Abstract

The effect of periodic loss on the performance of refractive-index gratings has been studied in detail. It is shown that the loss periodicity and relative phase strongly affects the symmetry of the reflection, transmission, and loss spectra. This asymmetry is explained successfully through consideration of the overlap between the standing-wave intensity distribution and the periodic loss pattern.

© 2002 Optical Society of America

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References

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  1. Y. J. Chen, A. W. Snyder, D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
    [Crossref]
  2. Special issue on fiber gratings, photosensitivity, and poling, J. Lightwave Technol. 15, 1261–1512 (1997).
  3. A. V. Kavokin, M. A. Kaliteevski, “Light-absorption effect on Bragg interference in multilayer semiconductor heterostructures,” J. Appl. Phys. 79, 595–598 (1996).
    [Crossref]
  4. Y. Boucher, “Influence of a localized scattering center upon the spectral characteristics of a distributed-feedback structure,” IEEE Photon. Technol. Lett. 9, 1454–1456 (1997).
    [Crossref]
  5. P. S. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
    [Crossref]
  6. T. Fessant, Y. Boucher, “Additional modal selectivity induced by a localized defect in quarter-wave-shifted DFB lasers,” IEEE J. Quantum Electron. 34, 602–608 (1998).
    [Crossref]
  7. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [Crossref]
  8. D. A. Cardimona, M. P. Sharma, V. Kovanis, A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
    [Crossref]
  9. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [Crossref] [PubMed]
  10. D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side- written fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 50–52.
  11. V. Grubsky, D. S. Starodubov, J. Feinberg, “Photochemical reaction of hydrogen with germanosilicate glass initiated by 3.4–5.4-eV ultraviolet light,” Opt. Lett. 24, 729–731 (1999).
    [Crossref]
  12. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
    [Crossref]
  13. M. J. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1988).
    [Crossref]
  14. R. B. Bylsma, D. H. Olson, A. M. Glass, “Photochromic gratings in photorefractive materials,” Opt. Lett. 13, 853–855 (1988).
    [Crossref] [PubMed]
  15. R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 15, 1326–1332 (1991).
    [Crossref]
  16. M. Liphardt, S. Ducharme, “Measurement of the photorefractive grating phase shift in a polimer by an ac phase-modulation technique,” J. Opt. Soc. Am. B 15, 2154–2160 (1998).
    [Crossref]
  17. M. Gehrtz, J. Pinsl, C. Bräuchle, “Sensitive detection of phase and absorption gratings: phase-modulated, homodyne detected holography,” Appl. Phys. B 43, 61–77 (1987).
    [Crossref]
  18. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  19. L. Dong, W. F. Liu, L. Reekie, “Negative index gratings formed at 193-nm excimer laser,” Opt. Lett. 21, 2032–2034 (1996).
    [Crossref] [PubMed]
  20. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [Crossref]
  21. D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorbance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
    [Crossref]
  22. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

1999 (1)

1998 (3)

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

M. Liphardt, S. Ducharme, “Measurement of the photorefractive grating phase shift in a polimer by an ac phase-modulation technique,” J. Opt. Soc. Am. B 15, 2154–2160 (1998).
[Crossref]

T. Fessant, Y. Boucher, “Additional modal selectivity induced by a localized defect in quarter-wave-shifted DFB lasers,” IEEE J. Quantum Electron. 34, 602–608 (1998).
[Crossref]

1997 (3)

Special issue on fiber gratings, photosensitivity, and poling, J. Lightwave Technol. 15, 1261–1512 (1997).

