## Plasmon enhanced upconversion luminescence near gold nanoparticles – simulation and analysis of the interactions: Errata |

Optics Express, Vol. 21, Issue 9, pp. 10606-10611 (2013)

http://dx.doi.org/10.1364/OE.21.010606

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### Abstract

The procedure used in our previous publication [Opt. Express **20**, 271, (2012)] to calculate how coupling to a spherical gold nanoparticle changes the upconversion luminescence of Er^{3+} ions contained several errors. The errors are corrected here.

© 2013 OSA

1. S. Fischer, F. Hallermann, T. Eichelkraut, G. von Plessen, K. W. Krämer, D. Biner, H. Steinkemper, M. Hermle, and J. C. Goldschmidt, “Plasmon enhanced upconversion luminescence near gold nanoparticles-simulation and analysis of the interactions,” Opt. Express **20**(1), 271–282 (2012). [CrossRef] [PubMed]

^{3+}ions. We considered two effects of the metal nanoparticle on the upconversion processes: first, the local electric field enhancement, quantified by an enhancement factor

*γ*

_{E}, and second the change of transitions rates within the upconverter, described by the Einstein coefficients. The Er

^{3+}ions of the upconverter were approximated as dipole emitters and their coupling to an adjacent spherical gold nanoparticle was modeled using Mie theory. The local-field enhancement and the nanoparticle-induced changes to the Einstein coefficients were then used in a rate equation model of the upconverter material

*β-*NaYF

_{4}: 20% Er

^{3+}.

1. S. Fischer, F. Hallermann, T. Eichelkraut, G. von Plessen, K. W. Krämer, D. Biner, H. Steinkemper, M. Hermle, and J. C. Goldschmidt, “Plasmon enhanced upconversion luminescence near gold nanoparticles-simulation and analysis of the interactions,” Opt. Express **20**(1), 271–282 (2012). [CrossRef] [PubMed]

## Criticisms of the treatment presented in [11. S. Fischer, F. Hallermann, T. Eichelkraut, G. von Plessen, K. W. Krämer, D. Biner, H. Steinkemper, M. Hermle, and J. C. Goldschmidt, “Plasmon enhanced upconversion luminescence near gold nanoparticles-simulation and analysis of the interactions,” Opt. Express **20**(1), 271–282 (2012). [CrossRef] [PubMed]

]

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

**1)**In [1

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

*u*

_{plasmon}(

*ω*) and u(

_{if}*ω*) are the spectral energy densities of the light field with and without the nanoparticle, respectively. The nanoparticle-induced changes in the Einstein

_{if}*A*coefficients for spontaneous de-excitation processes in the Er

^{3+}ions were described by factors and so thatwhere

*γ*

_{rad}is the factor for the nanoparticle-induced enhancement of the spontaneous emission rate and

*γ*

_{nonrad}describes the relative change in the

*A*due to non-radiative losses in the nanoparticle. Moreover, the well-known equationwhich relates the Einstein coefficients for stimulated emission Bif to Aif, was assumed to equally hold in the presence of the nanoparticle,However, it turns out that Eq. (4) leads to unphysical conclusions. When combined with Eq. (2), Eq. (4) would result in the expressionimplying that the probability of stimulated transitions (absorption and stimulated emission) would be enhanced by non-radiative processes in the metal nanoparticle. Therefore, we must reject Eq. (4).

_{if}*B*

_{if,plasmon}to the Einstein coefficients in the absence of a nanoparticle isThis means that coupling to the nanoparticle will influence the rates for stimulated processes only through the local-field enhancement contained in Eq. (1), and not through a change in the Einstein

*B*coefficients of the Er

^{3+}ions. It is this choice that we adopt in the improved calculations that we present below.

**2)**Another important aspect concerns the orientation of the optical dipoles of the Er

^{3+}ions. The dipole orientation is of great importance for the rates of the spontaneous emission processes in our rate-equation system because the factors

*γ*

_{rad}and

*γ*

_{nonrad}from Eq. (2) depend on the orientation of the emitting dipole relative to the surface of the gold nanoparticle, i.e. either parallel (PPOL) or perpendicular (SPOL). In the absorption processes, the dipoles excited in the ions by the local optical field are essentially oriented along the field and, in general, have components along both the SPOL and PPOL directions. In [1

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

2. A. Rokhmin, N. Nikonorov, A. Przhevuskii, A. Chukharev, and A. Ul’yashenko, “Study of polarized luminescence in erbium-doped laser glasses,” Opt. Spectrosc. **96**(2), 168–174 (2004). [CrossRef]

*γ*

_{rad}and

*γ*

_{nonrad}from Eq. (2) should be averaged over the PPOL and SPOL orientations

*before*entering them in the rate-equation system for the upconverter.

