Abstract

We investigate how the index profile of a few-mode fiber (FMF) can be designed so that group velocities of the two lowest-order modes can be equalized at a normalized frequency, which is below the cut-off frequency of the LP21 mode. This can be achieved using a single-clad power-law profile with a sufficiently large profile exponent or a double-clad profile consisting of a graded-core surrounded by a sufficiently thick depressed inner cladding without index jump at their interface. The fabrication tolerances, effective index differences, intramodal dispersion differences, and effective mode areas of various single- and double-clad profiles are compared. The results show that, in comparison to single-clad fibers, double-clad fibers are capable of producing higher fabrication tolerances and reduced sensitivity of group delay difference to wavelength by three and two orders of magnitude, respectively. Our analyses provide insights into the design of FMFs, which will facilitate future development of high-capacity mode division long-haul transmission systems.

© 2014 Optical Society of America

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    [CrossRef]
  3. Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6, 522–528 (2009).
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  5. A. A. Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4 × 4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19, 16672–16679 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. K. Kitayama, Y. Kato, S. Seikai, and N. Uchida, “Structural optimization for two-mode fiber: theory and experiment,” IEEE J. Quantum Electron. 17, 1057–1063 (1981).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (4)

2010 (1)

2009 (1)

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6, 522–528 (2009).

2008 (2)

2007 (1)

E. Ip and J. M. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” IEEE J. Lightwave Technol. 25, 2033–2043 (2007).
[CrossRef]

2006 (1)

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18, 2359–2361 (2006).
[CrossRef]

2004 (1)

F. Buchali and H. Bülow, “Adaptive PMD compensation by electrical and optical techniques,” IEEE J. Lightwave Technol. 22, 1116–1126 (2004).
[CrossRef]

2000 (1)

H. Rosenfeldt, R. Ulrich, U. Feiste, R. Ludwig, H. G. Weber, and A. Ehrhardt, “PMD compensation in 10  Gb/s NRZ field experiment using polarimetric error signal,” Electron. Lett. 36, 448–450 (2000).
[CrossRef]

1999 (1)

1987 (1)

M. M. Cvijetic and G. Lukatela, “Design considerations of dispersion-free dual-mode optical fibers: 1.55  μm wavelength operation,” IEEE J. Quantum Electron. 23, 469–472 (1987).
[CrossRef]

1982 (2)

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE Trans. Microw. Theory Tech. 30, 381–388 (1982).
[CrossRef]

Y. Kata, K. Kitayama, and S. Seikai, “Design consideration on broad-band w-type two-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 30, 1–5 (1982).
[CrossRef]

1981 (1)

K. Kitayama, Y. Kato, S. Seikai, and N. Uchida, “Structural optimization for two-mode fiber: theory and experiment,” IEEE J. Quantum Electron. 17, 1057–1063 (1981).
[CrossRef]

1978 (1)

J. Sakai, K. Kitayama, M. Ikeda, Y. Kato, and T. Kumura, “Design considerations of broadband dual-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 26, 658–665 (1978).
[CrossRef]

1976 (1)

S. Kawakami, S. Nishida, and M. Sumi, “Transmission characteristics of W-type optical fibres,” Proc. IEEE 123, 586–590 (1976).

1974 (1)

S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[CrossRef]

1971 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Amin, A. A.

A. A. Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4 × 4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19, 16672–16679 (2011).
[CrossRef]

A. A. Amin, A. Li, X. Chen, and W. Shieh, “LP01/LP11 dual-mode and dual-polarisation CO-OFDM transmission on two-mode fibre,” Electron. Lett. 47, 606–608 (2011).
[CrossRef]

A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” in Optical Fiber Communication Conference, OFC 2011, OSA Technical Digest Series (Optical Society of America, 2011), paper PDPB8.

Bai, N.

Barros, D. J. F.

Bolle, C. A.

Buchali, F.