Y. Boucher, “Influence of a localized scattering center upon the spectral characteristics of a distributed-feedback structure,” IEEE Photon. Technol. Lett. 9, 1454–1456 (1997).
[Crossref]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

1996 (2)

L. Dong, W. F. Liu, L. Reekie, “Negative index gratings formed at 193-nm excimer laser,” Opt. Lett. 21, 2032–2034 (1996).
[Crossref] [PubMed]

A. V. Kavokin, M. A. Kaliteevski, “Light-absorption effect on Bragg interference in multilayer semiconductor heterostructures,” J. Appl. Phys. 79, 595–598 (1996).
[Crossref]

1995 (1)

D. A. Cardimona, M. P. Sharma, V. Kovanis, A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[Crossref]

1992 (2)

Y. J. Chen, A. W. Snyder, D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorbance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[Crossref]

1991 (2)

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 15, 1326–1332 (1991).
[Crossref]

P. S. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[Crossref]

1989 (1)

1988 (2)

M. J. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1988).
[Crossref]

R. B. Bylsma, D. H. Olson, A. M. Glass, “Photochromic gratings in photorefractive materials,” Opt. Lett. 13, 853–855 (1988).
[Crossref] [PubMed]

1987 (1)

M. Gehrtz, J. Pinsl, C. Bräuchle, “Sensitive detection of phase and absorption gratings: phase-modulated, homodyne detected holography,” Appl. Phys. B 43, 61–77 (1987).
[Crossref]

1972 (1)

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

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Babic, D. I.

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorbance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[Crossref]

Bacher, G. D.

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 15, 1326–1332 (1991).
[Crossref]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

M. J. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1988).
[Crossref]

Boucher, Y.

T. Fessant, Y. Boucher, “Additional modal selectivity induced by a localized defect in quarter-wave-shifted DFB lasers,” IEEE J. Quantum Electron. 34, 602–608 (1998).
[Crossref]

Y. Boucher, “Influence of a localized scattering center upon the spectral characteristics of a distributed-feedback structure,” IEEE Photon. Technol. Lett. 9, 1454–1456 (1997).
[Crossref]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

Bräuchle, C.

M. Gehrtz, J. Pinsl, C. Bräuchle, “Sensitive detection of phase and absorption gratings: phase-modulated, homodyne detected holography,” Appl. Phys. B 43, 61–77 (1987).
[Crossref]

Brinkmeyer, E.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side- written fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 50–52.

Bylsma, R. B.

Cardimona, D. A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[Crossref]

Chen, Y. J.

Y. J. Chen, A. W. Snyder, D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Corzine, S. W.

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorbance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[Crossref]

Cudney, R. S.

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 15, 1326–1332 (1991).
[Crossref]

Dong, L.

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

Ducharme, S.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

Feinberg, J.

V. Grubsky, D. S. Starodubov, J. Feinberg, “Photochemical reaction of hydrogen with germanosilicate glass initiated by 3.4–5.4-eV ultraviolet light,” Opt. Lett. 24, 729–731 (1999).
[Crossref]

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 15, 1326–1332 (1991).
[Crossref]

Fessant, T.

T. Fessant, Y. Boucher, “Additional modal selectivity induced by a localized defect in quarter-wave-shifted DFB lasers,” IEEE J. Quantum Electron. 34, 602–608 (1998).
[Crossref]

Gavrielides, A.

D. A. Cardimona, M. P. Sharma, V. Kovanis, A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[Crossref]

Gehrtz, M.

M. Gehrtz, J. Pinsl, C. Bräuchle, “Sensitive detection of phase and absorption gratings: phase-modulated, homodyne detected holography,” Appl. Phys. B 43, 61–77 (1987).
[Crossref]

Glass, A. M.

Glenn, W. H.

Grubsky, V.

Johlen, D.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side- written fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 50–52.

Kaliteevski, M. A.

A. V. Kavokin, M. A. Kaliteevski, “Light-absorption effect on Bragg interference in multilayer semiconductor heterostructures,” J. Appl. Phys. 79, 595–598 (1996).
[Crossref]

Kavokin, A. V.

A. V. Kavokin, M. A. Kaliteevski, “Light-absorption effect on Bragg interference in multilayer semiconductor heterostructures,” J. Appl. Phys. 79, 595–598 (1996).
[Crossref]

Knappe, F.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side- written fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 50–52.

