## Theoretical analysis of enhanced nonlinear conversion from metallo-dielectric nano-structures |

Optics Express, Vol. 20, Issue 15, pp. 16258-16268 (2012)

http://dx.doi.org/10.1364/OE.20.016258

Acrobat PDF (1833 KB)

### Abstract

A new configuration of sub-wavelength silver coaxial apertures filled with Lithium Niobate (LN) is proposed to enhance the Second Harmonic Generation (SHG) in transmission mode. The chosen geometrical parameters allows having both *TE*_{11} guided mode excitation for local field confinement of the fundamental signal and Fabry-Perot high transmission of the SH wave. Furthermore, an implementation of the three-dimensional Finite Difference Time Domain (3D-FDTD) method for nonlinear optical simulation is described. This method provides a direct calculation of the nonlinear polarizations before calculating the nonlinear electric and magnetic fields. FDTD studies shows that by embedding metallic nano-structures, for exciting *TE*_{11} like-mode inside a nonlinear material (LN), we achieve a SH signal 27 times higher than that generated on unpatterned LN.

© 2012 OSA

## 1. Introduction

16. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. **7**, 118–119 (1961). [CrossRef]

*ω*through a crystal that exhibits a second order nonlinearity. This will induce a conversion of light into a double frequency 2

*ω*or half the wavelength. In this paper, we suggest a novel device that enhances this nonlinear conversion by combining the EOT to a large nonlinear dielectric. Our concept consists on filling the cavities of a silver AAA by LN by lithium niobate. The suggested configuration is schematically illustrated in Fig. 1. The choice of the geometrical parameters and the metal in use is described in our previous work [17

17. E. Barakat, M.-P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express **18**, 6530–6536 (2010). [CrossRef] [PubMed]

*TE*

_{11}-like mode at, first, its cutoff wavelength and, second, at the first FP harmonic. Briefly, their position varied as a function of the geometrical parameters. The radii have been chosen to locate the first peak at the fundamental wavelength. The silver thickness

*h*is optimized for enhancing the SH response by locating the Fabry-Perot-like resonance at the second harmonic wavelength.

17. E. Barakat, M.-P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express **18**, 6530–6536 (2010). [CrossRef] [PubMed]

19. M. N.O. Sadiku, *Numerical Techniques in Electromagnetics* (CRC Press, 2000). [CrossRef]

*χ*

^{2}of the metal is smaller than that of LN. While the largest second order nonlinear coefficient is

*d*

_{33}= 33 × 10

^{−12}

*m*/

*V*(

*χ*

^{(2)}= 2

*d*) for LN [20], the surface susceptibility is estimated to

*χ*

^{(2)}= 1.13 × 10

^{−20}

*m/V*for silver [21

21. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-Field Second-Harmonic Generation Induced by Local Field Enhancement,” Phys. Rev. Lett. **90**, 13903–13907 (2003). [CrossRef]

## 2. NL-FDTD algorithm

*χ*

^{2}(

*t*), the relationship between the electric displacement field

*D⃗*, the vacuum permittivity

*ε*

_{0}and the electric field

*E⃗*is given by the electromagnetic constitutive relation for a dispersive medium:

*P⃗*is the induced polarization,

*χ*

^{(n)}is the nth order nonlinear susceptibility and

*E⃗*is the electric field vector of the incident light.

*χ*

^{(1)}

*E⃗*describes the linear response of the material;

*χ*

^{(2)}

*E⃗E⃗*defines the SHG, sum and the difference frequency generation;

*χ*

^{(3)}

*E⃗E⃗E⃗*is both four-wave mixing and the third harmonic generation. Our work aims enhancing the second harmonic generation thus we limit the study to the second order nonlinear effect. The nonlinear FDTD method discussed here is based on the extension of the original Yee’s algorithm [22

22. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

*n*, Δ

*t*, Δ

*y*and Δ

*z*are the time index, time step, grid spacing in y and in z directions, respectively. Based on the x equations, we can calculate the two other components of

*E*and

*H*(

*E*,

_{y}*E*,

_{z}*H*and

_{y}*H*). The nonlinear polarization is calculated prior to the nonlinear electric field as follows:

_{z}*χ*

^{(2)}can be expressed in time domain by the linear convolution:

*χ*

^{(2)}is the material time domain susceptibility. To get around the convolution product difficulty, we use a continuous source at a fixed wavelength.

