Abstract

We present analytical formulae for the polarization dependent first- and second-order dispersion of a quarter-wave Bragg reflection waveguide (QtW-BRW). Using these formulae, we develop several qualitative properties of the QtW-BRW. In particular, we show that the birefringence of these waveguides changes sign at the QtW wavelength. Regimes of total dispersion corresponding to predominantly material-dominated and waveguide-dominated dispersion are identified. Using this concept, it is shown that the QtW-BRW can be designed so as to provide anomalous group velocity dispersion of large magnitude, or very small GVD of either sign, simply by an appropriate chose of layer thicknesses. Implications on nonlinear optical devices in compound semiconductors are discussed.

© 2006 Optical Society of America

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References

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  1. A. S. Helmy and B. R. West, "Phase matching using Bragg reflector waveguides," in Proceedings of 18th Annual Meeting of the IEEE Lasers and Electro-Optics Society (Institute of Electrical and Electronics Engineers, Sydney, 2005), pp. 424-425.
  2. A. S. Helmy, "Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications," Opt. Express 14, 1243-1252 (2006)
    [CrossRef] [PubMed]
  3. Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. 43,L631-L633 (2004).
    [CrossRef]
  4. E. Simova and I. Golub, "Polarization splitter/combiner in high index contrast Bragg reflector waveguides," Opt. Express 11, 3425-3430 (2003),
    [CrossRef] [PubMed]
  5. A. Mizrahi and L. Schächter, "Optical Bragg accelerators," Phys. Rev. E. 70, 016505 (2004).
    [CrossRef]
  6. S. Nakamura, K. Tajima, "Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch," Jpn. J. Appl. Phys. 35, L1426-L1429 (1996).
    [CrossRef]
  7. K. Cheng, ed., Handbook of Optical Components and Engineering (Wiley Interscience, 2003).
  8. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).
  9. U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995).
    [CrossRef]
  10. Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, "Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides," J. Appl. Phys. 98, 113102 (2005).
    [CrossRef]
  11. M. A. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, "Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires," Opt. Express 13, 6848-6855 (2005),
    [CrossRef] [PubMed]
  12. E. Valentinuzzi, "Dispersive properties of Kerr-like nonlinear optical structures," J. Lightwave Technol. 16, 152-155 (1998).
    [CrossRef]
  13. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005),
    [CrossRef] [PubMed]
  14. T. D. Engeness, M. Ibanescu, S. G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, "Dispersion tailoring and compensation by modal interactions in OmniGuide fibers," Opt. Express 11, 1175-1196 (2003)
    [CrossRef] [PubMed]
  15. A. Mizrahi and L. Schächter, "Bragg reflection waveguides with a matching layer," Opt. Express 12, 3156-3170 (2004).
    [CrossRef] [PubMed]
  16. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  17. Y. Sakurai and F. Koyama, "Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors," Jpn. J. Appl. Phys. 43,5828-5831 (2004).
    [CrossRef]
  18. P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
    [CrossRef]
  19. P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media: I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
    [CrossRef]
  20. B. R. West and A. S. Helmy, "Properties of the quarter-wave Bragg reflection waveguide: Theory," J. Opt. Soc. Am. B (to be published).
  21. A. S. Deif, Advanced Matrix Theory for Scientists and Engineers (Routledge, 1987).
  22. S. Adachi, "GaAs, AlAs, and AlxGa1-xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
    [CrossRef]
  23. M. A. Afromowitz, "Refractive index of Ga1-xAlxAs," Solid State Commun. 15, 59-63 (1974).
    [CrossRef]
  24. A. N. Pikhtin and A. D. Yas’kov, "Dispersion of refractive-index of semiconductors with diamond and zincblende structures," Sov. Phys. Semicond. 12, 622-626 (1978).
  25. S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
    [CrossRef]
  26. T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
    [CrossRef]
  27. B. R. West and A. S. Helmy, "Analysis and design equations for phase matching using Bragg reflector waveguides," IEEE J. Sel. Top. Quantum Electron. (to be published).

