Abstract

Scattering from tilted fiber gratings into free space is useful in various filtering and monitoring applications. We develop a waveguide-scattering analysis based on the coupled-mode theory and new sets of radiation modes that resemble the HE and EH guided modes of an optical fiber. By simplifying the calculation of modal coupling coefficients and the scattered field, this approach allows us to verify the results from free-space perturbation models such as the volume-current method. The HE and EH radiation modes defined here are suitable for scattering analysis of cylindrical waveguides in general.

© 2006 Optical Society of America

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References

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  1. G. Meltz, W. W. Morey, and W. H. Glenn, "In-fiber Bragg grating tap," in Optical Fiber Communication Conference, Vol. 1 of 1990 OSA Technical Digest Series (Optical Society of America, 1990), paper TuG1.
  2. R. Kashyap, R. Wyatt, and R. J. Campbell, "Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating," Electron. Lett. 29, 154-156 (1993).
    [CrossRef]
  3. J. L. Wagener, T. A. Strasser, J. R. Pedrazanni, J. DeMarco, and D. J. DiGiovanni, "Fiber grating optical spectrum analyzer tap," in Proceedings of the European Conference on Optical Communication (ECOC'97), IEE Conference Publication (ECOC, 1997), Vol. 448, pp. 65-68
  4. K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, "In-fiber spectrometer using tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 933-935 (2003).
    [CrossRef]
  5. S. Wielandy and S. C. Dunn, "Tilted superstructure fiber grating used as a Fourier-transform spectrometer," Opt. Lett. 29, 1614-1616 (2004).
    [CrossRef] [PubMed]
  6. R. Kashyap, R. Wyatt, and P. F. McKee, "Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings," Electron. Lett. 29, 1025-1026 (1993).
    [CrossRef]
  7. M. J. Holmes, R. Kashyap, R. Wyatt, and R. P. Smith, "Ultra narrow-band optical fibre side-tap," in Proceedings of the European Conference on Optical Communication (ECOC'98), IEEE Catalog 98TH8398 (ECOC, 1998), pp. 137-138.
  8. P. S. Westbrook, T. A. Strasser, and T. Erdogan, "In-line polarimeter using blazed fiber gratings," IEEE Photon. Technol. Lett. 12, 1352-1354 (2000).
    [CrossRef]
  9. J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, "Fibre-polarimeter based on grating taps," Electron. Lett. 38, 1248-1250 (2002).
    [CrossRef]
  10. S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, "Fabrication of tilted fibre-grating polarisation-dependent loss equalizer," Electron. Lett. 37, 284-286 (2001).
    [CrossRef]
  11. P. I. Reyes and P. S. Westbrook, "Tunable PDL of twisted-tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 828-830 (2003).
    [CrossRef]
  12. K. M. Zhou, G. Simpson, X. F. Chen, L. Zhang, and I. Bennion, "High extinction ratio in-fiber polarizers based on 45 degrees tilted fiber Bragg gratings," Opt. Lett. 30, 1285-1287 (2005).
    [CrossRef] [PubMed]
  13. T. Erdogan and J. E. Sipe, "Tilted fiber phase gratings," J. Opt. Soc. Am. A 13, 296-313 (1996).
    [CrossRef]
  14. A. W. Snyder, "Radiation on losses due to variations of radius dielectric or optical fibers," IEEE Trans. Microwave Theory Tech. MTT-18, 608-615 (1970).
    [CrossRef]
  15. Y. Li, M. Froggatt, and T. Erdogan, "Volume current method for analysis of tilted fiber gratings," J. Lightwave Technol. 19, 1580-1591 (2001).
    [CrossRef]
  16. R. B. Walker, S. J. Mihailov, P. Lu, D. Grobnic, "Shaping the radiation field of tilted fiber Bragg gratings," J. Opt. Soc. Am. B 22, 962-974 (2005).
    [CrossRef]
  17. M. J. Holmes, R. Kashyap, and R. Wyatt, "Physical properties of optical fiber sidetap grating filters: free-space model," IEEE J. Sel. Top. Quantum Electron. 5, 1353-1365 (1999).
    [CrossRef]
  18. A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720-727 (1971).
    [CrossRef]
  19. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).
  20. D. Marcuse, "Radiation losses of the dominant mode in round dielectric waveguides," Bell Syst. Tech. J. 49, 1665-1693 (1970).
  21. Y. Li, S. Wielandy, G. E. Carver, P. I. Reyes, and P. S. Westbrook, "Scattering from nonuniform tilted fiber gratings," Opt. Lett. 29, 1330-1332 (2004).
    [CrossRef] [PubMed]
  22. Leopold B. Felsen and Nathan Marcuvitz, Radiation and Scattering of Waves, (Prentice-Hall, 1973).
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994), Section 8.530.
  24. Y. Li, S. Wielandy, G. E. Carver, H. L. Durko, and P. S. Westbrook, "Influence of longitudinal mode field in grating scattering from weakly guided optical fiber waveguides," Opt. Lett. 29, 691-693 (2004).
    [CrossRef] [PubMed]

