Abstract

A new approximate theory was developed and applied to analysis of the second-order Bragg diffraction by a thick reflection grating formed in a medium with and without optical gain. To derive the general system of equations describing the optical wave interaction with a grating, the method of variation of constants was used, which allowed us to obtain the analytical formulas for the electric-field strength of transmitted and reflected waves. The proposed approach was extended to the case of grating formed in a material with nonlinear response to the recording field when dielectric permittivity modulation of a medium includes higher spatial harmonics.

© 2012 Optical Society of America

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    [CrossRef]
  2. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  10. T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105–109 (2001).
    [CrossRef]
  11. Y. Oki, S. Miyamoto, M. Maeda, and N. J. Vasa, “Multiwavelength distributed-feedback dye laser array and its application to spectroscopy,” Opt. Lett. 27, 1220–1222 (2002).
    [CrossRef]
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  13. F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
    [CrossRef]
  14. T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
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    [CrossRef]
  23. V. M. Fitio and T. N. Smirnova, “Analysis of light wave diffraction and amplification by reflection grating operating in the second-order Bragg regime. II. Reflectivity and spectral characteristics of a grating,”J. Opt. Soc. Am. B, under revision.
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2011

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

2009

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

M. V. Vasnetsov, V.Yu. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684–690 (2009).
[CrossRef]

2008

V. M. Fitio, O. V Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

2006

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

2005

2003

V. M. Fitio and Y. V. Bobitski, “Diffraction on TE and TM polarization optical waves in non-absorptive medium with periodical variation of the dielectric constant,” Ukr. Phys. J. 48, 1046–1054 (2003).

2002

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Y. Oki, S. Miyamoto, M. Maeda, and N. J. Vasa, “Multiwavelength distributed-feedback dye laser array and its application to spectroscopy,” Opt. Lett. 27, 1220–1222 (2002).
[CrossRef]

2001

G. A. Turnbul, P. Andrew, M. J. Jori, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122 (2001).
[CrossRef]

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105–109 (2001).
[CrossRef]

1999

1997

M. Maeda, Y. Oki, and K. Imamura, “Ultrashort pulse generation from an integrated single-chip dye laser,” IEEE J. Quantum Electron. 33, 2146–2149 (1997).
[CrossRef]

1983

1982

1974

S. Wang, “Principles of distributed feedback and distributed Bragg reflector lasers,” IEEE J. Quantum Electron. 10, 413–427(1974).
[CrossRef]

1972

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306–308 (1972).
[CrossRef]

1971

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
[CrossRef]

Abbate, G.

Andrew, P.

G. A. Turnbul, P. Andrew, M. J. Jori, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122 (2001).
[CrossRef]

Balslev, S.

Barnes, W. L.

G. A. Turnbul, P. Andrew, M. J. Jori, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122 (2001).
[CrossRef]

Bazhenov, V.Yu.

Bjorkholm, J. E.

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306–308 (1972).
[CrossRef]

Bobitski, Y. V.

V. M. Fitio and Y. V. Bobitski, “Diffraction on TE and TM polarization optical waves in non-absorptive medium with periodical variation of the dielectric constant,” Ukr. Phys. J. 48, 1046–1054 (2003).

Efendiev, T.S.

T.S. Efendiev, V. M. Katarkevich, A. N. Rubinov, and V. A. Zaporozhchenko, “A compact picosecond DFB laser on dye-doped jelly-like gelatin,” in Collected articles “Laser and opto-electronic engineering” (Minsk, 2002), pp. 26–30.

Fitio, V. M.

V. M. Fitio, O. V Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

V. M. Fitio and Y. V. Bobitski, “Diffraction on TE and TM polarization optical waves in non-absorptive medium with periodical variation of the dielectric constant,” Ukr. Phys. J. 48, 1046–1054 (2003).

V. M. Fitio and T. N. Smirnova, “Analysis of light wave diffraction and amplification by reflection grating operating in the second-order Bragg regime. II. Reflectivity and spectral characteristics of a grating,”J. Opt. Soc. Am. B, under revision.

Galstian, T. V.

Gaylord, T. K.

Gersborg-Hansen, M.

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

Imamura, K.

M. Maeda, Y. Oki, and K. Imamura, “Ultrashort pulse generation from an integrated single-chip dye laser,” IEEE J. Quantum Electron. 33, 2146–2149 (1997).
[CrossRef]

Iskra, K. F.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Jori, M. J.

