Analysis of leaky-mode losses for optical fibers
JOSA B, Vol. 25, Issue 4, pp. 545-554 (2008)
http://dx.doi.org/10.1364/JOSAB.25.000545
Acrobat PDF (193 KB)
Abstract
We present a direct, rigorous, and fast numerical method for obtaining leaky-mode losses in optical fibers by solely solving complex propagation constants of the characteristic equation of leaky modes. Both the modified Bessel function and the Hankel function of the second kind are individually used to express the field component of leaky modes in the outermost cladding. The characteristic equation of cylindrically symmetric fiber structures, which consist of uniform and graded layers, is derived by combining the Runge–Kutta method and the exact solution of a homogeneous layer. Since complex root searching is the key technique in this method, we also present a numerical algorithm for solving the characteristic equation of optical fibers. Moreover, because for both guided and leaky modes the field distributions in the outermost cladding region have the same expression, the leaky mode can be easily obtained by choosing an improper solution, and therefore the calculation of leaky modes demonstrates the simplicity of this method. An approximation rule of branch choices for lossy material is also derived. The approach we present is consistent with the results of previously published papers.
© 2008 Optical Society of America
1. INTRODUCTION
1. K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, “Accurate numerical method for the calculation of bending loss in optical waveguides using a matrix approach,” Opt. Lett. 12, 296–298 (1987). [CrossRef] [PubMed]
2. K. Thyagarajan, S. Diggavi, and A. K. Ghatak, “Analytical investigation of leaky and absorbing planar structures,” Opt. Quantum Electron. 19, 131–137 (1987). [CrossRef]
3. S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974). [CrossRef]
6. A. W. Snyder and D. J. Mitchell, “Leaky mode analysis of circular optical waveguides,” Opto-electronics (London) 6, 287–296 (1974). [CrossRef]
8. M. Maeda and S. Yamada, “Leaky modes on W-fibers: mode structure and attenuation,” Appl. Opt. 16, 2198–2203 (1977). [CrossRef] [PubMed]
9. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef]
10. H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31–32 (1991). [CrossRef]
11. M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. 6, 1285–1291 (1988). [CrossRef]
12. B. P. Pal and V. Priye, “The effect of an axial dip and ripples in the inner cladding on the leakage loss of
13. B. P. Pal, R. L. Gallawa, and I. C. Goyal, “
14. K. Thyagarajan, S. Diggavi, A. Taneja, and A. K. Ghatak, “Simple numerical technique for the analysis of cylindrically symmetric refractive-index profile optical fibers,” Appl. Opt. 30, 3877–3879 (1991). [CrossRef] [PubMed]
9. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef]
10. H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31–32 (1991). [CrossRef]
8. M. Maeda and S. Yamada, “Leaky modes on W-fibers: mode structure and attenuation,” Appl. Opt. 16, 2198–2203 (1977). [CrossRef] [PubMed]
9. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef]
10. H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31–32 (1991). [CrossRef]
11. M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. 6, 1285–1291 (1988). [CrossRef]
11. M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. 6, 1285–1291 (1988). [CrossRef]
12. B. P. Pal and V. Priye, “The effect of an axial dip and ripples in the inner cladding on the leakage loss of
13. B. P. Pal, R. L. Gallawa, and I. C. Goyal, “
14. K. Thyagarajan, S. Diggavi, A. Taneja, and A. K. Ghatak, “Simple numerical technique for the analysis of cylindrically symmetric refractive-index profile optical fibers,” Appl. Opt. 30, 3877–3879 (1991). [CrossRef] [PubMed]
15. R. Singh and E. K. Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. 37, 635–640 (2001). [CrossRef]
16. X. Qian and A. C. Boucouvalas, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles,” IEEE J. Quantum Electron. 40, 771–777 (2004). [CrossRef]
15. R. Singh and E. K. Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. 37, 635–640 (2001). [CrossRef]
16. X. Qian and A. C. Boucouvalas, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles,” IEEE J. Quantum Electron. 40, 771–777 (2004). [CrossRef]
15. R. Singh and E. K. Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. 37, 635–640 (2001). [CrossRef]
16. X. Qian and A. C. Boucouvalas, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles,” IEEE J. Quantum Electron. 40, 771–777 (2004). [CrossRef]
17. S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995). [CrossRef]
18. Y. Z. Lin, J. H. Zhan, and S. M. Tseng, “A new method of analyzing the light transmission in leaky and absorbing planar waveguides,” IEEE Photon. Technol. Lett. 9, 1241–1243 (1997). [CrossRef]
19. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999). [CrossRef]
20. J. Petracek and K. Singh, “Determination of leaky modes in planar multilayer waveguides,” IEEE Photon. Technol. Lett. 14, 810–812 (2002). [CrossRef]
17. S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995). [CrossRef]
18. Y. Z. Lin, J. H. Zhan, and S. M. Tseng, “A new method of analyzing the light transmission in leaky and absorbing planar waveguides,” IEEE Photon. Technol. Lett. 9, 1241–1243 (1997). [CrossRef]
19. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999). [CrossRef]
20. J. Petracek and K. Singh, “Determination of leaky modes in planar multilayer waveguides,” IEEE Photon. Technol. Lett. 14, 810–812 (2002). [CrossRef]
21. N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. 21, 1005–1012 (2003). [CrossRef]
21. N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. 21, 1005–1012 (2003). [CrossRef]
19. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999). [CrossRef]
17. S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995). [CrossRef]
22. N. H. Sun, C. C. Chou, H. W. Chang, J. K. Butler, and G. A. Evans, “Radiation loss of grating-assisted directional couplers using the Floquet-Bloch theory,” J. Lightwave Technol. 24, 2409–2415 (2006). [CrossRef]
10. H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31–32 (1991). [CrossRef]
14. K. Thyagarajan, S. Diggavi, A. Taneja, and A. K. Ghatak, “Simple numerical technique for the analysis of cylindrically symmetric refractive-index profile optical fibers,” Appl. Opt. 30, 3877–3879 (1991). [CrossRef] [PubMed]
22. N. H. Sun, C. C. Chou, H. W. Chang, J. K. Butler, and G. A. Evans, “Radiation loss of grating-assisted directional couplers using the Floquet-Bloch theory,” J. Lightwave Technol. 24, 2409–2415 (2006). [CrossRef]
24. N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, and P. Congdon, “Analysis of grating-assisted directional couplers using the Floquet-Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997). [CrossRef]
9. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef]
13. B. P. Pal, R. L. Gallawa, and I. C. Goyal, “
2. PROBLEM FORMULATION
26. M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics (London) 6, 271–286 (1974). [CrossRef]
26. M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics (London) 6, 271–286 (1974). [CrossRef]
3. BRANCH CHOICES FOR TRANSVERSE WAVE VECTORS
8. M. Maeda and S. Yamada, “Leaky modes on W-fibers: mode structure and attenuation,” Appl. Opt. 16, 2198–2203 (1977). [CrossRef] [PubMed]
10. H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31–32 (1991). [CrossRef]
4. RESULTS
9. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef]
9. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef]
9. L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef]
13. B. P. Pal, R. L. Gallawa, and I. C. Goyal, “
13. B. P. Pal, R. L. Gallawa, and I. C. Goyal, “
13. B. P. Pal, R. L. Gallawa, and I. C. Goyal, “
5. APPROXIMATION RULE OF BRANCH CHOICES FOR LOSSY MATERIAL
6. ALGORITHM FOR SEARCHING SOLUTIONS OF THE CHARACTERISTIC EQUATION
7. CONCLUSION
APPENDIX A
ACKNOWLEDGMENT
1. | K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, “Accurate numerical method for the calculation of bending loss in optical waveguides using a matrix approach,” Opt. Lett. 12, 296–298 (1987). [CrossRef] [PubMed] |
2. | K. Thyagarajan, S. Diggavi, and A. K. Ghatak, “Analytical investigation of leaky and absorbing planar structures,” Opt. Quantum Electron. 19, 131–137 (1987). [CrossRef] |
3. | S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974). [CrossRef] |
4. | Y. Akasaka, R. Sugizaki, S. Arai, Y. Suzuki, and T. Kamiya, “Dispersion flat compensation fiber for dispersion shifted fiber,” in 22nd European Conference on Optical Communication (IEEE, 1996), pp. 221–224. |
5. | T. Tsuda, Y. Akasaka, S. Sentsui, K. Aiso, Y. Suzuki, and T. Kamiya, “Broad band dispersion slope compensation of dispersion shifted fiber using negative slope fiber,” in 24th European Conference on Optical Communication (IEEE, 1998), pp. 233–234. |
6. | A. W. Snyder and D. J. Mitchell, “Leaky mode analysis of circular optical waveguides,” Opto-electronics (London) 6, 287–296 (1974). [CrossRef] |
7. | S. Kawakami and S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-11, 131–138 (1975). |
8. | M. Maeda and S. Yamada, “Leaky modes on W-fibers: mode structure and attenuation,” Appl. Opt. 16, 2198–2203 (1977). [CrossRef] [PubMed] |
9. | L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467–1472 (1982). [CrossRef] |
10. | H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31–32 (1991). [CrossRef] |
11. | M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. 6, 1285–1291 (1988). [CrossRef] |
12. | B. P. Pal and V. Priye, “The effect of an axial dip and ripples in the inner cladding on the leakage loss of |
13. | B. P. Pal, R. L. Gallawa, and I. C. Goyal, “ |
14. | K. Thyagarajan, S. Diggavi, A. Taneja, and A. K. Ghatak, “Simple numerical technique for the analysis of cylindrically symmetric refractive-index profile optical fibers,” Appl. Opt. 30, 3877–3879 (1991). [CrossRef] [PubMed] |
15. | R. Singh and E. K. Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. 37, 635–640 (2001). [CrossRef] |
16. | X. Qian and A. C. Boucouvalas, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles,” IEEE J. Quantum Electron. 40, 771–777 (2004). [CrossRef] |
17. | S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995). [CrossRef] |
18. | Y. Z. Lin, J. H. Zhan, and S. M. Tseng, “A new method of analyzing the light transmission in leaky and absorbing planar waveguides,” IEEE Photon. Technol. Lett. 9, 1241–1243 (1997). [CrossRef] |
19. | E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999). [CrossRef] |
20. | J. Petracek and K. Singh, “Determination of leaky modes in planar multilayer waveguides,” IEEE Photon. Technol. Lett. 14, 810–812 (2002). [CrossRef] |
21. | N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. 21, 1005–1012 (2003). [CrossRef] |
22. | N. H. Sun, C. C. Chou, H. W. Chang, J. K. Butler, and G. A. Evans, “Radiation loss of grating-assisted directional couplers using the Floquet-Bloch theory,” J. Lightwave Technol. 24, 2409–2415 (2006). [CrossRef] |
23. | R. E. Collin and F. J. Zucker, eds., Antenna Theory (McGraw-Hill, 1969), p. 203. |
24. | N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, and P. Congdon, “Analysis of grating-assisted directional couplers using the Floquet-Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997). [CrossRef] |
25. | Y. Yokoyama, T. Kato, M. Hirano, M. Onishi, E. Sasaoka, Y. Makio, and M. Nishimura, “Practically feasible dispersion flattened fibers produced by VCD technique,” in 24th European Conference on Optical Communication (IEEE, 1998), pp. 131–132. |
26. | M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics (London) 6, 271–286 (1974). [CrossRef] |
27. | R. L. Burden and J. D. Faires, Numerical Analysis (PWS-KENT, 1984). |
28. | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972). |
OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2310) Fiber optics and optical communications : Fiber optics
ToC Category:
Fiber Optics and Optical Communications
History
Original Manuscript: July 9, 2007
Revised Manuscript: November 17, 2007
Manuscript Accepted: January 21, 2008
Published: March 21, 2008
Citation
Chih-Cheng Chou and Nai-Hsiang Sun, "Analysis of leaky-mode losses for optical fibers," J. Opt. Soc. Am. B 25, 545-554 (2008)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-25-4-545
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References
- K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, “Accurate numerical method for the calculation of bending loss in optical waveguides using a matrix approach,” Opt. Lett. 12, 296-298 (1987). [CrossRef] [PubMed]
- K. Thyagarajan, S. Diggavi, and A. K. Ghatak, “Analytical investigation of leaky and absorbing planar structures,” Opt. Quantum Electron. 19, 131-137 (1987). [CrossRef]
- S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-10, 879-887 (1974). [CrossRef]
- Y. Akasaka, R. Sugizaki, S. Arai, Y. Suzuki, and T. Kamiya, “Dispersion flat compensation fiber for dispersion shifted fiber,” in 22nd European Conference on Optical Communication (IEEE, 1996), pp. 221-224.
- T. Tsuda, Y. Akasaka, S. Sentsui, K. Aiso, Y. Suzuki, and T. Kamiya, “Broad band dispersion slope compensation of dispersion shifted fiber using negative slope fiber,” in 24th European Conference on Optical Communication (IEEE, 1998), pp. 233-234.
- A. W. Snyder and D. J. Mitchell, “Leaky mode analysis of circular optical waveguides,” Opto-electronics (London) 6, 287-296 (1974). [CrossRef]
- S. Kawakami and S. Nishida, “Perturbation theory of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. QE-11, 131-138 (1975).
- M. Maeda and S. Yamada, “Leaky modes on W-fibers: mode structure and attenuation,” Appl. Opt. 16, 2198-2203 (1977). [CrossRef] [PubMed]
- L. G. Cohen, D. Marcuse, and W. L. Mammel, “Radiating leaky-mode losses in single-mode lightguides with depressed-index claddings,” IEEE J. Quantum Electron. QE-18, 1467-1472 (1982). [CrossRef]
- H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31-32 (1991). [CrossRef]
- M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fibers using matrix approach,” J. Lightwave Technol. 6, 1285-1291 (1988). [CrossRef]
- B. P. Pal and V. Priye, “The effect of an axial dip and ripples in the inner cladding on the leakage loss of LP01 mode in depressed index clad fibre,” IEE Proc.: Optoelectron. 137, 311-314 (1990). [CrossRef]
- B. P. Pal, R. L. Gallawa, and I. C. Goyal, “LP11-mode leakage loss in coated depressed clad fibers,” IEEE Photon. Technol. Lett. 4, 376-378 (1992). [CrossRef]
- K. Thyagarajan, S. Diggavi, A. Taneja, and A. K. Ghatak, “Simple numerical technique for the analysis of cylindrically symmetric refractive-index profile optical fibers,” Appl. Opt. 30, 3877-3879 (1991). [CrossRef] [PubMed]
- R. Singh, Sunanda, and E. K. Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. 37, 635-640 (2001). [CrossRef]
- X. Qian and A. C. Boucouvalas, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles,” IEEE J. Quantum Electron. 40, 771-777 (2004). [CrossRef]
- S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790-1802 (1995). [CrossRef]
- Y. Z. Lin, J. H. Zhan, and S. M. Tseng, “A new method of analyzing the light transmission in leaky and absorbing planar waveguides,” IEEE Photon. Technol. Lett. 9, 1241-1243 (1997). [CrossRef]
- E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929-941 (1999). [CrossRef]
- J. Petracek and K. Singh, “Determination of leaky modes in planar multilayer waveguides,” IEEE Photon. Technol. Lett. 14, 810-812 (2002). [CrossRef]
- N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. 21, 1005-1012 (2003). [CrossRef]
- N. H. Sun, C. C. Chou, H. W. Chang, J. K. Butler, and G. A. Evans, “Radiation loss of grating-assisted directional couplers using the Floquet-Bloch theory,” J. Lightwave Technol. 24, 2409-2415 (2006). [CrossRef]
- R. E. Collin and F. J. Zucker, eds., Antenna Theory (McGraw-Hill, 1969), p. 203.
- N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, and P. Congdon, “Analysis of grating-assisted directional couplers using the Floquet-Bloch theory,” J. Lightwave Technol. 15, 2301-2315 (1997). [CrossRef]
- Y. Yokoyama, T. Kato, M. Hirano, M. Onishi, E. Sasaoka, Y. Makio, and M. Nishimura, “Practically feasible dispersion flattened fibers produced by VCD technique,” in 24th European Conference on Optical Communication (IEEE, 1998), pp. 131-132.
- M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics (London) 6, 271-286 (1974). [CrossRef]
- R. L. Burden and J. D. Faires, Numerical Analysis (PWS-KENT, 1984).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
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