2005 (2)

2004 (3)

2003 (2)

P. I. Reyes and P. S. Westbrook, "Tunable PDL of twisted-tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 828-830 (2003).
[CrossRef]

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, "In-fiber spectrometer using tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 933-935 (2003).
[CrossRef]

2002 (1)

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, "Fibre-polarimeter based on grating taps," Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

2001 (2)

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, "Fabrication of tilted fibre-grating polarisation-dependent loss equalizer," Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Y. Li, M. Froggatt, and T. Erdogan, "Volume current method for analysis of tilted fiber gratings," J. Lightwave Technol. 19, 1580-1591 (2001).
[CrossRef]

2000 (1)

P. S. Westbrook, T. A. Strasser, and T. Erdogan, "In-line polarimeter using blazed fiber gratings," IEEE Photon. Technol. Lett. 12, 1352-1354 (2000).
[CrossRef]

1999 (1)

M. J. Holmes, R. Kashyap, and R. Wyatt, "Physical properties of optical fiber sidetap grating filters: free-space model," IEEE J. Sel. Top. Quantum Electron. 5, 1353-1365 (1999).
[CrossRef]

1996 (1)

1993 (2)

R. Kashyap, R. Wyatt, and P. F. McKee, "Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings," Electron. Lett. 29, 1025-1026 (1993).
[CrossRef]

R. Kashyap, R. Wyatt, and R. J. Campbell, "Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating," Electron. Lett. 29, 154-156 (1993).
[CrossRef]

1971 (1)

A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720-727 (1971).
[CrossRef]

1970 (2)

D. Marcuse, "Radiation losses of the dominant mode in round dielectric waveguides," Bell Syst. Tech. J. 49, 1665-1693 (1970).

A. W. Snyder, "Radiation on losses due to variations of radius dielectric or optical fibers," IEEE Trans. Microwave Theory Tech. MTT-18, 608-615 (1970).
[CrossRef]

Bandemer, A.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, "Fibre-polarimeter based on grating taps," Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Bennion, I.

Campbell, R. J.

R. Kashyap, R. Wyatt, and R. J. Campbell, "Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating," Electron. Lett. 29, 154-156 (1993).
[CrossRef]

Carver, G. E.

Chen, X. F.

DeMarco, J.

J. L. Wagener, T. A. Strasser, J. R. Pedrazanni, J. DeMarco, and D. J. DiGiovanni, "Fiber grating optical spectrum analyzer tap," in Proceedings of the European Conference on Optical Communication (ECOC'97), IEE Conference Publication (ECOC, 1997), Vol. 448, pp. 65-68

DiGiovanni, D. J.

J. L. Wagener, T. A. Strasser, J. R. Pedrazanni, J. DeMarco, and D. J. DiGiovanni, "Fiber grating optical spectrum analyzer tap," in Proceedings of the European Conference on Optical Communication (ECOC'97), IEE Conference Publication (ECOC, 1997), Vol. 448, pp. 65-68

Dunn, S. C.

Durko, H. L.

Erdogan, T.

Feder, K. S.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, "In-fiber spectrometer using tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 933-935 (2003).
[CrossRef]

Felsen, Leopold B.

Leopold B. Felsen and Nathan Marcuvitz, Radiation and Scattering of Waves, (Prentice-Hall, 1973).

Froggatt, M.

Ging, J.

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, "In-fiber spectrometer using tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 933-935 (2003).
[CrossRef]

Glenn, W. H.

G. Meltz, W. W. Morey, and W. H. Glenn, "In-fiber Bragg grating tap," in Optical Fiber Communication Conference, Vol. 1 of 1990 OSA Technical Digest Series (Optical Society of America, 1990), paper TuG1.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994), Section 8.530.

Grobnic, D.

Holmes, M. J.

M. J. Holmes, R. Kashyap, and R. Wyatt, "Physical properties of optical fiber sidetap grating filters: free-space model," IEEE J. Sel. Top. Quantum Electron. 5, 1353-1365 (1999).
[CrossRef]

M. J. Holmes, R. Kashyap, R. Wyatt, and R. P. Smith, "Ultra narrow-band optical fibre side-tap," in Proceedings of the European Conference on Optical Communication (ECOC'98), IEEE Catalog 98TH8398 (ECOC, 1998), pp. 137-138.

Johnson, D. C.

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, "Fabrication of tilted fibre-grating polarisation-dependent loss equalizer," Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Kashyap, R.