*P⃗*(

_{NL}*t*) = 2

*ε*

_{0}

*d̃E⃗*(

*t*)

*E⃗*(

*t*) where

*d̃*(

*χ*

^{(2)}= 2

*d̃*) is the simplified nonlinear tensor of the material in use. Finally, the nonlinear electric field components are obtained through this equation:

23. J. Dahdah, J. Hoblos, and F. I. Baida, “Nano-coaxial waveguide grating as quarter-wave plates in the visible range,” IEEE Photon. J. **4**, 87–94 (2011). [CrossRef]

## 3. Numerical simulations and results

*δ*=

_{z}*δ*= 2.5

_{y}*nm*, a non-uniform meshing is applied in

*x*direction to faithfully describe the structure. This latter varies continuously between 2.5

*nm*and 15

*nm*over 160 nm (20 spatial grids). The temporal step is set to

*δt*=

*δ*/2

_{z}*c*where

*c*is the speed of light in vacuum. We use the Perfectly Matched Layer (PML) as an artificial absorbing boundary for both linear and nonlinear signals. A Drude model is used to simulate the dispersion of the dielectric permittivity of silver. As shown on Fig. 1, the period is fixed to 300

*nm*and the dielectric constant of the LN is set to

*ε*=

*n*

^{2}= 2.138

^{2}at

*λ*= 1550

*nm*and

*ε*=

*n*

^{2}= 2.179

^{2}at

*λ*= 775

*nm*[24]. The thickness of the metallic film and the aperture radii are kept the same as in [17

17. E. Barakat, M.-P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express **18**, 6530–6536 (2010). [CrossRef] [PubMed]

*h*= 120

*nm R*= 65

_{i}*nm*and

*R*= 135

_{o}*nm*). Since the thickness of our structure is smaller than the coherence length of lithium niobate, no phase matching is needed to generate an optimum depletion of the nonlinear signal.

*d*of the substrate is an even multiple of the coherence length

*L*(

_{c}*d*= 2

*m*×

*L*,

_{c}*m*∈ IN). This length represents the distance over which the nonlinear polarization and the generated SH signal interfere constructively. By choosing

*d*= 2

*m*×

*L*, the SHG signal generated from the substrate is avoided and only the SHG generated by the LN located inside the 120

_{c}*nm*-thick cavities contributes in the enhancement factor.

*ω*. In Fig. 2(a), the zero-order transmission spectrum of the embedded structure shows two resonances. The main peak located at the fundamental wavelength (1550

*nm*) corresponds to the excitation and the propagation of the

*TE*

_{11}-like mode inside the apertures at its cutoff wavelength. This induces a light confinement inside and nearby the cavities. Consequently, by filling out the cavity with a nonlinear material i.e. lithium niobate, we improve the nonlinear conversion, in the present case the SHG. The second peak, which corresponds to the first harmonic Fabry-Perot resonance of the same guided mode, is placed at the SH wavelength (775

*nm*). This is obtained by adjusting the metal thickness [17

**18**, 6530–6536 (2010). [CrossRef] [PubMed]

*d̃*tensor that links the

*P⃗*to the electric field at the pump wavelength. This relationship is given by [20]:

_{NL}*d*

_{22}= 3

*pm/V*,

*d*

_{31}= 5

*pm/V*and

*d*

_{33}= 33

*pm/V*. Note that in Eq. (9) the crystal axis of the LN is along the z direction, consequently, a basis change is necessary to take into account the wafer cut.

