Abstract

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 OSA

1. Introduction

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 [1], 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 [24]. 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 [11] and Rh800 in different solvents [1214]. 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]:

N0=τ101N1f30N1=τ211N2τ101N1f21N2=τ321N3τ211N2+f21N3=τ321N3+f30
where primes denote temporal derivatives, and the driving terms for ij ∈ {30, 21} are fij = (h̄ωij)−1 E · (Pij + P ijΔωij/2). As shown below (cf. Fig. 2), we found the P ijΔωij/2-terms to only have a minor contribution to the simulated transmission. The following polarization terms
P30+Δω30P30+ω302P30=κ30(N0N3)EP21+Δω21P21+ω212P21=κ21(N1N2)E
are used to couple Maxwell’s equations with SRE [Eq. (1)], where Ni are the corresponding occupation densities, P ij are the transition polarizations, Δω30=τ321+τ301+2T2,301 and Δω21=τ211+τ101+2T2,211 are the transition line-widths, and κij = 6πɛ 0 c 3 γr,ij/(ij 2) are the coupling coefficients. Further the total lifetime of each level is given by τij=γij1=(1/τr,ij+1/τnr,ij)1, being the inverse of the total energy decay rate [15].

 

Fig. 1 (a) Generic 4-level atomic system, (b) Experimental pump-probe setup.

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Fig. 2 Transmission results performed with 2 ps (a–d) and 103 fs (e–f) pulse duration at different probing wavelengths. (blue dots: experiment; solid red line: simulation; dashed green line: reduced simulation without the b1p/2-term in the SRE).

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Because the total population is conserved, ΣNi = NΣ = 0, we eliminate the zero-level equation, obtaining N 0 via N0=NΣΣi=13Ni. The above formalism is further simplified utilizing a matrix notation (we assume E and P ij being column vectors) and normalized functions n=NΣ1[N1,N2,N3]T, p = (ɛ 0 Ep)−1 [P 30|P 21]T, e = E/Ep, with Ep being the magnitude of the pumping electric field:

n=gn+w(p+b1p/2)e,n(0)=[0,0,0]T
p+b1p+b02p=k(a0+a1n)eT,p(0)=p(0)=0
Here the constants are defined as
g=[τ101τ21100τ211τ32100τ321],w=[0w210w21w300],wij=ɛ0Ep2h¯NΣωij,b0=diag(ω30,ω21)b1=diag(Δω30,Δω21),
a0=[10],a1=[112110]andk=6πc3NΣn1diag(γr,30ω302,γr,12ω212).

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, i as e i +1 + e i, p i +1 + p i, and use the Crank-Nicolson scheme (n i +1n i)/τ = 1/2 gn̄ i + 1/2 w (dp i/τ + b 1 i/4) ē i.

Since, ē i, i and dp i = p i +1p i are known, we get

n3i+1=2γ32τ2+γ32τn3i+W30dp30i+p¯30iΔω30τ/42+γ32τe¯i,n2i+1=2γ21τ2+γ21τn2i+γ32τ2+γ21τn¯3i+W21dp21i+p¯21iΔω21τ/42+γ21τe¯i,n1i+1=2γ10τ2+γ10τn1i+γ21τ2+γ10τn¯2iW21dp21i+p¯21iΔω21τ/42+γ10τe¯i
with γij=τij1 and τ being the time step; clearly sub-indices 30 and 21 applied to matrices dp and p correspond to taking the first and the second row respectively.

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 [16], however to avoid additional complications, here we take the free term at i-th time step:

pi+12pi+pi1τ2+b1pi+1pi12τ+b02pi=k(a0+a1ni)eT,i

Explicit evaluation of dp and can be obtained with

dpi+1=1/2(β1β01)dpi+1/2(β1+β01)p¯i+α(a0+a1ni+1)eT,i+1,p¯i+1=p¯i+dpi+dpi+1
where β 2 = b 1 τ +2I, α=2β21kτ2, β0=β21(b1τ2I) and β1=2β21(2Ib02τ2). Furthermore, I is the 2x2 identity matrix.

