Abstract

Optical fibers and specialty waveguides are the bases of the majority of today's telecommunication, biomedical, sensing, and light-delivery applications. Modal analysis plays an important role in optimizing the optical performance of these fibers when they are integrated with optical systems. We present a full vectorial modal theoretical analysis of specialty cylindrical symmetric fibers with arbitrary index profiles, using a staircase approximation and scattering matrix approach with no constraints on the refractive index profile. We demonstrate the generality of this method by investigating the modal characteristics of two specialty fibers: graded-index fiber and concentric-shell multicore fiber. The calculated modal effective indices for the graded-index fiber are compared with those calculated by the WKB method, stressing the main differences between the scalar and vectorial approaches. Using the same approach, we calculate the Bragg grating response of a holographic grating written in the guiding regions of a concentric-shell fiber and compared them with experimental measurements.

© 2006 Optical Society of America

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References

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  1. R. Yamauchi, "Specialty fibers," in Lasers and Electro-Optics Society Annual Meeting (Institute of Electrical and Electronics Engineers, 1994), pp. 228-229.
  2. G. Jacobsen, "Multimode graded-index optical fibers: comparison of two Wentzel-Kramers-Brillouin formulations," J. Opt. Soc. Am. 71, 1492-1496 (1981).
    [CrossRef]
  3. G. Jacobsen, "Evanescent-wave analysis of general clad graded index fibers," J. Opt. Soc. Am. 72, 699-710 (1982).
    [CrossRef]
  4. J. H. Povisen, P. Danielsen, and G. Jacobsen, "Modal propagation constants, group delays, and eigenfields for practical multimode graded-index fibers," J. Opt. Soc. Am. 72, 1506-1513 (1982).
    [CrossRef]
  5. H. Ikuno and H. Watanabe, "Mode dispersion in graded-index optical fiber with near parabolic-index profiles," in MTT-S International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, 1979), pp. 472-474.
    [CrossRef]
  6. A. Kowalski, "On the analysis of optical fibers described in terms of Chebyshev polynomials," J. Lightwave Technol. 8, 164-167 (1990).
    [CrossRef]
  7. G. Jacobsen and J. J. R. Hansen, "Propagation constants and group delays of guided modes in graded index fibers: a comparison of three theories," Appl. Opt. 18, 2837-2842 (1979).
    [CrossRef] [PubMed]
  8. W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," IEEE Photon. Technol. Lett. 17, 1441-1443 (2005).
    [CrossRef]
  9. T. Szkopek, V. Pasupathy, J. E. Sipe, and P. W. E. Smith, "Novel multimode fiber for narrow-band Bragg gratings," IEEE J. Sel. Top. Quantum Electron. 7, 425-433 (2001).
    [CrossRef]
  10. T. Okoshi, Optical Fibers (Academic, 1984).
  11. N. Kishi, K. Tayama, and E. Yamashita, "Modal analysis of optical fiber with symmetrically distributed nonuniform cores," J. Lightwave Technol. 14, 1794-1800 (1996).
    [CrossRef]
  12. W. S. Mohammed, L. Vaissié, and E. G. Johnson, "Hybrid mode calculations for novel photonic crystal fibers," Opt. Eng. 42, 2311-2317 (2003).
    [CrossRef]
  13. Y. Sun, T. Szkopek, and P. W. E. Smith, "Demonstration of narrowband high-reflectivity Bragg gratings in a novel multimode fiber," Opt. Commun. 223, 91-95 (2003).
    [CrossRef]

2005 (1)

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," IEEE Photon. Technol. Lett. 17, 1441-1443 (2005).
[CrossRef]

2003 (2)

W. S. Mohammed, L. Vaissié, and E. G. Johnson, "Hybrid mode calculations for novel photonic crystal fibers," Opt. Eng. 42, 2311-2317 (2003).
[CrossRef]

Y. Sun, T. Szkopek, and P. W. E. Smith, "Demonstration of narrowband high-reflectivity Bragg gratings in a novel multimode fiber," Opt. Commun. 223, 91-95 (2003).
[CrossRef]

2001 (1)