Kogelnik, H.

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

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Kovanis, V.

D. A. Cardimona, M. P. Sharma, V. Kovanis, A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[Crossref]

Liphardt, M.

Liu, W. F.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

Meltz, G.

Morey, W. W.

Olson, D. H.

Payne, D. N.

Y. J. Chen, A. W. Snyder, D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Pethel, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

Pierce, R. M.

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 15, 1326–1332 (1991).
[Crossref]

Pinsl, J.

M. Gehrtz, J. Pinsl, C. Bräuchle, “Sensitive detection of phase and absorption gratings: phase-modulated, homodyne detected holography,” Appl. Phys. B 43, 61–77 (1987).
[Crossref]

Reekie, L.

Renner, H.

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side- written fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 50–52.

Russell, P. S. J.

P. S. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[Crossref]

Scalora, M.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

M. J. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1988).
[Crossref]

Shank, C. V.

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

Sharma, M. P.

D. A. Cardimona, M. P. Sharma, V. Kovanis, A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[Crossref]

Snyder, A. W.

Y. J. Chen, A. W. Snyder, D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Starodubov, D. S.

Appl. Phys. B (1)

M. Gehrtz, J. Pinsl, C. Bräuchle, “Sensitive detection of phase and absorption gratings: phase-modulated, homodyne detected holography,” Appl. Phys. B 43, 61–77 (1987).
[Crossref]

Appl. Phys. Lett. (1)

M. J. Bloemer, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1988).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

IEEE J. Quantum Electron. (4)

Y. J. Chen, A. W. Snyder, D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

T. Fessant, Y. Boucher, “Additional modal selectivity induced by a localized defect in quarter-wave-shifted DFB lasers,” IEEE J. Quantum Electron. 34, 602–608 (1998).
[Crossref]

D. A. Cardimona, M. P. Sharma, V. Kovanis, A. Gavrielides, “Dephased index and gain coupling in distributed feedback lasers,” IEEE J. Quantum Electron. 31, 60–66 (1995).
[Crossref]

D. I. Babic, S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorbance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28, 514–524 (1992).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Boucher, “Influence of a localized scattering center upon the spectral characteristics of a distributed-feedback structure,” IEEE Photon. Technol. Lett. 9, 1454–1456 (1997).
[Crossref]

J. Appl. Phys. (3)

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

A. V. Kavokin, M. A. Kaliteevski, “Light-absorption effect on Bragg interference in multilayer semiconductor heterostructures,” J. Appl. Phys. 79, 595–598 (1996).
[Crossref]

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[Crossref]

J. Lightwave Technol. (2)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

Special issue on fiber gratings, photosensitivity, and poling, J. Lightwave Technol. 15, 1261–1512 (1997).

J. Mod. Opt. (1)

P. S. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[Crossref]

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

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. Am. B 15, 1326–1332 (1991).
[Crossref]

M. Liphardt, S. Ducharme, “Measurement of the photorefractive grating phase shift in a polimer by an ac phase-modulation technique,” J. Opt. Soc. Am. B 15, 2154–2160 (1998).
[Crossref]

Opt. Lett. (4)

Other (2)

D. Johlen, F. Knappe, H. Renner, E. Brinkmeyer, “UV-induced absorption, scattering and transition losses in UV side- written fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 50–52.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

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Figures (13)

Fig. 1
Fig. 1

Schematic illustration of the refractive-index and background-loss Λ-periodic perturbation patterns discussed in the text.

Fig. 2
Fig. 2

Spectral characteristics of a uniform-loss grating (solid curves) with n = 1.4, n 1 = 10-4, L = 2 cm, λB = 1550 nm, α = 0.023 cm-1, and α1 = 0: (a) reflectivity, (b) transmissivity, (c) reflection time delay, (d) transmission time delay, and (e) loss. The dotted line represents the average single-pass loss level; the lossless case is shown as a dashed curve for comparison. All plots are expressed as a function of the wavelength detuning Δλ = λ - λB.