F. Buchali and H. Bülow, “Adaptive PMD compensation by electrical and optical techniques,” IEEE J. Lightwave Technol. 22, 1116–1126 (2004).
[CrossRef]

Bülow, H.

F. Buchali and H. Bülow, “Adaptive PMD compensation by electrical and optical techniques,” IEEE J. Lightwave Technol. 22, 1116–1126 (2004).
[CrossRef]

Chen, M. Y.

M. Y. Chen, Y. R. Li, Y. Zhang, Y. F. Zhu, Y. K. Zhang, and J. Zhou, “Design of dual-mode optical fibres for the FTTH applications,” J. Opt. 13, 015402 (2011).
[CrossRef]

Chen, S.

Chen, X.

A. A. Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4 × 4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19, 16672–16679 (2011).
[CrossRef]

A. A. Amin, A. Li, X. Chen, and W. Shieh, “LP01/LP11 dual-mode and dual-polarisation CO-OFDM transmission on two-mode fibre,” Electron. Lett. 47, 606–608 (2011).
[CrossRef]

A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” in Optical Fiber Communication Conference, OFC 2011, OSA Technical Digest Series (Optical Society of America, 2011), paper PDPB8.

Cohen, L. G.

L. G. Cohen, W. G. French, and C. Lin, “Propagation characteristics of double-mode fibers,” in Optical Fiber Communication Conference (OFC 1979), OSA Technical Digest Series (Optical Society of America, 1979), paper ThC2.

Cvijetic, M. M.

M. M. Cvijetic and G. Lukatela, “Design considerations of dispersion-free dual-mode optical fibers: 1.55  μm wavelength operation,” IEEE J. Quantum Electron. 23, 469–472 (1987).
[CrossRef]

Ehrhardt, A.

H. Rosenfeldt, R. Ulrich, U. Feiste, R. Ludwig, H. G. Weber, and A. Ehrhardt, “PMD compensation in 10  Gb/s NRZ field experiment using polarimetric error signal,” Electron. Lett. 36, 448–450 (2000).
[CrossRef]

Essiambre, R.-J.

Feiste, U.

H. Rosenfeldt, R. Ulrich, U. Feiste, R. Ludwig, H. G. Weber, and A. Ehrhardt, “PMD compensation in 10  Gb/s NRZ field experiment using polarimetric error signal,” Electron. Lett. 36, 448–450 (2000).
[CrossRef]

Fischer, G.

French, W. G.

L. G. Cohen, W. G. French, and C. Lin, “Propagation characteristics of double-mode fibers,” in Optical Fiber Communication Conference (OFC 1979), OSA Technical Digest Series (Optical Society of America, 1979), paper ThC2.

Gao, G.

Glingener, C.

Gloge, D.

Gnauck, A. H.

Gottwald, E.

Haase, W.

Hanzawa, N.

N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10  km two-mode fiber withmode coupler,” in Optical Fiber Communication Conference (OFC 2011), OSA Technical Digest Series (Optical Society of America, 2011), paper OWA4.

Hinz, S.

Huijskens, F. M.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18, 2359–2361 (2006).
[CrossRef]

Ikeda, M.

J. Sakai, K. Kitayama, M. Ikeda, Y. Kato, and T. Kumura, “Design considerations of broadband dual-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 26, 658–665 (1978).
[CrossRef]

Imamura, K.

K. Imamura, K. Mukasa, and T. Yagi, “Multi-core holey fibers for the long-distance (>100  km) ultra large capacity transmission,” in Optical Fiber Communication Conference (OFC 2009), OSA Technical Digest Series (Optical Society of America, 2009), paper OTuC3.

Ip, E.

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16, 753–791 (2008).
[CrossRef]

E. Ip and J. M. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” IEEE J. Lightwave Technol. 25, 2033–2043 (2007).
[CrossRef]

Kahn, J. M.

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16, 753–791 (2008).
[CrossRef]

E. Ip and J. M. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” IEEE J. Lightwave Technol. 25, 2033–2043 (2007).
[CrossRef]

Kata, Y.