T. Szkopek, V. Pasupathy, J. E. Sipe, and P. W. E. Smith, "Novel multimode fiber for narrow-band Bragg gratings," IEEE J. Sel. Top. Quantum Electron. 7, 425-433 (2001).
[CrossRef]

1996 (1)

N. Kishi, K. Tayama, and E. Yamashita, "Modal analysis of optical fiber with symmetrically distributed nonuniform cores," J. Lightwave Technol. 14, 1794-1800 (1996).
[CrossRef]

1990 (1)

A. Kowalski, "On the analysis of optical fibers described in terms of Chebyshev polynomials," J. Lightwave Technol. 8, 164-167 (1990).
[CrossRef]

1982 (2)

1981 (1)

1979 (1)

Danielsen, P.

Hansen, J. J. R.

Ikuno, H.

H. Ikuno and H. Watanabe, "Mode dispersion in graded-index optical fiber with near parabolic-index profiles," in MTT-S International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, 1979), pp. 472-474.
[CrossRef]

Jacobsen, G.

Johnson, E. G.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," IEEE Photon. Technol. Lett. 17, 1441-1443 (2005).
[CrossRef]

W. S. Mohammed, L. Vaissié, and E. G. Johnson, "Hybrid mode calculations for novel photonic crystal fibers," Opt. Eng. 42, 2311-2317 (2003).
[CrossRef]

Kishi, N.

N. Kishi, K. Tayama, and E. Yamashita, "Modal analysis of optical fiber with symmetrically distributed nonuniform cores," J. Lightwave Technol. 14, 1794-1800 (1996).
[CrossRef]

Kowalski, A.

A. Kowalski, "On the analysis of optical fibers described in terms of Chebyshev polynomials," J. Lightwave Technol. 8, 164-167 (1990).
[CrossRef]

Mehta, A.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," IEEE Photon. Technol. Lett. 17, 1441-1443 (2005).
[CrossRef]

Mohammed, W. S.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," IEEE Photon. Technol. Lett. 17, 1441-1443 (2005).
[CrossRef]

W. S. Mohammed, L. Vaissié, and E. G. Johnson, "Hybrid mode calculations for novel photonic crystal fibers," Opt. Eng. 42, 2311-2317 (2003).
[CrossRef]

Okoshi, T.

T. Okoshi, Optical Fibers (Academic, 1984).

Pasupathy, V.

T. Szkopek, V. Pasupathy, J. E. Sipe, and P. W. E. Smith, "Novel multimode fiber for narrow-band Bragg gratings," IEEE J. Sel. Top. Quantum Electron. 7, 425-433 (2001).
[CrossRef]

Pitchumani, M.

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," IEEE Photon. Technol. Lett. 17, 1441-1443 (2005).
[CrossRef]

Povisen, J. H.

Sipe, J. E.

T. Szkopek, V. Pasupathy, J. E. Sipe, and P. W. E. Smith, "Novel multimode fiber for narrow-band Bragg gratings," IEEE J. Sel. Top. Quantum Electron. 7, 425-433 (2001).
[CrossRef]

Smith, P. W. E.

Y. Sun, T. Szkopek, and P. W. E. Smith, "Demonstration of narrowband high-reflectivity Bragg gratings in a novel multimode fiber," Opt. Commun. 223, 91-95 (2003).
[CrossRef]

T. Szkopek, V. Pasupathy, J. E. Sipe, and P. W. E. Smith, "Novel multimode fiber for narrow-band Bragg gratings," IEEE J. Sel. Top. Quantum Electron. 7, 425-433 (2001).
[CrossRef]

Sun, Y.

Y. Sun, T. Szkopek, and P. W. E. Smith, "Demonstration of narrowband high-reflectivity Bragg gratings in a novel multimode fiber," Opt. Commun. 223, 91-95 (2003).
[CrossRef]

Szkopek, T.

Y. Sun, T. Szkopek, and P. W. E. Smith, "Demonstration of narrowband high-reflectivity Bragg gratings in a novel multimode fiber," Opt. Commun. 223, 91-95 (2003).
[CrossRef]

T. Szkopek, V. Pasupathy, J. E. Sipe, and P. W. E. Smith, "Novel multimode fiber for narrow-band Bragg gratings," IEEE J. Sel. Top. Quantum Electron. 7, 425-433 (2001).
[CrossRef]

Tayama, K.

