Abstract

The Bragg fibers have been extensively developed for the last 20  years. However, to improve understanding of their properties, a simple analytical theory of them is necessary. A theory based on the approximation of thin fuzzy layers by delta functions in the scalar wave equation has been developed. Two models are discussed: a plane waveguide model, and a realistic cylindrical fiber model. The former leads to simple formulas for the guided (Bragg) modes, which can be used to estimate qualitively the main properties of the Bragg fibers (including radiative losses) and optimize their structure. The latter is to be applied to calculations of parameters of realistic fiber structures. The developed theory can be useful for designing Bragg fibers, because it is simpler and in many respects closer to real fibers than the widely used model with sharply bounded homogeneous layers.

© 2009 Optical Society of America

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  1. P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
    [CrossRef]
  2. P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427-430 (1976).
    [CrossRef]
  3. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196-1201 (1978).
    [CrossRef]
  4. F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Single mode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelength,” Electron. Lett. 36, 514-515 (2000).
    [CrossRef]
  5. M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
    [CrossRef]
  6. J. Sakai, “Optical loss estimation in a Bragg fiber,” J. Opt. Soc. Am. B 24, 763-772 (2007).
    [CrossRef]
  7. E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060-1069 (2006).
    [CrossRef]
  8. D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Scalar theory of low-contrast Bragg waveguides,” Quantum Electron. 37, 873-880 (2007).
    [CrossRef]
  9. T. P. Horikis and W. L. Kath, “Modal analysis of circular Bragg fibers with arbitrary index profiles,” Opt. Lett. 31, 34-19 (2006).
    [CrossRef]
  10. J. A. M. Rojas, J. Alpuente, P. Lopez, and R. Sanchez, “Study of leaky modes in high contrast Bragg fibres,” J. Opt. A, Pure Appl. Opt. 9, 833-837 (2007).
    [CrossRef]
  11. Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
    [CrossRef] [PubMed]

2007 (4)

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Scalar theory of low-contrast Bragg waveguides,” Quantum Electron. 37, 873-880 (2007).
[CrossRef]

J. A. M. Rojas, J. Alpuente, P. Lopez, and R. Sanchez, “Study of leaky modes in high contrast Bragg fibres,” J. Opt. A, Pure Appl. Opt. 9, 833-837 (2007).
[CrossRef]

J. Sakai, “Optical loss estimation in a Bragg fiber,” J. Opt. Soc. Am. B 24, 763-772 (2007).
[CrossRef]

Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
[CrossRef] [PubMed]

2006 (4)

T. P. Horikis and W. L. Kath, “Modal analysis of circular Bragg fibers with arbitrary index profiles,” Opt. Lett. 31, 34-19 (2006).
[CrossRef]

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060-1069 (2006).
[CrossRef]

2000 (1)

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Single mode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelength,” Electron. Lett. 36, 514-515 (2000).
[CrossRef]

1978 (1)

1976 (1)

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

Alpuente, J.

J. A. M. Rojas, J. Alpuente, P. Lopez, and R. Sanchez, “Study of leaky modes in high contrast Bragg fibres,” J. Opt. A, Pure Appl. Opt. 9, 833-837 (2007).
[CrossRef]

Auguste, J.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Blondy, J.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Brechet, F.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Single mode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelength,” Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Bubnov, M. M.

Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
[CrossRef] [PubMed]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Desfarges-Berthelemot, A.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Dianov, E. M.

Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
[CrossRef] [PubMed]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Février, S.

Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
[CrossRef] [PubMed]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Golant, E. I.

E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060-1069 (2006).
[CrossRef]

Golant, K. M.

E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060-1069 (2006).
[CrossRef]

Gurjanov, A. N.

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Gurjanov, M. A.

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Hilaire, S.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Horikis, T. P.

T. P. Horikis and W. L. Kath, “Modal analysis of circular Bragg fibers with arbitrary index profiles,” Opt. Lett. 31, 34-19 (2006).
[CrossRef]

Jamier, R.

Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
[CrossRef] [PubMed]

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Kath, W. L.

T. P. Horikis and W. L. Kath, “Modal analysis of circular Bragg fibers with arbitrary index profiles,” Opt. Lett. 31, 34-19 (2006).
[CrossRef]

Kermene, V.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Khopin, V. F.

