Abstract

We present an analysis of amplification and lasing in one-dimensional isotropic nonlinear photonic crystal (1D PC), which is based on a generalized (multiwavelength) transfer matrix method. This approach was used for modeling a Raman signal amplification in 1D PC and in an homogenous structure, showing advantages of a stratified medium. Moreover, the threshold operation of a 1D PC Raman laser is studied, assuming both strong as well as depleted pump. The normalized threshold gain characteristics for various end reflections and photonic crystal laser length were calculated.

© 2009 Optical Society of America

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  1. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation(Wiley-Interscience, 2002).
  2. 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]
  3. B. G. Kim and E. Garmire, “Comparison between the matrix method and the coupled-wave method in the analysis of Bragg reflector structures,” J. Opt. Soc. Am. A 9, 132-136 (1992).
    [CrossRef]
  4. S. K. Srivastava and S. P. Ojha, “Reflection and anomalous behavior of refractive index in defect photonic band gap structure,” Microw. Opt. Technol. Lett. 38, 293-297 (2003).
    [CrossRef]
  5. P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. A 67, 423-438 (1977).
    [CrossRef]
  6. M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, S184 (2001).
    [CrossRef]
  7. J. M. Bendickson, J. P. Dowling, and M. Scarola, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107 (1996).
    [CrossRef]
  8. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910-916 (1989).
    [CrossRef]
  9. 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]
  10. M. F. Saleh, L. D. Negro, and B. E. A. Saleh, “Second-order parametric interactions in 1-D photonic-crystal microcavity structures,” Opt. Express 16, 5261-5276 (2008).
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  11. P. Szczepański, T. Osuch, Z. Jaroszewicz, and M. Buryk, ““Color” transfer matrix method in nonlinear medium,” in Frontier in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing on CD-ROM (Optical Society of America, 2008), paper JWA54.
    [PubMed]
  12. L. Florescu and X. Zhang, “Semiclassical model of stimulated Raman scattering in photonic crystals,” Phys. Rev. E 72, 016611 (2005).
    [CrossRef]
  13. R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer, 2004).
  14. T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343 (2004).
    [CrossRef]
  15. T. K. Liang, L. R. Nunes, H. K. Tsang, and M. Tsuchiya, “Theoretical analysis of continuous-wave Raman gain/lasing in silicon wire waveguides without carrier extraction scheme,” IEICE Electronics Express 2, 440-445 (2005).
    [CrossRef]
  16. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261-4268 (2004).
    [CrossRef] [PubMed]
  17. H. Ghafouri-Shiraz, Distributed Feedback Laser Diodes and Optical Tunable Filters (Wiley, 2003).
    [CrossRef]

2008 (1)

2007 (1)

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]

2005 (2)

L. Florescu and X. Zhang, “Semiclassical model of stimulated Raman scattering in photonic crystals,” Phys. Rev. E 72, 016611 (2005).
[CrossRef]

T. K. Liang, L. R. Nunes, H. K. Tsang, and M. Tsuchiya, “Theoretical analysis of continuous-wave Raman gain/lasing in silicon wire waveguides without carrier extraction scheme,” IEICE Electronics Express 2, 440-445 (2005).
[CrossRef]

2004 (2)

2003 (1)

S. K. Srivastava and S. P. Ojha, “Reflection and anomalous behavior of refractive index in defect photonic band gap structure,” Microw. Opt. Technol. Lett. 38, 293-297 (2003).
[CrossRef]

2001 (1)

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, S184 (2001).
[CrossRef]

1999 (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]

1996 (1)

J. M. Bendickson, J. P. Dowling, and M. Scarola, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107 (1996).
[CrossRef]

1992 (1)

1989 (1)

1977 (1)

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. A 67, 423-438 (1977).
[CrossRef]

Bayindir, M.

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, S184 (2001).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scarola, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107 (1996).
[CrossRef]

Bethune, D. S.

Buryk, M.

P. Szczepański, T. Osuch, Z. Jaroszewicz, and M. Buryk, ““Color” transfer matrix method in nonlinear medium,” in Frontier in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing on CD-ROM (Optical Society of America, 2008), paper JWA54.
[PubMed]

Cohen, O.

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scarola, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107 (1996).
[CrossRef]

Eggleton, B. J.

R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer, 2004).

Florescu, L.

L. Florescu and X. Zhang, “Semiclassical model of stimulated Raman scattering in photonic crystals,” Phys. Rev. E 72, 016611 (2005).
[CrossRef]

Garmire, E.

Ghafouri-Shiraz, H.

H. Ghafouri-Shiraz, Distributed Feedback Laser Diodes and Optical Tunable Filters (Wiley, 2003).
[CrossRef]

Hak, D.

Hong, C. S.

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. A 67, 423-438 (1977).
[CrossRef]

Jaroszewicz, Z.

P. Szczepański, T. Osuch, Z. Jaroszewicz, and M. Buryk, ““Color” transfer matrix method in nonlinear medium,” in Frontier in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing on CD-ROM (Optical Society of America, 2008), paper JWA54.
[PubMed]

Jeong, Y.