M. J. Holmes, R. Kashyap, and R. Wyatt, "Physical properties of optical fiber sidetap grating filters: free-space model," IEEE J. Sel. Top. Quantum Electron. 5, 1353-1365 (1999).
[CrossRef]

R. Kashyap, R. Wyatt, and R. J. Campbell, "Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating," Electron. Lett. 29, 154-156 (1993).
[CrossRef]

R. Kashyap, R. Wyatt, and P. F. McKee, "Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings," Electron. Lett. 29, 1025-1026 (1993).
[CrossRef]

M. J. Holmes, R. Kashyap, R. Wyatt, and R. P. Smith, "Ultra narrow-band optical fibre side-tap," in Proceedings of the European Conference on Optical Communication (ECOC'98), IEEE Catalog 98TH8398 (ECOC, 1998), pp. 137-138.

Krause, E.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, "Fibre-polarimeter based on grating taps," Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Li, Y.

Lu, P.

Marcuse, D.

D. Marcuse, "Radiation losses of the dominant mode in round dielectric waveguides," Bell Syst. Tech. J. 49, 1665-1693 (1970).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

Marcuvitz, Nathan

Leopold B. Felsen and Nathan Marcuvitz, Radiation and Scattering of Waves, (Prentice-Hall, 1973).

McKee, P. F.

R. Kashyap, R. Wyatt, and P. F. McKee, "Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings," Electron. Lett. 29, 1025-1026 (1993).
[CrossRef]

Meltz, G.

G. Meltz, W. W. Morey, and W. H. Glenn, "In-fiber Bragg grating tap," in Optical Fiber Communication Conference, Vol. 1 of 1990 OSA Technical Digest Series (Optical Society of America, 1990), paper TuG1.

Mihailov, S. J.

R. B. Walker, S. J. Mihailov, P. Lu, D. Grobnic, "Shaping the radiation field of tilted fiber Bragg gratings," J. Opt. Soc. Am. B 22, 962-974 (2005).
[CrossRef]

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, "Fabrication of tilted fibre-grating polarisation-dependent loss equalizer," Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Morey, W. W.

G. Meltz, W. W. Morey, and W. H. Glenn, "In-fiber Bragg grating tap," in Optical Fiber Communication Conference, Vol. 1 of 1990 OSA Technical Digest Series (Optical Society of America, 1990), paper TuG1.

Pedrazanni, J. R.

J. L. Wagener, T. A. Strasser, J. R. Pedrazanni, J. DeMarco, and D. J. DiGiovanni, "Fiber grating optical spectrum analyzer tap," in Proceedings of the European Conference on Optical Communication (ECOC'97), IEE Conference Publication (ECOC, 1997), Vol. 448, pp. 65-68

Peupelmann, J.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, "Fibre-polarimeter based on grating taps," Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Reyes, P. I.

Y. Li, S. Wielandy, G. E. Carver, P. I. Reyes, and P. S. Westbrook, "Scattering from nonuniform tilted fiber gratings," Opt. Lett. 29, 1330-1332 (2004).
[CrossRef] [PubMed]

P. I. Reyes and P. S. Westbrook, "Tunable PDL of twisted-tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 828-830 (2003).
[CrossRef]

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, "In-fiber spectrometer using tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 933-935 (2003).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994), Section 8.530.

Schaffer, C.

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, "Fibre-polarimeter based on grating taps," Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

Simpson, G.

Sipe, J. E.

Smith, R. P.

M. J. Holmes, R. Kashyap, R. Wyatt, and R. P. Smith, "Ultra narrow-band optical fibre side-tap," in Proceedings of the European Conference on Optical Communication (ECOC'98), IEEE Catalog 98TH8398 (ECOC, 1998), pp. 137-138.

Snyder, A. W.

A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720-727 (1971).
[CrossRef]

A. W. Snyder, "Radiation on losses due to variations of radius dielectric or optical fibers," IEEE Trans. Microwave Theory Tech. MTT-18, 608-615 (1970).
[CrossRef]

Stocki, T. J.

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, "Fabrication of tilted fibre-grating polarisation-dependent loss equalizer," Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Strasser, T. A.

P. S. Westbrook, T. A. Strasser, and T. Erdogan, "In-line polarimeter using blazed fiber gratings," IEEE Photon. Technol. Lett. 12, 1352-1354 (2000).
[CrossRef]

J. L. Wagener, T. A. Strasser, J. R. Pedrazanni, J. DeMarco, and D. J. DiGiovanni, "Fiber grating optical spectrum analyzer tap," in Proceedings of the European Conference on Optical Communication (ECOC'97), IEE Conference Publication (ECOC, 1997), Vol. 448, pp. 65-68

Wagener, J. L.

J. L. Wagener, T. A. Strasser, J. R. Pedrazanni, J. DeMarco, and D. J. DiGiovanni, "Fiber grating optical spectrum analyzer tap," in Proceedings of the European Conference on Optical Communication (ECOC'97), IEE Conference Publication (ECOC, 1997), Vol. 448, pp. 65-68

Walker, R. B.

R. B. Walker, S. J. Mihailov, P. Lu, D. Grobnic, "Shaping the radiation field of tilted fiber Bragg gratings," J. Opt. Soc. Am. B 22, 962-974 (2005).
[CrossRef]

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, "Fabrication of tilted fibre-grating polarisation-dependent loss equalizer," Electron. Lett. 37, 284-286 (2001).
[CrossRef]

Westbrook, P. S.