As we deal with normalized units, we also use a normalized magnetic field h=H/(Epɛ0/μ0). The normalized Ampere’s law is (omitting spatial indices since each polarization is solved locally for each cell) ei+1=ei+cτɛr1[×hi+1/2]ɛr1Σjk{30,21}dpjkiT, with c being the speed of light. Normalized Faraday’s law reads similarly, hi+1/2=hi1/2cτμr1[×ei].

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.

Tables Icon

Table 1. Experimental Setup Parameters

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 [17].

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 [13]. 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) [15], 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.

Tables Icon

Table 2. Kinetic 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 [14]. 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 [14]. 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. [14] are compared in Table 3 along with the derived values of the non-radiative decay rates.

Tables Icon

Table 3. Fitted Kinetic Parameters For Rh800/Solid Film Vs. The Reference Data For Methanol Solution

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.

4. Conclusion

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 [10].

Similar to the dispersive models of metals [18], 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.

Acknowledgments

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.

References and links

1. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010). [CrossRef]   [PubMed]  

2. M. P. Hatlo Andresen, A. V. Skaldebo, M. W. Haakestad, H. E. Krogstad, and J. Skaar, “Effect of gain saturation in a gain compensated perfect lens,” J. Opt. Soc. Am. B 27, 1610–1616 (2010). [CrossRef]  

3. Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

4. T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006). [CrossRef]  

5. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302 (1966). [CrossRef]  

6. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

7. S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515 (2007). [CrossRef]  

8. A. S. Nagra and R. A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Propag. 46, 334 (1998). [CrossRef]  

9. A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009). [CrossRef]  

10. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010). [CrossRef]   [PubMed]  

11. M. Lieberherr, p.39–46 (1988), “Laser induced fluorescence and scattering near interfaces,” Ph.D. thesis, ETH Zurich (1991).

12. P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988). [CrossRef]  

13. B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994). [CrossRef]  

14. D. P. Benfey, D. C. Brown, S. J. Davis, L. G. Piper, and R. F. Foutter, “Diode-pumped dye laser analysis and design,” Appl. Opt. 31, 7034–7041 (1992). [CrossRef]   [PubMed]  

15. A. E. Siegman, Lasers (University Science Books, 1986).

16. L. Zhili and L. Thylen, “On the accuracy and stability of several widely used FDTD approaches for modeling lorentz dielectrics,” IEEE Trans. Antennas Propag. 57, 3378 (2009). [CrossRef]  

17. E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).

18. L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47, 1150–1153 (2011). [CrossRef]  

References

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  1. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
    [Crossref] [PubMed]
  2. M. P. Hatlo Andresen, A. V. Skaldebo, M. W. Haakestad, H. E. Krogstad, and J. Skaar, “Effect of gain saturation in a gain compensated perfect lens,” J. Opt. Soc. Am. B 27, 1610–1616 (2010).
    [Crossref]
  3. Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).
  4. T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
    [Crossref]
  5. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302 (1966).
    [Crossref]
  6. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  7. S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515 (2007).
    [Crossref]
  8. A. S. Nagra and R. A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Propag. 46, 334 (1998).
    [Crossref]
  9. A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
    [Crossref]
  10. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
    [Crossref] [PubMed]
  11. M. Lieberherr, p.39–46 (1988), “Laser induced fluorescence and scattering near interfaces,” Ph.D. thesis, ETH Zurich (1991).
  12. P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988).
    [Crossref]
  13. B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
    [Crossref]
  14. D. P. Benfey, D. C. Brown, S. J. Davis, L. G. Piper, and R. F. Foutter, “Diode-pumped dye laser analysis and design,” Appl. Opt. 31, 7034–7041 (1992).
    [Crossref] [PubMed]
  15. A. E. Siegman, Lasers (University Science Books, 1986).
  16. L. Zhili and L. Thylen, “On the accuracy and stability of several widely used FDTD approaches for modeling lorentz dielectrics,” IEEE Trans. Antennas Propag. 57, 3378 (2009).
    [Crossref]
  17. E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).
  18. L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47, 1150–1153 (2011).
    [Crossref]

2011 (1)

L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47, 1150–1153 (2011).
[Crossref]

2010 (4)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

M. P. Hatlo Andresen, A. V. Skaldebo, M. W. Haakestad, H. E. Krogstad, and J. Skaar, “Effect of gain saturation in a gain compensated perfect lens,” J. Opt. Soc. Am. B 27, 1610–1616 (2010).
[Crossref]