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]

Kim, B. G.

Kural, C.

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, S184 (2001).
[CrossRef]

Lee, B.

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]

Li, J. J.

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]

Li, Z. Y.

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]

Liang, T. K.

T. K. Liang, L. R. Nunes, H. K. Tsang, and M. Tsuchiya, “Theoretical analysis of continuous-wave Raman gain/lasing in silicon wire waveguides without carrier extraction scheme,” IEICE Electronics Express 2, 440-445 (2005).
[CrossRef]

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343 (2004).
[CrossRef]

Liu, A.

Negro, L. D.

Nunes, L. R.

T. K. Liang, L. R. Nunes, H. K. Tsang, and M. Tsuchiya, “Theoretical analysis of continuous-wave Raman gain/lasing in silicon wire waveguides without carrier extraction scheme,” IEICE Electronics Express 2, 440-445 (2005).
[CrossRef]

Ojha, S. P.

S. K. Srivastava and S. P. Ojha, “Reflection and anomalous behavior of refractive index in defect photonic band gap structure,” Microw. Opt. Technol. Lett. 38, 293-297 (2003).
[CrossRef]

Osuch, T.

P. Szczepański, T. Osuch, Z. Jaroszewicz, and M. Buryk, ““Color” transfer matrix method in nonlinear medium,” in Frontier in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing on CD-ROM (Optical Society of America, 2008), paper JWA54.
[PubMed]

Ozbay, E.

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, S184 (2001).
[CrossRef]

Paniccia, M.

Rong, H.

Saleh, B. E. A.

Saleh, M. F.

Scarola, M.

J. M. Bendickson, J. P. Dowling, and M. Scarola, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107 (1996).
[CrossRef]

Slusher, R. E.

R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer, 2004).

Srivastava, S. K.

S. K. Srivastava and S. P. Ojha, “Reflection and anomalous behavior of refractive index in defect photonic band gap structure,” Microw. Opt. Technol. Lett. 38, 293-297 (2003).
[CrossRef]

Szczepanski, P.

P. Szczepański, T. Osuch, Z. Jaroszewicz, and M. Buryk, ““Color” transfer matrix method in nonlinear medium,” in Frontier in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing on CD-ROM (Optical Society of America, 2008), paper JWA54.
[PubMed]

Tsang, H. K.

T. K. Liang, L. R. Nunes, H. K. Tsang, and M. Tsuchiya, “Theoretical analysis of continuous-wave Raman gain/lasing in silicon wire waveguides without carrier extraction scheme,” IEICE Electronics Express 2, 440-445 (2005).
[CrossRef]

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343 (2004).
[CrossRef]

Tsuchiya, M.

T. K. Liang, L. R. Nunes, H. K. Tsang, and M. Tsuchiya, “Theoretical analysis of continuous-wave Raman gain/lasing in silicon wire waveguides without carrier extraction scheme,” IEICE Electronics Express 2, 440-445 (2005).
[CrossRef]

Yariv, A.

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. A 67, 423-438 (1977).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation(Wiley-Interscience, 2002).

Yeh, P.

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. A 67, 423-438 (1977).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation(Wiley-Interscience, 2002).

Zhang, D. Z.

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]

Zhang, X.

L. Florescu and X. Zhang, “Semiclassical model of stimulated Raman scattering in photonic crystals,” Phys. Rev. E 72, 016611 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343 (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]

IEICE Electronics Express (1)

T. K. Liang, L. R. Nunes, H. K. Tsang, and M. Tsuchiya, “Theoretical analysis of continuous-wave Raman gain/lasing in silicon wire waveguides without carrier extraction scheme,” IEICE Electronics Express 2, 440-445 (2005).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one-dimensional photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, S184 (2001).
[CrossRef]

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

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. A 67, 423-438 (1977).
[CrossRef]

B. G. Kim and E. Garmire, “Comparison between the matrix method and the coupled-wave method in the analysis of Bragg reflector structures,” J. Opt. Soc. Am. A 9, 132-136 (1992).
[CrossRef]

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

Microw. Opt. Technol. Lett. (1)

S. K. Srivastava and S. P. Ojha, “Reflection and anomalous behavior of refractive index in defect photonic band gap structure,” Microw. Opt. Technol. Lett. 38, 293-297 (2003).
[CrossRef]

Opt. Express (2)

Phys. Rev. E (3)

L. Florescu and X. Zhang, “Semiclassical model of stimulated Raman scattering in photonic crystals,” Phys. Rev. E 72, 016611 (2005).
[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]

J. M. Bendickson, J. P. Dowling, and M. Scarola, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107 (1996).
[CrossRef]

Other (4)

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation(Wiley-Interscience, 2002).

R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer, 2004).

P. Szczepański, T. Osuch, Z. Jaroszewicz, and M. Buryk, ““Color” transfer matrix method in nonlinear medium,” in Frontier in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing on CD-ROM (Optical Society of America, 2008), paper JWA54.
[PubMed]

H. Ghafouri-Shiraz, Distributed Feedback Laser Diodes and Optical Tunable Filters (Wiley, 2003).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Propagation of M wavelengths through the multilayer structure.

