## Second harmonic generation based on strong field enhancement in nanostructured THz materials |

Optics Express, Vol. 19, Issue 8, pp. 7262-7273 (2011)

http://dx.doi.org/10.1364/OE.19.007262

Acrobat PDF (1196 KB)

### Abstract

The THz response of slit structures and split-ring resonators (SRRs) featuring extremely small gaps on the micro- or nanoscale is investigated numerically. Both structures exhibit strong field enhancement in the gap region due to light-induced current flows and capacitive charging across the gap. Whereas nanoslits allow for broadband enhancement the resonant behavior of the SRRs leads to narrowband amplification and results in significantly higher field enhancement factors reaching several 10,000. This property is particularly beneficial for the realization of nonlinear THz experiments which is exemplarily demonstrated by a second harmonic generation process in a nonlinear substrate material. Positioning nanostructures on top of the substrate is found to result in a significant increase of the generation efficiency for the frequency doubled component.

© 2011 OSA

## 1. Introduction

1. G. P. Williams, “Filling the THz gap-high power sources and applications,” Rep. Prog. Phys. **69**, 301 (2006). [CrossRef]

3. T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. Elsaesser, “Generation of single-cycle THz transients with high electric-field amplitudes,” Opt. Lett. **30**, 2805–2807 (2005). [CrossRef] [PubMed]

4. K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. **90**, 171121 (2007). [CrossRef]

7. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microwave Theory Tech. **47**, 2075–2084 (1999). [CrossRef]

8. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science **306**, 1351–1353 (2004). [CrossRef] [PubMed]

## 2. Simulations

11. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. **24**, 4493–4499 (1985). [CrossRef] [PubMed]

12. N. Laman and D. Grischkowsky, “Terahertz conductivity of thin metal films,” Appl. Phys. Lett. **93**, 051105 (2008). [CrossRef]

*ν*was scanned parametrically. For all other boundaries scattering boundary conditions were chosen. In order to include nonlinear effects such as SHG, however, we analyze the structure’s response in the time-domain. For this purpose a multi-cycle THz transient was employed. Also, for the examination of the SHG process we chose to investigate the interaction of the THz pulse with slit and SRR arrays rather than single structures. For this purpose the corresponding boundaries were set to periodic conditions and the size of the simulation domain was chosen according to the periodicity.

## 3. Results

### 3.1. Obtainable field enhancement

#### 3.1.1. Nanoslit structures

*h*= 60 nm, the slit width

*D*was varied from 50 μm down to 40 nm, and the incident wave was polarized perpendicular to the slit’s long axis. From the frequency dependent simulations we determine the field enhancement factor where

*E*

_{gap}(

*ν*) denotes the absolute amplitude of the electric field in mid-gap and

*E*

_{inc}(

*ν*) the one of the incident electric field. We find that this is an adequate measure for the field enhancement allowing for convenient comparison between different geometries even though small inhomogeneities across the gap are observed. Seo and coworkers [5

5. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics **3**, 152–156 (2009). [CrossRef]

*λ*and the slit width

*D*as shown in Fig. 2(a). We find that for slits with

*D*≪

*λ*the field enhancement scales approximately linear with the ratio

*λ /D*. This is shown in Fig. 2(b) where we plot

*F*as a function of this ratio for four different frequency components. The different curves overlap which shows that the funneling is a purely geometric effect and that the smaller the slit on the scale of the wavelength the larger is the obtainable field enhancement. For the smallest considered slit width and frequency (

*D*= 40 nm,

*ν*= 100 GHz) a maximum enhancement factor approaching 3000 is obtained.

