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

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References

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  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 LP01 mode in depressed index clad fibre,” IEE Proc.: Optoelectron. 137, 311-314 (1990).
    [Crossref]
  13. 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]
  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, 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]
  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).

2006 (1)

2004 (1)

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]

2003 (1)

2002 (1)

J. Petracek and K. Singh, “Determination of leaky modes in planar multilayer waveguides,” IEEE Photon. Technol. Lett. 14, 810-812 (2002).
[Crossref]

2001 (1)

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]

1999 (1)

1997 (2)

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. 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]

1995 (1)

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]

1992 (1)

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]

1991 (2)

1990 (1)

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]

1988 (1)

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]

1987 (2)

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]

1982 (1)

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]

1977 (1)

1975 (1)

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).

1974 (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]

A. W. Snyder and D. J. Mitchell, “Leaky mode analysis of circular optical waveguides,” Opto-electronics (London) 6, 287-296 (1974).
[Crossref]

M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics (London) 6, 271-286 (1974).
[Crossref]

Appl. Opt. (2)

IEE Proc.: Optoelectron. (1)

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]

IEEE J. Quantum Electron. (6)

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).

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]

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]

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]

IEEE Photon. Technol. Lett. (4)

H. Renner, “Leaky-mode loss in coated depressed-cladding fibers,” IEEE Photon. Technol. Lett. 3, 31-32 (1991).
[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]

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]

J. Petracek and K. Singh, “Determination of leaky modes in planar multilayer waveguides,” IEEE Photon. Technol. Lett. 14, 810-812 (2002).
[Crossref]

J. Lightwave Technol. (5)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

K. Thyagarajan, S. Diggavi, and A. K. Ghatak, “Analytical investigation of leaky and absorbing planar structures,” Opt. Quantum Electron. 19, 131-137 (1987).
[Crossref]

Opto-electronics (London) (2)

A. W. Snyder and D. J. Mitchell, “Leaky mode analysis of circular optical waveguides,” Opto-electronics (London) 6, 287-296 (1974).
[Crossref]

M. O. Vassell, “Calculation of propagating modes in a graded-index optical fibre,” Opto-electronics (London) 6, 271-286 (1974).
[Crossref]

Other (6)

R. L. Burden and J. D. Faires, Numerical Analysis (PWS-KENT, 1984).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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.

R. E. Collin and F. J. Zucker, eds., Antenna Theory (McGraw-Hill, 1969), p. 203.

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.

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Figures (9)

Fig. 1
Fig. 1

Index profile of an optical fiber with four types of interfaces, which has been studied to determine its applicability to dispersion flattened fibers [25]. The four types of interfaces are the step–step interface between layers 1 and 2, the step-graded interface between layers 2 and 3, the graded–graded interface between layers 3 and 4, and the graded-step interface between layers 4 and 5.

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Fig. 2
Fig. 2

Arrows on the solid curves indicate the locus of α chosen by the rule as a function of ω. The attenuation constants correspond to guided modes when ω is greater than ω 0 , while the attenuation constants represent leaky modes when ω is less than ω 0 .

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Fig. 3
Fig. 3

Leaky losses of the fundamental mode as a function of the wavelength for the DIC fiber with a = 3.75   μ m and various values of Δ and b ∕ a . (a) Curves apply to Δ = 0 % , 0.2%, 0.23%, 0.25%, and 0.27% for b ∕ a = 6 and 7. (b) Curves apply to b ∕ a = 5.5 – 8 for Δ = 0 . The solid curves correspond to the results obtained by using the approximate analytical formula of [9], while the dashed curves correspond to our method.

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Fig. 4
Fig. 4

Attenuation as a function of the wavelength for a DIC optical fiber with a material loss of 0.176   dB ∕ km in each layer. The radii of the core and inner cladding are a = 3.75   μ m and b = 22.5   μ m , respectively.

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Fig. 5
Fig. 5

Leaky losses of the LP 11 mode as a function of the wavelength for the coated DIC fiber. The solid curve corresponds to the results obtained by using our method, while the dots correspond to the matrix method [13].

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Fig. 6
Fig. 6

Leaky losses of the fundamental mode as a function of the wavelength for a multilayer optical fiber with graded index profile, where the radii of each layer are 1.3754, 5.98, 8.74, 11.5, and 23   μ m , respectively. The refractive indices n 1 , n 2 , n 3 , and n 4 are 1.4689, 1.4593, 1.4651, and 1.4593, respectively. The curves apply to Δ = 0 % , 0.52%, 0.53%, 0.535%, and 0.54%.

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Fig. 7
Fig. 7

Attenuation as a function of the wavelength for a multilayer optical fiber with a graded index profile and material loss, where the radii of each layer are 1.3754, 5.98, 8.74, 11.5, and 23   μ m , respectively. The material loss is 0.176   dB ∕ km in each layer. The curves apply to Δ = 0.53 % .

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Fig. 8
Fig. 8

Absolute value of the characteristic equation as a function of the effective index for the case of Fig. 3, where n 1 , n 2 , and n 3 are 1.46, 1.4527, and 1.46, respectively. The radii of the core and inner cladding are 3.75 and 22.5   μ m , respectively.

