Abstract

We develop a generalized model for studying second-order parametric interactions in 1-D multilayered photonic structures, accounting for collinear oblique waves and partial pump depletion. This model is used to assess the performance of parametric devices in photonic-crystal microcavity (PCM) structures. Our model shows dramatic enhancement of nonlinear interactions at frequencies for which the waves are localized. Also, we demonstrate the exponential dependence of the conversion efficiency of second harmonic generation (SHG) on the number of layers as was recently pointed out. In addition, in optical parametric amplification (OPA), we find that the gain has a resonance-like dependence on the pump intensity, turning large peak gain into strong attenuation at greater intensities, which suggests that the device can operate as an optical switch.

© 2008 Optical Society of America

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  1. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures," Phys. Rev. A 56, 3166-3174 (1997).
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  23. W. E. Angerer, N. Yang, A. G. Yodh, M. A. Khan, and C. J. Sun, "Ultrafast second-harmonic generation spectroscopy of GaN thin films on sapphire," Phys. Rev. B 59, 2932-2946 (1999).
    [CrossRef]
  24. A. R. Cowan and J. F. Young, "Optical bistability involving photonic crystal microcavities and Fano line shapes," Phys. Rev. E 68, 046606 (2003).
    [CrossRef]
  25. M. Ghulinyan C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
    [CrossRef]

2007 (3)

2006 (1)

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]

2005 (2)

T. Ochiai and K. Sakoda, "Scaling law of enhanced second harmonic generation in finite Bragg stacks," Opt. Express 13, 9094-9114 (2005).
[CrossRef] [PubMed]

M. Centini, J. Peina, Jr., L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, "Entangled photon pair generation by spontaneous parametric down-conversion in finite-length one-dimensional photonic crystals," Phys. Rev. A 72, 033806 (2005).
[CrossRef]

2004 (4)

MSoljacic and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature (London) 3, 211-219 (2004).
[CrossRef]

M. Cherchi, "Exact analytic expressions for electromagnetic propagation and optical nonlinear generation in finite one-dimensional periodic multilayers," Phys. Rev. E 69, 066602 (2004).
[CrossRef]

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

M. Liscidinia and L. Andreani, "Highly efficient second-harmonic generation in doubly resonant planar microcavities," Appl. Phys. Lett. 85, 1883-1885 (2004).
[CrossRef]

2003 (2)

A. R. Cowan and J. F. Young, "Optical bistability involving photonic crystal microcavities and Fano line shapes," Phys. Rev. E 68, 046606 (2003).
[CrossRef]

M. Ghulinyan C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

2002 (3)

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and 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]

T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, A. A. Fedyanin, O. A. Aktsipetrov, G. Marowsky, V. A. Yakovlev, and G. Mattei, "Giant microcavity enhancement of second-harmonic generation in all-silicon photonic crystals," Appl. Phys. Lett. 81, 2725-2727 (2002).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Generalized coupled-mode theory for |(2) interactions in finite multilayered structures," J. Opt. Soc. Am. B 19, 2111-2121 (2002).
[CrossRef]

2001 (1)

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to Χ(2) interactions," Phys. Rev. E 64, 016609 (2001).
[CrossRef]

1999 (2)

Y. Jeong and B. Lee, "Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion," IEEE J. Quantum Electron. 35, 162-172 (1999).
[CrossRef]

W. E. Angerer, N. Yang, A. G. Yodh, M. A. Khan, and C. J. Sun, "Ultrafast second-harmonic generation spectroscopy of GaN thin films on sapphire," Phys. Rev. B 59, 2932-2946 (1999).
[CrossRef]

1998 (2)

J.W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimei, "Enhanced second-harmonic generation in media with a weak periodicity," Phys. Rev. A 57, 2120-2128 (1998).
[CrossRef]

A. Arraf and C. M. de Sterke, "Coupled-mode equations for quadratically nonlinear deep gratings," Phys. Rev. E 58, 7951-7958 (1998).
[CrossRef]