*d*

_{33}which means that the z-component of the pump electric field plays the main role in the observed SH signal. As the field inside the cavities is dominated by the component parallel to the incident polarization, we chose to direct this electric field along the crystal axis. Therfore, we have chosen an X-cut LN wafer in our simulations.

*TE*

_{11}wavelength), the patterned LN sample generates an enhanced SH signal, that is 27 times higher than that generated from the unpatterned lithium niobate layer. This value is higher than 17, the one predicted in ref [17

**18**, 6530–6536 (2010). [CrossRef] [PubMed]

*nm*(

*λ*= 750

_{SH}*nm*) from its expected position (775

*nm*) corresponding to the half of the main linear peak transmission wavelength (

*λ*= 1550

_{fund}*nm*). Unlike previously published results by Fan

*et al.*[25

25. W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. **6**, 1027–1030 (2006). [CrossRef]

*TE*

_{11}guided mode is the main reason of this observed blue-shift. That was numerically demonstrated by increasing the structure’s period allowing a redshift of the plasmon resonance far from the transmission peak of the guided mode. Thereby, the nonlinear signal transmission reaches its maximum at exactly half the fundamental wavelength.

*nm*which is due to the perturbation of the guided mode by the surface plasmon resonance excited at the metal-LN interface. This was verified by monitoring the surface plasmon resonance position as a function of the period. For large period values, the Surface Plasmon Resonance (SPR) occurs far from the guided mode position and no interferences are observed. Thus, the SPR follows a quite linear behavior given by its conventional dispersion relation. Contrarily, by decreasing the period, the SPR and the guided mode remain closer and a degeneracy breaking occurs inducing a redshift of the SPR position. That is why the SPR is located at

*λ*∼ 800

_{SPR}*nm*instead of 710

*nm*found from the dispersion relationship.

*TE*

_{11}guided mode inside the apertures. Let us emphasize that the SPR generated at the metal-LN interface also disturbs the transmitted signal from the metal-vacuum side. Consequently, a dip appears in the transmission spectrum around the SPR wavelength. Figures 3(b) and 3(c) show the nonlinear electric intensity distributions inside the cavity at these two wavelengths (

*λ*). For

_{SPR}*λ*=

_{fund}*λ*= 1620

_{SPR}*nm*is confined at the lithium niobate side. In this latter case, the reflected optical signal remains smaller than the transmitted one as it is confirmed in Fig. 3(a).

*zOy*plane, 5

*nm*far from the surface, with the z-polarized incident wave at

*λ*= 1500

*nm*. In Fig. 4(a), a localized electric field is observed in the cavity corresponding to the excitation of the

*TE*

_{11}guided mode. The SH field in Fig. 4(b) is also confined nearby the cavities and follows the fundamental distribution of the electric field shown in Fig. 4(a). Figure 4(c) shows the intensity enhancement in the structure (

*zOx*plane) at the pump resonance frequency of

8. F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B **74**, 205419–205426 (2006). [CrossRef]

*λ*= 775

_{fund}*nm*), the distribution of the linear electric intensity shows a node inside the apertures (Fig. 4(d)) corresponding to interferences between forward and backward propagating waves between the two metallic interfaces. Meanwhile the distribution of the SH signal presented in Fig. 4(e) is attributed to the excitation of the fundamental

*TE*

_{11}mode inside the cavity and is consistent with the uniform distribution of the fundamental field of Fig. 4(c).