Y. Li, S. Wielandy, G. E. Carver, P. I. Reyes, and P. S. Westbrook, "Scattering from nonuniform tilted fiber gratings," Opt. Lett. 29, 1330-1332 (2004).
[CrossRef] [PubMed]

Y. Li, S. Wielandy, G. E. Carver, H. L. Durko, and P. S. Westbrook, "Influence of longitudinal mode field in grating scattering from weakly guided optical fiber waveguides," Opt. Lett. 29, 691-693 (2004).
[CrossRef] [PubMed]

P. I. Reyes and P. S. Westbrook, "Tunable PDL of twisted-tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 828-830 (2003).
[CrossRef]

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, "In-fiber spectrometer using tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 933-935 (2003).
[CrossRef]

P. S. Westbrook, T. A. Strasser, and T. Erdogan, "In-line polarimeter using blazed fiber gratings," IEEE Photon. Technol. Lett. 12, 1352-1354 (2000).
[CrossRef]

Wielandy, S.

Wyatt, R.

M. J. Holmes, R. Kashyap, and R. Wyatt, "Physical properties of optical fiber sidetap grating filters: free-space model," IEEE J. Sel. Top. Quantum Electron. 5, 1353-1365 (1999).
[CrossRef]

R. Kashyap, R. Wyatt, and P. F. McKee, "Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings," Electron. Lett. 29, 1025-1026 (1993).
[CrossRef]

R. Kashyap, R. Wyatt, and R. J. Campbell, "Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating," Electron. Lett. 29, 154-156 (1993).
[CrossRef]

M. J. Holmes, R. Kashyap, R. Wyatt, and R. P. Smith, "Ultra narrow-band optical fibre side-tap," in Proceedings of the European Conference on Optical Communication (ECOC'98), IEEE Catalog 98TH8398 (ECOC, 1998), pp. 137-138.

Zhang, L.

Zhou, K. M.

Bell Syst. Tech. J. (1)

D. Marcuse, "Radiation losses of the dominant mode in round dielectric waveguides," Bell Syst. Tech. J. 49, 1665-1693 (1970).

Electron. Lett. (4)

R. Kashyap, R. Wyatt, and R. J. Campbell, "Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating," Electron. Lett. 29, 154-156 (1993).
[CrossRef]

R. Kashyap, R. Wyatt, and P. F. McKee, "Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings," Electron. Lett. 29, 1025-1026 (1993).
[CrossRef]

J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, "Fibre-polarimeter based on grating taps," Electron. Lett. 38, 1248-1250 (2002).
[CrossRef]

S. J. Mihailov, R. B. Walker, T. J. Stocki, and D. C. Johnson, "Fabrication of tilted fibre-grating polarisation-dependent loss equalizer," Electron. Lett. 37, 284-286 (2001).
[CrossRef]

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

M. J. Holmes, R. Kashyap, and R. Wyatt, "Physical properties of optical fiber sidetap grating filters: free-space model," IEEE J. Sel. Top. Quantum Electron. 5, 1353-1365 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

P. S. Westbrook, T. A. Strasser, and T. Erdogan, "In-line polarimeter using blazed fiber gratings," IEEE Photon. Technol. Lett. 12, 1352-1354 (2000).
[CrossRef]

P. I. Reyes and P. S. Westbrook, "Tunable PDL of twisted-tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 828-830 (2003).
[CrossRef]

K. S. Feder, P. S. Westbrook, J. Ging, P. I. Reyes, and G. E. Carver, "In-fiber spectrometer using tilted fiber gratings," IEEE Photon. Technol. Lett. 15, 933-935 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

A. W. Snyder, "Continuous mode spectrum of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720-727 (1971).
[CrossRef]

A. W. Snyder, "Radiation on losses due to variations of radius dielectric or optical fibers," IEEE Trans. Microwave Theory Tech. MTT-18, 608-615 (1970).
[CrossRef]

J. Lightwave Technol. (1)

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

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

Opt. Lett. (4)

Other (6)

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

Leopold B. Felsen and Nathan Marcuvitz, Radiation and Scattering of Waves, (Prentice-Hall, 1973).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, 1994), Section 8.530.

G. Meltz, W. W. Morey, and W. H. Glenn, "In-fiber Bragg grating tap," in Optical Fiber Communication Conference, Vol. 1 of 1990 OSA Technical Digest Series (Optical Society of America, 1990), paper TuG1.

J. L. Wagener, T. A. Strasser, J. R. Pedrazanni, J. DeMarco, and D. J. DiGiovanni, "Fiber grating optical spectrum analyzer tap," in Proceedings of the European Conference on Optical Communication (ECOC'97), IEE Conference Publication (ECOC, 1997), Vol. 448, pp. 65-68

M. J. Holmes, R. Kashyap, R. Wyatt, and R. P. Smith, "Ultra narrow-band optical fibre side-tap," in Proceedings of the European Conference on Optical Communication (ECOC'98), IEEE Catalog 98TH8398 (ECOC, 1998), pp. 137-138.