G. A. Turnbul, P. Andrew, M. J. Jori, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122 (2001).
[CrossRef]

Katarkevich, V. M.

T.S. Efendiev, V. M. Katarkevich, A. N. Rubinov, and V. A. Zaporozhchenko, “A compact picosecond DFB laser on dye-doped jelly-like gelatin,” in Collected articles “Laser and opto-electronic engineering” (Minsk, 2002), pp. 26–30.

Kavc, T.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Kern, W.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
[CrossRef]

Kranzelbinder, G.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Kriensen, A.

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

Kristensen, A.

Kzianzou, V.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

Langer, G.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

LaRoshelle, S.

Maeda, M.

Y. Oki, S. Miyamoto, M. Maeda, and N. J. Vasa, “Multiwavelength distributed-feedback dye laser array and its application to spectroscopy,” Opt. Lett. 27, 1220–1222 (2002).
[CrossRef]

M. Maeda, Y. Oki, and K. Imamura, “Ultrashort pulse generation from an integrated single-chip dye laser,” IEEE J. Quantum Electron. 33, 2146–2149 (1997).
[CrossRef]

Marom, E.

Maschke, D.

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Miyamoto, S.

Moharam, M. G.

Monguzzi, A.

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Neger, T.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Oki, Y.

Y. Oki, S. Miyamoto, M. Maeda, and N. J. Vasa, “Multiwavelength distributed-feedback dye laser array and its application to spectroscopy,” Opt. Lett. 27, 1220–1222 (2002).
[CrossRef]

M. Maeda, Y. Oki, and K. Imamura, “Ultrashort pulse generation from an integrated single-chip dye laser,” IEEE J. Quantum Electron. 33, 2146–2149 (1997).
[CrossRef]

Ozin, G. A.

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Pogantsch, A.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Puzzo, D. P.

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Richardson, K.

Rubinov, A. N.

T.S. Efendiev, V. M. Katarkevich, A. N. Rubinov, and V. A. Zaporozhchenko, “A compact picosecond DFB laser on dye-doped jelly-like gelatin,” in Collected articles “Laser and opto-electronic engineering” (Minsk, 2002), pp. 26–30.

Sakhno, O. V

V. M. Fitio, O. V Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

Sakhno, O. V.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

Saliminia, A.

Samuel, I. D. W.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

G. A. Turnbul, P. Andrew, M. J. Jori, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122 (2001).
[CrossRef]

Schade, W.

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105–109 (2001).
[CrossRef]

Scheel, D.

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105–109 (2001).
[CrossRef]

Schrader, S.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

Scotognella, F.

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306–308 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
[CrossRef]

Slussarenko, S. S.

Smirnova, T. N.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

V. M. Fitio, O. V Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

V. M. Fitio and T. N. Smirnova, “Analysis of light wave diffraction and amplification by reflection grating operating in the second-order Bragg regime. II. Reflectivity and spectral characteristics of a grating,”J. Opt. Soc. Am. B, under revision.

Stumpe, J.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

Toussaere, E.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Tubino, R.

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Turnbul, G. A.

G. A. Turnbul, P. Andrew, M. J. Jori, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122 (2001).
[CrossRef]

Turnbull, G. A.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

Vasa, N. J.

Vasnetsov, M. V.

Villeneuve, A.

Voss, T.

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105–109 (2001).
[CrossRef]

Wang, S.

S. Wang, “Principles of distributed feedback and distributed Bragg reflector lasers,” IEEE J. Quantum Electron. 10, 413–427(1974).
[CrossRef]

Wiersma, D. S.

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Yariv, A.

A. Yariv and R. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Yeh, R.

A. Yariv and R. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Zaporozhchenko, V. A.

T.S. Efendiev, V. M. Katarkevich, A. N. Rubinov, and V. A. Zaporozhchenko, “A compact picosecond DFB laser on dye-doped jelly-like gelatin,” in Collected articles “Laser and opto-electronic engineering” (Minsk, 2002), pp. 26–30.

Zylberberg, Z.

Appl. Phys. B

T. Voss, D. Scheel, and W. Schade, “A microchip-laser-pumped DFB-polymer-dye laser,” Appl. Phys. B 73, 105–109 (2001).
[CrossRef]

Appl. Phys. Lett.

H. Kogelnik and C. V. Shank, “Stimulated emission in periodic structure,” Appl. Phys. Lett. 18, 152–154 (1971).
[CrossRef]

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306–308 (1972).
[CrossRef]

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

Chem. Mater.