2006 (2)

A. S. Helmy, "Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications," Opt. Express 14, 1243-1252 (2006)
[CrossRef] [PubMed]

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
[CrossRef]

2005 (3)

2004 (5)

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. 43,L631-L633 (2004).
[CrossRef]

A. Mizrahi and L. Schächter, "Optical Bragg accelerators," Phys. Rev. E. 70, 016505 (2004).
[CrossRef]

A. Mizrahi and L. Schächter, "Bragg reflection waveguides with a matching layer," Opt. Express 12, 3156-3170 (2004).
[CrossRef] [PubMed]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Y. Sakurai and F. Koyama, "Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors," Jpn. J. Appl. Phys. 43,5828-5831 (2004).
[CrossRef]

2003 (2)

2000 (1)

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

1998 (1)

1996 (1)

S. Nakamura, K. Tajima, "Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch," Jpn. J. Appl. Phys. 35, L1426-L1429 (1996).
[CrossRef]

1995 (1)

U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995).
[CrossRef]

1985 (1)

S. Adachi, "GaAs, AlAs, and AlxGa1-xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

1978 (1)

A. N. Pikhtin and A. D. Yas’kov, "Dispersion of refractive-index of semiconductors with diamond and zincblende structures," Sov. Phys. Semicond. 12, 622-626 (1978).

1977 (1)

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media: I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

1976 (1)

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

1974 (1)

M. A. Afromowitz, "Refractive index of Ga1-xAlxAs," Solid State Commun. 15, 59-63 (1974).
[CrossRef]

Adachi, S.

S. Adachi, "GaAs, AlAs, and AlxGa1-xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

Afromowitz, M. A.

M. A. Afromowitz, "Refractive index of Ga1-xAlxAs," Solid State Commun. 15, 59-63 (1974).
[CrossRef]

Aitchison, J. S.

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
[CrossRef]

Bigot, L.

Boardman, A. D.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Bouwmans, G.

Cao, Q.

Douay, M.

Egan, P.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Engeness, T. D.

Fink, Y.

Foster, M. A.

Gaeta, A. L.

Gehrsitz, S.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Golub, I.

Gourgon, C.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Helmy, A. S.

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
[CrossRef]

A. S. Helmy, "Phase matching using Bragg reflection waveguides for monolithic nonlinear optics applications," Opt. Express 14, 1243-1252 (2006)
[CrossRef] [PubMed]

B. R. West and A. S. Helmy, "Analysis and design equations for phase matching using Bragg reflector waveguides," IEEE J. Sel. Top. Quantum Electron. (to be published).

B. R. West and A. S. Helmy, "Properties of the quarter-wave Bragg reflection waveguide: Theory," J. Opt. Soc. Am. B (to be published).

Herres, N.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Hong, C.-S.

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media: I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

Hutchings, D. C.

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
[CrossRef]

Ibanescu, M.

Jacobs, S.

Johnson, S. G.

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Kleckner, T. C.

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
[CrossRef]

Koyama, F.

Y. Sakurai and F. Koyama, "Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors," Jpn. J. Appl. Phys. 43,5828-5831 (2004).
[CrossRef]

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. 43,L631-L633 (2004).
[CrossRef]

Lederer, F.

U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995).
[CrossRef]

Lee, Y.

Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, "Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides," J. Appl. Phys. 98, 113102 (2005).
[CrossRef]

Lopez, F.

Mizrahi, A.

Nakamura, S.

S. Nakamura, K. Tajima, "Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch," Jpn. J. Appl. Phys. 35, L1426-L1429 (1996).
[CrossRef]

Peschel, T.

U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995).
[CrossRef]

Peschel, U.

U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995).
[CrossRef]

Pikhtin, A. N.

A. N. Pikhtin and A. D. Yas’kov, "Dispersion of refractive-index of semiconductors with diamond and zincblende structures," Sov. Phys. Semicond. 12, 622-626 (1978).

Provino, L.

Quiquempois, Y.

Reinhart, F. K.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Sakurai, Y.

Y. Sakurai and F. Koyama, "Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors," Jpn. J. Appl. Phys. 43,5828-5831 (2004).
[CrossRef]

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. 43,L631-L633 (2004).
[CrossRef]

Schächter, L.

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Sigg, H.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Simova, E.

Skorobogatiy, M.

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Tajima, K.

S. Nakamura, K. Tajima, "Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch," Jpn. J. Appl. Phys. 35, L1426-L1429 (1996).
[CrossRef]

Takei, A.

Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, "Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides," J. Appl. Phys. 98, 113102 (2005).
[CrossRef]

Taniguchi, T.

Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, "Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides," J. Appl. Phys. 98, 113102 (2005).
[CrossRef]

Trebino, R.

Uchiyama, H.

Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, "Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides," J. Appl. Phys. 98, 113102 (2005).
[CrossRef]

Valentinuzzi, E.

Vonlanthen, A.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Weisberg, O.

West, B. R.

B. R. West and A. S. Helmy, "Analysis and design equations for phase matching using Bragg reflector waveguides," IEEE J. Sel. Top. Quantum Electron. (to be published).

B. R. West and A. S. Helmy, "Properties of the quarter-wave Bragg reflection waveguide: Theory," J. Opt. Soc. Am. B (to be published).

