Abstract

Photonic bandgaps are demonstrated in one-dimensional corrugated waveguides in the infrared range. A coupling grating is superimposed with single and double Bragg gratings on an azopolymer film by a simple optical process, which allows easy control of the grating spacing. Light is coupled to the TE0 resonant mode, and gaps in the dispersion curve are introduced by careful selection of the gratings. The analysis is carried out by measuring the transmission through the waveguide as a function of the wavelength and angle of incidence of a probe beam. This results in a direct measurement of the dispersion curves, which are in excellent agreement with theory.

© 2007 Optical Society of America

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  1. S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
    [CrossRef]
  2. A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
    [CrossRef]
  3. N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
    [CrossRef]
  4. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  5. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  6. I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
    [CrossRef]
  7. J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
    [CrossRef]
  8. L. Lévesque and P. Rochon, "Surface plasmon photonic bandgap in azopolymer gratings sputtered with gold," J. Opt. Soc. Am. A 22, 2564-2568 (2005).
    [CrossRef]
  9. F. Kneubuhl, "Diffraction grating spectroscopy," Appl. Opt. 8, 505-519 (1969).
    [CrossRef] [PubMed]
  10. J. E. Bjorkholm and C. V. Shank, "Distributed-feedback lasers in thin-film optical waveguides," IEEE J. Quantum Electron. QE-8, 833-838 (1972).
    [CrossRef]
  11. K. S. Pennington and L. Kuhn, "Bragg diffraction beam splitter for thin film optical guided waves," Opt. Commun. 3, 357-359 (1971).
    [CrossRef]
  12. M. Zirngibl, C. R. Doerr, and C. H. Joyner, "Demonstration of a splitter/router based on a chirped waveguide grating router," IEEE Photon. Technol. Lett. 10, 87-89 (1998).
    [CrossRef]
  13. H. M. Stoll, "Distributed Bragg deflector: a multifunctional integrated optical device," Appl. Opt. 17, 2562-2569 (1978).
    [PubMed]
  14. M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin films," Appl. Phys. Lett. 16, 523-525 (1970).
    [CrossRef]
  15. S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32, 2606-2613 (1993).
    [CrossRef] [PubMed]
  16. S. Suzuki, A. Himeno, Y. Tachikawa, and Y. Yamanda, "Multichannel optical wavelength selective switch with arrayed-waveguide grating multiplexer," Electron. Lett. 30, 1091-1092 (1994).
    [CrossRef]
  17. A. Yi-Yan, C. Wilkinson, and P. Laybourn, "Two-dimensional grating unit cell demultiplexer for thin film optical waveguides," IEEE J. Quantum Electron. QE-16, 1089-1092 (1980).
    [CrossRef]
  18. S. Suzuki, Y. Inoue, and Y. Ohmori, "Polarisation-insensitive arrayed waveguide grating multiplexer with SiO2-on-SiO2 structure," Electron. Lett. 30, 642-643 (1994).
    [CrossRef]
  19. A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
    [CrossRef]
  20. R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).
    [CrossRef]
  21. A. Yariv, "Coupled mode theory for guided wave optics," IEEE J. Quantum Electron. QE-9, 919-933 (1973).
    [CrossRef]
  22. M. G. Moharam and T. K. Gaylord, "Rigorous coupled wave analysis of planar grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
  23. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
    [CrossRef]
  24. L. J. Martinez, A. Garcia-Martin, and P. A. Postigo, "Photonic band gaps in a two-dimensional hybrid triangular-graphite lattice," Opt. Express 12, 5684-5689 (2004).
    [CrossRef] [PubMed]
  25. J. Ctyroký, "Waveguide Bragg grating as a 1D photonic bandgap structure," Proc. SPIE 4016, 92-96 (2000).
    [CrossRef]
  26. G. Ma, S. H. Tang, J. Shen, Z. Zhang, and Z. Hua, "Defect-mode dependence of two-photon-absorption enhancement in a one-dimensional photonic band gap structure," Opt. Lett. 29, 1769-1771 (2004).
    [CrossRef] [PubMed]
  27. R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
    [CrossRef]
  28. P. St. J. Russel, "Bragg resonances of light in optical superlattices," Phys. Rev. Lett. 56, 596-599 (1986).
    [CrossRef]
  29. C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
    [CrossRef]
  30. J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
    [CrossRef] [PubMed]
  31. J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
    [CrossRef]
  32. C. R. Polluck, Fundamentals of Optoelectronics (R. D. Irwin, 1995), pp. 49-74.
  33. GSolver was developed in 1984 by the Grating Solver Development Company, P.O. Box 353, Allen, Texas, 75013, http://www.gsolver.com.
  34. P. Rochon, E. Batalla, and A. Natansohn, "Optically induced surface gratings on azoaromatic polymer films," Appl. Phys. Lett. 66, 136-138 (1995).
    [CrossRef]
  35. X. Mai, R. Moshrefzadeh, U. J. Gibson, G. I. Stegeman, and C. T. Eaton, "Simple versatile method for fabricating guided wave gratings," Appl. Opt. 24, 3155-3161 (1985).
    [CrossRef]
  36. P. Rochon, J. Paterson, and A. Natansohn, "Efficiency of optically induced surface gratings on azo polymer films," in Applied Optics and Optoelectronics 1996, Proceedings of the Applied Optics Divisional Conference of the Institute of Physics, K.T. V.Gratten, ed. (Institute of Physics, 1996), pp. 116-119.
  37. R. J. Stockermans and P. Rochon, "Modeling of photonic bandgaps in resonant waveguide grating systems using a simple theory for 1-D photonic crystals," J. Lightwave Technol. 25, 952-956(2007).
    [CrossRef]