**3)**An important upconversion process included in our rate-equation system is energy transfer upconversion (ETU). This process is based on Förster energy transfer between neighboring excited Er

^{3+}ions. There is an ongoing discussion in the literature on whether the rate of Förster energy transfer is influenced by the local density of photon states and could thus be altered by suitable photonic or plasmonic environments [3

3. F. Reil, U. Hohenester, J. R. Krenn, and A. Leitner, “Förster-type resonant energy transfer influenced by metal nanoparticles,” Nano Lett. **8**, 4128–4133 (2008). [CrossRef] [PubMed]

5. C. Blum, N. Zijlstra, A. Lagendijk, M. Wubs, A. P. Mosk, V. Subramaniam, and W. L. Vos, “Nanophotonic control of the Förster resonance energy transfer efficiency,” Phys. Rev. Lett. **109**(20), 203601 (2012). [CrossRef] [PubMed]

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

*γ*

_{rad,}

*of the involved transitions in both the donor and acceptor of the Er*

_{if}^{3+}ion pair, and thus proportional to the square of the local density of electromagnetic states, in agreement with [6

6. T. Nakamura, M. Fujii, S. Miura, M. Inui, and S. Hayashi, “Enhancement and suppression of energy transfer from Si nanocrystals to Er ions through a control of the photonic mode density,” Phys. Rev. B **74**(4), 045302 (2006). [CrossRef]

7. M. J. A. de Dood, J. Knoester, A. Tip, and A. Polman, “Förster transfer and the local optical density of states in erbium-doped silica,” Phys. Rev. B **71**(11), 115102 (2005). [CrossRef]

9. U. Hohenester and A. Trugler, “Interaction of single molecules with metallic nanoparticles,” IEEE J. Sel. Top. Quantum Electron. **14**(6), 1430–1440 (2008). [CrossRef]

5. C. Blum, N. Zijlstra, A. Lagendijk, M. Wubs, A. P. Mosk, V. Subramaniam, and W. L. Vos, “Nanophotonic control of the Förster resonance energy transfer efficiency,” Phys. Rev. Lett. **109**(20), 203601 (2012). [CrossRef] [PubMed]

8. A. O. Govorov, J. Lee, and N. A. Kotov, “Theory of plasmon-enhanced Förster energy transfer in optically excited semiconductor and metal nanoparticles,” Phys. Rev. B **76**(12), 125308 (2007). [CrossRef]

^{3+}ions coupled to the spherical gold nanoparticle studied in [1

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

## Revised implementation of the rate-equation calculations

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

*γ*

_{E}. The probability per unit time for ground state (GSA) or excited state absorption (ESA) between the energy levels

*i*and

*f*is then determined bywith the spectral energy density of the excitation

*u*(

*ω*

_{if}), the speed of light in vacuum

*c*, the reduced Planck constant

*ħ*, the degeneracies of the initial

*g*and final state

_{i}*g*and the Einstein coefficient for spontaneous emission from state

_{f}*f*to

*i A*. The probability per unit time for stimulated emission (STE) is modified in the same way:The change factor of the radiative transition rate

_{if}*γ*

_{rad,}

*for the transition from state*

_{if}*i*to

*f*modifies the probability of spontaneous emission (SPE) per unit time toIn consequence, the luminescence

*L*of state

_{i}*i*is calculated by multiplication with the occupation of the corresponding state

*n*Due to the presence of the metal nanoparticle an additional loss channel appears. This loss channel depopulates excited states of the upconverter ions and can be implemented in analogy to the spontaneous emissionFor multi-phonon relaxation and energy transfer processes, we assume that no changes are induced by the metal nanoparticle. These processes are implemented into the rate-equation system as described in [1

_{i}**20**(1), 271–282 (2012). [CrossRef] [PubMed]

10. S. Fischer, H. Steinkemper, P. Löper, M. Hermle, and J. C. Goldschmidt, “Modeling upconversion of erbium doped microcrystals based on experimentally determined Einstein coefficients,” J. Appl. Phys. **111**(1), 013109 (2012). [CrossRef]

## Results

*x-z*-plane at

*y*= 0 nm. As a consequence of the corrections discussed above, the calculated upconversion luminescence enhancement due to a single spherical gold nanoparticle with a diameter of 200 nm and for an incident irradiance of 1000 Wm

^{−2}at a monochromatic wavelength of 1523 mn, is much lower than presented in [1

**20**(1), 271–282 (2012). [CrossRef] [PubMed]

^{4}

*I*

_{11/2}to

^{4}

*I*

_{15/2}with a center emission wavelength of 980 nm.

^{4}

*I*

_{11/2}→

^{4}

*I*

_{15/2}and

^{4}

*I*

_{9/2}→

^{4}

*I*

_{15/2}, respectively, were determined by averaging over a distance range from 20 nm to 25 nm to the surface of the gold nanoparticle.