Yariv, A.

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media: I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Yas’kov, A. D.

A. N. Pikhtin and A. D. Yas’kov, "Dispersion of refractive-index of semiconductors with diamond and zincblende structures," Sov. Phys. Semicond. 12, 622-626 (1978).

Yeh, P.

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media: I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Zeaiter, K.

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
[CrossRef]

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

U. Peschel, T. Peschel, and F. Lederer, "A compact device for highly efficient dispersion compensation in fiber transmission," Appl. Phys. Lett. 67, 2111-2113 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. C. Kleckner, A. S. Helmy, K. Zeaiter, D. C. Hutchings, and J. S. Aitchison, "Dispersion and modulation of the linear optical properties of GaAs-AlAs superlattice waveguides using quantum-well intermixing," IEEE J. Quantum Electron. 42, 280-286 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

B. R. West and A. S. Helmy, "Analysis and design equations for phase matching using Bragg reflector waveguides," IEEE J. Sel. Top. Quantum Electron. (to be published).

J Appl. Phys. (1)

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

J. Appl. Phys. (2)

S. Adachi, "GaAs, AlAs, and AlxGa1-xAs material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

Y. Lee, A. Takei, T. Taniguchi, and H. Uchiyama, "Temperature tuning of dispersion compensation using semiconductor asymmetric coupled waveguides," J. Appl. Phys. 98, 113102 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media: I. General theory," J. Opt. Soc. Am. 67, 428-438 (1977).
[CrossRef]

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

B. R. West and A. S. Helmy, "Properties of the quarter-wave Bragg reflection waveguide: Theory," J. Opt. Soc. Am. B (to be published).

Jpn. J. Appl. Phys. (3)

Y. Sakurai and F. Koyama, "Control of group delay and chromatic dispersion in tunable hollow waveguide with highly reflective mirrors," Jpn. J. Appl. Phys. 43,5828-5831 (2004).
[CrossRef]

S. Nakamura, K. Tajima, "Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch," Jpn. J. Appl. Phys. 35, L1426-L1429 (1996).
[CrossRef]

Y. Sakurai and F. Koyama, "Proposal of tunable hollow waveguide distributed Bragg reflectors," Jpn. J. Appl. Phys. 43,L631-L633 (2004).
[CrossRef]

Opt. Commun. (1)

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Opt. Express (6)

Phys. Rev. E (1)

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Phys. Rev. E. (1)

A. Mizrahi and L. Schächter, "Optical Bragg accelerators," Phys. Rev. E. 70, 016505 (2004).
[CrossRef]

Solid State Commun. (1)

M. A. Afromowitz, "Refractive index of Ga1-xAlxAs," Solid State Commun. 15, 59-63 (1974).
[CrossRef]

Sov. Phys. Semicond. (1)

A. N. Pikhtin and A. D. Yas’kov, "Dispersion of refractive-index of semiconductors with diamond and zincblende structures," Sov. Phys. Semicond. 12, 622-626 (1978).

Other (4)

A. S. Helmy and B. R. West, "Phase matching using Bragg reflector waveguides," in Proceedings of 18th Annual Meeting of the IEEE Lasers and Electro-Optics Society (Institute of Electrical and Electronics Engineers, Sydney, 2005), pp. 424-425.

K. Cheng, ed., Handbook of Optical Components and Engineering (Wiley Interscience, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).

A. S. Deif, Advanced Matrix Theory for Scientists and Engineers (Routledge, 1987).

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Figures (7)

Fig. 1.
Fig. 1.

Refractive index profile of a BRW.

Fig. 2.
Fig. 2.

Effective index for a BRW that is quarter-wave at 1.55 μm. (Blue: TE, green: TM).

Fig. 3.
Fig. 3.

Normalized first-order dispersion for the waveguide of Fig. 2. (Blue: TE, green: TM). oe-14-9-4073-i001 Analytical solution from Eq. (23) (TE), oe-14-9-4073-i002 Analytical solution (TM).

Fig. 4.
Fig. 4.

Index dispersion of AlxGa1-xAs at 1.55 μm using three different models. (a) refractive index (b) normalized first-order dispersion (c) second-order dispersion parameter β 2. Blue: Afromowitz [23], Green: Adachi [22], Red: Gehrsitz et al. [25].

Fig. 5.
Fig. 5.

Normalized effective index vs. core thickness for various QtW-BRWs with core and cladding compositions identical to the waveguide analyzed in Figs. 2 and 3.

Fig. 6.
Fig. 6.

