## A numerical approach for analyzing higher harmonic generation in multilayer nanostructures

JOSA B, Vol. 27, Issue 5, pp. 1118-1130 (2010)

http://dx.doi.org/10.1364/JOSAB.27.001118

Enhanced HTML Acrobat PDF (364 KB)

### Abstract

We introduce a computational scheme for analyzing higher harmonic generation in nonlinear optical periodic nanostructures that are composed of multiple layers. Such nanostructures, i.e., photonic crystals, plasmonic nanostructures, or metamaterials, are currently in the focus of interest to enhance the efficiency of various nonlinear processes. We exploit an adapted Fourier modal method combined with a modified scattering-matrix algorithm to numerically model these processes. We explicitly present a numerically stable formulation of the scheme. The strength and the applicability of the algorithm are outlined at some selected examples.

© 2010 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(050.1950) Diffraction and gratings : Diffraction gratings

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4160) Nonlinear optics : Multiharmonic generation

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 4, 2009

Revised Manuscript: March 12, 2010

Manuscript Accepted: March 15, 2010

Published: April 29, 2010

**Citation**

Thomas Paul, Carsten Rockstuhl, and Falk Lederer, "A numerical approach for analyzing higher harmonic generation in multilayer nanostructures," J. Opt. Soc. Am. B **27**, 1118-1130 (2010)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-5-1118

Sort: Year | Journal | Reset

### References

- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localisation of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]
- Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990). [CrossRef] [PubMed]
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]
- E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991). [CrossRef] [PubMed]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]
- V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Experimental verification of a negative index of refraction,” Opt. Lett. 30, 3356–3358 (2005). [CrossRef]
- J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008). [CrossRef] [PubMed]
- C. Helgert, C. Menzel, C. Rockstuhl, E. Pshenay-Severin, E.-B. Kley, A. Chipouline, A. Tünnermann, F. Lederer, and T. Pertsch, “Polarization-independent negative-index metamaterial in the near infrared,” Opt. Lett. 34, 704–706 (2009). [CrossRef] [PubMed]
- E. Pshenay-Severin, U. Hübner, C. Menzel, C. Helgert, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Double-element metamaterial with negative index at near-infrared wavelengths,” Opt. Lett. 34, 1678–1680 (2009). [CrossRef] [PubMed]
- M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006). [CrossRef] [PubMed]
- M. W. Klein, N. Feth, M. Wegener, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15, 5238–5247 (2007). [CrossRef] [PubMed]
- E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78, 113102 (2008). [CrossRef]
- W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, Jr., K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. 6, 1027–1030 (2006). [CrossRef]
- F. B. P. Niesler, N. Feth, S. Linden, J. Niegemann, J. Gieseler, K. Busch, and M. Wegener, “Second-harmonic generation from split-ring resonators on a GaAs substrate,” Opt. Lett. 34, 1997–1999 (2009). [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. 7, 1251–1255 (2007). [CrossRef] [PubMed]
- B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, “Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,” J. Opt. A, Pure Appl. Opt. 7, S110–S117 (2005). [CrossRef]
- A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003). [CrossRef] [PubMed]
- I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006). [CrossRef]
- R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008). [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. B 79, 235109 (2009). [CrossRef]
- M. Nevière, P. Vincent, D. Maystre, R. Reinisch, and J. L. Coutaz, “Differential theory for metallic gratings in nonlinear optics: second-harmonic generation,” J. Opt. Soc. Am. B 5, 330–337 (1988). [CrossRef]
- R. Reinisch and M. Nevière , “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983). [CrossRef]
- D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988). [CrossRef]
- K. Busch, J. Niegemann, M. Pototschnig, and L. Tkeshelashvili, “A Krylov-subspace based solver for the linear and nonlinear Maxwell equations,” Phys. Status Solidi B 244, 3479–3496 (2007). [CrossRef]
- B. Maes, P. Bienstman, and R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B 22, 1378–1383 (2005). [CrossRef]
- M. F. Saleh, L. D. Negro, and B. E. A. Saleh, “Second-order parametric interactions in 1-D photonic-crystal microcavity structures,” Opt. Express 16, 5261–5276 (2008). [CrossRef] [PubMed]
- M. Cherchi, “Exact analytic expressions for electromagnetic propagation and optical nonlinear generation in finite one-dimensional periodic multilayers,” Phys. Rev. E 69, 066602 (2004). [CrossRef]
- E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994). [CrossRef]
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]
- L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997). [CrossRef]
- T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061 (2009). [CrossRef] [PubMed]
- W. Nakagawa, R.-C. Tyan, and Y. Fainman, “Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation,” J. Opt. Soc. Am. A 19, 1919–1928 (2002). [CrossRef]
- B. Bai and J. Turunen, “Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials,” J. Opt. Soc. Am. B 24, 1105–1112 (2007). [CrossRef]
- L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A, Pure Appl. Opt. 5, 345–355 (2003). [CrossRef]
- R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).
- N. Bonod, E. Popov, and M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005). [CrossRef]
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
- L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]
- N. P. K. Cotter, T. W. Preist, and J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995). [CrossRef]
- T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94, 6447–6455 (2003). [CrossRef]
- The amplitude transmission coefficients are normalized to the input pump field which has an intensity of |E|2=1012 V2/m2.
- E. Popov and B. Bozhkov, “Corrugated waveguides as resonance optical filters—advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001). [CrossRef]
- E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986). [CrossRef]
- S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]
- Because of the limited discretization distance in the numerical calculations the amplitude transmission coefficient does not reach a value of zero.
- In the chosen orientation of the GaAs crystal all susceptibility tensor elements of form χ1ab and χ2ab with a,b=1,2(x,y) are vanishing. Illuminating the structure with a plane wave under normal incidence does not allow for any x or y polarized SHG. Only a z-polarized SH will appear (χ312=χ321≠0). Therefore, no zeroth order SH plane wave is able to propagate.
- R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. Lett. 37, 191–203 (1985).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.