Abstract

From recent experimental results, the usual fiber model with infinite outer cladding fails to give an accurate description of the mode behavior beyond cutoff. The finite dimensions of the actual fiber must be taken into account; the leaky mode formalism is then replaced with a theory of coupling between a core mode and discrete, lossy cladding modes. The excitation and attenuation of the modes resulting from this coupling are considered in some detail; this more realistic model allows a better agreement with experiment.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Miya, K. Okamoto, Y. Ohmori, Y. Sasaki, IEEE J. Quantum Electron. QE-17, 858 (1981).
    [CrossRef]
  2. P. L. François, J. Lightwave Technol. LT-1, 26 (1983).
    [CrossRef]
  3. S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
    [CrossRef]
  4. L. G. Cohen, D. Marcuse, W. L. Mammel, IEEE J. Quantum Electron. QE-18, 1467 (1982).
    [CrossRef]
  5. P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
    [CrossRef]
  6. P. L. François, C. Vassallo, Electron. Lett. 19, 173 (1983).
    [CrossRef]
  7. C. Vassallo, Opt. Quantum Electron. 14, 280 (1982).
    [CrossRef]
  8. F. Alard, L. Jeunhomme, P. Sansonetti, Electron. Lett. 17, 958 (1981).
    [CrossRef]
  9. D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
    [CrossRef]
  10. D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).
  11. K. Petermann, AEU, 30, 337 (1976).
  12. T. Kanaka, Y. Suematsu, Trans. IECE Jpn. E59, 1 (1976).
  13. C. Vassallo, Opt. Quantum Electron. 15, 349 (1983).
    [CrossRef]
  14. F. N. H. Robinson, Macroscopic Electromagnetism (Pergamon, New York, 1973), Chap. 5.
  15. A. W. Snyder, W. R. Young, J. Opt. Soc. Am. 68, 297 (1978).
    [CrossRef]

1983 (3)

P. L. François, J. Lightwave Technol. LT-1, 26 (1983).
[CrossRef]

P. L. François, C. Vassallo, Electron. Lett. 19, 173 (1983).
[CrossRef]

C. Vassallo, Opt. Quantum Electron. 15, 349 (1983).
[CrossRef]

1982 (3)

C. Vassallo, Opt. Quantum Electron. 14, 280 (1982).
[CrossRef]

L. G. Cohen, D. Marcuse, W. L. Mammel, IEEE J. Quantum Electron. QE-18, 1467 (1982).
[CrossRef]

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

1981 (2)

F. Alard, L. Jeunhomme, P. Sansonetti, Electron. Lett. 17, 958 (1981).
[CrossRef]

T. Miya, K. Okamoto, Y. Ohmori, Y. Sasaki, IEEE J. Quantum Electron. QE-17, 858 (1981).
[CrossRef]

1978 (1)

1977 (1)

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

1976 (2)

K. Petermann, AEU, 30, 337 (1976).

T. Kanaka, Y. Suematsu, Trans. IECE Jpn. E59, 1 (1976).

1974 (1)

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

1973 (1)

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

Alard, F.

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

F. Alard, L. Jeunhomme, P. Sansonetti, Electron. Lett. 17, 958 (1981).
[CrossRef]

Cohen, L. G.

L. G. Cohen, D. Marcuse, W. L. Mammel, IEEE J. Quantum Electron. QE-18, 1467 (1982).
[CrossRef]

François, P. L.

P. L. François, J. Lightwave Technol. LT-1, 26 (1983).
[CrossRef]

P. L. François, C. Vassallo, Electron. Lett. 19, 173 (1983).
[CrossRef]

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

Jeunhomme, L.

F. Alard, L. Jeunhomme, P. Sansonetti, Electron. Lett. 17, 958 (1981).
[CrossRef]

Kammerer, P.

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

Kanaka, T.

T. Kanaka, Y. Suematsu, Trans. IECE Jpn. E59, 1 (1976).

Kawakami, S.

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

Keck, D. B.

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

Mammel, W. L.

L. G. Cohen, D. Marcuse, W. L. Mammel, IEEE J. Quantum Electron. QE-18, 1467 (1982).
[CrossRef]

Marcuse, D.