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Figures (7)

Fig. 1
Fig. 1

Coefficients of Bessel function Z ν ( K 2 r ) for HE and EH radiation modes. (a) A 1 A and B 1 B , ν = 1 ; (b) A 2 A and B 2 B , ν = 1 ; (c) A 1 A and B 1 B , ν = 10 ; (d) A 2 A and B 2 B , ν = 10 . Note that A 1 A B 1 B , A 2 A B 2 B for all the modes.

Fig. 2
Fig. 2

The power normalization factor A 1 A 2 + A 2 A 2 stays roughly constant under nonparaxial scattering condition. (a) HE and EH modes with ν = 1 ; (b) HE and EH modes with ν = 10 .

Fig. 3
Fig. 3

Illustration of wave parameters related to grating scattering. β 01 , propagation constant of LP 01 mode; K 1 , transverse wave vector of the scattered field in the core; K 2 , transverse wave vector of the scattered field in the cladding; K z , longitudinal grating wave number; and K t , transverse grating wave number. The transverse phase matching diagram, shown on the right side, can be derived from CMT with suitable approximations (see Section 4).

Fig. 4
Fig. 4

Coupling coefficients of radiation modes at different grating tilt ζ for a single-mode fiber with a = 4 μ m and Δ = 0.4 % . (a) HE modes, S-polarized input; (b) HE modes, P-polarized input; (c) EH modes, S-polarized input; (d) EH modes, P-polarized input. Note the anomaly for EH mode coefficients at ζ = 45 ° .

Fig. 5
Fig. 5

Radial component of the scattered field ( E r ) and the power flow density ( S r ) calculated by VCM and CMT, for a 10°-tilted grating and S-polarized input. (a) E r at distance r = 100 μ m , (b) E r at distance r = 1 mm , (c) S r at distance r = 100 μ m , and (d) S r at distance r = 1 mm .

Fig. 6
Fig. 6

Radial component of the scattered field ( E r ) and the power flow density ( S r ) calculated by VCM and CMT, for a 2°-tilted grating and S-polarized input. (a) E r at distance r = 100 μ m , (b) E r at distance r = 1 mm , (c) S r at distance r = 100 μ m , and (d) S r at distance r = 1 mm .

Fig. 7
Fig. 7

Radial component of the scattered field ( E r ) and the power flow density ( S r ) calculated by VCM and CMT, for a 45°-tilted grating and S-polarized input. (a) E r at distance r = 100 μ m , (b) E r at distance r = 1 mm , (c) S r at distance r = 100 μ m , and (d) S r at distance r = 1 mm . The large difference in E r between VCM and CMT is a near-field effect.

Equations (59)