Normalized first-order dispersion vs. normalized effective index of the TE mode including (blue) and omitting (green) the contribution of material dispersion. Inset: region around B ≈ 1. The curves are indistinguishable for B < 0.9.

Fig. 7.
Fig. 7.

Second-order dispersion parameter -β 2 vs. normalized effective index of the TE mode including (blue) and omitting (green) the contribution of material dispersion. Inset: region around B ≈ 1 (linear scale). Negative β 2 is shown here only to facilitate the use of the logarithmic scale.

Tables (1)

Tables Icon

Table 1. Material dispersion data for the waveguides analyzed in Figs. 2-3, 5-7, λ=1.55 μm (from [25])

Equations (50)

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k i = ω c n i 2 n eff 2
1 k co cot ( k co t co 2 ) = i k 1 e i K TE Λ A TE + B TE e i K TE Λ A TE B TE , ( TE )
1 k co cot ( k co t co 2 ) = i k 1 ( n 1 n co ) 2 e i K TM Λ A TM + B TM e i K TM Λ A TM B TM , ( TM , n 1 2 k 2 < n 2 2 k 1 )
k co cot ( k co t co 2 ) = i k 1 ( n co n 1 ) 2 e i K TM Λ A TM B TM e i K TM Λ A TM + B TM , ( TM , n 1 2 k 2 > n 2 2 k 1 )
A TE = e i k 1 a [ cos k 2 b + i 2 ( k 2 k 1 + k 1 k 2 ) sin k 2 b ]
A TM = e i k 1 a [ cos k 2 b + i 2 ( n 2 2 k 1 n 1 2 k 2 + n 1 2 k 2 n 2 2 k 1 ) sin k 2 b ]
B TE = e i k 1 a [ i 2 ( k 2 k 1 + k 1 k 2 ) sin k 2 b ]
B TM = e i k 1 a [ i 2 ( n 2 2 k 1 n 1 2 k 2 n 1 2 k 2 n 2 2 k 1 ) sin k 2 b ] .
M TE ( TM ) FT = [ A TE ( TM ) B TE ( TM ) B TE ( TM ) * A TE ( TM ) * ]
K TE ( TM ) Λ = cos 1 [ Re ( A TE ( TM ) ) ] .
ρ n 1 2 k 2 n 2 2 k 1
k 1 a = k 2 a = π 2 .
n eff = n co 2 ( πc ω 0 t co ) 2
A TE = 1 2 ( k 2 k 1 + k 1 k 2 ) B TE = 1 2 ( k 2 k 1 k 1 k 2 )
A TM = 1 2 ( 1 ρ + ρ ) B TM = 1 2 ( 1 ρ ρ )
e i K TE Λ = ( k 2 k 1 ) e i K TM Λ = { ρ EVEN 1 ρ ODD .
1 k co + Δ k co cot ( ( k co + Δ k co ) t co 2 )
= i k 1 + Δ k 1 ( [ e i K TE Λ + Δ ( e i K TE Λ ) ] [ A TE + Δ A TE ] + [ B TE + Δ B TE ] [ e i K TE Λ + Δ ( e i K TE Λ ) ] [ A TE + Δ A TE ] [ B TE + Δ B TE ] ) .
Δ k co t co 2 k co i k 1 ( Δ ( e i K TE Λ ) Δ A TE + Δ B TE e i K TE Λ A TE B TE )
A ͂ TE = A TE + Δ A TE
= e i ( k 1 + Δ k 1 ) a [ cos [ ( k 2 + Δ k 2 ) b ] + i 2 ( k 2 k 1 + Δ k 2 Δ k 1 + k 1 k 2 + Δ k 1 Δ k 2 ) sin [ ( k 2 + Δ k 2 ) b ] ]
i ( 1 + i Δ k 1 a ) [ Δ k 2 b + i 2 ( k 2 k 1 + k 1 k 2 ) ( 1 + 2 k 2 Δ k 2 + 2 k 1 Δ k 1 k 2 2 + k 1 2 Δ k 1 k 1 Δ k 2 k 2 ) ]
Δ A TE = Δ k 1 [ 1 2 k 1 ( k 2 k 1 + k 1 k 2 ) 1 k 2 i ( π 4 k 1 ) ( k 2 k 1 + k 1 k 2 ) ]
