Abstract

We analyze second harmonic generation in a generic medium composed of Lorentz oscillators for both the electrical permittivity and magnetic permeability. We discuss conditions where a fundamental beam is tuned in a positive index region and generates second and/or third harmonics in a negative index region. Phase matching condition, phase locking, inhibition of absorption at the harmonic wavelengths, and second and third harmonic lenses are also investigated.

© 2010 Optical Society of America

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  1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewar, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  2. V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  3. R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef] [PubMed]
  4. V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
    [CrossRef]
  5. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
    [CrossRef] [PubMed]
  6. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
    [CrossRef]
  7. A. K. Popov, V. V. Slabko, and V. M. Shalaev, “Second harmonic generation in left-handed metamaterials,” Laser Phys. Lett. 3, 293–297 (2006).
    [CrossRef]
  8. A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: Second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
    [CrossRef]
  9. N. M. Litchinitser and V. M. Shalaev, “Photonic metamaterials,” Laser Phys. Lett. 5, 411–420 (2008).
    [CrossRef]
  10. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
    [CrossRef]
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  12. A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  13. I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
    [CrossRef]
  14. M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Yu. S. Kivshar, “Dynamics of short pulses and phase matched second harmonic generation in negative index materials,” Opt. Express 14, 4746–4756 (2006).
    [CrossRef] [PubMed]
  15. V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
    [CrossRef]
  16. M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
    [CrossRef] [PubMed]
  17. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
    [CrossRef]
  18. V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
    [CrossRef]
  19. M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
    [CrossRef]
  20. S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
    [CrossRef]
  21. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
    [CrossRef] [PubMed]
  22. Q. Thommen and P. Mandel, “Electromagnetically induced left handedness in optically excited four-level atomic media,” Phys. Rev. Lett. 96, 053601 (2006).
    [CrossRef] [PubMed]
  23. A. Kussow, A. Akyurtlu, and N. Angkawisittpan, “Optically isotropic negative index of refraction metamaterial,” Phys. Status Solidi B 245, 992–997 (2008).
    [CrossRef]
  24. G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Transmission properties and field localization in a negative index Fabry–Perot etalon,” Laser Phys. 15, 590–593 (2005).
  25. M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
    [CrossRef]
  26. Y. R. Shen, Principles of Nonlinear Optics (Wiley, 2003).
  27. E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
    [CrossRef] [PubMed]
  28. N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.
  29. V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
    [CrossRef]
  30. A. Zharov, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
    [CrossRef]
  31. C. Ciracì and E. Centeno, “Focusing of second-harmonic signals with nonlinear metamaterial lenses: A biophotonic microscopy approach,” Phys. Rev. Lett. 103, 063901 (2009).
    [CrossRef] [PubMed]

2009 (4)

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

C. Ciracì and E. Centeno, “Focusing of second-harmonic signals with nonlinear metamaterial lenses: A biophotonic microscopy approach,” Phys. Rev. Lett. 103, 063901 (2009).
[CrossRef] [PubMed]

E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
[CrossRef] [PubMed]

2008 (5)

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

A. Kussow, A. Akyurtlu, and N. Angkawisittpan, “Optically isotropic negative index of refraction metamaterial,” Phys. Status Solidi B 245, 992–997 (2008).
[CrossRef]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[CrossRef]

N. M. Litchinitser and V. M. Shalaev, “Photonic metamaterials,” Laser Phys. Lett. 5, 411–420 (2008).
[CrossRef]

2007 (1)

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

2006 (6)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef] [PubMed]

Q. Thommen and P. Mandel, “Electromagnetically induced left handedness in optically excited four-level atomic media,” Phys. Rev. Lett. 96, 053601 (2006).
[CrossRef] [PubMed]

A. K. Popov, V. V. Slabko, and V. M. Shalaev, “Second harmonic generation in left-handed metamaterials,” Laser Phys. Lett. 3, 293–297 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: Second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
[CrossRef]

M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Yu. S. Kivshar, “Dynamics of short pulses and phase matched second harmonic generation in negative index materials,” Opt. Express 14, 4746–4756 (2006).
[CrossRef] [PubMed]

2005 (4)

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

A. Zharov, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Transmission properties and field localization in a negative index Fabry–Perot etalon,” Laser Phys. 15, 590–593 (2005).

2004 (2)

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

2003 (2)

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

2001 (1)

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewar, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1994 (1)

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

1962 (1)

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

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Akozbek, N.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Akozbeke, N.

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

Akyurtlu, A.

A. Kussow, A. Akyurtlu, and N. Angkawisittpan, “Optically isotropic negative index of refraction metamaterial,” Phys. Status Solidi B 245, 992–997 (2008).
[CrossRef]

Alonzo, M.