Fig. 3
Fig. 3

Spectral characteristics of a uniform refractive-index grating with fully modulated loss (α = α1 = 0.023 cm-1 and ϕ = 0). The other parameters are the same as in Fig. 2. (a) Reflectivity, (b) transmissivity, (c) reflection time delay, (d) transmission time delay, and (e) loss. The average single-pass loss level is shown by the dotted line. All plots are expressed as a function of the wavelength detuning Δλ = λ - λB.

Fig. 4
Fig. 4

Comparison between the periodic (p) and uniform (u) loss cases: (a) ΔR = R p - R u , (b) Δτ t = τ t p - τ t u , and (c) Δαloss = αloss p - αloss u are plotted as a function of the wavelength detuning Δλ (in the case of periodic loss, ϕ = 0).

Fig. 5
Fig. 5

Total field intensity (solid curve) and refractive-index pattern (gray area) versus grating length in five different zones (specified by the z coordinate written on the top of each column) at the five wavelengths: (a–e) λII -, (f–j) λI -, (k–o) λB, (p–t) λI +, and (u–y) λII +. The field intensity is expressed in arbitrary units, to point out the dephasing between the total-field-intensity and refractive-index patterns.

Fig. 6
Fig. 6

Spatial-dephasing dependence on grating position, for the wavelengths (a) λB, (b) λI - (dashed curve) and λI + (solid curve), and (c) λII - (dashed curve) and λII + (solid curve). The corresponding normalized power envelope is also plotted below each case.

Fig. 7
Fig. 7

Spectral loss as a function of the wavelength detuning for different dephasing angles: (a) ϕ = 0 (solid curve) and ϕ = π (dashed curve) and (b) ϕ = π/2 (dotted-dashed curve), ϕ = 3π/2 (dotted-dotted-dashed curve); the uniform-loss case is also shown by the dotted curve. All the other parameters are as in Fig. 3. The dotted line that appears in both (a) and (b) represents the average single-pass loss level.

Fig. 8
Fig. 8

Reflectivity spectra versus wavelength detuning corresponding to the four full modulated-loss and the uniform-loss cases illustrated in Fig. 7. The reflectivity peak has been magnified in the inset.

Fig. 9
Fig. 9

Intensity distribution (thick curve), refractive-index pattern (thin curve), and background-loss pattern (shaded area) versus grating length over five periods, at the Bragg wavelength and for a dephasing angle equal to (a) ϕ = 0, (b) ϕ = π/2, (c) ϕ = π, and (d) ϕ = 3π/2 (the three quantities are expressed in arbitrary units).

Fig. 10
Fig. 10

Reflectivity peak R B versus κL for n 1 = 10-3 (upper curve), n 1 = 10-4 (middle curve), and n 1 = 10-5 (lower curve); α p = 0.2 dB/cm, α1 = 0 (uniform loss); κ is the real part of κ c .

Fig. 11
Fig. 11

Reflectivity peak R B versus κL for four ϕ values (solid curves), with n 1 = 10-5, α p = 0.2 dB/cm, and α1 = α (fully modulated loss); the dotted curve corresponds to the lossless case (α = 0); κ is the real part of κ c .

Fig. 12
Fig. 12

Amplitude difference Δαpeaks of the two loss peaks (shown in the inset) as a function of the refractive-index modulation, for a (lossless) grating reflectivity of 0.999 and ϕ = 0 and for four different values of the power loss coefficient: α p = 0.1 dB/cm (solid curve), α p = 0.2 dB/cm (long-dashed curve), α p = 0.3 dB/cm (medium-dashed curve), and α p = 0.4 dB/cm (short-dashed curve).