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

2 ψ r 2 + 1 r ψ r + 1 r 2 2 ψ ϕ 2 + K j 2 ψ = 0.
d 2 F d r 2 + 1 r d F d r + ( K j 2 ν 2 r 2 ) F = 0.
E z = { i K 1 β A J ν ( K 1 r ) exp ( i ν ϕ ) , r < a i K 2 β [ A 1 J ν ( K 2 r ) + A 2 N ν ( K 2 r ) ] exp ( i ν ϕ ) , r > a , }
H z = { K 1 k 0 ε 0 μ 0 B J ν ( K 1 r ) exp ( i ν ϕ ) , r < a K 2 k 0 ε 0 μ 0 [ B 1 J ν ( K 2 r ) + B 2 N ν ( K 2 r ) ] exp ( i ν ϕ ) , r > a . }
E r = [ A J ν 1 ( K 1 r ) + B A K 1 r ν J ν ( K 1 r ) ] exp ( i ν ϕ ) ,
E ϕ = [ i B J ν 1 ( K 1 r ) + i A B K 1 r ν J ν ( K 1 r ) ] exp ( i ν ϕ ) ,
H r = [ i β k 0 B J ν 1 ( K 1 r ) + i β 2 B k 0 2 n 1 2 A k 0 β K 1 r ν J ν ( K 1 r ) ] ε 0 μ 0 exp ( i ν ϕ ) ,
H ϕ = [ k 0 n 1 2 β A J ν 1 ( K 1 r ) + β 2 B k 0 2 n 1 2 A k 0 β K 1 r ν J ν ( K 1 r ) ] ε 0 μ 0 exp ( i ν ϕ ) .
[ A 1 A A 2 A ] = M [ K 1 K 2 J ν ( K 1 a ) n 1 2 n 2 2 J ν 1 ( K 1 a ) ] ,
[ B 1 B B 2 B ] = M [ K 1 K 2 J ν ( K 1 a ) J ν 1 ( K 1 a ) ] ,
[ A 1 A A 2 A ] = M [ K 1 K 2 J ν ( K 1 a ) n 1 2 n 2 2 ( J ν 1 ( K 1 a ) + ( n 1 2 n 2 2 ) ( β 2 + k 0 2 n 1 2 ) ν J ν ( K 1 a ) n 1 2 K 2 2 K 1 a ) ] ,
[ B 1 B B 2 B ] = M [ K 1 K 2 J ν ( K 1 a ) J ν 1 ( K 1 a ) + ( n 1 2 n 2 2 ) ( β 2 + k 0 2 n 1 2 ) ν J ν ( K 1 a ) n 1 2 K 2 2 K 1 a ] .
M [ J ν ( K 2 a ) N ν ( K 2 a ) J ν 1 ( K 2 a ) N ν 1 ( K 2 a ) ] 1 = π K 2 a 2 [ N ν 1 ( K 2 a ) N ν ( K 2 a ) J ν 1 ( K 2 a ) J ν ( K 2 a ) ] .
Z ν ( K 2 r ) = A 1 A J ν ( K 2 r ) + A 2 A N ν ( K 2 r ) B 1 B J ν ( K 2 r ) + B 2 B N ν ( K 2 r ) .
{ e ν H E [ Z ν 1 ( K 2 r ) ( r ̂ + i ϕ ̂ ) + i K 2 β Z ν ( K 2 r ) z ̂ ] exp ( i ν ϕ ) , h ν H E ε μ 0 [ 0 0 ] i p Z ν 1 ( K 2 r ) ( r ̂ + i ϕ ̂ ) i q Z ν + 1 ( K 2 r ) × ( r ̂ i ϕ ̂ ) + [ K 2 k 0 n Z ν ( K 2 r ) z ̂ ] exp ( i ν ϕ ) , }
{ e ν E H [ p Z ν + 1 ( K 2 r ) ( r ̂ i ϕ ̂ ) + q Z ν 1 ( K 2 r ) ( r ̂ + i ϕ ̂ ) 0 0 ] [ i K 2 k 0 n Z ν ( K 2 r ) z ̂ ] exp ( i ν ϕ ) , h ν E H ε μ 0 [ i Z ν + 1 ( K 2 r ) ( r ̂ i ϕ ̂ ) + K 2 β Z ν ( K 2 r ) z ̂ ] × exp ( i ν ϕ ) . }
p 1 2 ( k 0 n β + β k 0 n ) , q 1 2 ( k 0 n β β k 0 n ) .
1 2 0 2 π d ϕ 0 [ e ν μ ( r , ϕ , β ) × h ν μ * ( r , ϕ , β ) ] z r d r P ν μ δ ν ν δ μ μ δ ( K 2 K 2 ) ,
P ν μ = π ( β 2 + k 0 2 n 2 ) k 0 β K 2 ε 0 μ 0 ( A 1 A 2 + A 2 A 2 ) .
J ν 1 ( K 1 a ) + n 1 2 n 2 2 n 1 2 β 2 + k 0 2 n 1 2 K 2 2 ν J ν ( K 1 a ) K 1 a J ν 1 ( K 1 a ) .
{ A 1 A π a 2 [ K 1 J ν ( K 1 a ) N ν 1 ( K 2 a ) K 2 J ν 1 ( K 1 a ) N ν ( K 2 a ) ] , A 2 A π a 2 [ K 1 J ν ( K 1 a ) J ν 1 ( K 2 a ) K 2 J ν 1 ( K 1 a ) J ν ( K 2 a ) ] . }
J ν ( x ) 2 π x cos ( x ν 2 π 1 4 π ) ,
N ν ( x ) 2 π x sin ( x ν 2 π 1 4 π ) .
A 1 A cos ( K 1 a K 2 a ) , A 2 A sin ( K 1 a K 2 a ) .