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Lavoute, L.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Leproux, P.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Likhachev, M. E.

Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
[CrossRef] [PubMed]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Lopez, P.

J. A. M. Rojas, J. Alpuente, P. Lopez, and R. Sanchez, “Study of leaky modes in high contrast Bragg fibres,” J. Opt. A, Pure Appl. Opt. 9, 833-837 (2007).
[CrossRef]

Marcou, J.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Single mode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelength,” Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Marom, E.

Pagnoux, D.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Single mode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelength,” Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Popov, A. V.

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Scalar theory of low-contrast Bragg waveguides,” Quantum Electron. 37, 873-880 (2007).
[CrossRef]

Prokopovich, D. V.

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Scalar theory of low-contrast Bragg waveguides,” Quantum Electron. 37, 873-880 (2007).
[CrossRef]

Restoin, C.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Rojas, J. A. M.

J. A. M. Rojas, J. Alpuente, P. Lopez, and R. Sanchez, “Study of leaky modes in high contrast Bragg fibres,” J. Opt. A, Pure Appl. Opt. 9, 833-837 (2007).
[CrossRef]

Roy, P.

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Single mode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelength,” Electron. Lett. 36, 514-515 (2000).
[CrossRef]

Sakai, J.

Sanchez, R.

J. A. M. Rojas, J. Alpuente, P. Lopez, and R. Sanchez, “Study of leaky modes in high contrast Bragg fibres,” J. Opt. A, Pure Appl. Opt. 9, 833-837 (2007).
[CrossRef]

Semjonov, S. L.

Yu. A. Uspenskii, E. E. Uzorin, A. V. Vinogradov, M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, R. Jamier, and S. Février, “Effect of polymer coating on leakage losses in Bragg fibers,” Opt. Lett. 32, 1202-1204 (2007).
[CrossRef] [PubMed]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Uspenskii, Yu. A.

Uzorin, E. E.

Viale, P.

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Vinogradov, A. V.

Yariv, A.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196-1201 (1978).
[CrossRef]

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

Yeh, P.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196-1201 (1978).
[CrossRef]

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

Yu. Salganskii, M.

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

C. R. Phys. (1)

P. Roy, P. Leproux, S. Février, D. Pagnoux, J. Auguste, J. Blondy, S. Hilaire, L. Lavoute, R. Jamier, A. Desfarges-Berthelemot, V. Kermene, and C. Restoin, “Photonic crystal fibres for lasers and amplifiers,” C. R. Phys. 7, 224-232 (2006).
[CrossRef]

Electron. Lett. (1)

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Single mode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelength,” Electron. Lett. 36, 514-515 (2000).
[CrossRef]

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

J. A. M. Rojas, J. Alpuente, P. Lopez, and R. Sanchez, “Study of leaky modes in high contrast Bragg fibres,” J. Opt. A, Pure Appl. Opt. 9, 833-837 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

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

Opt. Lett. (2)

Quantum Electron. (2)

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Scalar theory of low-contrast Bragg waveguides,” Quantum Electron. 37, 873-880 (2007).
[CrossRef]

M. E. Likhachev, S. L. Semjonov, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Yu. Salganskii, M. A. Gurjanov, A. N. Gurjanov, R. Jamier, P. Viale, S. Février, and J. Blondy, “Development and study of Bragg fibres with a large mode field and low optical losses,” Quantum Electron. 36, 581-586 (2006).
[CrossRef]

Tech. Phys. (1)

E. I. Golant and K. M. Golant, “New method for calculating the spectra and radiation losses of leaky waves in multilayer optical waveguides,” Tech. Phys. 51, 1060-1069 (2006).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Loss decrements of the following modes of the Bragg fiber: (1,0) (thick black curve) and (2,0) (thin black curve), (1,1) (thick black dashed curve) and (2,1) (thin black dashed curve), (1,2) (thick black dotted-dashed curve) and (2,2) (thin black dotted-dashed curve).

Fig. 2
Fig. 2

Group speed dispersions of the following modes of the Bragg fiber: (1,0) (thick black curve) and (2,0) (thin black curve), (1,1) (thick black dashed curve) and (2,1) (thin black dashed curve), (1,2) (thick black dotted-dashed curve) and (2,3) (thin black dotted-dashed curve).