Fig. 13
Fig. 13

(a) Amplitude difference Δαpeaks, (b) peak wavelength shift |λ p - λB|, and (c) normalized peak wavelength shift |λ p - λB|/Δλpeaks as a function of the refractive-index-modulation amplitude, for L = 2 cm and five different values of the ratio R n = α/n 1. The value of α p at n 1 = 3 × 10-4 is equal to 0.3 dB/cm (solid curve), 0.2 dB/cm (long-dashed curve), 0.1 dB/cm (medium-dashed curve), 0.05 dB/cm (short-dashed curve), and 0.01 dB/cm (dotted curve).

Equations (46)

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

nz=n+n1 cos 2βBz,
αz=α+α1 cos2βBz+ϕ,
kz=k0nz+jαz,
n1  n, α  β0=k0n, α1  β0,
k2zβ02+j2β0α+4β0κc cos 2βz-κs sin 2βz,
κc=πλ n1+j α12cos ϕ,
κs=j α12sin ϕ
2E+k2zEz=0
Ez=Efz+Ebz=Azexpjβz+Bzexp-jβz,
β2=β0+jα2β02+j2β0α.
Az=jκc+jκsexpj2ΔzB,
Bz=-jκc-jκsexp-j2ΔzA,
Efz=Eincp coshpz-L-jΔ sinhpz-Lp coshpL+jΔ sinhpL×expjβBz,
Ebz=-jEincκc-jκssinhpz-Lp coshpL+jΔ sinhpL×exp-jβBz,
r=Eb0Ef0=jκc-jκssinhpLp coshpL+jΔ sinhpL,
t=EfLEf0=p expjβBLp coshpL+jΔ sinhpL,
R=|r|2,
T=|t|2.
αloss=-10 log10T+R.
τr,t=-λ22πcdθr,tdλ,
κf=κc+jκs=πλ n1+j α12expjϕ,
κb=κc-jκs=πλ n1+j α12exp-jϕ.
Iz=|Ez|2=|Efz+Ebz|2,
Iz=|Ez|2=|Az|2+|Bz|2+2|Az| |Bz|cos2βz+ΔϕABz,
Fz=2|Az| |Bz||Az|2+|Bz|2=2|Efz| |Ebz||Efz|2+|Ebz|2.
Φz=2βz+ΔϕAB=0z2πΛswζdζ,
2πΛswz=dΦzdz=2β+dϕAzdz-dϕBzdz.
Λsw=λ/2n=λn/2,
Iz=|Ez|2=|Efz|2+|Ebz|2+2|Efz| |Ebz|cos2βBz+Δϕfbz.
ϕfz=arctan-p¯Δ sinp¯zΔ2 cosp¯z-κ2 cosp¯L cosp¯z-L+πδˆΔ2 cosp¯z-κ2 cosp¯L×cosp¯z-L<0,
ϕbz=arctanp¯Δcotp¯L+πδˆΔκ sinp¯L sinp¯z-L>0,
ϕf±zp¯z
ϕb±z±π2-p¯L+πδˆ±sinp¯L sinp¯z-L>0,
ϕf±zm+=p¯L+mπ,
ϕb±zm+=±π2 ±p¯L+m+1mπ,
ϕf±zm-=p¯L+mπ,
ϕb±zm-=±π2p¯L+mm+1π,
ϕf±z¯m=p¯L±2m+1π2,
ϕb±z¯m=±π2-p¯L+mm+1π.
ϕfz=arctan-2pΔ sinh pzp2-Δ2cosh pz+κ2cosh pz-2L,
ϕbz=arctanpΔcoth pL+πδˆΔ<0.
ϕfL-=-arctanΔptanh pL,
ϕbL-=arctanpΔ1tanh pL+πδˆΔ<0,
dIlossλ=2αzIz, λIincλdz,
Ilossλ=2|Einc|20L αz|Ez, λ|2dz,
Ilossλ2α|Einc|20L|Efz, λ|2+|Ebz, λ|2dz+2α1|Einc|20L |Efz, λ| |Ebz, λ|×cosϕ-Δϕfbz, λdz.

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