*zOy*plane. This means that there is no energy flow in this transversal plane. However, in the

*zOx*plane (the polarization plane), most energy flow is in the apertures and appears as a uniform vertical current (Fig. 5(c)). The Poynting vector of the nonlinear signal at

*λ*= 750

*nm*is depicted in Fig. 5(d). Indeed, a nonlinear source is created inside the cavity and propagates in both directions of the

*Ox*-axis. However it is clear that it is more transmitted on the air side than reflected in the substrate side. Note that the Poynting vectors in the vertical direction are not evenly spaced, this is due to the nonuniform meshing used in our calculations. In our older paper [17

**18**, 6530–6536 (2010). [CrossRef] [PubMed]

## 4. Polarization

*θ*= 0°. In fact the dominant polarization component (parallel to the crystal axis) follows a cosine square relationship that is in accordance with the polarization dependence of the SHG. The fundamental electric field is multiplied by the

*d*

_{33}element of the simplified nonlinear tensor

*d̃*, consequently the resulting nonlinear polarization component is predominant with regard to the two other components. Furthermore, for a

*z*–cut nano-patterned LN wafer, we do not observe a similar polarization dependence. In fact, there is an additional effect of anisotropy and the SH strength is an order of magnitude lower than the SH generated from the nano-patterned

*x*–cut LN. The amplitudes of the three

*P*components are quite similar and the resulting behavior of the nonlinear polarization can be mathematically described by a sum of cosine square and sine×cosine terms. Indeed, this can be attributed to the fact that the

_{NL}*d*

_{33}component does not intervene significantly in the case of

*z*–cut LN.

## 5. Conclusion

*TE*

_{11}guided mode improves the nonlinear conversion, the first Fabry-Perot harmonic helps to transmit the SH signal in the far field. Moreover, we show that the SH signal strength reaches its maximum value for an z-polarized incident light. We would like to also point out that in our structure, no phase matching is needed to generate the second harmonic signal. A forthcoming paper will detail the fabrication and the experimental results of this structure.

## Acknowledgment

## References and links

1. | R. Raad, E. Inaty, P. Fortier, and H. M. H. Shalaby, “Optical S-ALOHA/CDMA Systems for Multirate Applications: Architecture, Performance Evaluation, and System Stability,” J. Lightwave Technol. |

2. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

3. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

4. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) |

5. | H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature (London) |

6. | F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. |

7. | F. I. Baida and D. Van Labeke, “Three-dimensional structures for enhanced transmission through a metallic film: Annular aperture arrays,” Phys. Rev. B |

8. | F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B |

9. | J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong Modification of the Nonlinear Optical Response of Metallic Subwavelength Hole Arrays,” Phys. Rev. B |

10. | S. Park, J. W. Hahn, and J. Y. Lee, “Doubly resonant metallic nanostructure for high conversion efficiency of second harmonic generation,” Opt. Express |

11. | E. A. Mamonov, T. V. Murzina, I. A. Kolmychek, A. I. Maydykovsky, V. K. Valev, A. V. Silhanek, E. Ponizovskaya, A. Bratkovsky, T. Verbiest, V. V. Moshchalkov, and O. A. Aktsipetrov, “Coherent and incoherent second harmonic generation in planar G-shaped nanostructures,” Opt. Lett. |

12. | B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local Field Asymmetry Drives Second-Harmonic Generation in Noncentrosymmetric Nanodimers,” Nano Lett. |

13. | R. Zhou, H. Lu, X. Liu, Y. Gong, and D. Mao, “Second-harmonic generation from a periodic array of noncentrosymmetric nanoholes,” J. Opt. Soc. Am. B |

14. | Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B |

15. | M. Roussey, M.-P. Bernal, N. Courjal, and F. I. Baida, “Experimental and theoretical characterization of a lithium niobate photonic crystal,” Appl. Phys. Lett. |

16. | P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. |

17. | E. Barakat, M.-P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express |

18. | A. Taflove and S. C. Hagness, |

19. | M. N.O. Sadiku, |

20. | Y. R. Shen, |

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

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

23. | J. Dahdah, J. Hoblos, and F. I. Baida, “Nano-coaxial waveguide grating as quarter-wave plates in the visible range,” IEEE Photon. J. |

24. | W. Sellmeier, “Zur Erklrung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. Chem. |

25. | W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. |

**OCIS Codes**

(130.3730) Integrated optics : Lithium niobate

(130.4310) Integrated optics : Nonlinear

(190.2620) Nonlinear optics : Harmonic generation and mixing

(160.3918) Materials : Metamaterials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 15, 2012