2007 (1)

2005 (3)

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
[CrossRef]

L. Lévesque and P. Rochon, "Surface plasmon photonic bandgap in azopolymer gratings sputtered with gold," J. Opt. Soc. Am. A 22, 2564-2568 (2005).
[CrossRef]

2004 (2)

2003 (1)

J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
[CrossRef]

2002 (1)

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

2001 (1)

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

2000 (1)

J. Ctyroký, "Waveguide Bragg grating as a 1D photonic bandgap structure," Proc. SPIE 4016, 92-96 (2000).
[CrossRef]

1999 (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

1998 (1)

M. Zirngibl, C. R. Doerr, and C. H. Joyner, "Demonstration of a splitter/router based on a chirped waveguide grating router," IEEE Photon. Technol. Lett. 10, 87-89 (1998).
[CrossRef]

1995 (2)

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

P. Rochon, E. Batalla, and A. Natansohn, "Optically induced surface gratings on azoaromatic polymer films," Appl. Phys. Lett. 66, 136-138 (1995).
[CrossRef]

1994 (3)

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

S. Suzuki, Y. Inoue, and Y. Ohmori, "Polarisation-insensitive arrayed waveguide grating multiplexer with SiO2-on-SiO2 structure," Electron. Lett. 30, 642-643 (1994).
[CrossRef]

S. Suzuki, A. Himeno, Y. Tachikawa, and Y. Yamanda, "Multichannel optical wavelength selective switch with arrayed-waveguide grating multiplexer," Electron. Lett. 30, 1091-1092 (1994).
[CrossRef]

1993 (1)

1992 (1)

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

1987 (3)

R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
[CrossRef]

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

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

1986 (1)

P. St. J. Russel, "Bragg resonances of light in optical superlattices," Phys. Rev. Lett. 56, 596-599 (1986).
[CrossRef]

1985 (1)

1981 (1)

1980 (1)

A. Yi-Yan, C. Wilkinson, and P. Laybourn, "Two-dimensional grating unit cell demultiplexer for thin film optical waveguides," IEEE J. Quantum Electron. QE-16, 1089-1092 (1980).
[CrossRef]

1978 (1)

1977 (1)

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
[CrossRef]

1973 (1)

A. Yariv, "Coupled mode theory for guided wave optics," IEEE J. Quantum Electron. QE-9, 919-933 (1973).
[CrossRef]

1972 (1)

J. E. Bjorkholm and C. V. Shank, "Distributed-feedback lasers in thin-film optical waveguides," IEEE J. Quantum Electron. QE-8, 833-838 (1972).
[CrossRef]

1971 (1)

K. S. Pennington and L. Kuhn, "Bragg diffraction beam splitter for thin film optical guided waves," Opt. Commun. 3, 357-359 (1971).
[CrossRef]

1970 (1)

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin films," Appl. Phys. Lett. 16, 523-525 (1970).
[CrossRef]

1969 (1)

Batalla, E.

P. Rochon, E. Batalla, and A. Natansohn, "Optically induced surface gratings on azoaromatic polymer films," Appl. Phys. Lett. 66, 136-138 (1995).
[CrossRef]

Bauer, C.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Benisty, H.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Biswas, R.