N. Kishi, K. Tayama, and E. Yamashita, "Modal analysis of optical fiber with symmetrically distributed nonuniform cores," J. Lightwave Technol. 14, 1794-1800 (1996).
[CrossRef]

Vaissié, L.

W. S. Mohammed, L. Vaissié, and E. G. Johnson, "Hybrid mode calculations for novel photonic crystal fibers," Opt. Eng. 42, 2311-2317 (2003).
[CrossRef]

Watanabe, H.

H. Ikuno and H. Watanabe, "Mode dispersion in graded-index optical fiber with near parabolic-index profiles," in MTT-S International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, 1979), pp. 472-474.
[CrossRef]

Yamashita, E.

N. Kishi, K. Tayama, and E. Yamashita, "Modal analysis of optical fiber with symmetrically distributed nonuniform cores," J. Lightwave Technol. 14, 1794-1800 (1996).
[CrossRef]

Yamauchi, R.

R. Yamauchi, "Specialty fibers," in Lasers and Electro-Optics Society Annual Meeting (Institute of Electrical and Electronics Engineers, 1994), pp. 228-229.

Appl. Opt. (1)

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

T. Szkopek, V. Pasupathy, J. E. Sipe, and P. W. E. Smith, "Novel multimode fiber for narrow-band Bragg gratings," IEEE J. Sel. Top. Quantum Electron. 7, 425-433 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. S. Mohammed, A. Mehta, M. Pitchumani, and E. G. Johnson, "Selective excitation of the TE01 mode in hollow-glass waveguide using a subwavelength grating," IEEE Photon. Technol. Lett. 17, 1441-1443 (2005).
[CrossRef]

J. Lightwave Technol. (2)

A. Kowalski, "On the analysis of optical fibers described in terms of Chebyshev polynomials," J. Lightwave Technol. 8, 164-167 (1990).
[CrossRef]

N. Kishi, K. Tayama, and E. Yamashita, "Modal analysis of optical fiber with symmetrically distributed nonuniform cores," J. Lightwave Technol. 14, 1794-1800 (1996).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

Y. Sun, T. Szkopek, and P. W. E. Smith, "Demonstration of narrowband high-reflectivity Bragg gratings in a novel multimode fiber," Opt. Commun. 223, 91-95 (2003).
[CrossRef]

Opt. Eng. (1)

W. S. Mohammed, L. Vaissié, and E. G. Johnson, "Hybrid mode calculations for novel photonic crystal fibers," Opt. Eng. 42, 2311-2317 (2003).
[CrossRef]

Other (3)

T. Okoshi, Optical Fibers (Academic, 1984).

H. Ikuno and H. Watanabe, "Mode dispersion in graded-index optical fiber with near parabolic-index profiles," in MTT-S International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, 1979), pp. 472-474.
[CrossRef]

R. Yamauchi, "Specialty fibers," in Lasers and Electro-Optics Society Annual Meeting (Institute of Electrical and Electronics Engineers, 1994), pp. 228-229.

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Figures (10)

Fig. 1
Fig. 1

(a) Arbitrary continuous refractive index profile. (b) Quantized profile.

Fig. 2
Fig. 2

Graded-index fiber quantization schemes for (a) a fixed width, (b) a fixed refractive index difference, (c) a combination of both schemes.

Fig. 3
Fig. 3

Refractive index profile error as a function of radius r for various numbers of quantization regions (M) for (a) a fixed-width scheme, (b) a fixed-refractive-index-difference scheme, and (c) the combined scheme.

Fig. 4
Fig. 4

Convergence of modal calculations for the three quantization schemes when the number of quantizated layers was increased for (a) m = 1, l = 0 (fundamental) mode and (b) m = 2, l = 4 doublets.

Fig. 5
Fig. 5

Effective refractive indices of the guided modes in quadratic graded-index fiber calculations by the WKB (circles) and the full vectorial (crosses) methods.