1997 (1)

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures," Phys. Rev. A 56, 3166-3174 (1997).
[CrossRef]

1996 (1)

1989 (1)

1988 (1)

Appl. Opt. (1)

Appl. Phys. Lett. (3)

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]

T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, A. A. Fedyanin, O. A. Aktsipetrov, G. Marowsky, V. A. Yakovlev, and G. Mattei, "Giant microcavity enhancement of second-harmonic generation in all-silicon photonic crystals," Appl. Phys. Lett. 81, 2725-2727 (2002).
[CrossRef]

M. Liscidinia and L. Andreani, "Highly efficient second-harmonic generation in doubly resonant planar microcavities," Appl. Phys. Lett. 85, 1883-1885 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Jeong and B. Lee, "Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion," IEEE J. Quantum Electron. 35, 162-172 (1999).
[CrossRef]

J. Appl. Phys. (1)

M. Ghulinyan C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

J. Lightwave Technol. (1)

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

J. Opt. Soc. Am. B (3)

Nature (London) (1)

MSoljacic and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature (London) 3, 211-219 (2004).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (4)

J.W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimei, "Enhanced second-harmonic generation in media with a weak periodicity," Phys. Rev. A 57, 2120-2128 (1998).
[CrossRef]

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Theory of spontaneous parametric downconversion from photonic crystals," Phys. Rev. A 70, 043810 (2004).
[CrossRef]

M. Centini, J. Peina, Jr., L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, "Entangled photon pair generation by spontaneous parametric down-conversion in finite-length one-dimensional photonic crystals," Phys. Rev. A 72, 033806 (2005).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, "Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures," Phys. Rev. A 56, 3166-3174 (1997).
[CrossRef]

Phys. Rev. B (1)

W. E. Angerer, N. Yang, A. G. Yodh, M. A. Khan, and C. J. Sun, "Ultrafast second-harmonic generation spectroscopy of GaN thin films on sapphire," Phys. Rev. B 59, 2932-2946 (1999).
[CrossRef]

Phys. Rev. E (5)

A. R. Cowan and J. F. Young, "Optical bistability involving photonic crystal microcavities and Fano line shapes," Phys. Rev. E 68, 046606 (2003).
[CrossRef]

M. Cherchi, "Exact analytic expressions for electromagnetic propagation and optical nonlinear generation in finite one-dimensional periodic multilayers," Phys. Rev. E 69, 066602 (2004).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, "Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method," Phys. Rev. E 75, 056606 (2007).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, "Photonic band edge effects in finite structures and applications to Χ(2) interactions," Phys. Rev. E 64, 016609 (2001).
[CrossRef]

A. Arraf and C. M. de Sterke, "Coupled-mode equations for quadratically nonlinear deep gratings," Phys. Rev. E 58, 7951-7958 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and 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]

Other (1)

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 2007).

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Figures (6)

Fig. 1.
Fig. 1.

An N-layer photonic structure. The mth layer has refractive index nm , second-order nonlinear coefficient dm and thickness lm . The overall length is L. The forward waves are incident on the mth interface at an oblique angle θm . There are two sets of waves in the forward and backward directions: pump (p), signal (s) and idler (i). The superscripts + and - denote the forward and backward waves, respectively. The complex envelope at the left and right sides of the mth element are A m-1 and Am , respectively. E 0 and EN are the complex amplitudes at the left and right sides of the structure, respectively.

Fig. 2.
Fig. 2.

(a) A 1-D photonic-crystal microcavity structure made up of two alternating layers: a GaN layer with thickness l 1, refractive index n 1 and nonlinear coefficient d 1=10-22 F/V; and an air layer with thickness l 2, n 2=1 and d 2=0. The crystal is made up of a quarter wave stack, with 1.5µm as the reference wavelength. (b) Spectral characteristics of linear transmission through 50-layers structure (ωB is the Bragg frequency).

Fig. 3.
Fig. 3.