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and C. V. Shank, "Distributed-feedback lasers in thin-film optical waveguides," IEEE J. Quantum Electron. QE-8, 833-838 (1972).
[CrossRef]

Blum, R.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Böttger, G.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

Cheng, S. D.

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

Cross, A. W.

I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
[CrossRef]

Ctyroký, J.

J. Ctyroký, "Waveguide Bragg grating as a 1D photonic bandgap structure," Proc. SPIE 4016, 92-96 (2000).
[CrossRef]

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin films," Appl. Phys. Lett. 16, 523-525 (1970).
[CrossRef]

David, A.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

DenBaars, S. P.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Diana, F. S.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Doerr, C. R.

M. Zirngibl, C. R. Doerr, and C. H. Joyner, "Demonstration of a splitter/router based on a chirped waveguide grating router," IEEE Photon. Technol. Lett. 10, 87-89 (1998).
[CrossRef]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

Eaton, C. T.

Eich, M.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Elsner, H.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Garcia-Martin, A.

Gaylord, T. K.

Gibson, U. J.

Giessen, H.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin films," Appl. Phys. Lett. 16, 523-525 (1970).
[CrossRef]

Himeno, A.

S. Suzuki, A. Himeno, Y. Tachikawa, and Y. Yamanda, "Multichannel optical wavelength selective switch with arrayed-waveguide grating multiplexer," Electron. Lett. 30, 1091-1092 (1994).
[CrossRef]

Ho, K.-M.

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

Hu, E.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Hua, Z.

Inoue, Y.

S. Suzuki, Y. Inoue, and Y. Ohmori, "Polarisation-insensitive arrayed waveguide grating multiplexer with SiO2-on-SiO2 structure," Electron. Lett. 30, 642-643 (1994).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

John, S.

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Joyner, C. H.

M. Zirngibl, C. R. Doerr, and C. H. Joyner, "Demonstration of a splitter/router based on a chirped waveguide grating router," IEEE Photon. Technol. Lett. 10, 87-89 (1998).
[CrossRef]

Kim, P.

J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
[CrossRef]

Kley, E. B.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Kneubuhl, F.

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Konoplev, I. V.

I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
[CrossRef]

Kuhn, L.

K. S. Pennington and L. Kuhn, "Bragg diffraction beam splitter for thin film optical guided waves," Opt. Commun. 3, 357-359 (1971).
[CrossRef]

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin films," Appl. Phys. Lett. 16, 523-525 (1970).
[CrossRef]

Kuligk, A.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Kunert, J.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Laybourn, P.

A. Yi-Yan, C. Wilkinson, and P. Laybourn, "Two-dimensional grating unit cell demultiplexer for thin film optical waveguides," IEEE J. Quantum Electron. QE-16, 1089-1092 (1980).
[CrossRef]

Lee, G.

J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
[CrossRef]

Lévesque, L.

Liguda, C.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Ma, G.

Magnusson, R.

Mahrt, R. F.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Mai, X.

Martinez, L. J.

McCalmont, S.

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

McGrane, P.

I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
[CrossRef]

Meier, C.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Meyer, H. G.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Moharam, M. G.

Moll, N.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Morgenroth, W.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Moshrefzadeh, R.

Nakamura, M.

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
[CrossRef]

Nakamura, S.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Natansohn, A.

P. Rochon, E. Batalla, and A. Natansohn, "Optically induced surface gratings on azoaromatic polymer films," Appl. Phys. Lett. 66, 136-138 (1995).
[CrossRef]

P. Rochon, J. Paterson, and A. Natansohn, "Efficiency of optically induced surface gratings on azo polymer films," in Applied Optics and Optoelectronics 1996, Proceedings of the Applied Optics Divisional Conference of the Institute of Physics, K.T. V.Gratten, ed. (Institute of Physics, 1996), pp. 116-119.

Oh, C.

J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
[CrossRef]

Ohmori, Y.

S. Suzuki, Y. Inoue, and Y. Ohmori, "Polarisation-insensitive arrayed waveguide grating multiplexer with SiO2-on-SiO2 structure," Electron. Lett. 30, 642-643 (1994).
[CrossRef]

Ozbay, E.

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

Paterson, J.

P. Rochon, J. Paterson, and A. Natansohn, "Efficiency of optically induced surface gratings on azo polymer films," in Applied Optics and Optoelectronics 1996, Proceedings of the Applied Optics Divisional Conference of the Institute of Physics, K.T. V.Gratten, ed. (Institute of Physics, 1996), pp. 116-119.