L. G. Cohen, D. Marcuse, W. L. Mammel, IEEE J. Quantum Electron. QE-18, 1467 (1982).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

Maurer, R. D.

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

Miya, T.

T. Miya, K. Okamoto, Y. Ohmori, Y. Sasaki, IEEE J. Quantum Electron. QE-17, 858 (1981).
[CrossRef]

Monerie, M.

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

Nishida, S.

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

Ohmori, Y.

T. Miya, K. Okamoto, Y. Ohmori, Y. Sasaki, IEEE J. Quantum Electron. QE-17, 858 (1981).
[CrossRef]

Okamoto, K.

T. Miya, K. Okamoto, Y. Ohmori, Y. Sasaki, IEEE J. Quantum Electron. QE-17, 858 (1981).
[CrossRef]

Petermann, K.

K. Petermann, AEU, 30, 337 (1976).

Robinson, F. N. H.

F. N. H. Robinson, Macroscopic Electromagnetism (Pergamon, New York, 1973), Chap. 5.

Sansonetti, P.

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

F. Alard, L. Jeunhomme, P. Sansonetti, Electron. Lett. 17, 958 (1981).
[CrossRef]

Sasaki, Y.

T. Miya, K. Okamoto, Y. Ohmori, Y. Sasaki, IEEE J. Quantum Electron. QE-17, 858 (1981).
[CrossRef]

Schultz, P. C.

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

Snyder, A. W.

Suematsu, Y.

T. Kanaka, Y. Suematsu, Trans. IECE Jpn. E59, 1 (1976).

Vassallo, C.

C. Vassallo, Opt. Quantum Electron. 15, 349 (1983).
[CrossRef]

P. L. François, C. Vassallo, Electron. Lett. 19, 173 (1983).
[CrossRef]

C. Vassallo, Opt. Quantum Electron. 14, 280 (1982).
[CrossRef]

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

Young, W. R.

AEU (1)

K. Petermann, AEU, 30, 337 (1976).

Appl. Phys. Lett. (1)

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

Electron. Lett. (3)

P. Sansonetti, F. Alard, C. Vassallo, P. Kammerer, P. L. François, M. Monerie, Electron. Lett. 18, 989 (1982).
[CrossRef]

P. L. François, C. Vassallo, Electron. Lett. 19, 173 (1983).
[CrossRef]

F. Alard, L. Jeunhomme, P. Sansonetti, Electron. Lett. 17, 958 (1981).
[CrossRef]

IEEE J. Quantum Electron. (3)

T. Miya, K. Okamoto, Y. Ohmori, Y. Sasaki, IEEE J. Quantum Electron. QE-17, 858 (1981).
[CrossRef]

S. Kawakami, S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).
[CrossRef]

L. G. Cohen, D. Marcuse, W. L. Mammel, IEEE J. Quantum Electron. QE-18, 1467 (1982).
[CrossRef]

J. Lightwave Technol. (1)

P. L. François, J. Lightwave Technol. LT-1, 26 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Quantum Electron. (2)

C. Vassallo, Opt. Quantum Electron. 15, 349 (1983).
[CrossRef]

C. Vassallo, Opt. Quantum Electron. 14, 280 (1982).
[CrossRef]

Trans. IECE Jpn. (1)

T. Kanaka, Y. Suematsu, Trans. IECE Jpn. E59, 1 (1976).

Other (1)

F. N. H. Robinson, Macroscopic Electromagnetism (Pergamon, New York, 1973), Chap. 5.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Experimental (solid line) and theoretical (dotted line) transmission factor T (left scale). Experimental (+ symbol) and theoretical (dash-dotted line) values of the normalized HE11 mode spot size ω/a.

Fig. 2
Fig. 2

(a) Schematic geometry of an actual depressed inner-cladding fiber. (b) Usual model with infinite outer cladding.

Fig. 3
Fig. 3

Launching efficiency T(1) vs ω/a for (c2,σ2) = (2.1). Labeling parameter is Γ1, the fractional power traveling in the core. Γ1 decreases when cutoff is approached.

