## Enhanced nonlinear response from metal surfaces |

Optics Express, Vol. 19, Issue 3, pp. 1777-1785 (2011)

http://dx.doi.org/10.1364/OE.19.001777

Acrobat PDF (3930 KB)

### Abstract

While metals benefit from a strong nonlinearity at optical frequencies, its practical exploitation is limited by the weak penetration of the electric field within the metal and the screening by the surface charges. It is shown here that this limitation can be bypassed by depositing a thin dielectric layer on the metal surface or, alternatively, using a thin metal film. This strategy enables us to enhance four-wave mixing in metals by up to four orders of magnitude.

© 2011 Optical Society of America

4. F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. **14**, 1029–1031 (1965). [CrossRef]

5. H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B **44**, 1220–1224 (1991). [CrossRef]

6. A. Leitner, “Second-harmonic generation in metal island films consisting of oriented silver particles of low symmetry,” Mol. Phys. **70**, 197 (1990). [CrossRef]

7. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-Field Second Harmonic Generation Induced by Local Field Enhancement,” Phys. Rev. Lett. **90**, 013903 (2003). [CrossRef] [PubMed]

8. N. A. Papadogiannis, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. **166**, 133–139 (1999). [CrossRef]

10. H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO_{2} composite films on femtosecond and picosecond time scales,” Opt. Lett. **23**, 388–390 (1998). [CrossRef]

*single*nanoparticles [11

11. M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. **5**, 799–802 (2005). [CrossRef] [PubMed]

12. M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. **98**, 026104 (2007). [CrossRef] [PubMed]

13. N. K. Grady, M. W. Knight, R. Bardhan, and N. J. Halas, “Optically-driven collapse of a plasmonic nanogap self-monitored by optical frequency mixing,” Nano Lett. **10**, 1522–1528 (2010). [CrossRef] [PubMed]

14. H. Harutyunyan, S. Palomba, J. Renger, R. Quidant, and L. Novotny, “Nonlinear dark-field microscopy,” Nano Lett. **10**, 5076–5079 (2010). [CrossRef]

15. Y. Wang, C.-Y. Lin, A. Nikolaenko, V. Raghunathan, and E. O. Potma, “Four-wave mixing microscopy of nanostructures,” Adv. Opt. Photon. **3**, 1–52 (2011). [CrossRef]

10. H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO_{2} composite films on femtosecond and picosecond time scales,” Opt. Lett. **23**, 388–390 (1998). [CrossRef]

*LiNbO*

_{3}or

*KTP*by orders of magnitude [10

10. H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO_{2} composite films on femtosecond and picosecond time scales,” Opt. Lett. **23**, 388–390 (1998). [CrossRef]

17. J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. **104**, 046803 (2010). [CrossRef] [PubMed]

18. P. Genevet, J.-P. Tetienne, E. Gatzogiannis, R. Blanchard, M. Kats, M. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. **10**, 4880–4883 (2010). [CrossRef]

## 1. Four-wave mixing at a coated metal interface (theory)

*d*. The limit of a bare metal surface corresponds to

*d*→ 0. As illustrated in Fig. 1, two coherent incident laser beams with frequencies

*ω*

_{1}and

*ω*

_{2}are incident from angles

*θ*

_{1}and

*θ*

_{2}, respectively. The angles are measured from the surface normal in clockwise direction. The two beams induce a nonlinear polarization at frequencies: which in turn gives rise to two outgoing beams propagating at angles

*θ*

_{4wm1}and

*θ*

_{4wm2}, respectively. Equations (1) are statements of energy conservation and define the frequencies of the outgoing radiation. Similarly, the in-plane momentum conservation at a planar surface defines the outgoing propagation directions according to: After substituting Eqs. (1) it becomes evident, that

*real*solutions for

*θ*

_{4wm1}and

*θ*

_{4wm2}exist only for certain angular ranges of

*θ*

_{1}and

*θ*

_{2}. Solutions represented by

*imaginary*angles correspond to evanescent 4WM fields such as surface plasmon polaritons [19

19. J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. **103**, 266802 (2009). [CrossRef]

*ω*

_{4wm1}. The induced nonlinear polarization can be expressed as where we used

*ω*

_{4wm}=

*ω*

_{4wm1}.