Pennington, K. S.

K. S. Pennington and L. Kuhn, "Bragg diffraction beam splitter for thin film optical guided waves," Opt. Commun. 3, 357-359 (1971).
[CrossRef]

Petit, R.

R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).
[CrossRef]

Phelps, A. D. R.

I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
[CrossRef]

Polluck, C. R.

C. R. Polluck, Fundamentals of Optoelectronics (R. D. Irwin, 1995), pp. 49-74.

Postigo, P. A.

Rochon, P.

R. J. Stockermans and P. Rochon, "Modeling of photonic bandgaps in resonant waveguide grating systems using a simple theory for 1-D photonic crystals," J. Lightwave Technol. 25, 952-956(2007).
[CrossRef]

L. Lévesque and P. Rochon, "Surface plasmon photonic bandgap in azopolymer gratings sputtered with gold," J. Opt. Soc. Am. A 22, 2564-2568 (2005).
[CrossRef]

P. Rochon, E. Batalla, and A. Natansohn, "Optically induced surface gratings on azoaromatic polymer films," Appl. Phys. Lett. 66, 136-138 (1995).
[CrossRef]

P. Rochon, J. Paterson, and A. Natansohn, "Efficiency of optically induced surface gratings on azo polymer films," in Applied Optics and Optoelectronics 1996, Proceedings of the Applied Optics Divisional Conference of the Institute of Physics, K.T. V.Gratten, ed. (Institute of Physics, 1996), pp. 116-119.

Ronald, K.

I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
[CrossRef]

Roth, H.

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

Russel, P. St. J.

P. St. J. Russel, "Bragg resonances of light in optical superlattices," Phys. Rev. Lett. 56, 596-599 (1986).
[CrossRef]

Scherf, U.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Schnabel, B.

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin films," Appl. Phys. Lett. 16, 523-525 (1970).
[CrossRef]

Shank, C. V.

J. E. Bjorkholm and C. V. Shank, "Distributed-feedback lasers in thin-film optical waveguides," IEEE J. Quantum Electron. QE-8, 833-838 (1972).
[CrossRef]

Sharma, R.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Shen, J.

Song, S. H.

J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
[CrossRef]

Stegeman, G. I.

Stockermans, R. J.

Stoll, H. M.

Suzuki, S.

S. Suzuki, A. Himeno, Y. Tachikawa, and Y. Yamanda, "Multichannel optical wavelength selective switch with arrayed-waveguide grating multiplexer," Electron. Lett. 30, 1091-1092 (1994).
[CrossRef]

S. Suzuki, Y. Inoue, and Y. Ohmori, "Polarisation-insensitive arrayed waveguide grating multiplexer with SiO2-on-SiO2 structure," Electron. Lett. 30, 642-643 (1994).
[CrossRef]

Tachikawa, Y.

S. Suzuki, A. Himeno, Y. Tachikawa, and Y. Yamanda, "Multichannel optical wavelength selective switch with arrayed-waveguide grating multiplexer," Electron. Lett. 30, 1091-1092 (1994).
[CrossRef]

Tang, S. H.

Tuttle, G.

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Wang, S. S.

Weisbuch, C.

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

Wilkinson, C.

A. Yi-Yan, C. Wilkinson, and P. Laybourn, "Two-dimensional grating unit cell demultiplexer for thin film optical waveguides," IEEE J. Quantum Electron. QE-16, 1089-1092 (1980).
[CrossRef]

Yablonovitch, E.

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

Yamanda, Y.

S. Suzuki, A. Himeno, Y. Tachikawa, and Y. Yamanda, "Multichannel optical wavelength selective switch with arrayed-waveguide grating multiplexer," Electron. Lett. 30, 1091-1092 (1994).
[CrossRef]

Yariv, A.

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
[CrossRef]

A. Yariv, "Coupled mode theory for guided wave optics," IEEE J. Quantum Electron. QE-9, 919-933 (1973).
[CrossRef]

Yi-Yan, A.

A. Yi-Yan, C. Wilkinson, and P. Laybourn, "Two-dimensional grating unit cell demultiplexer for thin film optical waveguides," IEEE J. Quantum Electron. QE-16, 1089-1092 (1980).
[CrossRef]

Yoon, J.

J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
[CrossRef]

Zengerle, R.

R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
[CrossRef]

Zhang, Z.

Zirngibl, M.