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Fig. 9
Fig. 9

Absolute value of the characteristic equation as a function of the effective index for the case of Fig. 7. The refractive index of each layer is n 1 = 1.4689 − j 5.0 × 10 − 12 , n 2 = 1.4593 − j 5.0 × 10 − 12 , n 3 = 1.4651 − j 5.0 × 10 − 12 , n 4 = 1.4593 − j 5.0 × 10 − 12 , and n 5 = 1.4689 − j 5.0 × 10 − 12 .

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Tables (1)

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Table 1 Comparison of the Effective Index and Leaky Loss of the LP11 Mode in a Coated DIC Fiber a

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Equations (23)

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∂ 2 ∂ r 2 ( E z , i ( r ) H z , i ( r ) ) + 1 r ∂ ∂ r ( E z , i ( r ) H z , i ( r ) ) + [ ( k 0 2 n i 2 − k z 2 ) − ν 2 r 2 ] ( E z , i ( r ) H z , i ( r ) ) = 0 ,
E z , i ( r ) = { A i J ν ( k c i r ) + D i Y ν ( k c i r ) for n i ⩾ n eff A i I ν ( k c i r ) + D i K ν ( k c i r ) for n i ⩽ n eff } ,
H z , i ( r ) = { B i J ν ( k c i r ) + E i Y ν ( k c i r ) for n i ⩾ n eff B i I ν ( k c i r ) + E i K ν ( k c i r ) for n i ⩽ n eff } ,
k c i = k 0 2 n i 2 − k z 2 for n i ⩾ n eff ,
k c i = k z 2 − k 0 2 n i 2 for n i ⩽ n eff .
F i = [ E z , i ( r ) H z , i ( r ) E ϕ , i ( r ) H ϕ , i ( r ) ] = [ P ν ( k c i r ) 0 Q ν ( k c i r ) 0 0 P ν ( k c i r ) 0 Q ν ( k c i r ) k z ν k c i 2 r P ν ( k c i r ) j ω μ k c i P ν ′ ( k c i r ) k z ν k c i 2 r Q ν ( k c i r ) j ω μ k c i Q ν ′ ( k c i r ) − j ω ϵ i k c i P ν ′ ( k c i r ) k z ν k c i 2 r P ν ( k c i r ) − j ω ϵ i k c i Q ν ′ ( k c i r ) k z ν k c i 2 r Q ν ( k c i r ) ] [ A i B i D i E i ] = M i ( r ) ⋅ C i .
d d r F i ( r ) = d d r [ E z , i ( r ) H z , i ( r ) E ϕ , i ( r ) H ϕ , i ( r ) ] = [ 0 − j k z ν ω n i 2 ϵ 0 r 0 − j ω μ 0 ( k z 2 n i 2 k 0 2 − 1 ) j k z ν ω μ 0 r 0 j ω ϵ 0 ( k z 2 k 0 2 ) 0 0 j ω μ 0 ( ν 2 k 0 2 n i 2 r 2 − 1 ) − 1 r j k z ν ω n i 2 ϵ 0 r j ω ϵ 0 ( n i 2 − ν 2 k 0 2 r 2 ) 0 − j k z ν ω μ 0 r − 1 r ] [ E z , i ( r ) H z , i ( r ) E ϕ , i ( r ) H ϕ , i ( r ) ] ,
F i ( r i ) = Y i ⋅ F i ( r i − 1 ) ,
[ W ( k z ) ] ⋅ V = 0 ,
det [ W ( k z ) ] = 0 ,
( E z , n H z , n ) = ( D n H ν ( 2 ) ( k c n r ) E n H ν ( 2 ) ( k c n r ) ) ,
k c n = ± k 0 2 n n 2 − k z 2 .
( E z , n H z , n ) = ( D n K ν ( k c n r ) E n K ν ( k c n r ) ) ,
k c n = ± k z 2 − k 0 2 n n 2 .
Re ( k c n ) ⩾ 0 , if n eff > n n ( guided modes ) .
Re ( k c n ) ⩽ 0 , if n eff < n n ( leaky modes ) .
P loss = 20 ⋅ α ⋅ L ⋅ log ( exp ( 1 ) ) ( dB ) .
k c n 2 = k z 2 − n n 2 k 0 2 ≈ ( β 2 − n n r 2 k 0 2 ) + 2 j ( n n r n n i k 0 2 − α β ) .
Re ( k c n ) ⩽ 0 , if Re ( k c n 2 ) < 0 , Im ( k c n 2 ) < 0 .
k c n 2 = n n 2 k 0 2 − k z 2 ≈ ( n n r 2 k 0 2 − β 2 ) + 2 j ( α β − n n r n n i k 0 2 ) .
Re ( k c n ) ⩾ 0 , if Re ( k c n 2 ) > 0 , Im ( k c n 2 ) > 0 .
H ν ( 2 ) ( − j k c n r ) = A K ν ( k c n r ) ,
− j k c n = k 0 2 n n 2 − k z 2 ,

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