Y. Kata, K. Kitayama, and S. Seikai, “Design consideration on broad-band w-type two-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 30, 1–5 (1982).
[CrossRef]

Kato, Y.

K. Kitayama, Y. Kato, S. Seikai, and N. Uchida, “Structural optimization for two-mode fiber: theory and experiment,” IEEE J. Quantum Electron. 17, 1057–1063 (1981).
[CrossRef]

J. Sakai, K. Kitayama, M. Ikeda, Y. Kato, and T. Kumura, “Design considerations of broadband dual-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 26, 658–665 (1978).
[CrossRef]

Kawakami, S.

S. Kawakami, S. Nishida, and M. Sumi, “Transmission characteristics of W-type optical fibres,” Proc. IEEE 123, 586–590 (1976).

S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[CrossRef]

Kitayama, K.

Y. Kata, K. Kitayama, and S. Seikai, “Design consideration on broad-band w-type two-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 30, 1–5 (1982).
[CrossRef]

K. Kitayama, Y. Kato, S. Seikai, and N. Uchida, “Structural optimization for two-mode fiber: theory and experiment,” IEEE J. Quantum Electron. 17, 1057–1063 (1981).
[CrossRef]

J. Sakai, K. Kitayama, M. Ikeda, Y. Kato, and T. Kumura, “Design considerations of broadband dual-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 26, 658–665 (1978).
[CrossRef]

Kokubun, Y.

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6, 522–528 (2009).

Koonen, A. M. J.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18, 2359–2361 (2006).
[CrossRef]

Koshiba, M.

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6, 522–528 (2009).

N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10  km two-mode fiber withmode coupler,” in Optical Fiber Communication Conference (OFC 2011), OSA Technical Digest Series (Optical Society of America, 2011), paper OWA4.

Kumura, T.

J. Sakai, K. Kitayama, M. Ikeda, Y. Kato, and T. Kumura, “Design considerations of broadband dual-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 26, 658–665 (1978).
[CrossRef]

Lau, A. P. T.

Li, A.

A. A. Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4 × 4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19, 16672–16679 (2011).
[CrossRef]

A. A. Amin, A. Li, X. Chen, and W. Shieh, “LP01/LP11 dual-mode and dual-polarisation CO-OFDM transmission on two-mode fibre,” Electron. Lett. 47, 606–608 (2011).
[CrossRef]

A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” in Optical Fiber Communication Conference, OFC 2011, OSA Technical Digest Series (Optical Society of America, 2011), paper PDPB8.

Li, G.

Li, M. J.

Li, Y. R.

M. Y. Chen, Y. R. Li, Y. Zhang, Y. F. Zhu, Y. K. Zhang, and J. Zhou, “Design of dual-mode optical fibres for the FTTH applications,” J. Opt. 13, 015402 (2011).
[CrossRef]

Lin, C.

L. G. Cohen, W. G. French, and C. Lin, “Propagation characteristics of double-mode fibers,” in Optical Fiber Communication Conference (OFC 1979), OSA Technical Digest Series (Optical Society of America, 1979), paper ThC2.

Lingle, R.

Love, J. D.

N. Riesen and J. D. Love, “Dispersion equalization in few-mode fibres,” in 35th Australian Conference on Optical Fibre Technology (ACOFT) (2010), pp. 1–3.

Ludwig, R.

H. Rosenfeldt, R. Ulrich, U. Feiste, R. Ludwig, H. G. Weber, and A. Ehrhardt, “PMD compensation in 10  Gb/s NRZ field experiment using polarimetric error signal,” Electron. Lett. 36, 448–450 (2000).
[CrossRef]

Lukatela, G.

M. M. Cvijetic and G. Lukatela, “Design considerations of dispersion-free dual-mode optical fibers: 1.55  μm wavelength operation,” IEEE J. Quantum Electron. 23, 469–472 (1987).
[CrossRef]

Martinez, A.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18, 2359–2361 (2006).
[CrossRef]

Matsui, T.

