## Dynamics of short pulses and phase matched second harmonic generation in negative index materials

Optics Express, Vol. 14, Issue 11, pp. 4746-4756 (2006)

http://dx.doi.org/10.1364/OE.14.004746

Acrobat PDF (350 KB)

### Abstract

We study pulsed second harmonic generation in metamaterials under conditions of significant absorption. Tuning the pump in the negative index range, a second harmonic signal is generated in the positive index region, such that the respective indices of refraction have the same magnitudes but opposite signs. This insures that a forward-propagating pump is exactly phase matched to the backward-propagating second harmonic signal. Using peak intensities of ~500 MW/cm^{2}, assuming *χ*^{(2)}~80pm/V, we predict conversion efficiencies of 12% and 0.2% for attenuation lengths of 50 and 5µm, respectively.

© 2006 Optical Society of America

2. J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966 (2000). [CrossRef] [PubMed]

3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059 (1987). [CrossRef] [PubMed]

4. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486 (1987). [CrossRef] [PubMed]

5. Focus issue on: Negative Refraction and Metamaterials, Opt. Express11, .639–760 (2003). [PubMed]

7. 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]

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

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

10. A.K. Popov, V.V. Slabko, and V.M. Shalaev, “Second harmonic generation in left-handed materials,” Las. Phys. Lett., published online February 2006. [CrossRef]

11. M.V. Gorkunov, I.V. Shradivov, and Y.S. Kivshar, “Enhanced parametric processes in binary metamaterials,” Appl. Phys. Lett. **88**, 071912 (2006). [CrossRef]

**

_{ω}(

*z,t*)=2

_{ω}𝓔

_{2ω},

_{2ω}(

*z,t*)=

*𝔐*

_{ω}

*z,t*)=2

***

_{ω}

**

_{2ω}, and

*𝔐*

_{2ω}(

*z,t*)=

16. M. Scalora, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, J.W. Haus, and A.M. Zheltikov, “Negative refraction of ultrashort electromagnetic pulses,” Appl. Phys. B **81**, 393 (2005). [CrossRef]

18. M. Scalora, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. J. Bloemer, M. Centini, C. Sibilia, and M. Bertolotti, “Pulse propagation, dispersion, and energy in negative index materials,” Phys. Rev. E **72**, 066601 (2005). [CrossRef]

*λ*

_{r}=1µm as the reference wavelength, and have adopted the following scaling:

*ξ*=

*z*/

*λ*

_{r}is the scaled longitudinal coordinate,

*τ*=

*ctλ*

_{r}is the time in units of the optical cycle,

*β*=2

*π*ω ˜ is the scaled wave vector, and

*is the scaled frequency. Although they contain no approximations other than the assumption that the medium is isotropic, under most circumstances Eqs.(4) may be simplified. For instance, in ordinary dielectric materials the dispersion length (associated with the second order temporal derivative on the left hand sides of Eqs.(4)) varies from a few millimeters for few-cycle pulses, to a few meters for pulses several tens or a few hundred wave cycles in duration. NIMs, however, are more dispersive than ordinary materials, as described by either a Drude or a Lorentz model near a resonance condition. Nevertheless, typical dispersion lengths in regions of interest range from several hundred to several thousand wave cycles, depending on incident pulse width. In any case, the typical absorption length that we consider, which varies from 5µm to approximately 50µm, may be several orders of magnitude smaller than higher order dispersion lengths. This fact by itself obviates the need for the inclusion of higher order terms beyond the first order temporal derivative, which affects the group velocity of the pulse and should be preserved. As a result, Eqs.(4) may be simplified to read:*ω ˜
=ω/ω

_{r}19. A. Zharov, I.V. Shadrivov, and Y.S. Kivshar, “Nonlinear properties of left-handed materials,” Phys. Rev. Lett. **91**, 037401 (2003). [CrossRef] [PubMed]

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

*=10*γ ˜

^{-3}. This choice results in an attenuation length of approximately 50µm, so that the intensity of the peak of the pulse drops to 1/e of its input value within that distance, and it is down to approximately 1% at 200 microns from the surface. One may compare this to the degree of absorption (i.e. imaginary part of the index) exhibited by GaAs at λ~890nm.