Dependence of the total SHG conversion efficiency η ++η - on: (a) the pump intensity I 0 ( p + ) , and (b) the number of layers N. The straight (dotted) lines represent values computed under the iterative perturbative technique (undepleted pump approximation).

Fig. 4.
Fig. 4.

(Color online). Dependence of the forward and backward gain on the signal and pump frequencies in units of ωB . The dashed lines represent frequencies- and band-boundaries. The bright spots represent values of gain. The pump and signal intensities are 50 kW/cm2 and 0.1 kW/cm2, respectively, and the number of layers is 70. The forward and backward gain are enhanced at ωp =ωB , ωs =ωB and ωi =ωB .

Fig. 5.
Fig. 5.

Dependence of the forward and backward gain on the pump intensity I 0 ( p + ) for three values of the number of layers N. (a) ωs =ωB , ωp =1.3ωB . (b) ωs =0.5ωB , ωp =1.5ωB . (c) ωs =0.6ωB , ωp =ωB .

Fig. 6.
Fig. 6.

(Color online). Dependence of the forward and backward conversion efficiencies on the signal and pump frequencies in units of ωB . The dashed lines represent bandboundaries. The bright spots represent values of conversion efficiencies. The pump and signal intensities are 50 kW/cm2 and 0.1 kW/cm2, respectively, and the number of layers is 70. The forward and backward conversion efficiencies are enhanced at ωp =ωB , ωs =ωB and ωi =ωB .

Equations (52)