**E**

*are the electric field vectors associated with the incident beams of frequencies*

_{i}*ω*with

_{i}*i*=1,2, and

*χ*

^{(3)}is the third-order susceptibility, a tensor of rank four. We next assume that the nonlinear response is associated with the bulk and that the material is isotropic. In this case, the 81 components of

*χ*

^{(3)}can be reduced to only three non-zero and independent components, namely

**E**

*define the nonlinear polarization, which can be written as where*

_{i}**k**

_{4wm}= 2

**k**

_{1}–

**k**

_{2}. Here,

**k**

_{1}and

**k**

_{2}are the wavevectors of the exciting fields in the non-linear medium.

**E**

*can be represented in terms of the angle of incidence*

_{i}*θ*, polarization angle

_{i}*ϕ*, and the wavevectors

_{i}**k**

*of the incident waves in the nonlinear medium as where*

_{i}*t*and

_{s}*t*are the Fresnel transmission coefficients for

_{p}*s*(=

*TE*)- and

*p*(=

*TM*)-polarized incident light, respectively, and

*t*and

_{s}*t*depend on the material and the thickness of the dielectric layer deposited on top of the metal surface.

_{p}**P**defines a source current and gives rise to electromagnetic fields at the four-wave mixing frequency

*ω*

_{4wm}. Following the theory outlined by Bloembergen and Pershan [20

20. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. **128**, 606–622 (1962). [CrossRef]

**E**

_{4wm}can be calculated [17

17. J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. **104**, 046803 (2010). [CrossRef] [PubMed]

*ɛ*

_{1}=

*ɛ*and

_{air}*ɛ*

_{2}=

*ɛ*(

_{metal}*ω*

_{4wm}) and the z-components of the wavevector defined by

*k*

_{4wm,x}=

*k*

_{↓,x}=

*k*

_{↑,x}. Depending on which of the two four-wave mixing processes is being considered we further have

*k*

_{4wm,x}= 2

*k*

_{1,x}–

*k*

_{2,x}or

*k*

_{4wm,x}= 2

*k*

_{2,x}–

*k*

_{1,x}, which is a restatement of Eq. (2). Furthermore, since

*k*

_{1,y}=

*k*

_{2,y}= 0 for the incident waves we also have

*k*

_{4wm,y}=

*k*

_{↓,y}=

*k*

_{↑,y}= 0. The optical properties of the dielectric layer are contained in the transmission coefficients

*t*and

_{s}*t*. Notice, that

_{p}*t*and

_{s}*t*in Eq. (6) are evaluated at the nonlinear frequency

_{p}*ω*

_{4}

*, whereas in Eq. (5) they are evaluated at the frequencies of the incident radiation.*

_{wm}## 2. Four-wave mixing at a coated metal interface (experiment)

*λ*

_{2}= 800 nm, and an optical parametric oscillator (OPO) providing pulses of similar duration and wavelength

*λ*

_{1}= 707 nm. The beams are first expanded to 10 mm diameter and then focused by two lenses of focal length

*f*= 50 mm on the surface. The angle between the two laser beams is held fixed at

*θ*

_{2}–

*θ*

_{1}= 60° and the laser pulses are made to overlap in time by use of a delay line. The spot diameters at the surface are ∼ 4.5

*μ*m and are spatially overlapping. We use a detection angle that is fixed with respect to the angles of the excitation beams, namely

*θ*

_{det}=

*θ*

_{1}+ 26°. The radiation is collected and collimated by a

*f*= 75mm lens, filtered by optical stop-band filters to reject light at the two excitation frequencies, and then sent into a fiber-coupled spectrometer. Alternatively, the collected light is detected with a single-photon counting APD for intensity measurements.

*θ*

_{det}consists of peaks that correspond to the 4WM frequencies described by Eq. (1). These peaks are located to the blue and the red side of the excitation wavelengths

*λ*

_{1}and

*λ*

_{2}. The 4WM peaks are only observed if the angles

*θ*

_{1},

*θ*

_{2}, and

*θ*

_{det}fulfill the resonance condition defined by Eqs. (1) and (2). For

*λ*

_{4wm1}= 633 nm we used

*θ*

_{1}= 6° and

*θ*

_{2}= 66°, whereas for

*λ*

_{4wm2}= 920 nm we chose

*θ*

_{1}= −72.8° and

*θ*

_{2}= −12.8°. The 4WM peaks disappear when the pulses of the excitation beams are temporally or spatially detuned or if the sample rotation does not allow for momentum conservation of all contributing waves. It is important to notice that the spectra at planar metal samples are essentially background free. While optical four-wave mixing can also be measured on metal nanostructures such as particles [12