#### 3.1.2. Nanogap split-ring resonators

*L*[13

13. M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. **7**, S12–S22 (2005). [CrossRef]

14. A. Bitzer, J. Wallauer, H. Helm, H. Merbold, T. Feurer, and M. Walther, “Lattice modes mediate radiative coupling in metamaterial arrays,” Opt. Express **17**, 22108–22113 (2009). [CrossRef] [PubMed]

*L*= 300 μm to obtain resonances at the low frequency side of the THz range. The wire width was set to

*w*= 5 μm, the structure height was

*h*= 1 μm, and the gap width

*d*was varied from 100 μm down to 100 nm. The LC resonance can be excited through either an electric field component across the SRR gap, a magnetic field component perpendicular to the SRR plane, or by a combination of the previous two [15

15. P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. **92**, 2929–2936 (2002). [CrossRef]

16. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. **84**, 2943–2945 (2004). [CrossRef]

*E*-field excitation where the

*E*,

*H*, and

*k*triad of the incoming field is oriented along the

*x*,

*y*, and

*z*axes. In Fig. 3(a)–3(c) we exemplarily show the field enhancement

*F*(

*ν*) obtained for three SRRs with decreasing gap width.Within the considered frequency range the curves exhibit a single peak corresponding to the fundamental SRR resonance. With decreasing gap width the resonance is redshifted which can be qualitatively understood by the analogy between a SRR and a simple LC resonator [8

8. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science **306**, 1351–1353 (2004). [CrossRef] [PubMed]

*d*the in-gap field strength is found to be significantly enhanced. In analogy to the nanoslits we find that the field enhancement scales linearly with the ratio

*λ*/

_{r}*d*where

*λ*denotes the wavelength at resonance. This is shown in Fig. 3(d) where the red line is a linear fit to the data points. For the smallest considered gap width of

_{r}*d*= 100 nm giant enhancement factors approaching 40,000 are obtained which is more than an order of magnitude larger than the maximum field enhancement obtained for the nanoslits. This is explained by the resonant behavior where the oscillating currents excited in the SRR are much stronger than the non-resonant current flow induced on the metallic surface of the slit structures.

*h*and

*w*the aspect ratio of the gap volume becomes more and more extreme for decreasing gap width requiring an increasing amount of mesh points which in turn results in a significant increase of the computational demands. Simulating gap sizes smaller than 100 nm could therefore not be realized. We expect, however, that the inverse scaling of the field enhancement with

*d*extends to smaller gap widths so that even larger field strengths could be realized using e.g. SRRs with 40 nm gap width. Here, the threshold for breakdown fields (on the order of 10

^{8}V/cm) imposes an upper limit on the obtainable in-gap field strengths. The fabrication of nanostructures featuring such extreme aspect ratios, on the other hand, has already been demonstrated [17

17. S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology **21**, 295303 (2010). [CrossRef] [PubMed]

18. J. García-García, F. Martín, J. D. Baena, R. Marqués, and L. Jelinek, “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. **98**, 033103 (2005). [CrossRef]

*E*-field excitation the LC resonance may be excited more strongly using a combination of

*E*- and

*H*-field excitation. Using this excitation scheme, where the

*E*,

*H*, and

*k*triad is oriented along the

*x*,

*-z*, and

*y*axes, we find that the in-gap field enhancement can be increased even further. This is exemplarily shown in Fig. 3(e) for a SRR with 1 μm gap width. By simply rotating the structure the peak values of the in-gap fields can approximately be increased by another factor of 1.3.

### 3.2. Optimizing the geometry for maximum nonlinear response

#### 3.2.1. Slit structures arranged in one dimensional arrays

*p*as shown in Fig. 1(c). Since decreasing the periodicity increases the slit density the transmission is expected to scale inversely with

*p*. For small periodicities, however, neighboring slits start sharing the incident field energy so that, compared to single slits, the obtainable field enhancement is reduced [19]. The use of nanoslit arrays is consequently accompanied by a tradeoff between an increased volume and a decreased field enhancement. For the realization of nonlinear

*χ*

^{(2)}and

*χ*

^{(3)}effects the properties to be optimized are the ratios

*F*

^{2}(

*p*)/

*p*and

*F*

^{3}(

*p*)/

*p*, respectively. Their dependence on the periodicity is shown in Fig. 4(a) as obtained for 40 nm wide slits at a frequency of 500 GHz. For the two curves maxima are obtained at different values of

*p*indicating that the ideal periodicity depends on the order of the nonlinearity to be optimized. In addition, the behavior also exhibits a strong frequency dependence as exemplarily shown for a

*χ*

^{(2)}process in Fig. 4(b).