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T m = P m 1 B m P m C m ,
B m ( p ) = [ B m ( p + , p + ) B m ( p + , p ) B m ( p , p + ) B m ( p , p ) ] , B m ( s , i ) = [ B m ( s + , s + ) B m ( s + , s ) 0 0 B m ( s , s + ) B m ( s , s ) 0 0 0 0 B m ( i + , i + ) B m ( i + , i ) 0 0 B m ( i , i + ) B m ( i , i ) ] ,
B m ( q ± , q ± ) = { ( n m ( q ) cos θ m ( q ) + n m 1 ( q ) cos θ m 1 ( q ) ) 2 n m ( q ) cos θ m ( q )     TE polarization ( n m ( q ) sec θ m ( q ) + n m 1 ( q ) sec θ m 1 ( q ) ) cos θ m 1 ( q ) 2 n m ( q ) TM polarization ,
B m ( q ± , q ) = { ( n m ( q ) cos θ m ( q ) - n m 1 ( q ) cos θ m 1 ( q ) ) 2 n m ( q ) cos θ m ( q )     TE polarization ( n m ( q ) sec θ m ( q ) - n m 1 ( q ) sec θ m 1 ( q ) ) cos θ m 1 ( q ) 2 n m ( q ) TM polarization ,
n m ( q ) sin θ m ( q ) = n m 1 ( q ) sin θ m 1 ( q ) = n 0 ( q ) sin θ 0 ( q ) , q = p , s , i ,
P m ( p ) = [ Φ m ( p + ) Φ m ( p ) ] I 2 × 2 , P m ( s , i ) = [ Φ m ( s + ) Φ m ( s ) Φ m ( i + ) Φ m ( i ) ] I 4 × 4 ,
Φ m ( q ± ) = exp ( j 1 c k = 1 k = m n k ( q ) ω q l k cos θ k ) ,   q = p , s , i ,
d a m ( u ± ) d z = j g m ( u ) a m ( w ± ) a m ( v ± ) * exp ( j Δ k m z ) ,
d a m ( v ± ) d z = j g m ( v ) a m ( w ± ) a m ( u ± ) * exp ( j Δ k m z ) ,
d a m ( w ± ) d z = j g m ( w ) a m ( u ± ) a m ( v ± ) exp ( ± j Δ k m z ) ,
Δ k m = 1 c ( n m ( w ) ω w n m ( u ) ω u n m ( v ) ω v ) , g m ( q ) = 120 π d m ω q n m ( q ) , u , v , w , q = p , s , i ,
C m ( p ) = [ 1 0 0 1 ] , C m ( s , i ) = [ C m ( s + , s + ) 0 C m ( s + , i + ) 0 0 C m ( s , s ) 0 C m ( s , i ) C m ( i + , s + ) 0 C m ( i + , i + ) 0 0 C m ( i , s ) 0 C m ( i , i ) ] ,
A m ( q ) = T m ( q ) T m 1 ( q ) T 2 ( q ) T 1 ( q ) B 0 ( q ) E 0 ( q ) , { q = p q = s , i ,
E 0 ( p ) = [ E 0 ( p + ) E 0 ( p ) ] t , E 0 ( s , i ) = [ E 0 ( s + ) E 0 ( s ) E 0 ( i + ) E 0 ( i ) ] t ,
A m ( p ) = [ a m ( p + ) a m ( p ) ] t , A m ( s , i ) = [ a m ( s + ) a m ( s ) a m ( i + ) a m ( i ) ] t .
A m ( p ) = T m ( p ) T m 1 ( p ) T 2 ( p ) T 1 ( p ) B 0 ( p ) E 0 ( p ) + X m ( p ) ,
X m ( p ) = k = 1 m T m ( p ) T m 1 ( p ) T k ( p ) [ x k ( p + ) x k ( p ) ]
Δ = I output I input I input ,
n 1 ( ω B ) l 1 cos θ 1 = n 2 ( ω B ) l 2 cos θ 2 = λ B 4 = π c 2 ω B ,
d a k ( p ± ) d z = j g k ( p ) a k ( s ± ) a k ( p ± ) * exp ( j Δ k k z ) ,
d a k ( s ± ) d z = j 1 2 g k ( s ) ( a k ( p ± ) ) 2 exp ( ± j Δ k k z ) ,
a k ( s ± ) ( z k ) = a k ( s ± ) ( z k 1 ) + x k ( s ± ) ,
x k ( s ± ) = j α k ( s ± ) sinc ( Δ k k l k 2 ) exp [ ± j Δ k k ( l k + 2 z k 1 ) 2 ] ,   α k ( s ± ) = 1 2 g k ( s ) l k ( a ¯ k ( p ± ) ) 2 ,
a k ( p ± ) ( z k ) = a k ( p ± ) ( z k 1 ) + x k ( p ± ) ,
x k ( p ± ) = j [ α k ( p ± ) sinc ( Δ k k l k 2 ) exp ( j Δ k k ( l k + 2 z k 1 ) 2 ) + β k ( p ± ) ] ,