T. Kavc, G. Langer, W. Kern, G. Kranzelbinder, E. Toussaere, G. A. Turnbull, I. D. W. Samuel, K. F. Iskra, T. Neger, and A. Pogantsch, “Index and relief gratings in polymer films for organic distributed feedback lasers,” Chem. Mater. 14, 4178–4185 (2002).
[CrossRef]

IEEE J. Quantum Electron.

M. Maeda, Y. Oki, and K. Imamura, “Ultrashort pulse generation from an integrated single-chip dye laser,” IEEE J. Quantum Electron. 33, 2146–2149 (1997).
[CrossRef]

S. Wang, “Principles of distributed feedback and distributed Bragg reflector lasers,” IEEE J. Quantum Electron. 10, 413–427(1974).
[CrossRef]

J. Appl. Phys.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Lightwave Technol.

J. Opt.

T. N. Smirnova, O. V. Sakhno, J. Stumpe, V. Kzianzou, and S. Schrader, “Distributed feedback lasing in dye-doped nanocomposite holographic transmission gratings,” J. Opt. 13, 035709 (2011).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Optik

V. M. Fitio, O. V Sakhno, and T. N. Smirnova, “Analysis of the diffraction by the gratings generated in the materials with a nonlinear response,” Optik 119, 236–246 (2008).
[CrossRef]

Phys. Rev. B

G. A. Turnbul, P. Andrew, M. J. Jori, W. L. Barnes, and I. D. W. Samuel, “Relationship between photonic band structure and emission characteristics of a polymer distributed feedback laser,” Phys. Rev. B 64, 125122 (2001).
[CrossRef]

Small

F. Scotognella, D. P. Puzzo, A. Monguzzi, D. S. Wiersma, D. Maschke, R. Tubino, and G. A. Ozin, “Nanoparticle one-dimensional photonic-crystal dye laser,” Small 5, 2048–2052 (2009).
[CrossRef]

Ukr. Phys. J.

V. M. Fitio and Y. V. Bobitski, “Diffraction on TE and TM polarization optical waves in non-absorptive medium with periodical variation of the dielectric constant,” Ukr. Phys. J. 48, 1046–1054 (2003).

Other

V. M. Fitio and T. N. Smirnova, “Analysis of light wave diffraction and amplification by reflection grating operating in the second-order Bragg regime. II. Reflectivity and spectral characteristics of a grating,”J. Opt. Soc. Am. B, under revision.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Mechanics (Pergamon, 1960).

P. A. M. Dirac, The Principles of Quantum Mechanics(Clarendon, 1958).

L. S. Pontryagin, Normal Differential Equations (Nauka, 1974).

A. Yariv and R. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

T.S. Efendiev, V. M. Katarkevich, A. N. Rubinov, and V. A. Zaporozhchenko, “A compact picosecond DFB laser on dye-doped jelly-like gelatin,” in Collected articles “Laser and opto-electronic engineering” (Minsk, 2002), pp. 26–30.

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Figures (3)

Fig. 1.
Fig. 1.

Scheme of the wave propagation in the grating (a) and a vector diagram of Bragg diffraction for the first-order resonance (b) and the second-order resonance (c).

Fig. 2.
Fig. 2.

Dependences of reflectivity on the grating thickness (a) and on the propagation angle of an optical wave in the grating (b): curve 1, d = 360 Λ ; curve 2, d = 1000 Λ . Continuous curves are calculated by Eq. (15). The points are obtained by the FFE method while taking into account nine terms of Fourier expansion.

Fig. 3.
Fig. 3.

Modulus of the exact solution (continuous curve) and the approximate one (points).

Equations (47)