Angkawisittpan, N.

A. Kussow, A. Akyurtlu, and N. Angkawisittpan, “Optically isotropic negative index of refraction metamaterial,” Phys. Status Solidi B 245, 992–997 (2008).
[CrossRef]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Belardini, A.

Bertolotti, M.

E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

Bloembergen, N.

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

Bloemer, M.

Bloemer, M. J.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Transmission properties and field localization in a negative index Fabry–Perot etalon,” Laser Phys. 15, 590–593 (2005).

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Cai, W.

Centeno, E.

C. Ciracì and E. Centeno, “Focusing of second-harmonic signals with nonlinear metamaterial lenses: A biophotonic microscopy approach,” Phys. Rev. Lett. 103, 063901 (2009).
[CrossRef] [PubMed]

Centini, M.

E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
[CrossRef] [PubMed]

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Yu. S. Kivshar, “Dynamics of short pulses and phase matched second harmonic generation in negative index materials,” Opt. Express 14, 4746–4756 (2006).
[CrossRef] [PubMed]

Chauvet, M.

Chettiar, U. K.

Ciracì, C.

C. Ciracì and E. Centeno, “Focusing of second-harmonic signals with nonlinear metamaterial lenses: A biophotonic microscopy approach,” Phys. Rev. Lett. 103, 063901 (2009).
[CrossRef] [PubMed]

Cojocaru, C.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

Crenshaw, M. E.

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

D’Aguanno, G.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Yu. S. Kivshar, “Dynamics of short pulses and phase matched second harmonic generation in negative index materials,” Opt. Express 14, 4746–4756 (2006).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Transmission properties and field localization in a negative index Fabry–Perot etalon,” Laser Phys. 15, 590–593 (2005).

de Ceglia, D.

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Yu. S. Kivshar, “Dynamics of short pulses and phase matched second harmonic generation in negative index materials,” Opt. Express 14, 4746–4756 (2006).
[CrossRef] [PubMed]

Dolling, G.

Drachev, V. P.

Enkrich, C.

Fazio, E.

E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
[CrossRef] [PubMed]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

Foreman, J. V.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Gorkunov, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Halioua, Y.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

Haus, J. W.

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewar, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Kandidov, V. P.

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

Kildishev, A. V.

Kivshar, Yu. S.

Klein, M. W.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef] [PubMed]

Kosareva, O. G.

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

Kussow, A.

A. Kussow, A. Akyurtlu, and N. Angkawisittpan, “Optically isotropic negative index of refraction metamaterial,” Phys. Status Solidi B 245, 992–997 (2008).
[CrossRef]

Lapine, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Lederer, F.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Linden, S.

Litchinitser, N. M.

N. M. Litchinitser and V. M. Shalaev, “Photonic metamaterials,” Laser Phys. Lett. 5, 411–420 (2008).
[CrossRef]

Mandel, P.

Q. Thommen and P. Mandel, “Electromagnetically induced left handedness in optically excited four-level atomic media,” Phys. Rev. Lett. 96, 053601 (2006).
[CrossRef] [PubMed]

Mattiucci, N.

M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Yu. S. Kivshar, “Dynamics of short pulses and phase matched second harmonic generation in negative index materials,” Opt. Express 14, 4746–4756 (2006).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Transmission properties and field localization in a negative index Fabry–Perot etalon,” Laser Phys. 15, 590–593 (2005).

McPeake, D.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Menzel, C.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[CrossRef]

O’Brien, S.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Paul, T.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Pendry, J. B.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewar, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Pershan, P. S.

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

Pertsch, T.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Pettazzi, F.

E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
[CrossRef] [PubMed]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

Popov, A. K.

A. K. Popov, V. V. Slabko, and V. M. Shalaev, “Second harmonic generation in left-handed metamaterials,” Laser Phys. Lett. 3, 293–297 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: Second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
[CrossRef]

Powell, D. A.

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

Raineri, F.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

Raj, R.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

Ramakrishna, S. A.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Ringhofer, K. H.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewar, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Rockstuhl, C.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[CrossRef]

Roppo, V.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Sagnes, I.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

Sarychev, A. K.

Scalora, M.

E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
[CrossRef] [PubMed]

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

M. Scalora, G. D’Aguanno, M. Bloemer, M. Centini, D. de Ceglia, N. Mattiucci, and Yu. S. Kivshar, “Dynamics of short pulses and phase matched second harmonic generation in negative index materials,” Opt. Express 14, 4746–4756 (2006).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Transmission properties and field localization in a negative index Fabry–Perot etalon,” Laser Phys. 15, 590–593 (2005).