## Acknowledgment

## References and links

1. | S. Fischer, F. Hallermann, T. Eichelkraut, G. von Plessen, K. W. Krämer, D. Biner, H. Steinkemper, M. Hermle, and J. C. Goldschmidt, “Plasmon enhanced upconversion luminescence near gold nanoparticles-simulation and analysis of the interactions,” Opt. Express |

2. | A. Rokhmin, N. Nikonorov, A. Przhevuskii, A. Chukharev, and A. Ul’yashenko, “Study of polarized luminescence in erbium-doped laser glasses,” Opt. Spectrosc. |

3. | F. Reil, U. Hohenester, J. R. Krenn, and A. Leitner, “Förster-type resonant energy transfer influenced by metal nanoparticles,” Nano Lett. |

4. | M. Lessard-Viger, M. Rioux, L. Rainville, and D. Boudreau, “FRET enhancement in multilayer core-shell nanoparticles,” Nano Lett. |

5. | C. Blum, N. Zijlstra, A. Lagendijk, M. Wubs, A. P. Mosk, V. Subramaniam, and W. L. Vos, “Nanophotonic control of the Förster resonance energy transfer efficiency,” Phys. Rev. Lett. |

6. | T. Nakamura, M. Fujii, S. Miura, M. Inui, and S. Hayashi, “Enhancement and suppression of energy transfer from Si nanocrystals to Er ions through a control of the photonic mode density,” Phys. Rev. B |

7. | M. J. A. de Dood, J. Knoester, A. Tip, and A. Polman, “Förster transfer and the local optical density of states in erbium-doped silica,” Phys. Rev. B |

8. | A. O. Govorov, J. Lee, and N. A. Kotov, “Theory of plasmon-enhanced Förster energy transfer in optically excited semiconductor and metal nanoparticles,” Phys. Rev. B |

9. | U. Hohenester and A. Trugler, “Interaction of single molecules with metallic nanoparticles,” IEEE J. Sel. Top. Quantum Electron. |

10. | S. Fischer, H. Steinkemper, P. Löper, M. Hermle, and J. C. Goldschmidt, “Modeling upconversion of erbium doped microcrystals based on experimentally determined Einstein coefficients,” J. Appl. Phys. |

**OCIS Codes**

(190.7220) Nonlinear optics : Upconversion

(240.6680) Optics at surfaces : Surface plasmons

(260.3800) Physical optics : Luminescence

(350.6050) Other areas of optics : Solar energy

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 11, 2013

Published: April 23, 2013

**Citation**

Stefan Fischer, Florian Hallermann, Toni Eichelkraut, Gero von Plessen, Karl W. Krämer, Daniel Biner, Heiko Steinkemper, Martin Hermle, and Jan Christoph Goldschmidt, "Plasmon enhanced upconversion luminescence near gold nanoparticles – simulation and analysis of the interactions: Errata," Opt. Express **21**, 10606-10611 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10606

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### References

- S. Fischer, F. Hallermann, T. Eichelkraut, G. von Plessen, K. W. Krämer, D. Biner, H. Steinkemper, M. Hermle, and J. C. Goldschmidt, “Plasmon enhanced upconversion luminescence near gold nanoparticles-simulation and analysis of the interactions,” Opt. Express20(1), 271–282 (2012). [CrossRef] [PubMed]
- A. Rokhmin, N. Nikonorov, A. Przhevuskii, A. Chukharev, and A. Ul’yashenko, “Study of polarized luminescence in erbium-doped laser glasses,” Opt. Spectrosc.96(2), 168–174 (2004). [CrossRef]
- F. Reil, U. Hohenester, J. R. Krenn, and A. Leitner, “Förster-type resonant energy transfer influenced by metal nanoparticles,” Nano Lett.8, 4128–4133 (2008). [CrossRef] [PubMed]
- M. Lessard-Viger, M. Rioux, L. Rainville, and D. Boudreau, “FRET enhancement in multilayer core-shell nanoparticles,” Nano Lett.9(8), 3066–3071 (2009). [CrossRef] [PubMed]
- C. Blum, N. Zijlstra, A. Lagendijk, M. Wubs, A. P. Mosk, V. Subramaniam, and W. L. Vos, “Nanophotonic control of the Förster resonance energy transfer efficiency,” Phys. Rev. Lett.109(20), 203601 (2012). [CrossRef] [PubMed]
- T. Nakamura, M. Fujii, S. Miura, M. Inui, and S. Hayashi, “Enhancement and suppression of energy transfer from Si nanocrystals to Er ions through a control of the photonic mode density,” Phys. Rev. B74(4), 045302 (2006). [CrossRef]
- M. J. A. de Dood, J. Knoester, A. Tip, and A. Polman, “Förster transfer and the local optical density of states in erbium-doped silica,” Phys. Rev. B71(11), 115102 (2005). [CrossRef]
- A. O. Govorov, J. Lee, and N. A. Kotov, “Theory of plasmon-enhanced Förster energy transfer in optically excited semiconductor and metal nanoparticles,” Phys. Rev. B76(12), 125308 (2007). [CrossRef]
- U. Hohenester and A. Trugler, “Interaction of single molecules with metallic nanoparticles,” IEEE J. Sel. Top. Quantum Electron.14(6), 1430–1440 (2008). [CrossRef]
- S. Fischer, H. Steinkemper, P. Löper, M. Hermle, and J. C. Goldschmidt, “Modeling upconversion of erbium doped microcrystals based on experimentally determined Einstein coefficients,” J. Appl. Phys.111(1), 013109 (2012). [CrossRef]

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