12. M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. **98**, 026104 (2007). [CrossRef] [PubMed]

21. M. R. Beversluis, A. Bouhelier, and L. Novotny, “Continuum generation from single gold nanostructures through near-field mediated intraband transitions,” Phys. Rev. B **68**, 115433 (2003). [CrossRef]

17. J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. **104**, 046803 (2010). [CrossRef] [PubMed]

_{2}or SiO

_{2}and thin gold films have been fabricated by thermal deposition. The thicknesses

*d*of either metal film or dielectric layer are varied from sample to sample. For every sample, the 4WM intensity has been measured as a function of the excitation angles. Special care has been taken to nicely overlap the incident beams in space and time and letting the focus at the surface coincide with the axis of sample rotation.

*d*deposited on the metal’s surface is Here,

*k*is the perpendicular component of the wavevector in the dielectric, and

_{d,z}*t*(

_{p,s}*λ*,

*θ*,

*d*) influence the 4WM efficiency in several ways, namely through the in-coupling of the excitation fields in Eq. (5) at the excitation wavelengths

*λ*

_{1}and

*λ*

_{2}, and through the out-coupling of the 4WM field in Eq. (6) at the four-wave mixing wavelength

*λ*

_{4wm}.

*t*(

_{p,s}*λ*,

*θ*,

*d*) in Eqs. (5) and (6) yields the oscillatory intensity plot depicted in Fig. 2. The figure shows the 4WM enhancement as a function of layer thickness

*d*and index of refraction

*n*relative to a bare silver surface. The calculation assumes 4WM generation at

*λ*

_{4wm}= 633 nm and TM polarized excitation fields. According to this calculation, a 50 nm film with

*n*=3 yields a 4WM enhancement of more than two orders of magnitude. As discussed later on, considerably higher enhancements are found for TE incidence.

_{2}(

*n*≈ 1.5) leads to a behavior as plotted in Fig. 3. These curves depict the 4WM intensity as a function of thickness

*d*of a SiO

_{2}layer deposited on a silver surface. The top part of Fig. 3 shows the results for TM polarized incident fields and the bottom part for TE polarized incident fields. The curves have been normalized with the 4WM intensity calculated for a bare silver surface (

*d*→ 0). For TM polarized fields the maximum 4WM enhancement at

*λ*

_{4WM1}= 633 nm is predicted to be ≈ 6, whereas for TE polarized fields we obtain a maximum 4WM enhancement at

*λ*

_{4WM1}= 633 nm of ≈ 25.

_{2}layer has been adjusted during the deposition process. The experimental data points follow the theoretical curves reasonably well. The only adjustable parameter in our theory is the nonlinear susceptibility

*χ*

^{(3)}of silver, which can be estimated by a comparison of theory and experiment. We obtain a value that is a factor of ≈ 1.5 times larger than the value of

22. P. Ginzburg, A. Hayat, N. Berkovitch, and M. Orenstein, “Nonlocal ponderomotive nonlinearity in plasmonics,” Opt. Lett. **35**, 1551–1553 (2010). [CrossRef] [PubMed]

*λ*

_{4wm}= 920 nm. For a SiO

_{2}thickness of

*d*=120 nm and for TE polarized excitation we find a predicted 4WM enhancement of more than two orders of magnitude. We were not able to experimentally verify the curves for

*λ*

_{4wm}= 920 nm because of nearly grazing incidence of the

*ω*

_{1}beam. In particular, for TE incidence and for layer thicknesses smaller than 80 nm we did not observe any 4WM. Furthermore, for TM polarization and

*λ*

_{4wm}= 920 nm we measured a fluorescence background, which most likely originates from local imperfections in the SiO

_{2}layer.

_{2}can be neglected in our analysis. However, this is not the case for dielectrics with higher

*χ*

^{(3)}coefficients, such as TiO

_{2}(

_{2}(

*n*≈ 2.3) is significantly larger than the refractive index of SiO

_{2}, which gives rise to much stronger Fabry-Perot resonances (see Fig. 2). Our initial theoretical calculations followed the same steps as those outlined for the SiO

_{2}layer above, neglecting any nonlinear contributions from the TiO

_{2}layer. The maximum calculated 4WM enhancement factors turned out to be 450× for TE incidence and

*λ*

_{4wm}= 633 nm, 290× for TE incidence and

*λ*

_{4wm}= 920 nm, 50× for TM incidence and

*λ*

_{4wm}= 633 nm, and 20× for TM incidence and

*λ*

_{4wm}= 920 nm. While these values are larger than the values calculated and measured for the SiO

_{2}layer, they are significantly lower than the experimental data shown in Fig. 4. We therefore conclude that the TiO

_{2}layer itself contributes to the nonlinear response.

