Abstract

One-dimensional defective photonic crystals containing nonlinear material for coupled third harmonic generation (CTHG) have been designed. The general solution of the CTHG in such structure has been derived. The wavelengths of the fundamental wave (FW), second harmonic generation (SHG), and CTHG have been designed to lie at the defect states in different photonic band gaps by employing the simulated annealing (SA) method. Due to the strong location, low group velocity, and spatial phase locking, the conversion efficiencies of the SHG and CTHG have been greatly enhanced. Designed structure for multiple frequencies CTHG is also demonstrated.

© 2007 Optical Society of America

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References

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. F. Omenetto, A. Efimov, A. Taylor, J. Knight, W. Wadsworth, and P. Russell, "Polarization dependent harmonic generation in microstructured fibers," Opt. Express 11, 61-67 (2003).
    [CrossRef] [PubMed]
  3. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  4. S. Somekh and A. Yariv, "Phase matching by periodic modulation of the nonlinear optical properties," Opt. Commun. 6, 301-304 (1972).
    [CrossRef]
  5. 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. 56, 3166 -3174 (1997).
    [CrossRef]
  6. M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
    [CrossRef]
  7. F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
    [CrossRef]
  8. L. M. Zhao and B. Y. Gu, "Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material," Appl. Phys. Lett. 88, 122904 (2006).
    [CrossRef]
  9. L. M. Zhao and B. Y. Gu, "Enhanced second-harmonic generation for multiple wavelengths by defect modes in one-dimensional photonic crystals," Opt. Lett. 31, 1510-1512 (2006).
    [CrossRef] [PubMed]
  10. D. S. Smith and H. D. Riccius, "Refractive indices of lithium niobate," Opt. Commun. 17, 332-335 (1976).
    [CrossRef]

2006

L. M. Zhao and B. Y. Gu, "Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material," Appl. Phys. Lett. 88, 122904 (2006).
[CrossRef]

L. M. Zhao and B. Y. Gu, "Enhanced second-harmonic generation for multiple wavelengths by defect modes in one-dimensional photonic crystals," Opt. Lett. 31, 1510-1512 (2006).
[CrossRef] [PubMed]

2004

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

2003

2001

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

1997

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. 56, 3166 -3174 (1997).
[CrossRef]

1987

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

1976

D. S. Smith and H. D. Riccius, "Refractive indices of lithium niobate," Opt. Commun. 17, 332-335 (1976).
[CrossRef]

1972

S. Somekh and A. Yariv, "Phase matching by periodic modulation of the nonlinear optical properties," Opt. Commun. 6, 301-304 (1972).
[CrossRef]

Bertolotti, M.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

Bloemer, M. J.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

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. 56, 3166 -3174 (1997).
[CrossRef]

Bowden, C. M.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

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. 56, 3166 -3174 (1997).
[CrossRef]

Centini, M.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

Cheng, C.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

D’Aguanno, G.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

Dowling, J. P.

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. 56, 3166 -3174 (1997).
[CrossRef]

Efimov, A.

Gu, B. Y.

L. M. Zhao and B. Y. Gu, "Enhanced second-harmonic generation for multiple wavelengths by defect modes in one-dimensional photonic crystals," Opt. Lett. 31, 1510-1512 (2006).
[CrossRef] [PubMed]

L. M. Zhao and B. Y. Gu, "Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material," Appl. Phys. Lett. 88, 122904 (2006).
[CrossRef]

Haus, J. W.

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. 56, 3166 -3174 (1997).
[CrossRef]

Hirao, K.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

Knight, J.

Li, R.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

Manka, A. S.

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. 56, 3166 -3174 (1997).
[CrossRef]

Omenetto, F.

Qiu, J. R.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

Ren, F. F.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

Riccius, H. D.

D. S. Smith and H. D. Riccius, "Refractive indices of lithium niobate," Opt. Commun. 17, 332-335 (1976).
[CrossRef]

Russell, P.

Scalora, M.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

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. 56, 3166 -3174 (1997).
[CrossRef]

Si, J. H.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

Sibilia, C.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

Smith, D. S.

D. S. Smith and H. D. Riccius, "Refractive indices of lithium niobate," Opt. Commun. 17, 332-335 (1976).
[CrossRef]

Somekh, S.

