)] and normalized Maxwell’s equations are then solved numerically. To solve Eq. (3) with 2nd-order accuracy, we define ē i, p̄ i as e i +1 + e i, p i +1 + p i, and use the Crank-Nicolson scheme (n i +1 − n i)/τ = 1/2 gn̄ i + 1/2 w (dp i/τ + b 1 p̄ i/4) ē i.
Since, ē i, p̄ i and dp i = p i +1 − p i are known, we get
Standard central difference technique is applied to Eq. (4). Because of its nonrestrictive stability condition, a bilinear scheme is usually recommended for the free-term in the Lorentz oscillator , however to avoid additional complications, here we take the free term at i-th time step:
Explicit evaluation of dp and p̄ can be obtained with
As we deal with normalized units, we also use a normalized magnetic field . The normalized Ampere’s law is (omitting spatial indices since each polarization is solved locally for each cell) , with c being the speed of light. Normalized Faraday’s law reads similarly, .
In the experiment, two subsequent pulses (a pump and a much weaker probe pulse) hit the sample. The setup is shown in Fig. 1(b). The pump pulse from an optical parametric amplifier (OPA) gradually increases the sample irradiance, providing a saturation dependence of the probe signal amplification. The probe pulse was filtered with a band-pass filter from a white light continuum source generated in water under 800 nm laser pulse illumination. A set of filters was used with different central wavelengths, 710, 715, 720, 726 nm and FWHM of about 20 nm. Assuming a Gaussian profile of the laser beams, the setup is arranged to have the beam waist located at the sample. All experimental parameters are shown in Table 1.
3. Results and Discussion
We assume a uniform gain film, so that a 1-dimensional simulation is sufficient. We found that due to the coupled atomic system, a spatial stepsize Δ < λ/50 is necessary to ensure convergence (usually Δ ≈ λ/20 is chosen for classical FDTD). To minimize the computational effort, we only solve for the transmission into the glass substrate via FDTD and incorporate the effect of the glass-air interface at the backside by adjusting the transmission, T = 4[nglass/(1 + nglass)]2|et/ei|2 .
In the 4-level system, all transitions except for the “lasing” transition 2-1, are modeled with strictly non-radiative decays. To determine the important ratio, γ nr,21/γ r,21, we have performed additional measurements of the quantum yield, finding ηf = 0.04 for our solid film sample. The quantum yield was measured using a reference sample, 20 μM Rh800 in Ethanol solution with known quantum yield of about 0.2 . The absorption (A) and fluorescence signal (F) ratio for the film relative to the solution were measured under the same conditions. The quantum yield is then given by ηf = ηsol (Asol/Af) (Ff/Fsol) (nf/nsol)2.
Using the definition of the quantum yield η = (γ 32/γ 3)(γr ,21/γ 2) , we proceed with the ratio γ 32/γ 3, where γ 32 = γ nr,32 and γ 3 = γ nr,32 +γ r,30. Since the system is optically pumped, we include the stimulated 0–3 transition, where its strength is determined by the radiative decay γ r,30 (see κij). Yet, γ r,30 is by several orders of magnitude smaller than the non-radiative decay γ nr,32, thus γ 32/γ 3 yields unity. Level 3 is always rapidly depopulated into level 2, while other transitions do not affect the system kinetics and can be neglected in the rate equations. Since γ 2 = γ r,21 +γ nr,21 we finally write γ nr,21/γ r,21 ≈ (1 – η)/η.
To improve the fitting quality, the simulations and optical experiments have been performed with different durations and wavelengths of Gaussian laser pulses. A collection of measurements with corresponding simulations are shown in Fig. 2. The retrieved system parameters are given in Table 2. It is important to note that all simulations have been performed with identical system parameters.
Since no distinct features (e.g. resonances) are expected, the main fitting characteristics are the probe transmission without any pumping and its saturated value for high pump powers, along with the transmission spectra and the pulse-duration dependence. The final set of the fitting parameters (indices ij for the respective transition) was taken from a RMS difference comparison of the simulated transmission behavior to the measured data at λ = 720 nm. The optimal set of the fitting parameters is collected in Table 2.
While some parameters, such as the lasing transition lifetimes (and therefore the quantum yield) or the dephasing times, affect the system response more strongly with respect to amplitude and spectral behavior, other parameters, e.g. all non-lasing transition lifetimes, only have minor impact. The results obtained from the numerical model with the best-fit parameters of Table 2, match the experiment well. Saturation as well as transmission characteristics without pumping are in good agreement for several probing wavelengths. The pulse-duration dependence exhibits a very good match (2 ps vs. 103 fs).
The fitted radiative lifetime shows only moderate change induced by the epoxy as a host material, 6.3 ns relative to 8.2 ns for Rh800 in methanol solution . On the contrary, the fluorescence lifetime of the dye (corresponding to the total decay rate) in epoxy is strongly reduced, τf = 0.25 ns relative to 0.71 ns for the dye in methanol . The epoxy environment provides additional non-radiative decay resulting in a strong change in the quantum yield. The obtained parameters versus the known experimental data for Rh800/methanol solution from Benfey et al.  are compared in Table 3 along with the derived values of the non-radiative decay rates.
In conclusion, due to the solid film there is a slight change of radiative lifetime, but more importantly a strong change of the non-radiative decay rate. These observations confirm the hypothesis [12, 14] that the changes in the radiative processes are sensitive to the optical environment, while the non-radiative are mainly due to the chemical environment. Effects of the environment need to be carefully considered and optimized in the ongoing work, especially for the active compensation of losses in plasmonic elements.
Interestingly, both simulations and experiments demonstrate a similar trend - the saturation transmission for 103 fs pulse duration is slightly higher than for 2 ps (e.g. 720 nm-line). We believe this is due to the kinetics of the population processes involving the inherent dephasing times. As the pulse durations are getting closer to the dephasing time, its effect could become significantly weaker, hence resulting in a higher transmission.
To conclude, we have employed a generic 4-level atomic system coupled to Maxwell’s equations. We have obtained kinetic parameters of the 4-level model of Rh800 dye in solid epoxy film. The model fitted with the pump-probe experiment could be further coupled to additional material models. Among them are the dispersive models of noble metals. Such a coupling can lead to complex time domain simulations of advanced active metamaterials. These more complicated models are particularly instrumental for acquiring insight into the time-resolved physics of plasmonic nanostructures with gain .
Similar to the dispersive models of metals , the kinetic model shown here could be used within any time-domain multiphysics simulation environment with an appropriate full-wave Maxwell numerical solver (e.g., FETD or FVTD engine) and is not limited solely to the FDTD method.
This work was supported in part by U.S. Army Research Office under grant number 50372-CH-MUR, by ARO grant W911NF-04-1-0350, and by ARO-MURI award 50342-PH-MUR.
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