*χ*

_{1221}/

*χ*

_{1122}= 1/ – 0.1 for the nonlinear susceptibility components, whereas for TiO

_{2}we chose

*χ*

_{1221}=

*χ*

_{1122}and a value that is a factor 0.03exp(

*i*0.8

*π*) smaller than that of silver. This choice is justified because sputter deposition without post-annealing leads to an isotropic composition. The data shown in Fig. 4 indicates that a 70 nm TiO

_{2}layer enhances the nonlinear response by more than four orders of magnitude.

_{2}nonlinearity we deposited a 100-nm-thin TiO

_{2}film on a glass substrate and performed similar 4WM measurements. The measured 4WM intensity turned out to be only 3 (TM, 4WM @ 633 nm) or 8 (TM, 4WM @ 920 nm) times the 4WM intensity from a bare silver surface. Thus, the giant enhancement observed for a TiO

_{2}coated silver surface must be the result of a combined effect. Note that the differences between the calculations and measurements for TM incidence and

*λ*

_{4wm}= 920 nm and

*d*= 200..300 nm have to be attributed to experimental imperfections and fluorescence background generated inside TiO

_{2}.

## 3. Four-wave mixing at a thin metal film

*t*(

_{p,s}*λ*,

*θ*,

*d*) need to be replaced since we’re no longer interested in the energy transmitted through a film but in the energy deposited in a film.

*d*. For

*d*>50 nm the film behaves like bulk metal and no enhancement is observed. On the other hand, below 50 nm, when the thickness becomes comparable to the skin depth, the influence of the lower metal-glass interface comes into play and the field in the metal increases. The enhanced fields in the metal film can be seen as an interference effect: the wave reflected from the top air-gold interface and the wave emanating from the bottom gold-substrate interface are nearly out of phase, thereby lowering their combined intensity and leaving more energy in the metal film. As a result we find that the 4WM intensity of a 20 nm Au film can be enhanced by a factor of 6 over a thick gold film.

21. M. R. Beversluis, A. Bouhelier, and L. Novotny, “Continuum generation from single gold nanostructures through near-field mediated intraband transitions,” Phys. Rev. B **68**, 115433 (2003). [CrossRef]

## 4. Conclusions

_{2}by silicon or GaAs.

## Acknowledgments

## References and links

1. | R. Boyd, |

2. | Y. R. Shen, |

3. | T. Heinz, |

4. | F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. |

5. | H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B |

6. | A. Leitner, “Second-harmonic generation in metal island films consisting of oriented silver particles of low symmetry,” Mol. Phys. |

7. | A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-Field Second Harmonic Generation Induced by Local Field Enhancement,” Phys. Rev. Lett. |

8. | N. A. Papadogiannis, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. |

9. | B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and off-resonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. |

10. | H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO |

11. | M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. |

12. | M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. |

13. | N. K. Grady, M. W. Knight, R. Bardhan, and N. J. Halas, “Optically-driven collapse of a plasmonic nanogap self-monitored by optical frequency mixing,” Nano Lett. |

14. | H. Harutyunyan, S. Palomba, J. Renger, R. Quidant, and L. Novotny, “Nonlinear dark-field microscopy,” Nano Lett. |

15. | Y. Wang, C.-Y. Lin, A. Nikolaenko, V. Raghunathan, and E. O. Potma, “Four-wave mixing microscopy of nanostructures,” Adv. Opt. Photon. |

16. | C. Flytzanis, F. Hache, M. Klein, D. Ricard, and P. Roussignol, “1. Semiconductor and metal crystallites in dielectrics:,” in “ |

17. | J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. |

18. | P. Genevet, J.-P. Tetienne, E. Gatzogiannis, R. Blanchard, M. Kats, M. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. |

19. | J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. |

20. | N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. |

21. | M. R. Beversluis, A. Bouhelier, and L. Novotny, “Continuum generation from single gold nanostructures through near-field mediated intraband transitions,” Phys. Rev. B |

22. | P. Ginzburg, A. Hayat, N. Berkovitch, and M. Orenstein, “Nonlocal ponderomotive nonlinearity in plasmonics,” Opt. Lett. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(240.4350) Optics at surfaces : Nonlinear optics at surfaces