S. Somekh and A. Yariv, "Phase matching by periodic modulation of the nonlinear optical properties," Opt. Commun. 6, 301-304 (1972).
[CrossRef]

Taylor, A.

Viswanathan, R.

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. 56, 3166 -3174 (1997).
[CrossRef]

Wadsworth, W.

Wang, H. T.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

S. Somekh and A. Yariv, "Phase matching by periodic modulation of the nonlinear optical properties," Opt. Commun. 6, 301-304 (1972).
[CrossRef]

Zhao, L. M.

L. M. Zhao and B. Y. Gu, "Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material," Appl. Phys. Lett. 88, 122904 (2006).
[CrossRef]

L. M. Zhao and B. Y. Gu, "Enhanced second-harmonic generation for multiple wavelengths by defect modes in one-dimensional photonic crystals," Opt. Lett. 31, 1510-1512 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett.

L. M. Zhao and B. Y. Gu, "Giant enhancement of second harmonic generation in multiple photonic quantum well structures made of nonlinear material," Appl. Phys. Lett. 88, 122904 (2006).
[CrossRef]

Opt. Commun.

D. S. Smith and H. D. Riccius, "Refractive indices of lithium niobate," Opt. Commun. 17, 332-335 (1976).
[CrossRef]

S. Somekh and A. Yariv, "Phase matching by periodic modulation of the nonlinear optical properties," Opt. Commun. 6, 301-304 (1972).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A.

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. 56, 3166 -3174 (1997).
[CrossRef]

Phys. Rev. B.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, "Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized mode," Phys. Rev. B. 70, 245109 (2004).
[CrossRef]

Phys. Rev. E.

M. Centini, G. D’Aguanno, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Simultaneously phase-matched enhanced second and third harmonic generation," Phys. Rev. E. 64, 046606 (2001).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Other

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

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Figures (5)

Fig. 1.
Fig. 1.

The designed structure constituted with layers C and D between the band structure of the PC (AB)10, the C with black color and D with white color.

Fig. 2.
Fig. 2.

Transmission spectrum of the designed photonic quantum well. Three dashed lines represent wavelengths λ 1,1 = 1.500μm, λ 2,1= 0.750μm, and λ 3,1 = 0.500μm, respectively.

Fig. 3.
Fig. 3.

Conversion efficiency of the photonic quantum well for (a) forward SHG and (b) forward CTHG.

Fig. 4.
Fig. 4.

Transmission spectrum of the designed photonic quantum well structure for double wavelengths CTHG. The dashed lines represent λ 1,1 = 1.458μm, λ 1,2 =1.578μm, λ 2,1 = 0.729μm, λ 2,2 = 0.789μm, λ 3,1 = 0.486μm, and λ 3,2 = 0.526μm, respectively.

Fig. 5.
Fig. 5.

Conversion efficiency of the forward CTHG.

Equations (22)