^{4}Volts/m. We then calculated the total SH energy emitted in both directions, defined as

*(*

**W**_{T}*τ*=∫

^{∞}-

_{∞}

*U*(

*ξ*,

*τ*)

*d*

*ξ*, where the integrand represents the local, instantaneous, energy density, normalized by the initial pump energy. We estimate that the second order dispersion length, defined as

*τ*

_{p}is pulse duration, and

*λ*

_{r}for 20 wave cycle pulses, to about 6×10

^{4}

*λ*

_{r}for 200 wave cycle pulses. Therefore, we are well justified in taking the necessary steps to distill Eqs.(5) from Eqs.(4).

*ω̃*≈0.7905, which corresponds to the highest SH efficiency at twice that frequency, the index of refraction calculated from the Drude dispersion is

*n*=-0.600278+

_{ω˜}*i*0.002024. At the second harmonic frequency, 2

*ω̃*≈1.581, the index of refraction is positive,

*n*

_{2ω̃}=0.599929+

*i*0.000253. The phase matching condition between fundamental and

*k*

_{2ω}-2

*k*

_{ω})=0, is nearly exactly satisfied for the backward generated SH pulse [8

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

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

*k*

_{2ω}-2

*k*

_{ω})=2

*π*(-2

*ω̃*|

*n*

_{2ω|}+2

*ω̃*|

*n*

_{ω}|)~2×10

^{-3}

^{2}incident pulse (corresponding to a field ~3x10

^{7}V/m) crosses into a nonlinear NIM from vacuum. The pump is quickly attenuated, as a SH pulse is generated in the backward direction and rapidly exits the medium. The Drude dispersion reveals that the SH absorption length is approximately ten times larger compared to the pump absorption length, as can easily be deduced from the relative magnitudes of the imaginary part of the index.

*γ̃*=10

^{-3},

*η*

_{SH}varies from ~2.5% for pulses 200fs in duration, to ~12% for pulses ~2.5ps long. On the same figure we also show the conversion efficiency for

^{-2}. Changing

*γ̃*in this fashion leads to an order of magnitude increase in the imaginary part of the index, and reduces the attenuation depth down to 5µm, with marginal effects to the real part of the index. Nevertheless, the conversion efficiency still reaches ~0.2%. The figure also suggests that the efficiency improves by increasing pulse duration. This can be understood in terms of incident pulse bandwidth: as we increase pulse duration, more of the pulse comes into the phase matching condition, which is almost exactly satisfied at the carrier wavelength.

^{7}), because we have assumed that ε=µ, a restriction that can easily be removed.

*χ*

^{(2)}and incident peak power, it would be possible to obtain similar conversion efficiencies if, in the absence of absorption, two conditions could be satisfied: (i) exact phase and (ii) group velocity matching. For comparison purposes, if a 200fs, 500MW/cm

^{2}pulse were incident on a semi-infinite, ideally matched medium having n~1.42, one obtains a conversion efficiency of η~10% after the peak of the pulse propagates ~45µm. In contrast, the introduction of an index mismatch due to normal material dispersion (we consider a typical index variation of ~6% between ω (n~1.42)and 2ω (n~1.52)), along with an appropriate group velocity mismatch (

*V*/1.57;

^{ω˜}_{g}~c*c*/2.2), reduces the conversion efficiency dramatically down to η~0.01%, with further decreases if absorption is added. We should note that a group velocity mismatch has a much less pronounced impact on conversion efficiency compared to an index mismatch, as expected, but it can make its impact felt, particularly for shorter pulses. However, it is not feasible to simultaneously have phase and group velocity matching in a bulk medium, as natural material dispersion generally does not allow it.