Fig. 4
Fig. 4

Normalized HE11 mode spot size vs V according to Marcuse's (thick line) or Petermann's (dotted line) definition. Launching efficiency T (solid line) and fractional power Γ1 (dash-dotted line) traveling in the core (right scale).

Fig. 5
Fig. 5

Normalized propagation constant for the leaky HE11 mode. The solid lines correspond to an exact calculation and the dotted lines to Kawakami's perturbation technique.3 The curves are scaled by (1 − V/Vc) expressed in percent (see Appendix B for values of Vc).

Fig. 6
Fig. 6

Normalized propagation constant B for the modes of the whole structure (solid lines), of the cladding (dotted lines), and of the core (dash-dotted line). The modes are labeled by the number m of zeros in their field distribution.

Fig. 7
Fig. 7

Coupling between core and cladding modes: (a) Respective positions of the dispersion curves for the modes of the whole structure (solid line), of the cladding (dotted lines), and of the core (dash-dotted line), (b) Field distributions at points P1, P2, and P3 for the modes of the whole structure (solid lines) and of the cladding (dotted lines).

Fig. 8
Fig. 8

Normalized spot size ω/a (solid lines, left scale) of the best Gaussian approximation for the HE11 mode and corresponding launching efficiency T (dotted lines, right scale). The curves are labeled by the order m of the modes.

Fig. 9
Fig. 9

Launching efficiency into the fiber with flat outer cladding of a Gaussian beam with ω/a = 1: (c2,σ2) = (2,2), c3 = 16, Vc3 ≃ 1.78.

Fig. 10
Fig. 10

Launching efficiency into the fiber with flat outer cladding of a Gaussian beam with ω/a = 1: (c2,σ2) = (2,1), c3 = 16, Vc3 ≃ 1.55.

Fig. 11
Fig. 11

Launching efficiency into the fiber with flat outer cladding of a Gaussian beam with ω/a = 1: (c2,σ2) = (3,2), c3 = 17, Vc3 ≃ 1.79.

Fig. 12
Fig. 12

Fractional power Γ4 traveling in the substrate when σ2 = 2, c2 = 2, c3 = 6, c4 = 16.

Fig. 13
Fig. 13

Fractional power Γ4 traveling in the substrate vs V ̂ = V 1 + σ 2 for σ2 = 2 (solid lines), σ2 = 1 (dotted lines), and the values of c2 indicated. The figure in parentheses is the order m of the mode. The right scale gives the attenuation α = Γ4α4 with α4 = 200 dB/km. The upper wavelength abscissa corresponds to a core radius a = 3.5 μm and to Δ1(1 + σ2) = 0.5%.

Fig. 14
Fig. 14

Leaky mode attenuation αL when σ2 = 2 (solid lines) and σ2 = 1 (dotted lines), for c2 = 6,8, and 10 [core radius a = 3.5 μm, Δ1(1 + σ2) = 0.5%]. The thick lines (c2 = 6) are drawn from Fig. 13 and correspond to an attenuation through coupling to lossy cladding modes. For c2 = 8 or 10, the significant part of these curves would be <10−3 dB/km.

Fig. 15
Fig. 15

Normalized propagation constant B vs V = ka n 1 2 n 3 2 for the modes of the whole structure of Fig. 2(a) (solid lines), of the cladding (dotted lines), and of the core (dash-dotted lines). Vc3 and Vc4 correspond to the conditions neff = n3 and neff = n4, respectively. The modes are labeled by the number m of zeros in their field distributions.

Fig. 16
Fig. 16

Theoretical excitation (solid lines) and transmission (dotted lines) factor for the modes of the whole structure of Fig. 2(a); λ3 and λ4 correspond to the conditions neff = n3 and neff = n4, respectively.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