+ Δ k 2 [ 1 2 k 2 ( k 2 k 1 + k 1 k 2 ) 1 k 1 i ( π 2 k 2 ) ] .
Δ B TE = Δ k 1 [ 1 2 k 1 ( k 2 k 1 k 1 k 2 ) 1 k 2 i ( π 4 k 1 ) ( k 2 k 1 k 1 k 2 ) ]
+ Δ k 2 [ 1 2 k 2 ( k 2 k 1 k 1 k 2 ) + 1 k 1 ] .
Δ ( e i K TE Λ ) = u TE , Ξ TE u TE = Re ( Δ A TE Δ B TE )
( Δ k co ) TE = α TE ( Δ k 1 + Δ k 2 ) TE , α TE = π k co t co ( k 2 2 k 1 2 ) .
( Δ k co ) TM EVEN = α TM EVEN [ ( n 2 n 1 ) 2 Δ k 1 + Δ k 2 ] TM EVEN , α TM EVEN = π k co n 1 4 n 2 2 n co 2 t co ( n 1 4 k 2 2 n 2 4 k 1 2 )
( Δ k co ) TM ODD = α TM ODD [ ( n 1 k 2 n 2 k 1 ) 2 Δ k 1 + Δ k 2 ] TM ODD , α TM ODD = ( n co n 2 k 1 ) 2 ( n 1 4 k 2 2 n 2 4 k 1 2 ) .
k ͂ i = k i + Δ k i ( ω 0 + Δ ω c ) [ ( n i + n i Δ ω ) 2 ( n eff + n eff Δ ω ) 2 ] 1 / 2
k ͂ i ( ω 0 + Δ ω c ) ( n i 2 n eff 2 ) 1 / 2 [ 1 + ( n i n i n eff n eff ) n i 2 n eff 2 Δω ] .
Δ k i Δ ω c ( G i H i n eff )
G i = ( n i 2 n eff 2 ) 1 2 ( 1 + ω 0 n i n′ i n i 2 n eff 2 )
H i = ω 0 n eff ( n i 2 n eff 2 ) 1 2 .
n′ eff = { G co α TE [ G 1 + G 2 ] H co α TE [ H 1 + H 2 ] , ( TE ) G co α TE EVEN [ ( n 2 n 1 ) 2 G 1 + G 2 ] H co α TE EVEN [ ( n 2 n 1 ) 2 H 1 + H 2 ] , ( TM , EVEN ) G co α TE ODD [ ( n 1 k 2 n 2 k 1 ) 2 G 1 + G 2 ] H co α TE ODD [ ( n 1 k 2 n 2 k 1 ) 2 H 1 + H 2 ] , ( TM , ODD ) .
Δ n eff = n eff , TE n eff , TM
= [ n eff + ( n eff ω ) TE Δ ω ] [ n eff + ( n eff ω ) TM Δ ω ]
= [ ( n eff ω ) TE ( n eff ω ) TM ] Δ ω .
πc ω 0 n co < t co < πc ω 0 n co 2 n 2 2 , ( n co > n 2 )
πc ω 0 n co < t co , ( n co < n 2 ) .
k ˜ i ( ω 0 + Δ ω c ) [ ( n i + n′ i Δ ω + 1 2 n″ i ( Δ ω ) 2 ) 2 ( n eff + n′ eff Δ ω + 1 2 n″ eff ( Δ ω ) 2 ) 2 ] 1 2
k ˜ i ( ω 0 + Δ ω c ) ( n i 2 n eff 2 ) 1 2 [ 1 + J i Δ ω + P i ( Δ ω ) 2 1 2 n eff n″ eff n i 2 n eff 2 ( Δ ω ) 2 ]
J i = n i n′ i n eff n′ eff n i 2 n eff 2
P i = 1 2 [ n i n″ i + ( n′ i ) 2 ( n′ eff ) 2 n i 2 n eff 2 ( n i n′ i n eff n′ eff ) 2 ( n i 2 n eff 2 ) 2 ] .
Δ k i Δ ω c ( G i H i n′ eff ) + ( Δ ω ) 2 c ( S i T i n″ eff )
S i = ( n i 2 n eff 2 ) 1 2 ( J i + ω 0 P i )
T i = ω 0 n eff 2 ( n i 2 n eff 2 ) 1 2 .
n″ eff = { S co α TE [ S 1 + S 2 ] T co α TE [ T 1 + T 2 ] , ( TE ) S co α TM EVEN [ ( n 2 n 1 ) 2 S 1 + S 2 ] T co α TM EVEN [ ( n 2 n 1 ) 2 T 1 + T 2 ] , ( TM , EVEN ) S co α TM ODD [ ( n 1 k 2 n 2 k 1 ) 2 S 1 + S 2 ] T co α TM ODD [ ( n 1 k 2 n 2 k 1 ) 2 T 1 + T 2 ] , ( TM , ODD ) .
B n eff n co = 1 ( πc ω 0 t co n co ) 2 .

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