Fig. 3
Fig. 3

Fields of the modes of the Bragg fiber: (1,0) (thick black curve) and (2,0) (thin black curve), (1,1) (thick black dashed curve) and (2,1) (thin black dashed curve), (1,2) (thick black dotted-dashed curve) and (2,2) (thin black dotted-dashed curve). Vertical lines mark the positions of the delta functions. The wavelength is λ = 0.9 μ m , which corresponds to the minimum of losses in Fig. 3.

Fig. 4
Fig. 4

Comparison of the losses in the planar waveguide in Bragg fiber with n = 0 . The losses for the (1,0) (thick black curve) and (2,0) (thick black dashed curve) modes of the fiber, and for the first (thin black curve) and the second (thin black dashed curve) of the planar waveguide are shown.

Equations (72)

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

ε ( x ) = ε 1 + α m = 0 δ ( x m l ) ,
E ( x ) + k 2 ( ε ( x ) p 2 ) E ( x ) = 0 ,
E ( x ) = C 1 m exp ( i ϰ ( x m l ) ) + C 2 m exp ( i ϰ ( x m l ) ) ,
R = C 20 C 10 .
C m 1 = M T C m ,
M T = 1 cos φ ( exp ( i ( l ϰ + φ ) ) i exp ( i l ϰ ) sin φ i exp ( i l ϰ ) sin φ exp ( i ( l ϰ + φ ) ) , ) , a = k 2 α 2 ϰ , tan φ = a , 0 < φ < π 2 .
M T C = μ C ;
μ 1 , 2 = A ̃ ± A ̃ 2 1 , A ̃ = cos ( l ϰ + φ ) cos φ ,
x 1 = ( i a exp ( i l ϰ ) 1 + a 2 exp ( i ( l ϰ + φ ) ) μ 1 ) , x 2 = ( 1 + a 2 exp ( i ( l ϰ + φ ) ) μ 2 i a exp ( i l ϰ ) ) .
R = e i l ϰ [ A i 1 A 2 ] , A = sin ( l ϰ + φ ) sin φ ,
sin ( l ϰ + φ ) = ± sin φ ,
2 l ε 1 p 2 = n λ , 2 l ε 1 p 2 = ( 2 φ π + n ) λ .
2 l ε 1 p 2 = λ ( φ π + n ) .
μ min = a ± 1 + a 2 = ± cot ( φ 2 + π 4 ) ,
φ π λ α ε 1 p 2 .
2 l ε 1 p 2 = α ε 1 p 2 + n λ ,
2 l ε 1 p 2 = λ n , 2 l ε 1 p 2 = 2 α ε 1 p 2 + λ n .
2 l ε 1 p 2 = λ ( 1 2 + n ) ,
2 l ε 1 p 2 = λ n , 2 l ε 1 p 2 = λ ( 1 + n ) .
ε ( x ) = { ε 0 x < d 2 ε 1 + α m = 0 δ ( x b d 2 m l ) x d 2 } ,
E ( x ) = E 0 cos ( η x ) , η = k ε 0 p 2 , x < d 2 .
tan ( η d 2 ) = i ϰ η 1 R exp ( 2 i b ϰ ) 1 + R exp ( 2 i b ϰ ) ,