N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10  km two-mode fiber withmode coupler,” in Optical Fiber Communication Conference (OFC 2011), OSA Technical Digest Series (Optical Society of America, 2011), paper OWA4.

McCurdy, A.

Mirvoda, V.

Monerie, M.

M. Monerie, “Propagation in doubly clad single-mode fibers,” IEEE Trans. Microw. Theory Tech. 30, 381–388 (1982).
[CrossRef]

Mukasa, K.

K. Imamura, K. Mukasa, and T. Yagi, “Multi-core holey fibers for the long-distance (>100  km) ultra large capacity transmission,” in Optical Fiber Communication Conference (OFC 2009), OSA Technical Digest Series (Optical Society of America, 2009), paper OTuC3.

Nishida, S.

S. Kawakami, S. Nishida, and M. Sumi, “Transmission characteristics of W-type optical fibres,” Proc. IEEE 123, 586–590 (1976).

S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[CrossRef]

Noé, R.

Nolan, D. A.

Peckham, D. W.

Randel, S.

Riesen, N.

N. Riesen and J. D. Love, “Dispersion equalization in few-mode fibres,” in 35th Australian Conference on Optical Fibre Technology (ACOFT) (2010), pp. 1–3.

Rosenfeldt, H.

H. Rosenfeldt, R. Ulrich, U. Feiste, R. Ludwig, H. G. Weber, and A. Ehrhardt, “PMD compensation in 10  Gb/s NRZ field experiment using polarimetric error signal,” Electron. Lett. 36, 448–450 (2000).
[CrossRef]

Ryf, R.

Saitoh, K.

N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10  km two-mode fiber withmode coupler,” in Optical Fiber Communication Conference (OFC 2011), OSA Technical Digest Series (Optical Society of America, 2011), paper OWA4.

Sakai, J.

J. Sakai, K. Kitayama, M. Ikeda, Y. Kato, and T. Kumura, “Design considerations of broadband dual-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 26, 658–665 (1978).
[CrossRef]

Sakamoto, T.

N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10  km two-mode fiber withmode coupler,” in Optical Fiber Communication Conference (OFC 2011), OSA Technical Digest Series (Optical Society of America, 2011), paper OWA4.

Sandel, D.

Scheerer, C.

Schöpflin, A.

Seikai, S.

Y. Kata, K. Kitayama, and S. Seikai, “Design consideration on broad-band w-type two-mode optical fibers,” IEEE Trans. Microw. Theory Tech. 30, 1–5 (1982).
[CrossRef]

K. Kitayama, Y. Kato, S. Seikai, and N. Uchida, “Structural optimization for two-mode fiber: theory and experiment,” IEEE J. Quantum Electron. 17, 1057–1063 (1981).
[CrossRef]

Senior, J. M.

J. M. Senior, Optical Fiber Communications (Prentice Hall, 1992).

Shieh, W.

A. A. Amin, A. Li, X. Chen, and W. Shieh, “LP01/LP11 dual-mode and dual-polarisation CO-OFDM transmission on two-mode fibre,” Electron. Lett. 47, 606–608 (2011).
[CrossRef]

A. A. Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4 × 4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19, 16672–16679 (2011).
[CrossRef]

A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” in Optical Fiber Communication Conference, OFC 2011, OSA Technical Digest Series (Optical Society of America, 2011), paper PDPB8.

Sierra, A.

Sumi, M.

S. Kawakami, S. Nishida, and M. Sumi, “Transmission characteristics of W-type optical fibres,” Proc. IEEE 123, 586–590 (1976).

Tomita, S.

N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10  km two-mode fiber withmode coupler,” in Optical Fiber Communication Conference (OFC 2011), OSA Technical Digest Series (Optical Society of America, 2011), paper OWA4.

Tsekrekos, C. P.

C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. 18, 2359–2361 (2006).
[CrossRef]

Uchida, N.