#### 3.2.2. Split-ring resonators featuring extended capacitive faces

*s*to the gap [20

20. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science **314**, 977–980 (2006). [CrossRef] [PubMed]

*d*= 1 μm gap width. As in the case of nanoslits this implies a tradeoff between an increased volume and a decreased field enhancement. For the realization of

*χ*

^{(2)}and

*χ*

^{(3)}effects the integrals are the property to be optimized. Their dependence on the split length

*s*is shown in Fig. 5(b). For

*U*

^{(2)}and

*U*

^{(3)}maxima at different values of

*s*are obtained which again shows that the ideal split length depends on the order of the nonlinearity.

### 3.3. Nanostructure induced second harmonic generation in lithium tantalate

_{3}). The purpose of this study is not to optimize the conversion efficiency of the SHG process but to demonstrate that the structures under study can be used to induce THz nonlinearities in adjacent materials. We note, that similar concepts have recently been used to control the THz transmission through a VO

_{2}film [21

21. J. Kyoung, M. Seo, H. Park, S. Koo, H. sun Kim, Y. Park, B.-J. Kim, K. Ahn, N. Park, H.-T. Kim, and D.-S. Kim, “Giant nonlinear response of terahertz nanoresonators on VO_{2} thin film,” Opt. Express **18**, 16452–16459 (2010). [CrossRef] [PubMed]

22. S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature **453**, 757–760 (2008). [CrossRef] [PubMed]

_{3}is a trigonal 3

*m*crystal having one extraordinary axis. The incident field is polarized along this axis and, for simplicity, we neglect the birefringence in the simulations. According to [23

23. T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. **37**, 317–350 (2007). [CrossRef]

_{3}at THz frequencies can then be set to n = 6.15. The nonlinearity was included by introducing a nonlinear polarization

*P*

^{(2)}=

*ε*

_{0}

*χ*

^{(2)}

*E*

^{2}into the governing time dependent equation. For

*χ*

^{(2)}at THz frequencies a constant numerical value could be derived from the potential energy surface as shown in Appendix A. For simplicity the imaginary part has been neglected. The central frequency of the exciting multi-cycle pulse was 138 GHz and the peak field strength was set to

*E*

_{inc}= 10 kV/cm. Note, that this value is about an order of magnitude smaller than what has already been demonstrated using table-top THz sources [4

4. K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. **90**, 171121 (2007). [CrossRef]

#### 3.3.1. Nanoslit arrays

_{3}substrate. The slit width was 40 nm and a periodicity of 30 μm was used optimized for maximum SHG efficiency at the center frequency. Note, that the introduction of the LiTaO

_{3}substrate leads to a modification of the dielectric environment which results in an effectively smaller wavelength

*λ*

_{eff}=

*λ /n*

_{eff}where

*n*

_{eff}denotes the effective refractive index. According to our previous findings this leads to a decrease of the field enhancement as compared to a freestanding structure. The black curve in Fig. 6(a) shows a reference spectrum of the incident pulse transmitted through an unstructured LiTaO

_{3}substrate. The nonlinearity of LiTaO

_{3}together with the given field strength of the incident pulse is small so that hardly any SHG signal can be observed. The red curve, on the other hand, shows the corresponding spectrum of a pulse transmitted through the nanoslit array on the substrate. The induced field enhancement leads to a much stronger SHG signal even though the nonlinear substrate material was only present below and not in the gap. Figure 6(b) shows the field enhancement in a

*xz*-slice cutting through the gap. The high field strength region extends a few tens of nanometers into the LiTaO

_{3}substrate. Even though, the volume comprising the high field strengths within LiTaO

_{3}is rather small, the field enhancement within this volume is large enough to cause a significant increase of the SHG efficiency. The generation efficiency can be scaled up even further if the nonlinear material is directly inserted into the slit, for example by using liquids [24

24. M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. **95**, 231105 (2009). [CrossRef]

_{3}. The obtained transmission spectrum is included in Fig. 6(a) as the green curve. Compared to SRRs with air-filled gaps (red curve) the SHG signal is increased by another factor of 2.3.