Revised Manuscript: June 22, 2012

Manuscript Accepted: June 22, 2012

Published: July 2, 2012

**Citation**

Elsie Barakat, Maria-Pilar Bernal, and Fadi Issam Baida, "Theoretical analysis of enhanced nonlinear conversion from metallo-dielectric nano-structures," Opt. Express **20**, 16258-16268 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16258

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### References

- R. Raad, E. Inaty, P. Fortier, and H. M. H. Shalaby, “Optical S-ALOHA/CDMA Systems for Multirate Applications: Architecture, Performance Evaluation, and System Stability,” J. Lightwave Technol.24, 1968–1977 (2006). [CrossRef]
- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58, 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58, 2486–2489 (1987). [CrossRef] [PubMed]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London)391, 667–669 (1998). [CrossRef]
- H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature (London)452, 728–731 (2008). [CrossRef]
- F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun.209, 17–22 (2002). [CrossRef]
- F. I. Baida and D. Van Labeke, “Three-dimensional structures for enhanced transmission through a metallic film: Annular aperture arrays,” Phys. Rev. B67, 155314–155317 (2003). [CrossRef]
- F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: Role of the plasmonic modes,” Phys. Rev. B74, 205419–205426 (2006). [CrossRef]
- J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, “Strong Modification of the Nonlinear Optical Response of Metallic Subwavelength Hole Arrays,” Phys. Rev. B97, 146102–146106 (2006). [CrossRef]
- S. Park, J. W. Hahn, and J. Y. Lee, “Doubly resonant metallic nanostructure for high conversion efficiency of second harmonic generation,” Opt. Express4, 4856–4871 (2012). [CrossRef]
- E. A. Mamonov, T. V. Murzina, I. A. Kolmychek, A. I. Maydykovsky, V. K. Valev, A. V. Silhanek, E. Ponizovskaya, A. Bratkovsky, T. Verbiest, V. V. Moshchalkov, and O. A. Aktsipetrov, “Coherent and incoherent second harmonic generation in planar G-shaped nanostructures,” Opt. Lett.36, 3681–3683 (2011). [CrossRef] [PubMed]
- B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local Field Asymmetry Drives Second-Harmonic Generation in Noncentrosymmetric Nanodimers,” Nano Lett.20, 1251–1255 (2007). [CrossRef]
- R. Zhou, H. Lu, X. Liu, Y. Gong, and D. Mao, “Second-harmonic generation from a periodic array of noncentrosymmetric nanoholes,” J. Opt. Soc. Am. B27, 2405–2409 (2010). [CrossRef]
- Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B79, 235109–235118 (2009). [CrossRef]
- M. Roussey, M.-P. Bernal, N. Courjal, and F. I. Baida, “Experimental and theoretical characterization of a lithium niobate photonic crystal,” Appl. Phys. Lett.87, 241101–241104 (2005). [CrossRef]
- P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett.7, 118–119 (1961). [CrossRef]
- E. Barakat, M.-P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express18, 6530–6536 (2010). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2005).
- M. N.O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, 2000). [CrossRef]
- Y. R. Shen, The Principles of Nonlinear Optics (Wiley Interscience, 1984).
- A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-Field Second-Harmonic Generation Induced by Local Field Enhancement,” Phys. Rev. Lett.90, 13903–13907 (2003). [CrossRef]
- K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966). [CrossRef]
- J. Dahdah, J. Hoblos, and F. I. Baida, “Nano-coaxial waveguide grating as quarter-wave plates in the visible range,” IEEE Photon. J.4, 87–94 (2011). [CrossRef]
- W. Sellmeier, “Zur Erklrung der abnormen Farbenfolge im Spectrum einiger Substanzen,” Ann. Phys. Chem.219, 272–282 (1981).
- W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6, 1027–1030 (2006). [CrossRef]

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