P ν μ P β π ( β 2 + k 0 2 n 2 ) k 0 β K 2 ε 0 μ 0 .
Δ n ( r , ϕ , z ) = { Δ n 0 ( z ) cos ( K z z + K t r sin ϕ + Φ ) , r < a 0 , r a .
E = a 01 ( z ) e 01 exp ( i β 01 z ) + ν μ a ν μ ( z , β ) e ν μ exp ( i β z ) d K 2 .
a ν μ z = i ω ε 0 n Δ n 0 ( z ) 4 P ν μ c ν μ a 01 ( z ) exp [ i ( β + K z β 01 ) z + i Φ ] ,
c ν μ 0 2 π 0 a e 01 e ν μ * exp ( i K t r sin ϕ ) r d r d ϕ .
a ν μ ( z , β ) = i ω ε 0 n c ν μ 4 P ν μ exp ( i Φ ) z + Δ n 0 ( z ) a 01 ( z ) exp [ i ( β + K z β 01 ) z ] d z = i ω ε 0 n c ν μ 4 P ν μ exp ( i Φ ) F ( β + K z β 01 ) .
E s = i ω ε 0 n 4 exp ( i Φ ) ν μ 0 k 0 n 2 c ν μ P ν μ e ν μ F ( β + K z β 01 ) exp ( i β z ) d K 2 i ω ε 0 n β 4 K 2 exp ( i β z + i Φ ) ν μ c ν μ P ν μ e ν μ β = ( K z β 01 ) 0 k 0 n 2 F ( β + K z β 01 ) exp [ i ( β + K z β 01 ) z ] d β = i π ω ε 0 n β 2 K 2 exp ( i β z + i Φ ) ν μ c ν μ P ν μ e ν μ β = ( K z β 01 ) Δ n 0 ( z ) a 01 ( z ) .
d a 01 d z = ω 2 ε 0 2 n 2 16 P 01 Δ n 0 ( z ) ν μ 0 k 0 n 2 β K 2 c ν μ 2 P ν μ F ( β + K z β 01 ) exp [ i ( β + K z β 01 ) z ] d β π ω 2 ε 0 2 n 2 8 P 01 Δ n 0 2 ( z ) [ β K 2 ν μ c ν μ 2 P ν μ ] β = ( K z β 01 ) a 01 ( z ) .
e 01 = J 0 ( u r ) cos ( ϕ δ ) r ̂ J 0 ( u r ) sin ( ϕ δ ) ϕ ̂ + i u β 01 J 1 ( u r ) cos ( ϕ δ ) z ̂ .
c ν H E = 2 π [ e i δ G ( 0 , ν 1 , ν 1 ) + K 1 u 2 β β 01 e i δ G ( 1 , ν , ν 1 ) + K 1 u 2 β β 01 e i δ G ( 1 , ν , ν + 1 ) ] ,
c ν E H = 2 π [ p e i δ G ( 0 , ν + 1 , ν + 1 ) + q e i δ G ( 0 , ν 1 , ν 1 ) K 1 u 2 k 0 n β 01 e i δ G ( 1 , ν , ν + 1 ) K 1 u 2 k 0 n β 01 e i δ G ( 1 , ν , ν 1 ) . ]
G ( m , n , k ) 0 a J m ( u r ) J n ( K 1 r ) J k ( K t r ) r d r .
{ c ν H E 2 π e i δ G ( 0 , ν 1 , ν 1 ) , c ν E H 2 π [ p e i δ G ( 0 , ν + 1 , ν + 1 ) + q e i δ G ( 0 , ν 1 , ν 1 ) ] . }
Z ν ( K 2 r ) = A 1 A J ν ( K 2 r ) + A 2 A N ν ( K 2 r ) = A 1 i A 2 2 A H ν ( 1 ) ( K 2 r ) + A 1 + i A 2 2 A H ν ( 2 ) ( K 2 r ) .
Z ν ( K 2 r ) A 1 + i A 2 2 A H ν ( 2 ) ( K 2 r ) .
ν μ c ν μ P ν μ e ν μ 1 P β ( ν c ν H E e ν H E + ν c ν E H e ν E H ) .
A 1 + i A 2 2 A H ν ( 2 ) ( K 2 r ) 1 2 π K 2 r exp [ i ( K 2 r ν 2 π 1 4 π + θ β ) ] .
{ e ν H E 1 2 π K 2 r exp [ i ( K 2 r ν ϕ ν 2 π 1 4 π + θ β ) ] ( i r ̂ + ϕ ̂ + i K 2 β z ̂ ) , e ν E H 1 2 π K 2 r exp [ i ( K 2 r ν ϕ ν 2 π 1 4 π + θ β ) ] ( i ( p q ) r ̂ + ( p + q ) ϕ ̂ i K 2 k 0 n z ̂ ) . }
ν c ν H E e ν H E 2 π K 2 r exp [ i ( K 2 r 1 4 π + θ β + δ ) ] ( i r ̂ + ϕ ̂ + i K 2 β z ̂ ) ν G ( 0 , ν 1 , ν 1 ) exp [ i ν ( ϕ + π 2 ) ] .