(190.4223) Nonlinear optics : Nonlinear wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 29, 2010

Revised Manuscript: December 27, 2010

Manuscript Accepted: December 28, 2010

Published: January 14, 2011

**Citation**

Jan Renger, Romain Quidant, and Lukas Novotny, "Enhanced nonlinear response from metal surfaces," Opt. Express **19**, 1777-1785 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1777

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### References

- R. Boyd, Nonlinear Optics (Academic Press, San Diego, 2008), 3rd ed.
- Y. R. Shen, The Principles of Nonlinear Optics (J. Wiley & Sons, New York, 1984).
- T. Heinz, Nonlinear Surface Electromagnetic Phenomena (Elsevier, Amsterdam, 1991).
- F. Brown, R. E. Parks, and A. M. Sleeper, "Nonlinear optical reflection from a metallic boundary," Phys. Rev. Lett. 14, 1029-1031 (1965). [CrossRef]
- H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, "Reflected second-harmonic generation at a silver surface," Phys. Rev. B 44, 1220-1224 (1991). [CrossRef]
- A. Leitner, "Second-harmonic generation in metal island films consisting of oriented silver particles of low symmetry," Mol. Phys. 70, 197 (1990). [CrossRef]
- A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, "Near-Field Second Harmonic Generation Induced by Local Field Enhancement," Phys. Rev. Lett. 90, 013903 (2003). [CrossRef] [PubMed]
- N. A. Papadogiannis, P. A. Loukakos, and S. D. Moustaizis, "Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime," Opt. Commun. 166, 133-139 (1999). [CrossRef]
- B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, "Resonant and off-resonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation," Phys. Rev. Lett. 83, 4421-4424 (1999). [CrossRef]
- H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, "Origin of third-order optical nonlinearity in Au:SiO2 composite films on femtosecond and picosecond time scales," Opt. Lett. 23, 388-390 (1998). [CrossRef]
- M. Lippitz, M. A. van Dijk, and M. Orrit, "Third-harmonic generation from single gold nanoparticles," Nano Lett. 5, 799-802 (2005). [CrossRef] [PubMed]
- M. Danckwerts, and L. Novotny, "Optical frequency mixing at coupled gold nanoparticles," Phys. Rev. Lett. 98, 026104 (2007). [CrossRef] [PubMed]
- N. K. Grady, M. W. Knight, R. Bardhan, and N. J. Halas, "Optically-driven collapse of a plasmonic nanogap self-monitored by optical frequency mixing," Nano Lett. 10, 1522-1528 (2010). [CrossRef] [PubMed]
- H. Harutyunyan, S. Palomba, J. Renger, R. Quidant, and L. Novotny, "Nonlinear dark-field microscopy," Nano Lett. 10, 5076-5079 (2010). [CrossRef]
- Y. Wang, C.-Y. Lin, A. Nikolaenko, V. Raghunathan, and E. O. Potma, "Four-wave mixing microscopy of nanostructures," Adv. Opt. Photon. 3, 1-52 (2011). [CrossRef]
- C. Flytzanis, F. Hache, M. Klein, D. Ricard, and P. Roussignol, "1. Semiconductor and metal crystallites in dielectrics:" in "Nonlinear Optics in Composite Materials:" vol. 29 of Progress in Optics, E. Wolf, ed. (Elsevier, 1991), pp. 321-411.
- J. Renger, R. Quidant, N. van Hulst, and L. Novotny, "Surface-enhanced nonlinear four-wave mixing," Phys. Rev. Lett. 104, 046803 (2010). [CrossRef] [PubMed]
- P. Genevet, J.-P. Tetienne, E. Gatzogiannis, R. Blanchard, M. Kats, M. Scully, and F. Capasso, "Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings," Nano Lett. 10, 4880-4883 (2010). [CrossRef]
- J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, "Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing," Phys. Rev. Lett. 103, 266802 (2009). [CrossRef]
- N. Bloembergen, and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962). [CrossRef]
- M. R. Beversluis, A. Bouhelier, and L. Novotny, "Continuum generation from single gold nanostructures through near-field mediated intraband transitions," Phys. Rev. B 68, 115433 (2003). [CrossRef]
- P. Ginzburg, A. Hayat, N. Berkovitch, and M. Orenstein, "Nonlocal ponderomotive nonlinearity in plasmonics," Opt. Lett. 35, 1551-1553 (2010). [CrossRef] [PubMed]

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