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Schultz, S.

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Shadrivov, I. V.

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Shalaev, V. M.

N. M. Litchinitser and V. M. Shalaev, “Photonic metamaterials,” Laser Phys. Lett. 5, 411–420 (2008).
[CrossRef]

A. K. Popov, V. V. Slabko, and V. M. Shalaev, “Second harmonic generation in left-handed metamaterials,” Laser Phys. Lett. 3, 293–297 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: Second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

Shelby, R.

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Shen, Y. R.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Y. R. Shen, Principles of Nonlinear Optics (Wiley, 2003).

Sibilia, C.

E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17, 3141–3147 (2009).
[CrossRef] [PubMed]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

Slabko, V. V.

A. K. Popov, V. V. Slabko, and V. M. Shalaev, “Second harmonic generation in left-handed metamaterials,” Laser Phys. Lett. 3, 293–297 (2006).
[CrossRef]

Smith, D. R.

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

Stewar, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewar, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Thommen, Q.

Q. Thommen and P. Mandel, “Electromagnetically induced left handedness in optically excited four-level atomic media,” Phys. Rev. Lett. 96, 053601 (2006).
[CrossRef] [PubMed]

Trull, J.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vicenti, M. A.

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

Vilaseca, R.

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

Vincenti, M. A.

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Wegener, M.

Yuan, H.

Zakhidov, A. A.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Zharov, A.

A. Zharov, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
[CrossRef]

Zharov, A. A.

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Zhou, J. F.

Appl. Phys. B (1)

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: Second-harmonic generation, Manley–Rowe relations and parametric amplification,” Appl. Phys. B 84, 131–137 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear electric metamaterials,” Appl. Phys. Lett. 95, 084102 (2009).
[CrossRef]

A. Zharov, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewar, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Laser Phys. (1)

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Transmission properties and field localization in a negative index Fabry–Perot etalon,” Laser Phys. 15, 590–593 (2005).

Laser Phys. Lett. (2)

N. M. Litchinitser and V. M. Shalaev, “Photonic metamaterials,” Laser Phys. Lett. 5, 411–420 (2008).
[CrossRef]

A. K. Popov, V. V. Slabko, and V. M. Shalaev, “Second harmonic generation in left-handed metamaterials,” Laser Phys. Lett. 3, 293–297 (2006).
[CrossRef]

Metamaterials (1)

V. Roppo, M. Centini, D. de Ceglia, M. A. Vicenti, J. W. Haus, N. Akozbeke, M. J. Bloemer, and M. Scalora, “Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses,” Metamaterials 2, 135–144 (2008).
[CrossRef]

Opt. Commun. (1)

M. Scalora and M. E. Crenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (1)

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

Phys. Rev. A (2)

V. Roppo, C. Cojocaru, F. Raineri, G. D’Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, “Field localization and enhancement of phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities,” Phys. Rev. A 80, 043834 (2009).
[CrossRef]

V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media,” Phys. Rev. A 76, 033829 (2007).
[CrossRef]

Phys. Rev. B (3)

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Phys. Rev. E (1)

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

Q. Thommen and P. Mandel, “Electromagnetically induced left handedness in optically excited four-level atomic media,” Phys. Rev. Lett. 96, 053601 (2006).
[CrossRef] [PubMed]

M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905 (2008).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

C. Ciracì and E. Centeno, “Focusing of second-harmonic signals with nonlinear metamaterial lenses: A biophotonic microscopy approach,” Phys. Rev. Lett. 103, 063901 (2009).
[CrossRef] [PubMed]

Phys. Status Solidi B (1)

A. Kussow, A. Akyurtlu, and N. Angkawisittpan, “Optically isotropic negative index of refraction metamaterial,” Phys. Status Solidi B 245, 992–997 (2008).
[CrossRef]

Science (2)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef] [PubMed]

R. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (2)

N. Akozbek, V. Roppo, M. A. Vincenti, J. V. Foreman, M. J. Bloemer, J. W. Haus, and M. Scalora, “Extraordinary nonlinear optics in ordinary semiconductors,” arXiv:0904.4082.

Y. R. Shen, Principles of Nonlinear Optics (Wiley, 2003).

Supplementary Material (5)

» Media 1: MOV (351 KB)     
» Media 2: MOV (322 KB)     
» Media 3: MOV (47 KB)     
» Media 4: MOV (130 KB)     
» Media 5: MOV (332 KB)     

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Figures (8)

Fig. 1
Fig. 1

(Media 1) A snapshot of SHG in a material with ε ( ω ) = μ ( ω ) , ω 0 e = ω p e = ω 0 m = ω p m = 2.0 , and γ e = γ m = 10 6 . The fundamental (blue-thin dashed lines, out of scale) is tuned at 800 nm. (a) Fields in real space; (b) momentum densities; (c) k-spectra; (d) SH energy.

