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References

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  1. L. G. Cohen, D. Marcuse, W. L. Mammel, in Digest of Topical Meeting on Optical Fiber Communications (Optical Society of America, Washington, D.C., 1982), paper THCC1.
  2. H. Murata, N. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
    [CrossRef]
  3. B. J. Ainslie, K. J. Beales, J. D. Rush, IEEE J. Quantum Electron. QE-17, 845 (1981).
  4. P. D. Lazay, A. D. Pearson, IEEE J. Quantum Electron. QE-18, 504 (1982).
    [CrossRef]
  5. A. Munlich, K. Rau, S. Simmat, N. Treber, “A New Doped Synthetic Fused Silica as Bulk Material for Low-Loss Optical Fibers,” First European Conference on Optical Fiber Communication, London, 1975; postdeadline paper.
  6. J. Irven, A. P. Harrison, C. R. Smith, Electron. Lett. 17, 3 (1981).
    [CrossRef]
  7. S. Kawakami, M. Miyagi, S. Nishida, Appl. Opt. 14, 2588 (1975); Proc. IEE 123, 586 (1976).
    [CrossRef] [PubMed]
  8. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  9. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1964).
  10. D. Marcuse, J. Opt. Soc. Am. 66, 216 (1976).
    [CrossRef]

1982 (1)

P. D. Lazay, A. D. Pearson, IEEE J. Quantum Electron. QE-18, 504 (1982).
[CrossRef]

1981 (3)

J. Irven, A. P. Harrison, C. R. Smith, Electron. Lett. 17, 3 (1981).
[CrossRef]

H. Murata, N. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

B. J. Ainslie, K. J. Beales, J. D. Rush, IEEE J. Quantum Electron. QE-17, 845 (1981).

1976 (1)

1975 (1)

1971 (1)

Ainslie, B. J.

B. J. Ainslie, K. J. Beales, J. D. Rush, IEEE J. Quantum Electron. QE-17, 845 (1981).

Beales, K. J.

B. J. Ainslie, K. J. Beales, J. D. Rush, IEEE J. Quantum Electron. QE-17, 845 (1981).

Cohen, L. G.

L. G. Cohen, D. Marcuse, W. L. Mammel, in Digest of Topical Meeting on Optical Fiber Communications (Optical Society of America, Washington, D.C., 1982), paper THCC1.

Gloge, D.

Harrison, A. P.

J. Irven, A. P. Harrison, C. R. Smith, Electron. Lett. 17, 3 (1981).
[CrossRef]

Inagaki, N.

H. Murata, N. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

Irven, J.

J. Irven, A. P. Harrison, C. R. Smith, Electron. Lett. 17, 3 (1981).
[CrossRef]

Kawakami, S.

Lazay, P. D.

P. D. Lazay, A. D. Pearson, IEEE J. Quantum Electron. QE-18, 504 (1982).
[CrossRef]

Mammel, W. L.

L. G. Cohen, D. Marcuse, W. L. Mammel, in Digest of Topical Meeting on Optical Fiber Communications (Optical Society of America, Washington, D.C., 1982), paper THCC1.

Marcuse, D.

D. Marcuse, J. Opt. Soc. Am. 66, 216 (1976).
[CrossRef]

L. G. Cohen, D. Marcuse, W. L. Mammel, in Digest of Topical Meeting on Optical Fiber Communications (Optical Society of America, Washington, D.C., 1982), paper THCC1.

Miyagi, M.

Munlich, A.

A. Munlich, K. Rau, S. Simmat, N. Treber, “A New Doped Synthetic Fused Silica as Bulk Material for Low-Loss Optical Fibers,” First European Conference on Optical Fiber Communication, London, 1975; postdeadline paper.

Murata, H.

H. Murata, N. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

Nishida, S.

Pearson, A. D.

P. D. Lazay, A. D. Pearson, IEEE J. Quantum Electron. QE-18, 504 (1982).
[CrossRef]

Rau, K.

A. Munlich, K. Rau, S. Simmat, N. Treber, “A New Doped Synthetic Fused Silica as Bulk Material for Low-Loss Optical Fibers,” First European Conference on Optical Fiber Communication, London, 1975; postdeadline paper.

Rush, J. D.

B. J. Ainslie, K. J. Beales, J. D. Rush, IEEE J. Quantum Electron. QE-17, 845 (1981).

Simmat, S.

A. Munlich, K. Rau, S. Simmat, N. Treber, “A New Doped Synthetic Fused Silica as Bulk Material for Low-Loss Optical Fibers,” First European Conference on Optical Fiber Communication, London, 1975; postdeadline paper.

Smith, C. R.

J. Irven, A. P. Harrison, C. R. Smith, Electron. Lett. 17, 3 (1981).
[CrossRef]

Treber, N.

A. Munlich, K. Rau, S. Simmat, N. Treber, “A New Doped Synthetic Fused Silica as Bulk Material for Low-Loss Optical Fibers,” First European Conference on Optical Fiber Communication, London, 1975; postdeadline paper.

Appl. Opt. (2)

Electron. Lett. (1)

J. Irven, A. P. Harrison, C. R. Smith, Electron. Lett. 17, 3 (1981).
[CrossRef]

IEEE J. Quantum Electron. (3)

H. Murata, N. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

B. J. Ainslie, K. J. Beales, J. D. Rush, IEEE J. Quantum Electron. QE-17, 845 (1981).

P. D. Lazay, A. D. Pearson, IEEE J. Quantum Electron. QE-18, 504 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (3)

A. Munlich, K. Rau, S. Simmat, N. Treber, “A New Doped Synthetic Fused Silica as Bulk Material for Low-Loss Optical Fibers,” First European Conference on Optical Fiber Communication, London, 1975; postdeadline paper.

L. G. Cohen, D. Marcuse, W. L. Mammel, in Digest of Topical Meeting on Optical Fiber Communications (Optical Society of America, Washington, D.C., 1982), paper THCC1.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1964).

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Figures (7)

Fig. 1
Fig. 1

Refractive-index distribution of the doubly clad fiber with depressed-index inner cladding.

Fig. 2
Fig. 2

Leakage loss of the straight, doubly clad fiber as a function of wavelength for several values of the cladding-to-core radius ratio b/a. Fiber radius a = 3.5 μm, relative core–cladding refractive-index difference, (n1n2)/n1 = 0.005, outer cladding index equal to core index, n3 = n1.

Fig. 3
Fig. 3

Loss of the leaky mode in the curved fiber relative to the loss of the mode in the straight fiber is plotted as a function of R/a for several values of the cladding-to-core radius ratio b/a. The following parameter values are used: n3 = n1 = 1.457, (n1n2)/n1 = 0.005, λ = 1.3 μm, a = 3.5 μm. With these values we have V = 2.47. The dotted lines represent the curvature loss of the singly clad fiber [Eq. (24)] relative to the leakage loss of the straight, doubly clad fiber.

Fig. 4
Fig. 4

Power loss of the HE11 mode as a function of wavelength for several values of R. The solid curve represents the loss of the straight fiber; the dotted curves represent the curved fiber loss. It is b/a = 8; all other fiber parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Loss coefficient of the HE11 mode in a doubly clad fiber with a 10-cm radius of curvature plotted as a function of wavelength for several values of the relative refractive-index difference (n1n3)/n1 between the core and outer cladding.

Fig. 6
Fig. 6

Power loss of the HE11 mode as a function of the cladding-to-core radius ratio b/a for several values of R. The other fiber parameters are the same as in Fig. 3.

Fig. 7
Fig. 7

Radius of curvature required to obtain fixed loss values (broken line, 0.1 dB/km; solid line, 1 dB/km) plotted as functions of wavelength. Rdc denotes the radius of curvature of the doubly clad fiber. Rsc is the radius of curvature of the corresponding singly clad fibor (b → ∞).

Equations (38)

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2 E ρ 2 + 1 ρ E ρ + 1 ρ 2 2 E ψ 2 + 2 E z 2 + n 2 k 2 E = 0 .
2 E r 2 + 1 r E r + 1 r 2 2 E ϕ 2 + [ n 2 ( 1 + 2 r R cos ϕ ) k 2 β 2 ] E = 0
n eff 2 = n 2 ( 1 + 2 r R cos ϕ )  .
E = A J 0 ( κ r )  exp ( i β z ) for  r < a ,
E = [ B K 0 ( γ r ) + C I 0 ( γ r ) ] exp ( i β z ) for  a < r < b ,
E = D H 0 ( 2 ) ( σ r ) exp ( i β z ) for  b < r ,
κ 2 = n 1 2 k 2 β 2 ,
γ 2 = β 2 n 2 2 k 2 ,
σ 2 = n 3 2 k 2 β 2 .
E = { B K 0 [ 0 r γ ¯ ( r ) d r ] + C I 0 [ 0 r γ ¯ ( r ) d r ] } exp ( i β z ) for a < r < b ,
E = D H 0 ( 2 ) [ 0 r σ ¯ ( r ) d r ] exp ( i β z ) for  b < r ,
γ ¯ 2 ( r ) = β 2 n 2 2 ( 1 + 2 r k cos ϕ ) k 2 ,
σ ¯ 2 ( r ) = n 3 2 ( 1 + 2 r R cos ϕ ) k 2 β 2 .
D B = γ ¯ ( b ) u [ γ ¯ ( b ) H 0 ( 2 ) ( υ ) I 1 ( u ) + σ ¯ ( b ) H 1 ( 2 ) ( υ ) I 0 ( u ) ] 1 ,
u = 0 b γ ¯ ( r ) d r a n d υ = 0 b σ ¯ ( r ) d r.
B = J 0 ( κ a ) K 0 ( γ a ) A .
B 2 = 2 κ 2 P μ 0 / ε 0 π n 2 ( n 1 2 n 2 2 ) k 2 K 1 2 ( γ a ) .
2 α ( R ) = 2 b κ 2 π 2 β V 2 K 1 2 ( γ a ) π π σ ¯ ( b ) υ | D B | 2 d ϕ .
V 2 = k 2 a 2 ( n 1 2 n 2 2 ) .
2 α ( R ) = 4 b κ 2 β V 2 K 1 2 ( γ a ) 0 π σ ¯ ( b ) γ ¯ 2 ( b ) u [ γ ¯ 2 ( b ) + σ ¯ 2 ( b ) ] exp ( 2 u ) d ϕ .
u = 0 b γ ¯ ( r ) d r = { R  [ γ 3 γ ¯ 3 ( b ) ] 3 n 2 2 k 2 cos ϕ for  ϕ π / 2 γ b for  ϕ = π / 2.
cos ϕ c = R σ 2 2 b n 3 2 k 2 .
R c = 2 n 2 2 k 2 γ 2 b ,
2 α ( R ) = π 1 / 2 κ 2 exp [ 2 3 ( γ 3 / β 2 ) R ] 2 V 2 γ 3 / 2 R 1 / 2 K 1 2 ( γ a )
2 α ( ) = 4 π σ γ κ 2  exp ( 2 γ b ) β V 2 ( γ 2 + σ 2 ) K 1 2 ( γ a ) .
ρ 2 = R 2 + r 2 + 2 r R   cos ϕ R 2   ( 1 + 2 r R cos ϕ ) ,
z = r   sin ϕ ,
ψ = z R .
E ρ = E r r ρ + E ϕ ϕ ρ ,
2 E ρ 2 = 2 E r 2 ( r ρ ) 2 + 2 2 E r ϕ ϕ ρ r ρ + 2 E ϕ 2 ( ϕ ρ ) 2 + E r 2 r ρ 2 + E ϕ 2 ϕ ρ 2 ,
2 E z 2 = 2 E r 2 ( r z ) 2 + 2 2 E r ϕ r z ϕ z + 2 E ϕ 2 ( ