Abstract

A transfer-matrix method is developed for modeling a corrugated long-period fiber grating. Cladding-mode resonance in such a corrugated structure can be controlled by the applied tensile stress based on the photoelastic effect. A first-order vectorial perturbation expansion is used to derive the mode fields of the two basic regions under the strain-induced index perturbation. Because the etched cladding radius is much smaller than the unetched radius, the effect of the corrugated structure on cladding modes cannot be treated as a small perturbation. Thus the conventional coupled-mode theory is inadequate for the modeling of such a structure. Based on a self-consistent mode-matching technique, mode coupling within the corrugated structure can be described by a set of transfer matrices. We apply the formulation to the calculation of the transmission spectra of a corrugated long-period grating and compare the calculated with the experimental results. The transfer-matrix approach is found to account well for the features of the transmission spectra of the corrugated long-period gratings.

© 2001 Optical Society of America

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  1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [Crossref]
  2. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
    [Crossref] [PubMed]
  3. B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
    [Crossref]
  4. V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
    [Crossref] [PubMed]
  5. H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
    [Crossref]
  6. A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
    [Crossref]
  7. H. S. Kim, S. H. Yun, I. K. Kwang, B. Y. Kim, “All-fiber acousto-optic tunable notch filter with electronically controllable spectral profile,” Opt. Lett. 22, 1476–1478 (1997).
    [Crossref]
  8. C. Y. Lin, L. A. Wang, “Loss-tunable long period fibre grating made from etched corrugated structure,” Electron. Lett. 35, 1872–1873 (1999).
    [Crossref]
  9. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [Crossref]
  10. G. W. Chern, L. A. Wang, “Transfer-matrix method based on perturbation expansion for periodic and quasi-periodic binary long-period gratings,” J. Opt. Soc. Am. A 16, 2675–2689 (1999).
    [Crossref]
  11. M. Song, B. Lee, S. B. Lee, S. S. Choi, “Interferometric temperature-insensitive strain measurement with different-diameter fiber Bragg gratings,” Opt. Lett. 22, 790–792 (1997).
    [Crossref] [PubMed]
  12. J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1969).
  13. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.
  14. A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-15, 508–517 (1967).
    [Crossref]
  15. R. Kuszelewicz, G. Aubert, “Modal matrix theory for light propagation in laterally restricted stratified media,” J. Opt. Soc. Am. A 14, 3262–3272 (1997).
    [Crossref]
  16. C. Dragone, “Scattering at a junction of two waveguides with different surfaces impedences,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1328 (1984).
    [Crossref]
  17. W.-P. Huang, J. Hong, “A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbation,” J. Lightwave Technol. 11, 1367–1374 (1992).
    [Crossref]
  18. M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992).
  19. T. W. MacDougall, S. Pilevar, C. W. Haggans, M. A. Jackson, “Generalized expression for the growth of long period gratings,” IEEE Photon. Technol. Lett. 10, 1449–1451 (1998).
    [Crossref]

1999 (3)

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

C. Y. Lin, L. A. Wang, “Loss-tunable long period fibre grating made from etched corrugated structure,” Electron. Lett. 35, 1872–1873 (1999).
[Crossref]

G. W. Chern, L. A. Wang, “Transfer-matrix method based on perturbation expansion for periodic and quasi-periodic binary long-period gratings,” J. Opt. Soc. Am. A 16, 2675–2689 (1999).
[Crossref]

1998 (1)

T. W. MacDougall, S. Pilevar, C. W. Haggans, M. A. Jackson, “Generalized expression for the growth of long period gratings,” IEEE Photon. Technol. Lett. 10, 1449–1451 (1998).
[Crossref]

1997 (5)

1996 (4)

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[Crossref] [PubMed]

H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[Crossref]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[Crossref] [PubMed]

1992 (1)

W.-P. Huang, J. Hong, “A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbation,” J. Lightwave Technol. 11, 1367–1374 (1992).
[Crossref]

1984 (1)

C. Dragone, “Scattering at a junction of two waveguides with different surfaces impedences,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1328 (1984).
[Crossref]

1967 (1)

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-15, 508–517 (1967).
[Crossref]

Abramov, A. A.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

Aubert, G.

Bagratashvili, V. N.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Bergano, N. S.

Bhatia, V.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[Crossref] [PubMed]

Chern, G. W.

Choi, S. S.

Davidson, C. R.

de Sandro, J. P.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Dong, L.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Dragone, C.

C. Dragone, “Scattering at a junction of two waveguides with different surfaces impedences,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1328 (1984).
[Crossref]

Eggleton, B. J.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

Erdogan, T.

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[Crossref]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

Espindola, R. P.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

Haggans, C. W.

T. W. MacDougall, S. Pilevar, C. W. Haggans, M. A. Jackson, “Generalized expression for the growth of long period gratings,” IEEE Photon. Technol. Lett. 10, 1449–1451 (1998).
[Crossref]

Hale, A.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

Hong, J.

W.-P. Huang, J. Hong, “A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbation,” J. Lightwave Technol. 11, 1367–1374 (1992).
[Crossref]

Huang, W.-P.

W.-P. Huang, J. Hong, “A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbation,” J. Lightwave Technol. 11, 1367–1374 (1992).
[Crossref]

Jackson, M. A.

T. W. MacDougall, S. Pilevar, C. W. Haggans, M. A. Jackson, “Generalized expression for the growth of long period gratings,” IEEE Photon. Technol. Lett. 10, 1449–1451 (1998).
[Crossref]

Judkins, J. B.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[Crossref] [PubMed]

Kersey, A. D.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[Crossref]

Kim, B. Y.

Kim, H. S.

Kuszelewicz, R.

Kwang, I. K.

Laming, R. I.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Lee, B.

Lee, S. B.

Lemaire, P. J.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[Crossref] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

Lin, C. Y.

C. Y. Lin, L. A. Wang, “Loss-tunable long period fibre grating made from etched corrugated structure,” Electron. Lett. 35, 1872–1873 (1999).
[Crossref]

Liu, W. F.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.

MacDougall, T. W.

T. W. MacDougall, S. Pilevar, C. W. Haggans, M. A. Jackson, “Generalized expression for the growth of long period gratings,” IEEE Photon. Technol. Lett. 10, 1449–1451 (1998).
[Crossref]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1969).

Ortega, B.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Patrick, H. J.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[Crossref]

Pedrazzani, J. R.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[Crossref]

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[Crossref] [PubMed]

Pilevar, S.

T. W. MacDougall, S. Pilevar, C. W. Haggans, M. A. Jackson, “Generalized expression for the growth of long period gratings,” IEEE Photon. Technol. Lett. 10, 1449–1451 (1998).
[Crossref]

Reekie, L.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Rogers, J. A.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

Sadiku, M. N. O.

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992).

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.

Song, M.

Strasser, T. A.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

Tsypina, S. I.

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

Vengsarkar, A. M.

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 336–338 (1996).
[Crossref] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[Crossref]

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[Crossref] [PubMed]

Wang, L. A.

C. Y. Lin, L. A. Wang, “Loss-tunable long period fibre grating made from etched corrugated structure,” Electron. Lett. 35, 1872–1873 (1999).
[Crossref]

G. W. Chern, L. A. Wang, “Transfer-matrix method based on perturbation expansion for periodic and quasi-periodic binary long-period gratings,” J. Opt. Soc. Am. A 16, 2675–2689 (1999).
[Crossref]

Wexler, A.

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-15, 508–517 (1967).
[Crossref]

Williams, G. M.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[Crossref]

Windeler, R. S.

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

Yun, S. H.

Electron. Lett. (1)

C. Y. Lin, L. A. Wang, “Loss-tunable long period fibre grating made from etched corrugated structure,” Electron. Lett. 35, 1872–1873 (1999).
[Crossref]

IEEE Photon. Technol. Lett. (4)

B. Ortega, L. Dong, W. F. Liu, J. P. de Sandro, L. Reekie, S. I. Tsypina, V. N. Bagratashvili, R. I. Laming, “High-performance optical fiber polarizers based on long-period gratings in birefringent optical fibers,” IEEE Photon. Technol. Lett. 9, 1370–1372 (1997).
[Crossref]

H. J. Patrick, G. M. Williams, A. D. Kersey, J. R. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[Crossref]

A. A. Abramov, B. J. Eggleton, J. A. Rogers, R. P. Espindola, A. Hale, R. S. Windeler, T. A. Strasser, “Electrically tunable efficient broad-band fiber filter,” IEEE Photon. Technol. Lett. 11, 445–447 (1999).
[Crossref]

T. W. MacDougall, S. Pilevar, C. W. Haggans, M. A. Jackson, “Generalized expression for the growth of long period gratings,” IEEE Photon. Technol. Lett. 10, 1449–1451 (1998).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

C. Dragone, “Scattering at a junction of two waveguides with different surfaces impedences,” IEEE Trans. Microwave Theory Tech. MTT-32, 1319–1328 (1984).
[Crossref]

A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory Tech. MTT-15, 508–517 (1967).
[Crossref]

J. Lightwave Technol. (2)

W.-P. Huang, J. Hong, “A transfer matrix approach based on local normal modes for coupled waveguides with periodic perturbation,” J. Lightwave Technol. 11, 1367–1374 (1992).
[Crossref]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Lett. (4)

Other (3)

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1969).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1991), Secs. 31-1 and 31-8.

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 1992).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of a corrugated LPFG.

Fig. 2
Fig. 2

Experimental setup for applying tensile force and measuring the transmission spectra of a corrugated LPFG: ELED, electroluminescent (light-emitting) diode; OSA, optical spectrum analyzer.

Fig. 3
Fig. 3

(a) Schematic diagram of scattering of mode k as a result of the heterojunction from the unetched region to the etched region. Also shown is the definition of the corresponding scattering coefficients. (b) Same as (a), except for mode incidence from the etched region to the unetched region.

Fig. 4
Fig. 4

Dependence of cross-transmission coefficients σ ≡ σco–cl and σ′ ≡ σcl–co′ on the ratio of the cladding radii a cl (e)/a cl (u).

Fig. 5
Fig. 5

Transmission spectrum of a corrugated LPFG under applied strain s z = 530 µstrain.

Fig. 6
Fig. 6

Electric fields of the core mode and the first cladding mode in the unetched region.

Fig. 7
Fig. 7

Comparison of the electric fields of cladding modes in unetched and etched regions.

Fig. 8
Fig. 8

Dispersion curves of effective indices for core and cladding modes in unetched and etched regions.

Fig. 9
Fig. 9

Dispersions of overlapping integrals τco for the core mode and τcl for the cladding mode.

Fig. 10
Fig. 10

Dispersions of self-coupling constants κco–co for the core mode and κcl-clu and κcl-cle for the cladding modes in unetched and etched regions.

Fig. 11
Fig. 11

Dispersions of cross-coupling constants κco–cl (u) and κco–cl (e) in unetched (u) and etched (e) regions.

Fig. 12
Fig. 12

(a) Measured transmission spectra of the corrugated LPFG under various applied strains. (b) The corresponding calculated transmission spectra.

Equations (48)

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

szu=FYAcluΔLL=sz,
sze=FYAcle=acluacle2sz,
Δntr=-1/2p12-νp11-νp12n0r3szr=-1/2ptn0r3szr,
Δnzr=-1/2p11-2νp12n0r3szr=-1/2pzn0r3szr,
1/2 Aekr×hjr*·zdA=δkj,
etj=kajk+bjkētk, htj=kajk-bjkh¯tk.
ajk=k04β¯j-β¯k1Z0A 2n0Δntētj·ētk*dA, bjk=k04β¯j+β¯k1Z0A 2n0Δntētj·ētk*dA.
βj-β¯j=k04Z0A 2n0Δntetj·ētj*dA.
ej1-djsz2ēj-szkjκjkβ¯j-β¯kēk, hj1-djsz2h¯j-szkjκjkβ¯j-β¯kh¯k,
κjk=ptωε04An0r4ētj·ētk*dA,
βj=β¯j+szΔβj=β¯j-szκjj,
κjj=ptωε04An0r4|ētj|2dA.
A+Arkketku+jk Arkjetju=j Atkjetje,
A-Arkkhtku-jk Arkjhtju=j Atkjhtje.
Ikj=1/2Aeku×hje*·zdA, Jkj=1/2Aeje*×hku·zdA.
Ikj=akj+bkj, Jkj=akj-bkj,
akj=k04βku-βje1Z0Antu2-nte2etku·etje*dA, bkj=k04βku+βje1Z0Antu2-nte2etku·etje*dA.
1+rkketku=tkketke+kjtkj-rkjetje+δek,
1-rkkhtku=tkkhtke+jktkj+rkjhtje+δhk,
δek=jk rkjetje-etju, δhk=jk rkjhtju-htje.
tkk=Ikk1+rkk=Jkk1-rkk.
rkk=Jkk-IkkJkk+Ikk=-bkkakk,
rkk=βke-βkuβke+βku,
1+rkkIkj=tkj-rkj, 1-rkkJkj=tkj+rkj.
tkj=1/2Jkj+Ikj-rkkJkj-Ikj=akj+rkkbkj, rkj=1/2Jkj-Ikj-rkkJkj+Ikj=-bkj-rkkakj.
ĪkjJ¯kjδkjτk,
tkjδkj1-αksz2τk+szσkj.
αk=dku+dkeacluacle4,
σkj=acluacle2κkjeτkβ¯ke-β¯je - κkjuτjβ¯ku-β¯jekj0k=j.
Fe|usz=1-αcosz2τcoσsz-σsz1-αclsz2τcl,
Fu|esz=1-αcosz2τco-σszσsz1-αclsz2τcl.
Pr=expiθcor00expiθclr,
θcor=β¯cor-szrκcocorΛr1+szr, θclr=β¯clr-szrκclclrΛr1+szr.
θcou-θclu+θcoe-θcle=2π.
β¯co-β¯cl-s¯zκcoco-κclcl=1-s¯z2πΛ,
λ0=Λn¯co-n¯cl1+s¯z1-ΔκsΛ2π,
D=ε0n2·E,
n2=nt2000nt2000nz2.
E=ej expiβjz, H=hj expiβjz,
E=ek expiβkz, H=hk expiβkz,
Fc=E×H*+E*×H.
zAFc·zdA=A ·FcdA.
×E=iZ0k0H, ×H=-ik0/Z0n2·E,
×E*=-iZ0k0H*, ×H*=ik0/Z0n2·E*,
βj=βk+k0Z0Aej·n2-n2·ek*dAAej×hk*+ek*×hj·zdA=βk+k0Z0Aδnt2etj·etk*+δnz2ezjezk*dAAej×hk*+ek*×hj·zdA.
etj=kajk+bjketk, htj=kajk-bjkhtk.
ezj=nz2nz2kajk-bjkezk, hzj=kajk+bjkhzk.
ajk=k04βj-βk1Z0Aδnt2etj·etk*+δnz2ezjezk*dA, bjk=k04βj+βk1Z0Aδnt2etj·etk*-δnz2ezjezk*dA.

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