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ε 2 ( z ) = R ε 20 + i I ε 20 + ε ˜ 1 2 [ exp ( i K z ) + exp ( i K z ) ] + ε ˜ 2 2 [ exp ( i 2 K z ) + exp ( i 2 K z ) ] ,
k 1 k 0 + K ,
k 1 k 0 + 2 K ,
E ( x , z ) = E t ( x , z ) + E r ( x , z ) = t ( z ) exp [ i ( k x x k z z ) ] + r ( z ) exp [ i ( k x x + k z z ) ] .
2 E ( x , z ) d z 2 + 2 E ( x , z ) d x 2 + ( 2 π λ ) 2 R ε 20 E ( x , z ) = 0.
2 E ( x , z ) d z 2 + 2 E ( x , z ) d x 2 + ( 2 π λ ) 2 { R ε 20 + i I ε 20 + ε ˜ 1 2 [ exp ( i K z ) + exp ( i K z ) ] } E ( x , z ) = 0.
t ( z ) = t 1 ( z ) exp ( i K z ) + t 0 ( z ) + t 1 ( z ) exp ( i K z ) ,
r ( z ) = r 1 ( z ) exp ( i K z ) + r 0 ( z ) + r 1 ( z ) exp ( i K z ) ,
E t ( x , z ) = [ cosh ( s z ) 1 s ( Δ k z + i g 2 χ 1 ) sin ( s z ) ] × ( κ 1 k z 2 + i γ + exp ( i K z ) + κ 1 ( k z + K ) 2 k z 2 i γ exp ( i 2 K z ) ) exp ( i k x x ) ,
E r ( x , z ) = χ 2 s sinh ( s z ) × ( κ 1 k z 2 + i γ + exp ( i K z ) + κ 1 ( k z + K ) 2 k z 2 i γ exp ( i 2 K z ) ) exp ( i k x x ) ,
g = γ k z , χ 1 = κ 1 2 2 k z [ 1 k z 2 + i γ 1 ( k z + K ) 2 k z 2 i γ ] , χ 2 = κ 1 2 2 k z ( k z 2 + i γ ) , γ = ( 2 π / λ ) 2 I ε 20 .
r ¯ ( m Λ ) = E r ( x , m Λ ) E t ( x , m Λ ) = χ 2 sinh ( s m Λ ) s cosh ( s m Λ ) ( Δ k z + i g 2 χ 1 ) sinh ( s m Λ ) ,
t ¯ ( m Λ ) = E t ( 0 ) E t ( x , m Λ ) = s s cosh ( sm Λ ) ( Δ k z + i g 2 χ 1 ) sinh ( sm Λ ) ,
s = ± [ χ 2 2 ( Δ k z + i g 2 χ 1 ) 2 ] 1 / 2 .
T ( m Λ ) = | t ¯ ( m Λ ) | 2 , R ( m Λ ) = | r ¯ ( m Λ ) | 2 .
t ( z ) = j = 2 2 t j ( z ) exp ( i j K z ) , r ( z ) = j = 2 2 r j ( z ) exp ( i j K z ) .
2 E ( x , z ) d z 2 + 2 E ( x , z ) d x 2 + ( 2 π λ ) 2 { R ε 20 + i I ε 20 + ε ˜ 1 2 [ exp ( i K z ) + exp ( i K z ) ] + ε ˜ 2 2 [ exp ( i 2 K z ) + exp ( i 2 K z ) ] } E ( x , z ) = 0.
E t ( x , z ) = [ cosh ( sz ) 1 s ( Δ k z + i g 2 χ 1 ) sin ( sz ) ] × ( κ 1 k z 2 + i γ + exp ( iKz ) + κ 1 ( k z + K ) 2 k z 2 i γ exp ( i 2 K z ) ) exp ( i k x x ) ,
E r ( x , z ) = ( χ 2 χ ) s sinh ( sz ) × ( κ 1 k z 2 + i γ + exp ( i K z ) + κ 1 ( k z + K ) 2 k z 2 i γ exp ( i 2 K z ) ) exp ( i k x x ) ,
s = ± [ ( χ 2 χ ) 2 ( Δ k z + i g 2 χ 1 ) 2 ] 1 / 2 .
r ¯ ( m Λ ) = E r ( x , m Λ ) E t ( x , m Λ ) = ( χ 2 χ ) sinh ( s m Λ ) s cosh ( s m Λ ) ( Δ k z + i g 2 χ 1 ) sinh ( s m Λ ) ,
t ¯ ( m Λ ) = E t ( 0 ) E t ( x , m Λ ) = s s cosh ( s m Λ ) ( Δ k z + i g 2 χ 1 ) sinh ( s m Λ ) .