Fig. 6
Fig. 6

Intensity profile of the (v = 1, l = 0) mode.

Fig. 7
Fig. 7

Transversal polarization profiles of the two mode doublets: (a), (b) for the m = 0, l = 0 doublet; (c), (d) for the m = 2, l = 0 doublet.

Fig. 8
Fig. 8

Multishell fiber structure.

Fig. 9
Fig. 9

Modal calculations for the multishell fiber at λ = 1.55 μm when the designing wavelength is (a) 1.55 and (b) 0.633 μm.

Fig. 10
Fig. 10

Top, calculated Bragg wavelengths for a uniform grating with a 535 nm period written in the cores of a four-shell fiber designed for 0.633 μm. Bottom, measured transmittance of the fabricated grating.

Tables (1)

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Table 1 Optimal Fiber Parameters That Correspond to Narrow Bragg Grating Response

Equations (46)

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n ( r ) n ̃ = j = 0 N n j   rect ( r r j t j ) δ ( r r j ) ,
t     2 E j + ( k j     2 β 2 ) E j = 0 ,
t     2 H j + ( k j     2 β 2 ) H j = 0.
E j     z ( r , θ , z ) = [ a j     e Z m , j               ( 1 ) ( r ) + b j     e Z m , j              ( 2 ) ( r ) ] × exp ( i m θ ) exp ( i β z ) ,
H j     z ( r , θ , z ) = [ a j     h Z m , j               ( 1 ) ( r ) + b j     h Z m , j               ( 2 ) ( r ) ] × exp ( i m θ ) exp ( i β z ) .
Z m , j               ( 1 ) ( r ) = { J m ( χ j r ) k j β I m ( γ j r ) k j < β ,
Z m , j               ( 2 ) ( r ) = { Y m ( χ j r ) k j β K m ( γ j r ) k j < β .
χ j = k j     2 β 2 , γ = β 2 k j     2 .
E j     z ( r , θ , z ) = d e K m ( γ c r ) exp ( i m θ ) exp ( i β z ) ,
H j     z ( r , θ , z ) = d h K m ( γ c r ) exp ( i m θ ) exp ( i β z ) .
A 0 ( 1 ) a 0 = A 1 ( 1 ) a 1 + B 1 ( 1 ) b 1 ,
A j ( 1 ) a j + B j ( 1 ) b j = A j + 1 ( 1 ) a j + 1 + B j + 1 ( 1 ) b j + 1 ,
j = 1 , 2 , ,   N 1 ,
D ( 1 ) d = A N c           ( 1 ) a N + B N c           ( 1 ) b N .
[ E θ H θ ] = ( i β ω 2 ε 0 μ 0 n 2 β 2 ) [ i m r ω μ 0 β r ω ε 0 n j     2 β r i m r ] [ E z H z ] .
A 0     ( 2 ) a 0 = A 1     ( 2 ) a 1 + B 1     ( 2 ) b 1 ,
A j     ( 2 ) a j + B j     ( 2 ) b j = A j + 1             ( 2 ) a j + 1 + B j + 1             ( 2 ) b j + 1 ,
j = 1 , 2 , , N 1 ,
D ( 2 ) d = A Nc ( 2 ) a N + B Nc ( 2 ) b N .
[ A 0     ( 1 ) ϕ A 0     ( 2 ) ϕ ] [ a 0 d ] = [ A 1     ( 1 ) B 1     ( 1 ) A 1     ( 2 ) B 1     ( 2 ) ] [ a 1 b 1 ] ,
[ A j     ( 1 ) B j     ( 1 ) A j     ( 2 ) B j     ( 2 ) ] [ a j b j ] = [ A j + 1             ( 1 ) B j + 1             ( 1 ) A j + 1             ( 2 ) B j + 1           ( 2 ) ] [ a j + 1 b j + 1 ] ,