20. M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E **60**, 4891 (1999). [CrossRef]

23. Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and J. A. Levenson, “Phase matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. **89**, 043901 (2002). [CrossRef] [PubMed]

*χ*

^{(2)}~80pm/V, and peak pump intensity is ~500MW/cm

^{2}. The interaction is consumed entirely near the surface interface, where both fields are more intense. In the absence of absorption, similar conversion efficiencies are possible in bulk PIMs, only by imposing simultaneous phase and group velocity matching. Finally, our calculations show that conversion efficiencies of order 0.2% are also possible, even when the attenuation depth is approximately 5µm, and suggest that similarly efficient higher harmonic generation and parametric amplification may be also achievable in a multi-wave mixing environment under conditions of significant absorption.

## Acknowledgments

## References and links

1. | V.G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” |

2. | J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

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

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

5. | Focus issue on: Negative Refraction and Metamaterials, Opt. Express11, .639–760 (2003). [PubMed] |

6. | Focus issue on: Metamaterials, J. Opt. Soc. Am. B23, 386–583 (2006). |

7. | M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E |

8. | V.M. Agranovich, Y.R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative index metamaterials,” Phys. Rev. B |

9. | I.V. Shadrivov, A.A. Zharov, and Y.S. Kivshar, “Second harmonic generation in nonlinear left-handed materials,” J. Opt. Soc. Am. B |

10. | A.K. Popov, V.V. Slabko, and V.M. Shalaev, “Second harmonic generation in left-handed materials,” Las. Phys. Lett., published online February 2006. [CrossRef] |

11. | M.V. Gorkunov, I.V. Shradivov, and Y.S. Kivshar, “Enhanced parametric processes in binary metamaterials,” Appl. Phys. Lett. |

12. | Jensen Li, Lei Zhou, C.T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. |

13. | G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and M. Scalora, “Large enhancement of second harmonic generation near the zero-n gap of a negative index Bragg grating,” Phys. Rev. E |

14. | N. Mattiucci, G. D’Aguanno, M. J. Bloemer, and M. Scalora, “Second harmonic generation form a positive-negative index material heterostructure,” Phys. Rev E |

15. | R.A. Shelby, D.R. Smith, and S. SchultzC. G. Parazzoli, R.B. Greegor, K. Li, B.E.C. Koltenbah, and M. Tanielian, “Experimental verification of a negative index of refraction,” |

16. | M. Scalora, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, J.W. Haus, and A.M. Zheltikov, “Negative refraction of ultrashort electromagnetic pulses,” Appl. Phys. B |

17. | M. Scalora, M. Syrchin, N. Akozbek, E.Y. Poliakov, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, and A.M. Zheltikov, “Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials,” Phys. Rev. Lett. |

18. | M. Scalora, G. D’Aguanno, N. Mattiucci, N. Akozbek, M. J. Bloemer, M. Centini, C. Sibilia, and M. Bertolotti, “Pulse propagation, dispersion, and energy in negative index materials,” Phys. Rev. E |

19. | A. Zharov, I.V. Shadrivov, and Y.S. Kivshar, “Nonlinear properties of left-handed materials,” Phys. Rev. Lett. |

20. | M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E |

21. | M. Scalora, M.J. Bloemer, A.S. Manka, J.P. Dowling, C.M. Bowden, R. Viswanathan, and J.W. Haus, “Pulsed second harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A |

22. | Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, and M. Scalora, “Enhancement of second harmonic generation in one-dimensional semiconductor photonic band gap,” Appl. Phys. Lett. |

23. | Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and J. A. Levenson, “Phase matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: March 21, 2006

Revised Manuscript: May 8, 2006

Manuscript Accepted: May 10, 2006

Published: May 29, 2006

**Citation**

Michael Scalora, Giuseppe D'Aguanno, Mark Bloemer, Marco Centini, Domenico de Ceglia, Nadia Mattiucci, and Yuri S. Kivshar, "Dynamics of short pulses and phase matched second harmonic generation in negative index materials," Opt. Express **14**, 4746-4756 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4746