η tan η d 2 = ϰ cot ( b ϰ l ϰ 2 γ 2 + π 4 ) , cos ( l ϰ + φ ) < 0 ,
η tan η d 2 = ϰ cot ( b ϰ l ϰ 2 + γ 2 π 4 ) , cos ( l ϰ + φ ) > 0 ,
γ = arcsin ( sin ( l ϰ + φ ) sin φ ) , π 2 < γ < π 2 .
η tan η d 2 = ϰ cot ( b ϰ ) .
η tan η d 2 = η = π d + 2 π m d , m = 0 , 1 , 2 , .
ε 0 p 2 = ( λ d ) 2 ( 1 2 + m ) 2 .
ϰ = 2 π λ ε 1 ε 0 .
cos ( 2 b ϰ l ϰ ) sin φ = sin ( l ϰ + φ ) .
sin l ϰ = 0 l = π m ϰ , m = 1 , 2 , ,
ε ( x ) = { ε 0 x < d 2 ε 1 + α m = 0 N δ ( x b d 2 m l ) d 2 x d 2 + ( N 1 ) l + b , ε 1 x > d 2 + ( N 1 ) l + b }
η tan ( η d 2 ) = i ϰ 1 R N exp ( 2 i b ϰ ) 1 + R N exp ( 2 i b ϰ ) ,
( M T ) N = 1 R ̃ R ( R ̃ μ 1 N R μ 2 N ( μ 1 N μ 2 N ) R R ̃ ( μ 1 N μ 2 N ) R ̃ μ 2 N R μ 1 N ) ,
R ̃ = e i l ϰ ( A ± i 1 A 2 ) .
R N = R R ̃ 1 ξ N R ̃ R ξ N , ξ = μ 2 μ 1 ,
η tan ( η d 2 ) = i ϰ 1 R 1 + R 1 ξ N R R ̃ 1 R ̃ 1 R 1 ξ N R R ̃ 1 + R ̃ 1 + R .
η tan ( η d 2 ) i ϰ 1 R 1 + R [ 1 + ξ N 2 R ( R ̃ R ) R ̃ ( 1 R 2 ) ] .
δ η 2 π 2 m + 1 1 ϰ d 2 1 R 1 + R [ 1 ξ N 2 R ( R ̃ R ) R ̃ ( 1 R 2 ) ] ,
δ η 2 π 2 m + 1 ξ N ϰ d 2 Re [ 2 R ( R ̃ R ) R ̃ ( 1 R ) 2 ] ,
Re [ 2 R ( R ̃ R ) R ̃ ( 1 R ) 2 ] = 2 < 0 .
d 2 E d ρ 2 + 1 ρ d E d ρ + ( k 2 ε ( ρ ) p 2 n 2 ρ 2 ) E = 0 ,
ε = ε 1 + α δ ( ρ ρ 0 ) ,
E ( ρ ) = { C 1 H n ( 1 ) ( ϰ ρ ) + C 2 H n ( 2 ) ( ϰ ρ ) , ρ < ρ 0 C 1 H n ( 1 ) ( ϰ ρ ) + C 2 H n ( 2 ) ( ϰ ρ ) , ρ > ρ 0 } ϰ = k ε 0 p 2 ,
( C 1 C 2 ) = M ( C 1 C 2 ) .
E ( ρ 0 + 0 ) E ( ρ 0 0 ) = α k 2 E ( ρ 0 ) .
M = ( m 11 m 12 m 21 m 22 ) ,
m 11 = 1 + k 2 α ϰ 1 M 0 , m 12 = k 2 α ϰ 1 M 0 H n ( 2 ) ( ϰ ρ 0 ) H n ( 1 ) ( ϰ ρ 0 ) ,
m 21 = k 2 α ϰ 1 M 0 H n ( 1 ) ( ϰ ρ 0 ) H n ( 2 ) ( ϰ ρ 0 ) , m 22 = 1 k 2 α ϰ 1 M 0 .
M 0 = H n + 1 ( 1 ) ( ϰ ρ 0 ) H n ( 1 ) ( ϰ ρ 0 ) + H n + 1 ( 2 ) ( ϰ ρ 0 ) H n ( 2 ) ( ϰ ρ 0 ) .
R = m 21 m 11 = k 2 α ϰ H n ( 1 ) ( ϰ ρ 0 ) H n ( 2 ) ( ϰ ρ 0 ) H n + 1 ( 1 ) ( ϰ ρ 0 ) H n ( 1 ) ( ϰ ρ 0 ) + H n + 1 ( 2 ) ( ϰ ρ 0 ) H n ( 2 ) ( ϰ ρ 0 ) + k 2 α ϰ .
ε ( ρ ) = ε 1 + q = 0 N 1 α q δ ( ρ ρ q ) ,
