## Experimental retrieval of the kinetic parameters of a dye in a solid film |

Optics Express, Vol. 19, Issue 19, pp. 18253-18259 (2011)

http://dx.doi.org/10.1364/OE.19.018253

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### 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

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]

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]

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]

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]

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]

## 2. Model Description and Experimental Setup

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]

*ij*∈ {30, 21} are

*f*= (

_{ij}*h̄ω*)

_{ij}^{−1}

**E**· (

**P**′

*+*

_{ij}**P**

*Δ*

_{ij}*ω*/2). As shown below (cf. Fig. 2), we found the

_{ij}**P**

*Δ*

_{ij}*ω*/2-terms to only have a minor contribution to the simulated transmission. The following polarization terms are used to couple Maxwell’s equations with SRE [Eq. (1)], where

_{ij}*N*are the corresponding occupation densities,

_{i}**P**

*are the transition polarizations,*

_{ij}*κ*= 6

_{ij}*πɛ*

_{0}

*c*

^{3}

*γ*/(

_{r,ij}*nω*

_{ij}^{2}) are the coupling coefficients. Further the total lifetime of each level is given by

*N*′

*=*

_{i}*N*′

_{Σ}= 0, we eliminate the zero-level equation, obtaining

*N*

_{0}via

**E**and

**P**

*being column vectors) and normalized functions*

_{ij}**p**= (

*ɛ*

_{0}

*E*)

_{p}^{−1}[

**P**

_{30}|

**P**

_{21}]

*,*

^{T}**e**=

**E**/

*E*, with

_{p}*E*being the magnitude of the pumping electric field: Here the constants are defined as

_{p}**ē**

*,*

^{i}**p̄**

*as*

^{i}**e**

^{i}^{+1}+

**e**

*,*

^{i}**p**

^{i}^{+1}+

**p**

*, and use the Crank-Nicolson scheme (*

^{i}**n**

^{i}^{+1}−

**n**

*)/*

^{i}*τ*= 1/2

**gn̄**

*+ 1/2*

^{i}**w**(

**dp**

*/*

^{i}*τ*+

**b**

_{1}

**p̄**

*/4)*

^{i}**ē**

*.*

^{i}**ē**

*,*

^{i}**p̄**

*and*

^{i}**dp**

*=*

^{i}**p**

^{i}^{+1}−

**p**

*are known, we get*

^{i}**dp**and

**p**correspond to taking the first and the second row respectively.

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]

**dp**and

**p̄**can be obtained with

*β*_{2}=

**b**

_{1}

*τ*+2

**I**,

**I**is the 2x2 identity matrix.

*c*being the speed of light. Normalized Faraday’s law reads similarly,

## 3. Results and Discussion

*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[

*n*/(1 +

_{glass}*n*)]

_{glass}^{2}|

*e*/

_{t}*e*|

_{i}^{2}[17].

*γ*

_{nr,21}/

*γ*

_{r,21}, we have performed additional measurements of the quantum yield, finding

*η*= 0.04 for our solid film sample. The quantum yield was measured using a reference sample, 20

_{f}*μ*M Rh800 in Ethanol solution with known quantum yield of about 0.2 [13

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]

*η*=

_{f}*η*(

_{sol}*A*/

_{sol}*A*) (

_{f}*F*/

_{f}*F*) (

_{sol}*n*/

_{f}*n*)

_{sol}^{2}.

*η*= (

*γ*

_{32}/

*γ*

_{3})(

*γ*

_{r}_{,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

*κ*). Yet,

_{ij}*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*.

*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.

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]

*τ*= 0.25 ns relative to 0.71 ns for the dye in methanol [14

_{f}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]

**31**, 7034–7041 (1992). [CrossRef] [PubMed]

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]

**31**, 7034–7041 (1992). [CrossRef] [PubMed]

## 4. Conclusion

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]

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]

## Acknowledgments

## 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 |

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 |

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 |

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. |

5. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

6. | A. Taflove and S. Hagness, |

7. | S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B |

8. | A. S. Nagra and R. A. York, “FDTD Analysis of Wave Propagation in Nonlinear Absorbing and Gain Media,” IEEE Trans. Antennas Propag. |

9. | A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B |

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. |

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. |

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. |

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. |

15. | A. E. Siegman, |

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. |

17. | E. Hecht, |

18. | L. J. Prokopeva, J. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. |

**OCIS Codes**

(160.2540) Materials : Fluorescent and luminescent materials

(160.3918) Materials : Metamaterials

(160.4236) Materials : Nanomaterials

**ToC Category:**

Materials

**History**

Original Manuscript: May 4, 2011

Manuscript Accepted: July 19, 2011

Published: September 2, 2011

**Citation**

Jan Trieschmann, Shumin Xiao, Ludmila J. Prokopeva, Vladimir P. Drachev, and Alexander V. Kildishev, "Experimental retrieval of the kinetic parameters of a dye in a solid film," Opt. Express **19**, 18253-18259 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18253

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### References

- 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).
- 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]
- 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. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method , 3rd ed. (Artech House, 2005).
- S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515 (2007). [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]
- A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009). [CrossRef]
- 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]
- M. Lieberherr, p.39–46 (1988), “Laser induced fluorescence and scattering near interfaces,” Ph.D. thesis, ETH Zurich (1991).
- 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]
- 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]
- 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]
- A. E. Siegman, Lasers (University Science Books, 1986).
- 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]
- E. Hecht, Optics , 4th ed. (Addison Wesley, 2001).
- 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]

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