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[ d 2 d z 2 + k l ( 1 ) 2 ] E l ( 1 ) ( z ) = 0 ,
[ d 2 d z 2 + k l ( 2 ) 2 ] E l ( 2 ) ( z ) = k 20 ( 2 ) 2 χ l ( z ) E l ( 1 ) 2 ( z ) ,
[ d 2 d z 2 + k l ( 3 ) 2 ] E l ( 3 ) ( z ) = 2 k 30 2 χ l ( z ) E l ( 1 ) ( z ) E l ( 2 ) ( z ) ,
E l ( 1 ) ( z ) = A l ( 1 ) e i k l ( 1 ) ( z z l 1 ) + B l ( 1 ) e i k l ( 1 ) ( z z l 1 ) ,
E l ( 2 ) ( z ) = A l ( 2 ) e i k l ( 2 ) ( z z l 1 ) + B l ( 2 ) e i k l ( 2 ) ( z z l 1 ) + C 21 e i 2 k l ( 1 ) ( z z l 1 ) + C 22 e i 2 k l ( 1 ) ( z z l 1 ) 2 k 20 2 χ l k l ( 2 ) 2 A l ( 1 ) B l ( 1 ) ,
C 21 = k 20 2 χ l A l ( 1 ) 2 k l ( 2 ) 2 4 k l ( 1 ) 2 , C 22 = k 20 2 χ l B l ( 1 ) 2 k l ( 2 ) 2 4 k l ( 1 ) 2 .
E l ( 3 ) ( x ) = A l ( 3 ) e i k l ( 3 ) x + B l ( 3 ) e i k l ( 3 ) x + C l , 31 e i ( k l ( 1 ) + k l ( 2 ) ) x + C l , 32 e i ( k l ( 1 ) + k l ( 2 ) ) x
+ D l , 31 e i ( k l ( 1 ) k l ( 2 ) ) x + D l , 32 e i ( k l ( 1 ) k l ( 2 ) ) x + E l , 31 e i 3 k l ( 1 ) x + E l , 32 e i 3 k l ( 1 ) x
+ F l , 31 e i k l ( 1 ) x + F l , 32 e i k l ( 1 ) x ,
C l , 31 = k 30 2 χ l A l ( 1 ) A l ( 2 ) k l ( 3 ) 2 ( k l ( 1 ) + k l ( 2 ) ) 2 , C l , 32 = k 30 2 χ l B l ( 1 ) B l ( 2 ) k l ( 3 ) 2 ( k l ( 1 ) + k l ( 2 ) ) 2 ,
D l , 31 = k 30 2 χ l A l ( 1 ) B l ( 2 ) k l ( 3 ) 2 ( k l ( 1 ) k l ( 2 ) ) 2 , D l , 32 = k 30 2 χ l A l ( 2 ) B l ( 1 ) k l ( 3 ) 2 ( k l ( 1 ) k l ( 2 ) ) 2 ,
E l , 31 = k 30 2 χ l A l ( 1 ) C l , 21 k l ( 3 ) 2 9 k l ( 1 ) 2 , E l , 32 = k 30 2 χ l B l ( 1 ) C l , 22 k l ( 3 ) 2 9 k l ( 1 ) 2 ,
F l , 31 = k 30 2 χ l B l ( 1 ) C l , 21 k l ( 3 ) 2 k l ( 1 ) 2 + 2 k 30 2 k 20 2 χ l ( 2 ) χ l A l ( 1 ) 2 B l ( 1 ) k l ( 2 ) 2 ( k l ( 3 ) 2 k l ( 1 ) 2 ) ,
F l , 32 = k 30 2 χ l A l ( 1 ) C l , 22 k l ( 3 ) 2 k l ( 1 ) 2 + 2 k 30 2 k 20 2 χ l ( 2 ) χ l B l ( 1 ) 2 A l ( 1 ) k l ( 2 ) 2 ( k l ( 3 ) 2 k l ( 1 ) 2 ) .
η forth ( 2 ) = n N ( 2 ) A N ( 2 ) 2 n 1 ( 2 ) A 1 ( 1 ) 2 , η back ( 2 ) = B 1 ( 2 ) 2 A 1 ( 1 ) 2 ( for SHG ) ,
η forth ( 3 ) = n N ( 3 ) A N ( 3 ) 2 n 1 ( 3 ) A 1 ( 1 ) 2 , η back ( 3 ) = B 1 ( 3 ) 2 A 1 ( 1 ) 2 ( for CTHG ) ,
O = α s k [ λ α , s o λ α , s ( k ) + β 1 η o η 1 , s ( k ) ] + β 2 max ( { η 1 , s ( k ) } ) min ( { η 1 , s ( k ) } )
λ 1 , s o , λ 1 , s ( k ) [ λ 1 , s 1 , λ 1 , s ] [ λ a , λ b ] , λ 2 , s o , λ 2 , s ( k ) [ λ 2 , s 1 , λ 2 , s ] [ λ c , λ d ] ,
λ 3 , s o , λ 3 , s ( k ) [ λ 3 , s 1 , λ 3 , s ] [ λ e , λ f ] , α = 1,2,3 ; s = 1,2,3
λ 1,0 ( = λ a ) < λ 1,1 o < λ 1,1 < λ 1,2 o < λ 1,2 < < λ 1 , s o < λ 1 , s ( = λ b ) ;
λ 2,0 ( = λ c ) < λ 2,1 o < λ 2,1 < λ 2,2 o < λ 2,2 < < λ 2 , s o < λ 2 , s ( = λ d ) ;
λ 3,0 ( = λ e ) < λ 3,1 o < λ 3,1 < λ 3,2 o < λ 3,2 < < λ 3 , s o < λ 3 , s ( = λ f ) ,

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