K. Kitayama, Y. Kato, S. Seikai, and N. Uchida, “Structural optimization for two-mode fiber: theory and experiment,” IEEE J. Quantum Electron. 17, 1057–1063 (1981).
[CrossRef]

Ulrich, R.

H. Rosenfeldt, R. Ulrich, U. Feiste, R. Ludwig, H. G. Weber, and A. Ehrhardt, “PMD compensation in 10  Gb/s NRZ field experiment using polarimetric error signal,” Electron. Lett. 36, 448–450 (2000).
[CrossRef]

Wang, T.

Weber, H. G.

H. Rosenfeldt, R. Ulrich, U. Feiste, R. Ludwig, H. G. Weber, and A. Ehrhardt, “PMD compensation in 10  Gb/s NRZ field experiment using polarimetric error signal,” Electron. Lett. 36, 448–450 (2000).
[CrossRef]

Weyrauch, T.

Winzer, P. J.

Yagi, T.

K. Imamura, K. Mukasa, and T. Yagi, “Multi-core holey fibers for the long-distance (>100  km) ultra large capacity transmission,” in Optical Fiber Communication Conference (OFC 2009), OSA Technical Digest Series (Optical Society of America, 2009), paper OTuC3.

Yaman, F.

Yoshida-Dierolf, M.

Zhang, Y.

M. Y. Chen, Y. R. Li, Y. Zhang, Y. F. Zhu, Y. K. Zhang, and J. Zhou, “Design of dual-mode optical fibres for the FTTH applications,” J. Opt. 13, 015402 (2011).
[CrossRef]

Zhang, Y. K.

M. Y. Chen, Y. R. Li, Y. Zhang, Y. F. Zhu, Y. K. Zhang, and J. Zhou, “Design of dual-mode optical fibres for the FTTH applications,” J. Opt. 13, 015402 (2011).
[CrossRef]

Zhou, J.

M. Y. Chen, Y. R. Li, Y. Zhang, Y. F. Zhu, Y. K. Zhang, and J. Zhou, “Design of dual-mode optical fibres for the FTTH applications,” J. Opt. 13, 015402 (2011).
[CrossRef]

Zhu, B.

Zhu, Y. F.

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Figures (7)

Fig. 1.
Fig. 1.

Schematics of various index profiles considered for designing a FMF. (a)–(d) Single-clad power-law profiles. (h) Single-clad profile with an index dip at the core/cladding interface. (e)–(g), (i)–(l) Double-clad profiles with a depressed inner cladding. Note that (l) is similar to (f), except for the truncated core.

Fig. 2.
Fig. 2.

Dependence of Δτ on V along with the corresponding values of Vc for various index profiles.

Fig. 3.
Fig. 3.

Values of dΔτ/dV|V0 for index profiles producing V0<Vc, showing that the sensitivity of Δτ to V for profiles (e)–(g) are smaller than (b)–(d) by three orders of magnitude.

Fig. 4.
Fig. 4.

Dependence of effective index difference ΔN on V for various index profiles along with the corresponding values of Vc for profiles producing V0<Vc.

Fig. 5.
Fig. 5.

Dispersion characteristics for the LP01 and LP11 modes of different index profiles. The two modes are distinguished by the dark and gray line colors while the legends apply to both the modes.

Fig. 6.
Fig. 6.

Difference in dispersion between the LP01 and LP11 modes operating at V0 for index profiles producing V0<Vc. The double-clad profiles (e)–(g) feature smaller dispersion differences by two orders of magnitude compared with those for power-law profiles (b)–(d).

Fig. 7.
Fig. 7.

Effective mode areas and effective index differences for various profiles producing V0<Vc operating at V0.

Tables (1)

Tables Icon

Table 1. Fiber Parameters for the Index Profiles Shown in Fig. 1 along with Relevant Characteristics Calculated at V0

Equations (1)

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n(r)={n1[12Δρ(r/a)α]1/2,0ran1(12Δζ)1/2,a>rbn1(12Δ)1/2=n2,r>b,

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