M. Zirngibl, C. R. Doerr, and C. H. Joyner, "Demonstration of a splitter/router based on a chirped waveguide grating router," IEEE Photon. Technol. Lett. 10, 87-89 (1998).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (6)

P. Rochon, E. Batalla, and A. Natansohn, "Optically induced surface gratings on azoaromatic polymer films," Appl. Phys. Lett. 66, 136-138 (1995).
[CrossRef]

C. Liguda, G. Böttger, A. Kuligk, R. Blum, M. Eich, H. Roth, J. Kunert, W. Morgenroth, H. Elsner, and H. G. Meyer, "Polymer photonic crystal slab waveguides," Appl. Phys. Lett. 78, 2434-2436 (2001).
[CrossRef]

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, "Grating coupler for efficient excitation of optical guided waves in thin films," Appl. Phys. Lett. 16, 523-525 (1970).
[CrossRef]

S. D. Cheng, R. Biswas, E. Ozbay, S. McCalmont, G. Tuttle, and K.-M. Ho, "Optimized dipole antennas on photonic bandgap crystals," Appl. Phys. Lett. 67, 3399-3401 (1995).
[CrossRef]

A. David, C. Meier, R. Sharma, F. S. Diana, S. P. DenBaars, E. Hu, S. Nakamura, C. Weisbuch, and H. Benisty, "Photonic bands in two-dimensionally patterned multimode GaN waveguides for light extraction," Appl. Phys. Lett. 87, 101107-1-101107-3 (2005).
[CrossRef]

N. Moll, R. F. Mahrt, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, and U. Scherf, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

Electron. Lett. (2)

S. Suzuki, A. Himeno, Y. Tachikawa, and Y. Yamanda, "Multichannel optical wavelength selective switch with arrayed-waveguide grating multiplexer," Electron. Lett. 30, 1091-1092 (1994).
[CrossRef]

S. Suzuki, Y. Inoue, and Y. Ohmori, "Polarisation-insensitive arrayed waveguide grating multiplexer with SiO2-on-SiO2 structure," Electron. Lett. 30, 642-643 (1994).
[CrossRef]

IEEE J. Quantum Electron. (4)

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).
[CrossRef]

A. Yariv, "Coupled mode theory for guided wave optics," IEEE J. Quantum Electron. QE-9, 919-933 (1973).
[CrossRef]

A. Yi-Yan, C. Wilkinson, and P. Laybourn, "Two-dimensional grating unit cell demultiplexer for thin film optical waveguides," IEEE J. Quantum Electron. QE-16, 1089-1092 (1980).
[CrossRef]

J. E. Bjorkholm and C. V. Shank, "Distributed-feedback lasers in thin-film optical waveguides," IEEE J. Quantum Electron. QE-8, 833-838 (1972).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. Zirngibl, C. R. Doerr, and C. H. Joyner, "Demonstration of a splitter/router based on a chirped waveguide grating router," IEEE Photon. Technol. Lett. 10, 87-89 (1998).
[CrossRef]

J. Appl. Phys. (2)

I. V. Konoplev, P. McGrane, A. W. Cross, K. Ronald, and A. D. R. Phelps, "Wave interference and band gap control in multiconductor one-dimensional Bragg structures," J. Appl. Phys. 97, 073101-1-073101-7 (2005).
[CrossRef]

J. Yoon, G. Lee, S. H. Song, C. Oh, and P. Kim, "Surface-plasmon photonic band gaps in dielectric gratings on a flat metal surface," J. Appl. Phys. 94, 123-129 (2003).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

J. P. Dowling and C. M. Bowden, "Anomalous index of refraction in photonic bandgap materials," J. Mod. Opt. 41, 345-351 (1994).
[CrossRef]

R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

K. S. Pennington and L. Kuhn, "Bragg diffraction beam splitter for thin film optical guided waves," Opt. Commun. 3, 357-359 (1971).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

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

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

P. St. J. Russel, "Bragg resonances of light in optical superlattices," Phys. Rev. Lett. 56, 596-599 (1986).
[CrossRef]

Proc. SPIE (1)

J. Ctyroký, "Waveguide Bragg grating as a 1D photonic bandgap structure," Proc. SPIE 4016, 92-96 (2000).
[CrossRef]

Other (4)

C. R. Polluck, Fundamentals of Optoelectronics (R. D. Irwin, 1995), pp. 49-74.

GSolver was developed in 1984 by the Grating Solver Development Company, P.O. Box 353, Allen, Texas, 75013, http://www.gsolver.com.