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref] [PubMed]

2009 (2)

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[Crossref]

L. Zhili and L. Thylen, “On the accuracy and stability of several widely used FDTD approaches for modeling lorentz dielectrics,” IEEE Trans. Antennas Propag. 57, 3378 (2009).
[Crossref]

2007 (1)

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515 (2007).
[Crossref]

2006 (1)

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[Crossref]

1998 (1)

A. S. Nagra and R. A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Propag. 46, 334 (1998).
[Crossref]

1994 (1)

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

1992 (1)

1988 (1)

P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988).
[Crossref]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302 (1966).
[Crossref]

Arden-Jacob, J.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Bachteler, B.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Benfey, D. P.

Borneman, J.

L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47, 1150–1153 (2011).
[Crossref]

Brown, D. C.

Cao, J.-X.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

Chigrin, D. N.

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515 (2007).
[Crossref]

Davis, S. J.

Dong, Z.-G.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[Crossref]

Drexhage, K.-H.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Fang, A.

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[Crossref]

Foutter, R. F.

Haakestad, M. W.

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hamm, J. M.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref] [PubMed]

Han, K.-T.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Hatlo Andresen, M. P.

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).

Hess, O.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref] [PubMed]

Kildishev, A. V.

L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47, 1150–1153 (2011).
[Crossref]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[Crossref]

Klar, T. A.

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[Crossref]

Koellner, M.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Koschny, T.

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[Crossref]

Krogstad, H. E.

Li, T.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Lieberherr, M.

M. Lieberherr, p.39–46 (1988), “Laser induced fluorescence and scattering near interfaces,” Ph.D. thesis, ETH Zurich (1991).

Liu, H.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Meier, B.

P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988).
[Crossref]

Mueller, R.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Nagra, A. S.

A. S. Nagra and R. A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Propag. 46, 334 (1998).
[Crossref]

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

Penzkoffer, A.

P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988).
[Crossref]

Piper, L. G.

Prokopeva, L. J.

L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47, 1150–1153 (2011).
[Crossref]

Pusch, A.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref] [PubMed]

Sauer, M.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Seeger, S.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Shalaev, V. M.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Skaar, J.

Skaldebo, A. V.

Soukoulis, C. M.

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[Crossref]

Spangler, W.

P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988).
[Crossref]

Sperber, P.

P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988).
[Crossref]

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Thylen, L.

L. Zhili and L. Thylen, “On the accuracy and stability of several widely used FDTD approaches for modeling lorentz dielectrics,” IEEE Trans. Antennas Propag. 57, 3378 (2009).
[Crossref]

Tsakmakidis, K. L.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref] [PubMed]

Wang, S.-M.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Wegener, M.

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[Crossref]

Wolfrum, J.

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

Wuestner, S.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref] [PubMed]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302 (1966).
[Crossref]

York, R. A.

A. S. Nagra and R. A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Propag. 46, 334 (1998).
[Crossref]

Yuan, H.-K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

Zhang, X.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Zhili, L.

L. Zhili and L. Thylen, “On the accuracy and stability of several widely used FDTD approaches for modeling lorentz dielectrics,” IEEE Trans. Antennas Propag. 57, 3378 (2009).
[Crossref]

Zhu, S.-N.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Zhu, Z.-H.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

Zhukovsky, S. V.

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515 (2007).
[Crossref]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[Crossref]

IEEE Trans. Antennas Propag. (3)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302 (1966).
[Crossref]

A. S. Nagra and R. A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Propag. 46, 334 (1998).
[Crossref]

L. Zhili and L. Thylen, “On the accuracy and stability of several widely used FDTD approaches for modeling lorentz dielectrics,” IEEE Trans. Antennas Propag. 57, 3378 (2009).
[Crossref]

IEEE Trans. Magn. (1)

L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47, 1150–1153 (2011).
[Crossref]

J. Lumin. (1)

B. Bachteler, K.-H. Drexhage, J. Arden-Jacob, K.-T. Han, M. Koellner, R. Mueller, M. Sauer, S. Seeger, and J. Wolfrum, “Sensitive fluorescence detection in capillary electrophoresis using laser diodes and multiplex dyes,” J. Lumin. 62, 101 (1994).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (1)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref] [PubMed]

Opt. Quantum. Electron. (1)

P. Sperber, W. Spangler, B. Meier, and A. Penzkoffer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum. Electron. 20, 395 (1988).
[Crossref]

Phys. Rev. B (2)

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Phys. Rev. B 96, 044104 (2010).