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### References

- V.G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Sov. Phys. USPEKHI 10, 509 (1968).
- J.B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85,3966 (2000). [CrossRef] [PubMed]
- E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987). [CrossRef] [PubMed]
- Focus issue on: Negative Refraction and Metamaterials, Opt. Express 11, 639-760 (2003). [PubMed]
- Focus issue on: Metamaterials, J. Opt. Soc. Am. B 23, 386-583 (2006).
- 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]
- V.M. Agranovich, Y.R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative index metamaterials," Phys. Rev. B 69, 165112 (2004). [CrossRef]
- I.V. Shadrivov, A.A. Zharov, Y.S. Kivshar, "Second harmonic generation in nonlinear left-handed materials," J. Opt. Soc. Am. B 23, 529 (2006). [CrossRef]
- A.K. Popov, V.V. Slabko, and V.M. Shalaev, "Second harmonic generation in left-handed materials," Las. Phys. Lett., published online February 2006. [CrossRef]
- M.V. Gorkunov, I.V. Shradivov, Y.S. Kivshar, "Enhanced parametric processes in binary metamaterials," Appl. Phys. Lett. 88, 071912 (2006). [CrossRef]
- Jensen Li, Lei Zhou, C.T. Chan, and P. Sheng, "Photonic band gap from a stack of positive and negative index materials," Phys. Rev. Lett. 90, 083901, (2003). [CrossRef] [PubMed]
- G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and M. Scalora, "Large enhancement of second harmonic generation near the zero-n gap of a negative index Bragg grating," Phys. Rev. E 73, 036603 (2006). [CrossRef]
- N. Mattiucci, G. D’Aguanno, M. J. Bloemer, and M. Scalora, "Second harmonic generation form a positive-negative index material heterostructure," Phys.Rev E 72, 066612 (2005). [CrossRef]
- R.A. Shelby, D.R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001);C. G. Parazzoli, R.B. Greegor, K. Li, B.E.C. Koltenbah, M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003). [CrossRef] [PubMed]
- M. Scalora, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, J.W. Haus, A.M. Zheltikov, "Negative refraction of ultrashort electromagnetic pulses," Appl. Phys. B 81, 393 (2005). [CrossRef]
- M. Scalora, M. Syrchin, N. Akozbek, E.Y. Poliakov, G. D'Aguanno, N. Mattiucci, M.J. Bloemer, A.M. Zheltikov, "Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: applications to negative index materials," Phys. Rev. Lett. 95, 013902 (2005). [CrossRef] [PubMed]
- M. Scalora, G. D'Aguanno, N. Mattiucci, N. Akozbek, M. J. Bloemer, M. Centini, C. Sibilia, M. Bertolotti, "Pulse propagation, dispersion, and energy in negative index materials," Phys. Rev. E 72, 066601 (2005). [CrossRef]
- A. Zharov, I.V. Shadrivov, Y.S. Kivshar, "Nonlinear properties of left-handed materials," Phys. Rev. Lett. 91, 037401 (2003). [CrossRef] [PubMed]
- M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, "Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions," Phys. Rev. E 60, 4891 (1999). [CrossRef]
- M. Scalora, M.J. Bloemer, A.S. Manka, J.P. Dowling, C.M. Bowden, R. Viswanathan, J.W. Haus, "Pulsed second harmonic generation in nonlinear, one-dimensional, periodic structures," Phys. Rev. A 56, 3166-75 (1997). [CrossRef]
- Y. Dumeige, P. Vidakovic, S. Sauvage, I. Sagnes, J. A. Levenson, C. Sibilia, M. Centini, G. D’Aguanno, M. Scalora, "Enhancement of second harmonic generation in one-dimensional semiconductor photonic band gap," Appl. Phys. Lett. 78, 3021 (2001). [CrossRef]
- Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, J. A. Levenson, "Phase matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length," Phys. Rev. Lett. 89, 043901 (2002). [CrossRef] [PubMed]

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