#### 3.3.2. Microgap split-ring resonator arrays

_{3}substrate results in a redshift of the LC resonance [13

13. M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. **7**, S12–S22 (2005). [CrossRef]

*λ*

_{eff}. This redshift was compensated by choosing a correspondingly smaller side length of

*L*= 70 μm for which the SRR resonance coincides with the center frequency of the incident spectrum. The SRRs were arranged in a two dimensional array and the periodicity was set to 100 μm. Since time-dependent simulations in three dimensions are particularly time consuming an intermediate gap width of

*d*=1 μm was chosen as a compromise between obtainable field enhancement and a manageable amount of mesh points.

_{3}(red curve), and the SRR array on substrate where the gap volume was additionally filled with LiTaO

_{3}(green curve). The exhibited behavior qualitatively resembles the one of the nanoslits except that the amplitudes of the generated second harmonic spectra are much higher. The slight discrepancy between the frequency doubled components of the black curves in Fig. 6(a) and 6(c) is assigned to different numerical noise levels in the two and three dimensional simulations. In Fig. 7(a) we plot the peak amplitudes of the SHG spectra obtained with the unfilled structures as a function of the incident field strength. Here, the data points were referenced to the peak amplitude of the incident spectrum at the highest considered field strength. From the log-log plot we obtain a slope of 2 for both curves indicating a quadratic dependence on the field strength as expected for a

*χ*

^{(2)}process. Much higher SHG efficiencies are obtained for the SRRs even though an optimized nanoslit geometry was employed whereas the SRR conversion efficiency can still be increased by (i) using narrower gaps, (ii) adding extended capacitive faces, and (iii) using a combined

*E*- and

*H*-field excitation. Note, however, that the strong field concentration may cause photorefractive damage so that the natural damage threshold imposes an upper limit on the obtainable nonlinearites.

*λ*. The difference in the behavior exhibited by the two structures can consequently be used to realize either broadband or narrowband conversion. The SRRs are thereby of particular interest because the narrowband conversion property allows for tunability of the nonlinearity by adjusting the dielectric environment or the structural parameters. This behavior can be seen in Fig. 6(c) where the red and green curves exhibit a dent in the incident spectrum corresponding to the fundamental SRR resonance. Filling the gap with LiTaO

_{3}results in a small redshift which is reproduced in the second harmonic spectrum. For better visibility of this effect the corresponding SHG spectra are shown again in Fig. 7(c) where the curves were normalized. The redshift induced by the modification of the dielectric environment can clearly be seen. The frequency position and linewidth of SRR resonances can be tailored by adjusting the spilt-ring’s geometry [13

13. M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. **7**, S12–S22 (2005). [CrossRef]

14. A. Bitzer, J. Wallauer, H. Helm, H. Merbold, T. Feurer, and M. Walther, “Lattice modes mediate radiative coupling in metamaterial arrays,” Opt. Express **17**, 22108–22113 (2009). [CrossRef] [PubMed]

25. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B **76**, 073101 (2007). [CrossRef]

_{3}. The SRR side length

*L*was reduced by 10% which results in a blueshift of the resonance that is consequently reproduced in the SHG spectrum.