E ( ρ ) = C 1 ( q ) H n ( 1 ) ( ϰ ρ ) H n ( 1 ) ( ϰ ρ q ) + C 2 ( q ) H n ( 2 ) ( ϰ ρ ) H n ( 2 ) ( ϰ ρ q ) , ρ q 1 < ρ < ρ q ,
T q = ( H n ( 1 ) ( ϰ ρ q 1 ) H n ( 1 ) ( ϰ ρ q ) 0 0 H n ( 2 ) ( ϰ ρ q 1 ) H n ( 2 ) ( ϰ ρ q ) ) ,
M q = ( 1 i a q i a q i a q 1 + i a q ) , a q = i α q k 2 ϰ 1 M 0 q ,
M 0 q = H n + 1 ( 1 ) ( ϰ ρ q ) H n ( 1 ) ( ϰ ρ q ) + H n + 1 ( 2 ) ( ϰ ρ q ) H n ( 2 ) ( ϰ ρ q ) .
( C 1 ( q ) C 2 ( q ) ) = M q T q + 1 ( C 1 ( q + 1 ) C 2 ( q + 1 ) ) .
Q + = q = 0 N 1 M q T q + 1 ,
R + = C 2 ( 0 ) C 1 ( 0 ) = ( Q + ) 12 ( Q + ) 11 .
R q = C 2 ( q ) C 1 ( q ) .
R q = R 0 q + Q 0 q S q R q + 1 1 R 0 q S q R q + 1 ,
R 0 q = i a q 1 i a q , Q 0 q = 1 + i a q 1 i a q , S q = H n ( 2 ) ( ϰ ρ q ) H n ( 1 ) ( ϰ ρ q ) H n ( 1 ) ( ϰ ρ q + 1 ) H n ( 2 ) ( ϰ ρ q + 1 ) .
ε ( ρ ) = { ε 0 ρ < d 2 ε 1 + q = 0 N 1 α q δ ( ρ ρ q ) ρ d 2 ε 1 ρ > ρ N 1 } ρ 0 = d 2 + b .
E ( ρ ) = E 0 J n ( η ρ ) , ρ < d 2 , η = k ε 0 p 2 , E 0 = 1 .
E ( ρ ) = A H n ( 1 ) ( ϰ ρ ) H n ( 1 ) ( ϰ ( d 2 + b ) ) + B H n ( 2 ) ( ϰ ρ ) H n ( 2 ) ( ϰ ( d 2 + b ) ) , d 2 < ρ < d 2 + b .
B A = R + ,
η J n + 1 ( η d 2 ) J n ( η d 2 ) = ϰ 1 + R + S 0 [ H n + 1 ( 1 ) ( ϰ d 2 ) H n ( 1 ) ( ϰ d 2 ) + R + S 0 H n + 1 ( 2 ) ( ϰ d 2 ) H n ( 2 ) ( ϰ d 2 ) ] ,
S 0 = H n ( 2 ) ( ϰ d 2 ) H n ( 1 ) ( ϰ d 2 ) H n ( 1 ) ( ϰ ( d 2 + b ) ) H n ( 2 ) ( ϰ ( d 2 + b ) ) .
J n ( η d 2 ) = 0 ,
d B = k Im p = Im k 2 ε 0 η 2 ,
A = J n ( η d 2 ) H n ( 1 ) ( ϰ ( d 2 + b ) ) H n ( 1 ) ( ϰ d 2 ) H n + 1 ( 2 ) ( ϰ d 2 ) H n ( 2 ) ( ϰ d 2 ) η ϰ J n + 1 ( η d 2 ) J n ( η d 2 ) H n + 1 ( 1 ) ( ϰ d 2 ) H n ( 1 ) ( ϰ d 2 ) η ϰ H n + 1 ( 2 ) ( ϰ d 2 ) H n ( 2 ) ( ϰ d 2 ) ,
B = J n ( η d 2 ) H n ( 2 ) ( ϰ ( d 2 + b ) ) H n ( 2 ) ( ϰ d 2 ) H n + 1 ( 1 ) ( ϰ d 2 ) H n ( 1 ) ( ϰ d 2 ) η ϰ J n + 1 ( η d 2 ) J n ( η d 2 ) H n + 1 ( 1 ) ( ϰ d 2 ) H n ( 1 ) ( ϰ d 2 ) η ϰ H n + 1 ( 2 ) ( ϰ d 2 ) H n ( 2 ) ( ϰ d 2 ) .

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