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[Crossref]

Phys. Rev. Lett. (1)

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105, 127401 (2010).
[Crossref] [PubMed]

Phys. Stat. Solidi B (1)

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515 (2007).
[Crossref]

Other (4)

E. Hecht, Optics, 4th ed. (Addison Wesley, 2001).

A. E. Siegman, Lasers (University Science Books, 1986).

M. Lieberherr, p.39–46 (1988), “Laser induced fluorescence and scattering near interfaces,” Ph.D. thesis, ETH Zurich (1991).

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (2)

Fig. 1
Fig. 1

(a) Generic 4-level atomic system, (b) Experimental pump-probe setup.

Fig. 2
Fig. 2

Transmission results performed with 2 ps (a–d) and 103 fs (e–f) pulse duration at different probing wavelengths. (blue dots: experiment; solid red line: simulation; dashed green line: reduced simulation without the b1p/2-term in the SRE).

Tables (3)

Tables Icon

Table 1 Experimental Setup Parameters

Tables Icon

Table 2 Kinetic System Parameters

Tables Icon

Table 3 Fitted Kinetic Parameters For Rh800/Solid Film Vs. The Reference Data For Methanol Solution

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

N 0 = τ 10 1 N 1 f 30 N 1 = τ 21 1 N 2 τ 10 1 N 1 f 21 N 2 = τ 32 1 N 3 τ 21 1 N 2 + f 21 N 3 = τ 32 1 N 3 + f 30
P 30 + Δ ω 30 P 30 + ω 30 2 P 30 = κ 30 ( N 0 N 3 ) E P 21 + Δ ω 21 P 21 + ω 21 2 P 21 = κ 21 ( N 1 N 2 ) E
n = gn + w ( p + b 1 p / 2 ) e , n ( 0 ) = [ 0 , 0 , 0 ] T
p + b 1 p + b 0 2 p = k ( a 0 + a 1 n ) e T , p ( 0 ) = p ( 0 ) = 0
g = [ τ 10 1 τ 21 1 0 0 τ 21 1 τ 32 1 0 0 τ 32 1 ] , w = [ 0 w 21 0 w 21 w 30 0 ] , w i j = ɛ 0 E p 2 h ¯ N Σ ω i j , b 0 = diag ( ω 30 , ω 21 ) b 1 = diag ( Δ ω 30 , Δ ω 21 ) ,
a 0 = [ 1 0 ] , a 1 = [ 1 1 2 1 1 0 ] and k = 6 π c 3 N Σ n 1 diag ( γ r , 30 ω 30 2 , γ r , 12 ω 21 2 ) .
n 3 i + 1 = 2 γ 32 τ 2 + γ 32 τ n 3 i + W 30 d p 30 i + p ¯ 30 i Δ ω 30 τ / 4 2 + γ 32 τ e ¯ i , n 2 i + 1 = 2 γ 21 τ 2 + γ 21 τ n 2 i + γ 32 τ 2 + γ 21 τ n ¯ 3 i + W 21 d p 21 i + p ¯ 21 i Δ ω 21 τ / 4 2 + γ 21 τ e ¯ i , n 1 i + 1 = 2 γ 10 τ 2 + γ 10 τ n 1 i + γ 21 τ 2 + γ 10 τ n ¯ 2 i W 21 d p 21 i + p ¯ 21 i Δ ω 21 τ / 4 2 + γ 10 τ e ¯ i
p i + 1 2 p i + p i 1 τ 2 + b 1 p i + 1 p i 1 2 τ + b 0 2 p i = k ( a 0 + a 1 n i ) e T , i
d p i + 1 = 1 / 2 ( β 1 β 0 1 ) d p i + 1 / 2 ( β 1 + β 0 1 ) p ¯ i + α ( a 0 + a 1 n i + 1 ) e T , i + 1 , p ¯ i + 1 = p ¯ i + d p i + d p i + 1

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