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/cm2, 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

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  1. V.G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Sov. Phys. USPEKHI 10, 509 (1968).
  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. Express 11, 639-760 (2003).
    [PubMed]
  6. Focus issue on: Metamaterials, J. Opt. Soc. Am. B 23, 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 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, 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, Y.S. Kivshar, "Enhanced parametric processes in binary metamaterials," Appl. Phys. Lett. 88, 071912 (2006).
    [CrossRef]
  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. 90, 083901, (2003).
    [CrossRef] [PubMed]
  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 73, 036603 (2006).
    [CrossRef]
  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 72, 066612 (2005).
    [CrossRef]
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. A. Zharov, I.V. Shadrivov, Y.S. Kivshar, "Nonlinear properties of left-handed materials," Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  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]
  21. 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]
  22. 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]
  23. 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]

2006 (4)

Focus issue on: Metamaterials, J. Opt. Soc. Am. B 23, 386-583 (2006).

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]

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

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]

2005 (4)

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]

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]

2004 (1)

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]

2003 (4)

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

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]

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]

Focus issue on: Negative Refraction and Metamaterials, Opt. Express 11, 639-760 (2003).
[PubMed]

2002 (1)

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]

2001 (2)

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]

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]

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]

2000 (1)

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

1999 (1)

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]

1997 (1)

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]

1987 (2)

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]

1968 (1)

V.G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Sov. Phys. USPEKHI 10, 509 (1968).

Abram, I.

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]

Agranovich, V.M.

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]

Akozbek, N.

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]

Baughman, R. H.

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]

Bertolotti, M.

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]

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]

Bloemer, M. J.

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]

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]

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]

Bloemer, M.J.

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, 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.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]

Bowden, C. M.

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]

Bowden, C.M.

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]

Centini, M.

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]

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]

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]

D’Aguanno, G.

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]

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]

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]

D'Aguanno, G.

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]

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. 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]

Dowling, J.P.

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]

Dumeige, Y.

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]

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]

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]

Gorkunov, M.V.

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

Greegor, R.B.

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]

Haus, J.W.

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.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]

John, S.

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

Kivshar, Y.S.

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]

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

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

Koltenbah, B.E.C.

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]

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]

Levenson, J. A.

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]

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]

Li, K.

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]

Manka, A.S.

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]

Mattiucci, N.

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]

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]

Mériadec, C.

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]

Monnier, P.

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]

Nefedov, I.

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]

Parazzoli, C. G.

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]

Pendry, J.B.

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

Poliakov, E.Y.

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]

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]

Sagnes, I.

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]

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]

Sauvage, S.

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]

Scalora, M.

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]

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, M.J. Bloemer, J.W. Haus, A.M. Zheltikov, "Negative refraction of ultrashort electromagnetic pulses," Appl. Phys. B 81, 393 (2005).
[CrossRef]

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]

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]

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]

Schultz, S.

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]

Shadrivov, I.V.

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

Shelby, R.A.

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]

Shen, Y.R.

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]

Shradivov, I.V.

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

Sibilia, C.

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]

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]

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]

Smith, D.R.

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]

Syrchin, M.

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]

Tanielian, M.

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]

Veselago, V.G.

V.G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Sov. Phys. USPEKHI 10, 509 (1968).

Vidakovic, P.

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]

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]

Viswanathan, R.

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]

Yablonovitch, E.

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

Zakhidov, A. A.

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]

Zharov, A.

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

Zharov, A.A.

Zheltikov, A.M.

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]

Appl. Phys. B (1)

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]

Appl. Phys. Lett. (2)

M.V. Gorkunov, I.V. Shradivov, Y.S. Kivshar, "Enhanced parametric processes in binary metamaterials," Appl. Phys. Lett. 88, 071912 (2006).
[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]

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Phys. Rev. A (1)

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]

Phys. Rev. B (1)

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]

Phys. Rev. E (4)

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. 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]

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]

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]

Phys. Rev. Lett. (7)

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

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]

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]

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]

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]

Rev E (1)

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]

Science (1)

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]

Sov. Phys. USPEKHI (1)

V.G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Sov. Phys. USPEKHI 10, 509 (1968).

Other (1)

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]

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Figures (5)

Fig. 1.
Fig. 1.

Real (blue curve-left axis) and imaginary (red curve-right axis) parts of the index of refraction for the Drude model, with γ ˜ =10-3. The SH conversion efficiency, η, is also shown (solid black line with empty triangles-arbitrary scale). η sweeps across seven orders of magnitude from its minimum to its maximum, at the phase matching condition. ω̃ and 2ω̃ indicate the tuning ranges of the fundamental and SH fields. The arrows point to the magnitudes of the indices of refraction at the fundamental and SH frequencies where the phase matching condition is approximately satisfied.

Fig. 2.
Fig. 2.

A 600fs (200 wave cycles) pump (blue curve) is incident on a NIM (shaded region) from vacuum, and tuned to ω ˜ =0.7905. The choice γ̃=10-3 results in an attenuation depth of ~50µm. Most of the SH signal (red curve) is generated backward under nearly exact phase matching condition, at 2ω̃=1.581.

Fig. 3.
Fig. 3.