Fig. 2
Fig. 2

(Media 2) A snapshot of SHG in a material as in Fig. 1. The fundamental (blue-thin dashed lines, out of scale) is tuned at PM condition. (a) Fields in real space; (b) momentum densities; (c) k-spectra; (d) SH energy.

Fig. 3
Fig. 3

(Media 3) A FF beam tuned at the PM condition (881 nm) impinges at a 35° angle with respect to the entry surface. The material is as in Fig. 1. The negative values of the dispersion relation at the SH frequency cause a SH signal to be continuously generated backward inside the material and to be refracted negatively once it exits into vacuum.

Fig. 4
Fig. 4

Superposition of several snapshots of the FF beam propagation tuned at 800 nm and the generated SH signal. The impinging angle is 35° with respect to the entry surface. The material is as in Fig. 1: n ( ω ) = 2.64 and n ( 2 ω ) = 0.77 . Notice the presence of the inhomogeneous PL SH component bounded to the FF and the homogeneous SH component that refracts negatively.

Fig. 5
Fig. 5

Superposition of several snapshots of the FF beam propagation tuned at 1200 nm and the generated TH signal. The impinging angle is again 35° with respect to the entry surface. The material is as in Fig. 1, i.e., n ( ω ) = 2.21 and n ( 3 ω ) = 0.77 . Notice the presence of the inhomogeneous PL TH component that locks to pump in analogy to the SH signal and the homogeneous TH component that refracts negatively.

Fig. 6
Fig. 6

(Media 4): (Upper row) Snapshot of pulse propagation inside the material. The pulse was initially half-optical-cycles in diameter. Due to the strong diffraction the pulse mimics the dynamics of a point source having a wide angle space spectrum. (Lower row) The PL SH component keeps propagating locked to the fundamental while the homogeneous component focuses inside the medium.

Fig. 7
Fig. 7

A point source is placed in front of a metamaterial slab (monochromatic stationary waves). The SH field is focused by the lens into an image of the FF radiating object. (a) Real part of the electric FF. (b) The SH field intensity.

Fig. 8
Fig. 8

(Media 5) A snapshot of SHG in a material with the same parameters as in Fig. 1 but using γ e = γ m = 0.1 at the SH. The index of refraction is n ( 2 ω ) = 0.76 + i 0.2 . The fundamental (blue-thin dashed lines, out of scale) is tuned at 800 nm. (a) Fields in the real space; (b) momentum densities; (c) k-spectra; (d) SH energy. Notice how in this situation the homogeneous SH pulse is missing.

Equations (10)

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ε ( ω ) = 1 ω p e 2 ω 2 + i ω γ e ω 0 e 2 ,
μ ( ω ) = 1 ω p m 2 ω 2 + i ω γ m ω 0 m 2 ,
H = x ̂ = 1 ( H x ω ( z , y , t ) + c .c . ) = x ̂ = 1 ( H x ω ( z , y , t ) e i ( k z ω t ) + c .c . ) ,
E = y ̂ = 1 ( E y ω ( z , y , t ) + c .c . ) + z ̂ = 1 ( E z ω ( z , y , t ) + c .c . ) = y ̂ = 1 ( E y ω ( z , y , t ) e i ( k z ω t ) + c .c . ) + z ̂ = 1 ( E z ω ( z , y , t ) e i ( k z ω t ) + c .c . ) ,
H x ω τ = i β ( H x ω + E z ω   sin   θ i + E y ω   cos   θ i ) E z ω y ̃ + E y ω ξ ,
E y ω τ = i β ( E y ω + H x ω   cos   θ i ) 4 π ( J y ω i β P y ω ) + i 4 π β P y , NL ω 4 π P y , NL ω τ + H x ω ξ ,
E z ω τ = i β ( E z ω + H x ω   sin   θ i ) 4 π ( J z ω i β P z ω ) + i 4 π β P z , NL ω 4 π P z , NL ω τ H x ω y ̃ ,
J y ω τ = ( 2 i β γ ) J y ω + ( β 2 + i γ β β r 2 ) P y ω + π ω p 2 ω 0 2 E y ω ,
J z ω τ = ( 2 i β γ ) J z ω + ( β 2 + i γ β β r 2 ) P z ω + π ω p 2 ω 0 2 E z ω ,
J y ω = P y ω τ ,     J z ω = P z ω τ .

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