[ ϕ D ( 1 ) ϕ D ( 2 ) ] [ a 0 d ] = [ A N c          ( 1 ) B N c         ( 1 ) A N c          ( 2 ) B N c         ( 2 ) ] [ a N b N ] .
[ A j + 1             ( 1 ) B j + 1             ( 1 ) A j + 1             ( 2 ) B j + 1             ( 2 ) ] 1 [ A j ( 1 ) B j ( 1 ) A j ( 2 ) B j     ( 2 ) ] [ a j b j ] = [ a j + 1 b j + 1 ]
[ a j + 1 b j + 1 ] = M j [ a j b j ] .
[ a N b N ] = M N 1 M N 2 M 2 M 1 [ a 1 b 1 ]
[ a N b N ] = M [ a 1 b 1 ] .
[ A 0     ( 1 ) D ( 1 ) A 0     ( 2 ) D ( 2 ) ] [ a 0 d ] = [ [ A 1     ( 1 ) B 1     ( 1 ) A 1     ( 2 ) B 1     ( 2 ) ] + M ] [ a 1 b 1 ] .
[ [ A 1     ( 1 ) B 1     ( 1 ) A 1     ( 2 ) B 1     ( 2 ) ] + M ϕ ϕ [ A 0     ( 1 ) D ( 1 ) A 0     ( 2 ) D ( 2 ) ] ] [ a 1 b 1 a 0 d ] = 0.
S v = 0 ,
E x = E r sin θ E θ cos θ ,
E y = E r cos θ E θ sin θ .
n c < n eff < n max .
2 n c d < λ < 2 n max d .
H ( λ ) = v = 1 M c v h ( λ ) δ ( λ λ v ) .
n ( r ) = n 1 [ 1 ( r a ) 2 δ ] 1 / 2 ,
δ = n 1     2 n 2     2 2 n 1     2 .
P e j = r j t j / 2 r j + t j / 2 | n ( r ) n j | r d r ,
P e 0 = 0 r 1 t 1 / 2 | n ( r ) n 1 | r d r .
A j = [ Z v , j             ( 1 ) ( r j + 1 t j + 1 / 2 ) 0 0 Z v , j + 1                     ( 1 ) ( r j + 1 t j + 1 / 2 ) ] ,
A N c = [ Z v , j             ( 1 ) ( r N + t N / 2 ) 0 0 Z v , N                 ( 1 ) ( r N + t N / 2 ) ] ,
D ( 1 ) = [ K m [ γ c ( r N + t N / 2 ) ] 0 0 K m [ γ c ( r N + t N / 2 ) ] ] .
a j = [ a j     e a j     h ] , b j = [ b j     e b j     h ] , d = [ d e d h ] ,
[ E θ H θ ] = ( i β ω 2 ε 0 μ 0 n 2 β 2 ) [ i m r ω μ 0 β r ω ε 0 n j     2 β r i m r ] [ E z H z ] .
A j ( 2 ) = ( i β ω 2 ε 0 μ 0 n j     2 β 2 ) [ i m r Z v , j             ( 1 ) ( r j + 1 t j + 1 / 2 ) ω μ 0 β Z v , j             ( 1 ) ( r ) r | r = r j + 1 t j + 1 / 2 ω ε 0 n j     2 β Z v , j             ( 1 ) ( r ) r | r = r j + 1 t j + 1 / 2 i m r Z v , j             ( 1 ) ( r j + 1 t j + 1 / 2 ) ] ,
A Nc           ( 2 ) = ( i β ω 2 ε 0 μ 0 n N         2 β 2 ) [ i m r Z v , N               ( 1 ) ( r N + t N / 2 ) ω μ 0 β Z v , N                 ( 1 ) ( r ) r | r = r N + t N / 2 ω ε 0 n N       2 β Z v , N               ( 1 ) ( r ) r | r = r N + t N / 2 i m r Z v , N               ( 1 ) ( r N + t N / 2 ) ] .
D ( 2 ) = ( i β ω 2 ε 0 μ 0 n c     2 β 2 ) [ i m r K N [ γ c ( r N + t N / 2 ) ] ω μ 0 β K N ( γ c r ) r | r = r N + t N / 2 ω ε 0 n c 2 β K N ( γ c r ) r | r = r N + t N / 2 i m r K N [ γ c ( r N + t N / 2 ) ] ] .

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