T ( F ) = [ 0 2 F exp ( u 2 / F 2 ) ϕ ( u ) udu ] 2 0 ϕ 2 ( u ) udu
F 2 = ( ω a ) 2 = 2 0 ϕ 2 ( u ) u 3 du 0 ϕ 2 ( u ) udu .
( ω a ) 2 4 3 W 3 2 for V c < V < V d .
ϕ ( c 5 ) = 0 .
ϕ ( x ) = { A 1 J 0 ( U 1 x ) , for 0 < x 1 A 2 I 0 ( W 2 x ) + B 2 K 0 ( W 2 x ) , for 1 < x c 2 A 3 J 0 ( U 3 x ) + B 3 Y 0 ( U 3 x ) , for c 2 < x c 3 .
E core ( U 1 , W 2 ) = J ̂ 0 ( U 1 ) K ̂ 0 ( W 2 ) = C E ( W 2 , U 3 ) E clad ( W 2 , U 3 ) ,
[ E ( W 2 , U 3 ) E clad ( W 2 , U 3 ) ] = [ K ̂ 0 ( W 2 c 2 ) J ̂ 0 ( U 3 c 2 ) K ̂ 0 ( W 2 c 2 ) Y ̂ 0 ( U 3 c 2 ) I ̂ 0 ( W 2 c 2 ) + J ̂ 0 ( U 3 c 2 ) I ̂ 0 ( W 2 c 2 ) + Y ̂ 0 ( U 3 c 2 ) ] × [ J 0 ( U 3 c 2 ) Y 0 ( U 3 c 3 ) J 0 ( U 3 c 3 ) Y 0 ( U 3 c 2 ) ] , Z ̂ m ( x ) = x Z m + 1 ( x ) / Z m ( x ) with Z m = J m , Y m , I m , or K m .
E ( W 2 , U 3 ) E clad ( W 2 , U 3 ) = W 2 sin ( U 3 δ ) U 3 cos ( U 3 δ ) W 2 sin ( U 3 δ ) + U 3 cos ( U 3 δ ) ,
π 2 ( 2 m + 1 ) U 3 δ < π ( m + 1 ) .
{ W 2 sin ( U 3 δ ) + U 3 cos ( U 3 δ ) = 0 , W 2 2 = σ 2 V 2 U 3 2 .
α = 2 k n 3 Δ 1 Im ( B ) .
α = i Γ i α i ( here i = 4 , 5 ) ,
F 2 = ( ω a ) 2 = F core 2 + A 1 A 2 2 1 + A 1 ,
F core = 4 3 [ J 0 ( U 1 ) U 1 J 1 ( U 1 ) + 1 W 2 2 1 U 1 2 + 1 2 ] ,
A 2 = c 2 4 3 [ K 0 ( c 2 W 3 ) c 2 W 3 K 1 ( c 2 W 3 ) + 1 c 2 2 W 2 2 + 1 c 2 2 W 3 2 + 1 2 ] ,
A 1 = A 3 2 c 2 2 σ 2 1 + σ 2 [ K 1 ( c 2 W 3 ) K 0 ( c 2 W 3 ) ] 2 [ J 0 ( U 1 ) J 1 ( U 1 ) ] 2 ,
F 2 = A 2 2 4 3 W 3 2 .
J ̂ 0 ( U 1 ) K ̂ 0 ( W 2 ) = C K ̂ 0 ( c 2 W 3 ) K ̂ 0 ( c 2 W 2 ) K ̂ 0 ( c 2 W 3 ) + I ̂ 0 ( c 2 W 2 ) ,
C = I 0 ( W 2 ) K 0 ( c 2 W 2 ) I 0 ( c 2 W 2 ) K 0 ( W 2 ) [ J ̂ 0 ( U 1 ) + I ̂ 0 ( W 2 ) ] , Z ̂ m ( x ) = x Z m + 1 ( x ) / Z m ( x ) with Z m = J m , I m , or K m .
J ̂ 0 ( U 1 ) K ̂ 0 ( W 2 ) = C K ̂ 0 ( c 2 W 2 ) I ̂ 0 ( c 2 W 2 ) ,
{ U 1 = V c , W 2 = V c σ 2 .
C 2 W 2 exp [ 2 W 2 ( c 2 1 ) ] ,
W 2 = V B + σ 2 .
J ̂ 0 ( U 1 ) K ̂ 0 ( W 2 ) = 0 ,
B d = W d 2 / V c 2 , W d = 1.123 c 2 exp { 8 W 2 exp [ 2 W 2 ( c 2 1 ) ] + ( 1 + σ 2 ) W d 2 c 2 W 2 ( 1 + σ 2 ) W d 2 }

Metrics