α k ( p ± ) = g k ( p ) a ¯ k ( p ± ) * [ l k a k ( s ± ) ( z k 1 ) + α k ( s ± ) Δ k k exp ( ± j Δ k k z k 1 ) ] ,
β k ( p ± ) = g k ( p ) a ¯ k ( p ± ) * α k ( s ± ) Δ k k .
η + = I N ( s + ) I 0 ( p + ) , η = I 0 ( s ) I 0 ( p + ) ,
η + + η ζ I 0 ( p + ) exp ( N φ ) ,
C k ( s ± , s ± ) = exp ( σ k l k ) [ cosh ( γ k ± l k ) ± σ k γ k ± sinh ( γ k ± l k ) ] ,
C k ( s ± , i ± ) = j g k ( s ) a ¯ k ( p ± ) γ k ± exp [ σ k ( l k + 2 z k 1 ) ] sinh ( γ k ± l k ) ,
C k ( i ± , s ± ) = ± j g k ( i ) a ¯ k ( p ± ) * γ k ± exp [ ± σ k ( l k + 2 z k 1 ) ] sinh ( γ k ± l k ) ,
C k ( i ± , i ± ) = exp ( ± σ k l k ) [ cosh ( γ k ± l k ) σ k γ k ± sinh ( γ k ± l k ) ] ,
γ k ± = ( σ k 2 + g k ( s ) g k ( i ) a ¯ k p ± 2 ) 1 2 , σ k = j Δ k k 2 .
P k ( s , i ) = [ Φ k ( s + ) Φ k ( s ) Φ k ( i + ) * Φ k ( i ) * ] I 4 × 4 .
a k ( p ± ) ( z k ) = a k ( p ± ) ( z k 1 ) + x k ( p ± ) ,
α k ± = j ( g k ( i ) a ¯ k ( p ± ) * γ k ± * a k ( s ± ) 2 + g k ( s ) a ¯ k ( p ± ) γ k ± a k ( i ± ) 2 ) ± ρ k ± a k ( s ± ) a k ( i ± ) ( σ k γ k ± σ k * γ k ± * ) ,
β k ± = ± j ( g k ( i ) a ¯ k ( p ± ) * γ k ± * a k ( s ± ) 2 g k ( s ) a ¯ k ( p ± ) γ k ± a k ( i ± ) 2 ) ± ρ k ± a k ( s ± ) a k ( i ± ) ( σ k γ k ± + σ k * γ k ± * ) ,
ξ k ± = 1 γ k ± 2 [ ρ k ± a k ( s ± ) a k ( i ± ) ( γ k ± 2 σ k 2 ) ρ k ± * g k ( s ) g k ( i ) a k ( s ± ) * a k ( i ± ) * a ¯ k ( p ± ) 2
j g k ( i ) σ k a ¯ k ( p ± ) * a k ( s ± ) 2 + j g k ( s ) σ k * a ¯ k ( p ± ) a k ( i ± ) 2 ] ,
δ k ± = ρ k ± a k ( s ± ) a k ( i ± ) ξ k ± , ρ k ± = exp ( ± j Δ k k z k 1 ) ,
G + = I N ( s + ) I 0 ( s + ) , G = I 0 ( s ) I 0 ( s + ) ,
G + ( 1 1 I 0 ( p + ) I th ) 2 ,
C k ( s ± , i ± ) = j g k ( s ) a ¯ k ( p ± ) * γ k ± exp [ σ k ( l k + 2 z k 1 ) ] sinh ( γ k ± l k ) ,
C k ( i ± , s ± ) = j g k ( i ) a ¯ k ( p ± ) γ k ± exp [ ± σ k ( l k + 2 z k 1 ) ] sinh ( γ k ± l k ) ,
γ k ± = ( σ k 2 g k ( s ) g k ( i ) a ¯ k ( p ± ) 2 ) 1 2 .
α k ± = j ( g k ( i ) a ¯ k ( p ± ) γ k ± a k ( s ± ) 2 g k ( s ) a ¯ k ( p ± ) * γ k ± * a k ( i ± ) 2 ) ρ k ± * a k ( s ± ) * a k ( i ± ) ( σ k γ k ± σ k * γ k ± * ) ,
β k ± = j ( g k ( i ) a ¯ k ( p ± ) γ k ± a k ( s ± ) 2 + g k ( s ) a ¯ k ( p ± ) * γ k ± * a k ( i ± ) 2 ) ρ k ± * a k ( s ± ) * a k ( i ± ) ( σ k γ k ± + σ k * γ k ± * ) ,
ξ k ± = 1 γ k ± 2 [ ρ k ± * a k ( s ± ) * a k ( i ± ) ( γ k ± 2 σ k 2 ) + ρ k ± g k ( s ) g k ( i ) a k ( s ± ) a k ( i ± ) * a ¯ k ( p ± ) 2
j g k ( i ) σ k * a ¯ k ( p ± ) a k ( s ± ) 2 j g k ( s ) σ k a ¯ k ( p ± ) * a k ( i ± ) 2 ] ,
δ k ± = ρ k ± * a k ( s ± ) * a k ( i ± ) ξ k ± .
η + = I N ( i + ) I 0 ( s + ) , η = I 0 ( i ) I 0 ( s + ) ,

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