## 4. Conclusion

## Appendix A. Nonlinearity of LiTaO_{3} at THz frequencies

## A.1. Potential energy surface

_{3}may well be represented by [26

26. I. Inbar and R. E. Cohen, “Comparison of the electronic structures and energetics of ferroelectric LiNbO_{3} and LiTaO_{3},” Phys. Rev. B **53**, 1193 (1996). [CrossRef]

27. V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. **11**, 3559 (1999). [CrossRef]

*Q*is the net ionic displacement,

*M*is the oscillator mass, −

*is the frequency of the transverse optical phonon mode, and*

_{T}*V*

_{0}is the value of the potential at its minima (well depth).

## A.2. Expressions for the nonlinear susceptibilities

*P*and the electric field

*E*is usually expanded in terms of a power series [28] where

*ε*

_{0}is the vacuum permittivity and

*χ*

^{(}

^{n}^{)}with

*n*= 1,2,3, … denotes the susceptibility of different order. In order to include nonlinear effects of second and third order in our simulations we need to obtain numerical values for

*χ*

^{(2)}and

*χ*

^{(3)}of LiTaO

_{3}at THz frequencies. We start by assuming that the crystal consists of independent harmonic oscillators that couple to the electric field through their dipole moment [23

23. T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. **37**, 317–350 (2007). [CrossRef]

*Q*is the ionic displacement, Γ is a phenomenological damping constant and

*q*denotes the effective charge. Using a Taylor expansion of

*V*(

*Q*) where we consider only terms up to third order we obtain where We now follow the approach in [28] and use a procedure analogous to Rayleigh-Schrödinger perturbation theory. The electric field

*E*(

*t*) is replaced by

*λ E*(

*t*) and

*Q*is expanded in a power series

*Q*=

*λQ*

^{(1)}+

*λ*

^{2}

*Q*

^{(2)}+

*λ*

^{3}

*Q*

^{(3)}. The terms proportional to the different orders of

*λ*

^{(}

^{n}^{)}must satisfy Eq. (6) separately so that we obtain a set of three differential equations The macroscopic polarization of order

*n*is linked to the displacement

*Q*

^{(}

^{n}^{)}and the electric field

*E*through Assuming harmonic time dependence of the form we can obtain solutions for Eq. (8)–(10) which can be combined with Eq. (11) to obtain expressions for the susceptibilities of different order and where

## A.3. Numerical values of constants used for the calculation of the nonlinear susceptibilities

*χ*

^{(2)}and

*χ*

^{(3)}we need numerical values for

*ω*, Γ,

_{T}*V*

_{0},

*N*,

*M*, and

*q*. The first three are found in the literature and are given in Table 1 where we assume a polarization along the extraordinary axis. The oscillator density

*N*is calculated by considering the inverse of a 10-atom unit cell with hexagonal lattice in a rhombohedral setting [29] where the lattice coordinates

*a*and

*c*can be taken from x-ray and neutron scattering data [30

30. H. Boysen and F. Altorfer, “A neutron powder investigation of the high-temperature structure and phase transition in LiNbO_{3},” Acta Crystallogr., Sect. B **50**, 405 (1994). [CrossRef]

31. R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO_{3} and LiTaO_{3},” Acta Crystallogr., Sect. B **53**, 420 (1997). [CrossRef]

*M*is the reduced mass given by [29] where

*m*

*refers to the Li, Nb, and O atoms within a unit cell. The values calculated for*

_{j}*M*and

*N*are included in Table 1. In order to also obtain a value for

*q*we use the permittivity where denotes the oscillator strength. Combining the static permittivity

*ε*and

_{s}*ε*

_{∞}are found in the literature. The corresponding values along the extraordinary axis as well as the calculated value for

*q*are included in Table 1. The nonlinear susceptibilities are then calculated to

*χ*

^{(2)}= (1.75 –

*ι ·*0.08) · 10

^{−}

^{8}m/V and

*χ*

^{(3)}= (2.1 –

*ι ·*0.2) · 10

^{−18}m

^{2}/V

^{2}. Note, that the imaginary part of

*χ*

^{(2)}causes an absolute phase shift of the second harmonic field whereas the imaginary part of

*χ*

^{(3)}leads to two-photon absorption, i.e. loss.