Fig. 2
Fig. 2

Relative gain coefficient of Raman signal versus number of PC periods for various small signal gain G 0 = 10 , 20 , 50 , and 100 cm 1 at saturation parameter I S = 1 and normalized pump field amplitude a ω P = 1 .

Fig. 3
Fig. 3

Relative gain coefficient of Raman signal versus number of PC periods for various saturation parameter I S = 0.1 , 1 , 10 at small signal gain G 0 = 50 cm 1 and pump field amplitude a ω P = 1 .

Fig. 4
Fig. 4

Gain characteristics of pump and Raman signals versus number of PC periods for various pump field amplitude a ω P = 0.5 , 1 , 2 at small signal gain G 0 = 50 cm 1 and saturation param eter I S = 1 .

Fig. 5
Fig. 5

Gain characteristics of pump and Raman signals versus number of PC periods with and without gain saturation assumption at small signal gain G 0 = 30 cm 1 and pump field amplitude a ω P = 1 : a) Raman signal without pump saturation; b) Raman signal with pump saturation; c) pump with gain saturation; d) pump without saturation.

Fig. 6
Fig. 6

Raman gain as a function of the pump intensity for various length of homogenous structure. N = 20 , 50 , 100 means that the longitudinal size of the medium is equal to the length of the photonic crystal consisting 20, 50, or 100 unit cell correspondingly.

Fig. 7
Fig. 7

Raman gain as a function of the pump intensity for various number of PC length (number of periods).

Fig. 8
Fig. 8

Characteristics of threshold gain as a function of wavelength for various PC length (number of periods).

Fig. 9
Fig. 9

Characteristics of threshold gain as a function of wavelength for various reflection coefficients of laser mirrors.

Equations (14)

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( a ω 1 n 1 b ω 1 n 1 a ω M n 1 b ω M n 1 ) = ( A ω 1 n ( I ) B ω 1 n ( I ) 0 0 C ω 1 n ( I ) D ω 1 n ( I ) 0 0 0 0 A ω M n ( I ) B ω M n ( I ) 0 0 C ω M n ( I ) D ω M n ( I ) ) ( a ω 1 n b ω 1 n a ω M n b ω M n ) ,
n 1 , ω 1 n , , n 1 , ω M n , n 2 , ω 1 n , , n 2 , ω M n = f ( I ) = f ( I ω 1 n 1 , , I ω M n 1 ) = f ( a ω 1 n 1 , , a ω M n 1 , b ω 1 n 1 , , b ω M n 1 ) .
( a ω P n 1 b ω P n 1 a ω R n 1 b ω R n 1 ) = ( A ω P n ( I ) B ω P n ( I ) 0 0 C ω P n ( I ) D ω P n ( I ) 0 0 0 0 A ω R n ( I ) B ω R n ( I ) 0 0 C ω R n ( I ) D ω R n ( I ) ) ( a ω P n b ω P n a ω R n b ω R n ) ,
A ω m n = e i k 1 ω m n l 1 [ cos k 2 ω m n l 2 1 2 i ( p k 2 ω m n k 1 ω m n + k 1 ω m n p k 2 ω m n ) sin k 2 ω m n l 2 ] , B ω m n = e i k 1 ω m n l 1 [ 1 2 i ( p k 2 ω m n k 1 ω m n k 1 ω m n p k 2 ω m n ) sin k 2 ω m n l 2 ] , C ω m n = e i k 1 ω m n l 1 [ 1 2 i ( p k 2 ω m n k 1 ω m n k 1 ω m n p k 2 ω m n ) sin k 2 ω m n l 2 ] , D ω m n = e i k 1 ω m n l 1 [ cos k 2 ω m n l 2 + 1 2 i ( p k 2 ω m n k 1 ω m n + k 1 ω m n p k 2 ω m n ) sin k 2 ω m n l 2 ] ,
k α ω m n = ( ω m c n α n ) 2 β 2 .
d I ω R d x = ( 2 γ R + G m n I ω P ) I ω R ,
d I ω P d x = [ 2 γ P ( ω n ω m ) G m n I ω R ] I ω P ,
d I ω R d x = ( G m n I ω P ) I ω R = G ˜ m n , P I ω R ,
d I ω P d x = ( G m n I ω R ) I ω P = G ˜ m n , R I ω P ,
n 1 ω P = n 1 r , ω P i n 1 i , ω P = n 1 r , ω P + i G n m I ω R / k 0 P ,
n 1 ω R = n 1 r , ω R i n 1 i , ω R = n 1 r , ω R i G n m I ω P / k 0 R .
G m n = G 0 1 + ( | I ω P | 2 · | I ω R | 2 ) I S ,
( a ω R N b ω R N ) = ( A ω R B ω R C ω R D ω R ) N ( a ω R 0 b ω R 0 ) = ( A ˜ ω R B ˜ ω R C ˜ ω R D ˜ ω R ) ( a ω R 0 b ω R 0 ) ,
r 2 = C ˜ ω R r 1 + D ˜ ω R A ˜ ω R r 1 + B ˜ ω R .

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