{ 2 i k z d t 0 d z + i γ t 0 + κ 1 [ t 1 + t 1 + r 1 exp ( i 2 Δ k z z ) ] = 0 , 2 i k z d r 0 d z + i γ r 0 + κ 1 [ t 1 exp ( i 2 Δ k z z ) + r 1 + r 1 ] = 0 ,
{ ( k z + K ) 2 t 1 + k z 2 t 1 + i γ t 1 + κ 1 t 0 = 0 , ( k z + K ) 2 r 1 + k z 2 r 1 + i γ r 1 + κ 1 r 0 = 0 ,
( k z 2 t 1 + i γ t 1 + κ 1 t 0 ) + ( k z 2 r 1 + i γ r 1 + κ 1 r 0 ) exp ( i 2 Δ k z z ) = 0 ,
t 1 = κ 1 t 0 / [ ( k z + K ) 2 k z 2 i γ ] ,
r 1 = κ 1 r 0 / [ ( k z + K ) 2 k z 2 i γ ] .
t 1 + r 1 exp ( i 2 Δ k z z ) = κ 1 [ t 0 + r 0 exp ( i 2 Δ k z z ) ] / ( k z 2 + i γ ) ,
t 1 exp ( i 2 Δ k z z ) + r 1 = κ 1 [ t 0 exp ( i 2 Δ k z z ) + r 0 ] / ( k z 2 + i γ ) .
{ d t 0 d z + i g 2 t 0 + i χ 1 t 0 + i χ 2 r 0 exp ( i 2 Δ k z z ) = 0 , d r 0 d z i g 2 t 0 i χ 2 t 0 exp ( i 2 Δ k z z ) i χ 1 r 0 = 0 ,
{ d t ˜ 0 d z = i ( Δ k z + i g 2 χ 1 ) t ˜ 0 i χ 2 r ˜ 0 , d r ˜ 0 d z = i χ 2 t ˜ 0 i ( Δ k z + i g 2 χ 1 ) r ˜ 0 .
s 2 + ( Δ k z + i g / 2 χ 1 ) 2 χ 2 2 = 0 ,
s = ± [ χ 2 2 ( Δ k z + i g / 2 χ 1 ) 2 ] 1 / 2 .
t ˜ 0 ( z ) = C 1 exp ( s z ) + C 2 exp ( s z ) ,
r ˜ 0 ( z ) = C 3 exp ( s z ) + C 4 exp ( s z ) .
{ C 1 + C 2 = 1 , s C 1 s C 2 = i ( Δ k z + i g / 2 χ 1 ) ,
{ C 3 + C 4 = 0 , s C 3 s C 4 = i χ 2 .
{ 2 i k z d t 0 d z + i γ t 0 + κ 1 [ t 1 + t 1 + r 1 exp ( i 2 Δ k z z ) ] + κ 2 r 0 exp ( i 2 Δ k z z ) = 0 , 2 i k z d r 0 d z + i γ r 0 + κ 1 [ t 1 exp ( i 2 Δ k z z ) + r 1 + r 1 ] + κ 2 t 0 exp ( i 2 Δ k z z ) = 0 ,
{ ( k z + K ) 2 t 1 + k z 2 t 1 + i γ t 1 + κ 1 t 0 + κ 2 t 1 + κ 2 r 1 exp ( i 2 Δ k z ) = 0 , ( k z + K ) 2 r 1 + k z 2 r 1 + i γ r 1 + κ 1 r 0 + κ 2 r 1 + κ 2 t 1 exp ( i 2 Δ k z ) = 0 ,
( k z 2 t 1 + i γ t 1 + κ 1 t 0 + κ 2 t 1 ) + ( k z 2 r 1 + i γ r 1 + κ 1 r 0 + κ 2 r 1 ) exp ( i 2 Δ k z z ) = 0 ,
{ d t ˜ 0 d z = i ( Δ k z + i g 2 χ 1 ) t ˜ 0 i ( χ 2 χ ) r ˜ 0 , d r ˜ 0 d z = i ( χ 2 χ ) t ˜ 0 i ( Δ k z + i g 2 χ 1 ) r ˜ 0 ,
s 2 + ( Δ k z + i g / 2 χ 1 ) 2 ( χ 2 χ ) 2 = 0.
d t ( z ) d z = i Ω t ( z ) + F ( z ) ,
t ( z ) = i F ( z ) Ω .
t ( z ) = exp ( i Ω z ) [ C + 0 z exp ( i Ω z ) F ( z ) d z ] ,
t ( z ) = exp ( i Ω z ) [ exp ( i Ω z ) F ( z ) i Ω + 1 i Ω 0 z exp ( i Ω z ) d F ( z ) d z d z ] .
t ( z ) = exp ( i Ω z ) i Ω + exp ( s z ) s i Ω exp ( i Ω z ) s i Ω .

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