Abstract

Using an asymptotic analysis, we obtain an eigenvalue equation for the general mode dispersion in Bragg fibers. The asymptotic analysis is applied to calculate the dispersion relation and the field distribution of TE modes in a Bragg fiber. We compare the asymptotic results with exact solutions and find excellent agreement between them. This asymptotic approach greatly simplifies the analysis and design of Bragg fibers.

© 2000 Optical Society of America

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References

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  1. P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. 68, 1196 (1978).
  2. M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, Appl. Phys. Lett. 43, 431 (1983).
    [CrossRef]
  3. H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
    [CrossRef]
  4. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, J. Lightwave Technol. 17, 2039 (1999).
    [CrossRef]
  5. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
    [CrossRef] [PubMed]
  6. P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
    [CrossRef]
  7. J. Mathews and R. L. Walker, Mathematical Methods of Physics (Addison-Wesley, Reading, Mass., 1970).

1999 (2)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, J. Lightwave Technol. 17, 2039 (1999).
[CrossRef]

1995 (1)

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

1983 (1)

M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, Appl. Phys. Lett. 43, 431 (1983).
[CrossRef]

1978 (1)

1976 (1)

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

Aizawa, Y.

M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, Appl. Phys. Lett. 43, 431 (1983).
[CrossRef]

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Birks, T. A.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Chen, C.

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Fan, S.

Fink, Y.

Hongo, A.

M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, Appl. Phys. Lett. 43, 431 (1983).
[CrossRef]

Ito, H.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

Jhe, W.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

Joannopoulos, J. D.

Kawakami, S.

M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, Appl. Phys. Lett. 43, 431 (1983).
[CrossRef]

Knight, J. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Lee, K. I.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Marom, E.

Mathews, J.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Addison-Wesley, Reading, Mass., 1970).

Miyagi, M.

M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, Appl. Phys. Lett. 43, 431 (1983).
[CrossRef]

Nakata, T.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

Ohtsu, M.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

Ripin, D. J.

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Russell, P. St. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Sakaki, K.

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

Thomas, E. L.

Walker, R. L.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Addison-Wesley, Reading, Mass., 1970).

Yariv, A.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. 68, 1196 (1978).

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. 68, 1196 (1978).

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

Appl. Phys. Lett. (1)

M. Miyagi, A. Hongo, Y. Aizawa, and S. Kawakami, Appl. Phys. Lett. 43, 431 (1983).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

P. Yeh and A. Yariv, Opt. Commun. 19, 427 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, Phys. Rev. Lett. 76, 4500 (1995).
[CrossRef]

Science (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science 285, 1537 (1999).
[CrossRef] [PubMed]

Other (1)

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Addison-Wesley, Reading, Mass., 1970).

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Figures (3)

Fig. 1
Fig. 1

Schematic of the rz cross section of a Bragg fiber. The fiber core has refractive index nc and radius ρ1. The fiber cladding is composed of pairs of alternating layers of high- and low-index material. The high-index layer has refractive index n1 and thickness l1. The low-index layer has refractive index n2 and thickness l2.

Fig. 2
Fig. 2

Dispersion of the fundamental TE mode in an air-core Bragg fiber. The parameters of the Bragg fiber are nc=1, ρ1=1 µm, n1=3.0, l1=0.130 µm, n2=1.5, and l2=0.265 µm.

Fig. 3
Fig. 3

Hz and Eθ field of the guided TE mode at λ=1.55 µm in the same Bragg fiber as before. The dotted lines indicate cladding interfaces.

Equations (16)

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Ez=acJlkcr 0<r<ρ1,  Ez=an expik1r-ρn+bn exp-ik1r-ρnk1r,  ρn<r<ρn+l1,  Ez=an expik2r-ρn+bn exp-ik2r-ρnk2r,  ρn<r<ρn+l2,
Hz=ccJlkcr,  0<r<ρ1,  Hz=cn expik1r-ρn+dn exp-ik1r-ρnk1r,  ρn<r<ρn+l1,  Hz=cn expik2r-ρn+dn exp-ik2r-ρnk2r,  ρn<r<ρn+l2,
an+1bn+1=ATMBTMBTM*ATM*anbn,
cn+1dn+1=ATEBTEBTE*ATE*cndn,
ATE=expik1l1ik12+k222k1k2 sink2l2+cosk2l2,
BTE=i exp-ik1l1k12-k222k1k2 sink2l2,
ATM=expik1l1in24k12+n14k222n12n22k1k2 sink2l2+cosk2l2,
BTM=i exp-ik1l1n24k12-n14k222n12n22k1k2 sink2l2.
an+1bn+1=expiKTMΛanbn,  cn+1dn+1=expiKTEΛcndn,
expiKTMΛ=ReATM±ReATM2-11/2,
expiKTEΛ=ReATE±ReATE2-11/2.
ω2c2nc2Jlkcρ1Jlkcρ1+ikcn12 expiKTMΛ-ATM-BTMk1nc2 expiKTMΛ-ATM+BTM×Jlkcρ1Jlkcρ1+ikck1expiKTEΛ-ATE-BTEexpiKTEΛ-ATE+BTE=β2l2kc2ρ12.
J0kcρ1J0kcρ1+ikck1expiKTEΛ-ATE-BTEexpiKTEΛ-ATE+BTE=0.
cndn=C expin-1KTEΛBTEexpiKTEΛ-ATE,
cndn=12k2k1k1+k2k1 expik1l1k1-k2k1 exp-ik1l1k1-k2k1 expik1l1k1+k2k1 exp-ik1l1×cndn,
C=J0kcρ1k1ρ1expiKTEΛ-ATE+BTE.

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