P. Rochon, J. Paterson, and A. Natansohn, "Efficiency of optically induced surface gratings on azo polymer films," in Applied Optics and Optoelectronics 1996, Proceedings of the Applied Optics Divisional Conference of the Institute of Physics, K.T. V.Gratten, ed. (Institute of Physics, 1996), pp. 116-119.

R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Dispersion curve from Eq. (1) for a periodic layered media of dielectric slabs of index n 1 = 1.66 and air, n 2 = 1 . The widths of the layers are equal, and bandgaps form when k = m π d . The minimum frequency where a bandgap occurs is ω 0 .

Fig. 2
Fig. 2

Coupling condition, from Eq. (2), for a single grating waveguide, of grating vector G. The vertical dotted lines represent normal incidence, where light couples at a single frequency, ω 2 . At some incident angle θ i , light couples at two frequencies, ω 1 due to positive coupling and ω 3 due to negative coupling.

Fig. 3
Fig. 3

Coupling condition for two superimposed gratings. (a) When G = G B 2 , the bandgap occurs at normal incidence at frequencies ω 2 and ω 3 . At some incident angle, the light couples at two other frequencies, ω 1 and ω 4 . (b) When G < G B 2 , the bandgap occurs when β = G B 2 , and light couples at four frequencies due to the different coupling combinations of Eq. (3).

Fig. 4
Fig. 4

Atomic force microscope image of a doubly corrugated surface grating optically inscribed on the azopolymer waveguide. The pattern is the superposition of two gratings, of approximate period 700 and 375 nm .

Fig. 5
Fig. 5

Experimental setup for measuring the dispersion curves of the waveguide gratings with polarizer P 1 and a collimating mirror.

Fig. 6
Fig. 6

Transmission curves for a single coupling grating, Λ = 704 nm at 1° intervals. The branches labeled ± G correspond to the form of Eq. (2) used for the dispersion curve.

Fig. 7
Fig. 7

Dispersion curve for a single grating shown in Fig. 6. From Eq. (2), squares represent positive coupling ( m = + 1 ) and triangles represent negative coupling ( m = 1 ) . Dotted vertical lines represent the light line at normal incidence, and data from positive and negative angles are identified.

Fig. 8
Fig. 8

Double grating where G = G B 2 , Λ = 700 nm , Λ B = 350 nm . (a) Measured data, where the gap occurs at normal incidence. (b) Dispersion curve showing experimental and theoretical results. Squares represent positive coupling and triangles represent negative coupling in Eq. (2). The bandgap is seen at normal incidence (vertical dotted line), of 0.0073 eV .

Fig. 9
Fig. 9

Double grating where G < G B 2 , Λ = 703 nm , Λ B = 338 nm . (a) Measured data. (b) Dispersion curve with measured bandgap of 0.012 eV , showing theoretical and experimental results. The vertical dotted lines are labeled by their k values. Note that the coupling due to + G (squares) overlaps with that of + G G B (filled circles). The highest-frequency branch is not shown, as it is outside the measured frequency range.

Fig. 10
Fig. 10

Double grating where G > G B 2 , Λ = 703 nm , Λ B = 365 nm . (a) Measured data. (b) Dispersion curve with measured bandgap of 0.0096 eV , showing theoretical and experimental results. The vertical dotted lines are labeled by their k values. Now the coupling due to G (triangles) overlaps with that of G + G B (filled circles). The lowest-frequency branch is not shown, as it is outside the measured frequency range.

Fig. 11
Fig. 11

Triple grating where G B 1 2 > G > G B 2 , Λ = 703 nm , Λ B 1 = 338 nm , and Λ B 2 = 365 nm . (a) Measured data. Narrow gaps appear in both main branches ( ± G ) of the dispersion curve. The other secondary branches are very weak, but visible. (b) Dispersion curve showing experimental and theoretical results with measured bandgaps of 0.005 eV , which appear at G B 1 2 and G B 2 2 . Coupling due to + G G B 2 and G + G B 1 is not within the experimental frequency range.

Equations (3)

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

cos k d = cos ( ω c a n 1 ) cos ( ω c b n 2 ) n 1 2 + n 2 2 2 n 1 n 2 sin ( ω c a n 1 ) sin ( ω c b n 2 ) ,
k 0 sin θ i + m G = β ,
k 0 sin θ i + m 1 G + m 2 G B = β ,

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