SH conversion efficiency (%) as a function of incident pulse duration, for γ̃=10-3 and γ̃=10-2, assuming χ (2)~80pm/V and a peak intensity of ~500MW/cm2. For γ̃=10-3, the attenuation depth is ~50µm. For γ̃=10-2, the attenuation depth is drastically reduced to ~5µm. Quasi-monochromatic pulses, i.e. pulse duration greater than 2.5ps, yield conversion efficiencies of ~12% and ~0.2%, respectively.

Fig. 4.
Fig. 4.

Fourier decomposition of the incident (kω ) and scattered pump pulses. The wave number is dimensionless. The transmitted wave packet (kω̃n(ω)) is a superposition of negative wave vectors, even though the pulse propagates in the forward direction. There is a small reflected component propagating backward in free space (-k ω ˜ ).

Fig. 5.
Fig. 5.

Fourier decomposition of the generated SH field. The wave number is dimensionless, as in Fig. 4. The positive component identifies a mismatched signal propagating forward inside the medium (k2 ω ˜ n(2 ω ˜ )), that is eventually absorbed away. The negative components correspond to a signal that has already emerged in vacuum, and has twice the incident wave vector (-k 2 ω ˜ ), and a signal traveling in the negative direction, inside the medium (-k2 ω ˜ n(2ω̃)), that will eventually exit into the vacuum.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

P NL = χ P ( 2 ) E E ; M NL = χ M ( 2 ) H H ,
E = x ̂ ( 𝔐 ω ( z , t ) e i ( k z ω 0 t ) + c . c + 𝔐 2 ω ( z , t ) e 2 i ( k z ω 0 t ) + c . c )
H = y ̂ ( 𝓗 ω ( z , t ) e i ( k z ω 0 t ) + c . c + 𝓗 2 ω ( z , t ) e 2 i ( k z ω 0 t ) + c . c ) ,
P NL = x ̂ ( ω ( z , t ) e i ( k z ω 0 t ) + c . c + 2 ω ( z , t ) e 2 i ( k z ω 0 t ) + c . c )
M NL = y ̂ ( ω ( z , t ) e i ( k z ω 0 t ) + c . c + 2 ω ( z , t ) e 2 i ( k z ω 0 t ) + c . c ) ,
α ω ˜ 𝓔 ω ˜ τ + i α ω ˜ 4 π 2 𝓔 ω ˜ τ 2 α " ω ˜ 24 π 2 3 𝓔 ω ˜ τ 3 + = i β ( ε ω ˜ , ξ 𝓗 ω ˜ 𝓗 ω ˜ ) 𝓗 ω ˜ ξ + 4 π ( i β 𝒫 ω ˜ 𝒫 ω ˜ τ )
γ ω ˜ H ω ˜ τ + i γ ' ω ˜ 4 π 2 𝓗 ω ˜ τ 2 γ " ω ˜ 24 π 2 3 𝓗 ω ˜ τ 3 + = i β ( μ ω ˜ , ξ 𝓗 ω ˜ 𝓔 ω ˜ ) 𝓔 ω ˜ ξ + 4 π ( i β 𝔐 ω ˜ 𝔐 ω ˜ τ )
α 2 ω ˜ 𝓔 2 ω ˜ τ + i 2 ω ˜ 4 π 2 𝓔 2 ω ˜ τ 2 α " 2 ω ˜ 24 π 2 3 𝓔 2 ω ˜ τ 3 = 2 i β ( ε 2 ω ˜ , ξ 𝓔 2 ω ˜ 𝓗 2 ω ˜ ) 𝓗 2 ω ˜ ξ
+ 4 π ( i 2 β 𝒫 2 ω ˜ 𝒫 2 ω ˜ τ )
γ 2 ω ˜ H 2 ω ˜ τ + i γ ' 2 ω ˜ 4 π 2 𝓗 2 ω ˜ τ 2 γ " 2 ω ˜ 24 π 2 3 𝓗 2 ω ˜ τ 3 = 2 i β ( μ 2 ω ˜ , ξ 𝓗 2 ω ˜ 𝓔 2 ω ˜ ) 𝓔 2 ω ˜ ξ
+ 4 π ( i 2 β 𝔐 2 ω ˜ 𝔐 2 ω ˜ τ )
α ω ˜ 𝓔 ω ˜ τ i β ( ε ω ˜ , ξ 𝓔 ω ˜ 𝓗 ω ˜ ) 𝓗 ω ˜ ξ + 4 π ( i β 𝒫 ω ˜ 𝒫 ω ˜ τ )
γ ω ˜ H ω ˜ τ i β ( μ ω ˜ , ξ 𝓗 ω ˜ 𝓔 ω ˜ ) 𝓔 ω ˜ ξ + 4 π ( i β 𝔐 ω ˜ 𝔐 ω ˜ τ )
α 2 ω ˜ 𝓔 2 ω ˜ τ 2 i β ( ε 2 ω ˜ , ξ 𝓔 2 ω ˜ 𝓗 2 ω ˜ ) 𝓗 2 ω ˜ ξ + 4 π ( i 2 β 𝒫 2 ω ˜ 𝒫 2 ω ˜ τ )
γ 2 ω ˜ H 2 ω ˜ τ 2 i β ( μ 2 ω ˜ , ξ 𝓗 2 ω ˜ 𝓔 2 ω ˜ ) 𝓔 2 ω ˜ ξ + 4 π ( i 2 β 𝔐 2 ω ˜ 𝔐 2 ω ˜ τ )

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