## Acknowledgment

## References and links

1. | G. P. Williams, “Filling the THz gap-high power sources and applications,” Rep. Prog. Phys. |

2. | E. Budiarto, J. Margolies, S. Jeong, and J. Song, “High-intensity terahertz pulses at 1-kHz repetition rate,” IEEE J. Quantum Electron. |

3. | T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. Elsaesser, “Generation of single-cycle THz transients with high electric-field amplitudes,” Opt. Lett. |

4. | K. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. |

5. | M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics |

6. | H. R. Park, Y. M. Park, H. S. Kim, J. S. Kyoung, M. A. Seo, D. J. Park, Y. H. Ahn, K. J. Ahn, and D. S. Kim, “Terahertz nanoresonators: Giant field enhancement and ultrabroadband performance,” Appl. Phys. Lett. |

7. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microwave Theory Tech. |

8. | S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science |

9. | J. Jin, |

10. | COMSOL Multiphysics 3.5. |

11. | M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. |

12. | N. Laman and D. Grischkowsky, “Terahertz conductivity of thin metal films,” Appl. Phys. Lett. |

13. | M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A, Pure Appl. Opt. |

14. | A. Bitzer, J. Wallauer, H. Helm, H. Merbold, T. Feurer, and M. Walther, “Lattice modes mediate radiative coupling in metamaterial arrays,” Opt. Express |

15. | P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. |

16. | N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. |

17. | S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology |

18. | J. García-García, F. Martín, J. D. Baena, R. Marqués, and L. Jelinek, “On the resonances and polarizabilities of split ring resonators,” J. Appl. Phys. |

19. | M. Shalaby, H. Merbold, M. Peccianti, L. Razzari, G. Sharma, R. Morandotti, T. Ozaki, T. Feurer, A. Weber, L. Heyderman, H. Sigg, and B. Patterson, “Concurrent field enhancement and high transmission of THz radiation in nanoslit arrays,” in preparation (2011). |

20. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science |

21. | J. Kyoung, M. Seo, H. Park, S. Koo, H. sun Kim, Y. Park, B.-J. Kim, K. Ahn, N. Park, H.-T. Kim, and D.-S. Kim, “Giant nonlinear response of terahertz nanoresonators on VO |

22. | S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature |

23. | T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Satz, and K. A. Nelson, “Terahertz Polaritonics,” Annu. Rev. Mater. Res. |

24. | M. C. Hoffmann, N. C. Brandt, H. Y. Hwang, K. Yeh, and K. A. Nelson, “Terahertz Kerr effect,” Appl. Phys. Lett. |

25. | H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B |

26. | I. Inbar and R. E. Cohen, “Comparison of the electronic structures and energetics of ferroelectric LiNbO |

27. | V. Romero-Rochin, R. M. Koehl, C. J. Brennan, and K. A. Nelson, “Anharmonic phonon-polariton excitation through impulsive stimulated Raman scattering and detection through wave vector overtone spectroscopy: theory and comparison to experiments on lithium tantalate,” J. Chem. Phys. |

28. | R. W. Boyd, |

29. | D. W. Ward, “Polaritonics: An Intermediate Regime Between Electronics and Photonics,” Ph.D. thesis, Department of Chemistry – Massachusetts Institute of Technology (2005). |

30. | H. Boysen and F. Altorfer, “A neutron powder investigation of the high-temperature structure and phase transition in LiNbO |

31. | R. Hsu, E. N. Maslen, D. du Boulay, and N. Ishizawa, “Synchrotron X-ray Studies of LiNbO |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(190.4360) Nonlinear optics : Nonlinear optics, devices

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 17, 2011

Revised Manuscript: March 12, 2011

Manuscript Accepted: March 13, 2011

Published: March 31, 2011

**Citation**

Hannes Merbold, Andreas Bitzer, and Thomas Feurer, "Second harmonic generation based on strong field enhancement in nanostructured THz materials," Opt. Express **19**, 7262-7273 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7262

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