ν G ( 0 , ν 1 , ν 1 ) exp [ i ν ( ϕ + π 2 ) ] = 0 a J 0 ( u r ) { ν J ν 1 ( K 1 r ) J ν 1 ( K t r ) exp [ i ν ( ϕ + π 2 ) ] } r d r = 0 a J 0 ( u r ) { ν J ν ( K 1 r ) J ν ( K t r ) exp [ i ν ( ϕ + π 2 ) ] } r d r exp [ i ( ϕ + π 2 ) ] = 0 a J 0 ( u r ) J 0 ( K new r ) r d r exp [ i ( ϕ + π 2 ) ] .
ν c ν E H e ν E H 2 π K 2 r exp [ i ( K 2 r 1 4 π + θ β ) ] ( p e i φ + q e i φ ) 0 a J 0 ( u r ) J 0 ( K new r ) r d r [ i ( p q ) r ̂ + ( p + q ) ϕ ̂ i K 2 k 0 n z ̂ ] .
ν c ν μ e ν μ k 0 2 n 2 + β 2 k 0 2 n 2 β 2 2 π K 2 r exp [ i ( K 2 r 1 4 π + θ β ) ] 0 a J 0 ( u r ) J 0 ( K new r ) r d r [ β 2 sin φ r ̂ + k 0 2 n 2 2 cos φ ϕ ̂ β K 2 sin φ z ̂ ] ,
E s 1 2 n 2 π K 2 r exp ( i β z i K 2 r i π 4 + i Φ i θ β ) Δ n 0 ( z ) a 01 ( z ) 0 a J 0 ( u r ) J 0 ( K new r ) r d r β 2 cos ( ϕ δ ) r ̂ k 0 2 n 2 2 sin ( ϕ δ ) ϕ ̂ β K 2 cos ( ϕ δ ) z ̂ .
E r = A 1 J ν 1 ( K 2 r ) + A 2 N ν 1 ( K 2 r ) + B 1 A 1 K 2 r ν J ν ( K 2 r ) + B 2 A 2 K 2 r ν N ν ( K 2 r ) ,
E ϕ = i [ B 1 J ν 1 ( K 2 r ) + B 2 N ν 1 ( K 2 r ) + A 1 B 1 K 2 r ν J ν ( K 2 r ) + A 2 B 2 K 2 r ν N ν ( K 2 r ) ] ,
H r = i ε 0 μ 0 [ β k 0 B 1 J ν 1 ( K 2 r ) β k 0 B 2 N ν 1 ( K 2 r ) + β 2 B 1 k 0 2 n 2 2 A 1 k 0 β K 2 r ν J ν ( K 2 r ) + β 2 B 2 k 0 2 n 2 2 A 2 k 0 β K 2 r ν N ν ( K 2 r ) ] ,
H ϕ = ε 0 μ 0 [ k 0 n 2 2 β A 1 J ν 1 ( K 2 r ) + k 0 n 2 2 β A 2 N ν 1 ( K 2 r ) + β 2 B 1 k 0 2 n 2 2 A 1 k 0 β K 2 r ν J ν ( K 2 r ) + β 2 B 2 k 0 2 n 2 2 A 2 k 0 β K 2 r ν N ν ( K 2 r ) ] .
E r H ϕ * E ϕ H r * 1 2 ε 0 μ 0 [ ( A 1 B 1 ) ( k 0 n 2 2 β A 1 * β k 0 B 1 * ) J ν + 1 2 ( K 2 r ) ( A 1 + B 1 ) ( k 0 n 2 2 β A 1 * + β k 0 B 1 * ) J ν 1 2 ( K 2 r ) ] + 1 2 ε 0 μ 0 [ ( A 2 B 2 ) ( k 0 n 2 2 β A 2 * β k 0 B 2 * ) N ν + 1 2 ( K 2 r ) ( A 2 + B 2 ) ( k 0 n 2 2 β A 2 * + β k 0 B 2 * ) N ν 1 2 ( K 2 r ) ] .
k 0 n 2 β ( A 1 * B 1 + A 2 * B 2 ) + β k 0 n 2 ( A 1 B 1 * + A 2 B 2 * ) = 0.
{ ν G ( 1 , ν , ν 1 ) exp [ i ν ( ϕ + π 2 ) ] = exp [ i ( π 2 ϕ new ) ] 0 a J 1 ( u r ) J 1 ( K new r ) r d r , ν G ( 1 , ν , ν + 1 ) exp [ i ν ( ϕ + π 2 ) ] = exp [ i ( π 2 ϕ new ) ] 0 a J 1 ( u r ) J 1 ( K new r ) r d r . }
ν c ν H E e ν H E 2 π K 2 r exp ( i K 2 r + i π 4 i θ β ) ( I 0 e i φ + i K 1 β I 1 ) [ i r ̂ + ϕ ̂ + i K 2 β z ̂ ] ,
ν c ν E H e ν E H 2 π K 2 r exp ( i K 2 r + i π 4 i θ β ) [ I 0 ( p e i φ + q e i φ ) i K 1 k 0 n I 1 ] [ i ( p q ) r ̂ + ( p + q ) ϕ ̂ i K 2 k 0 n z ̂ ] .
I 0 0 a J 0 ( u r ) J 0 ( K new r ) r d r ,
I 1 u β 01 cos ( ϕ new δ ) 0 a J 1 ( u r ) J 1 ( K new r ) r d r .
E s 1 2 n 2 π K 2 r exp ( i β z i K 2 r i π 4 + i Φ i θ β ) Δ n 0 ( z ) a 01 ( z ) { [ I 0 β 2 cos ( ϕ δ ) + I 1 β K 1 ] r ̂ I 0 k 0 2 n 2 sin ( ϕ δ ) ϕ ̂ [ I 0 β K 2 cos ( ϕ δ ) + I 1 K 1 K 2 ] z ̂ } .

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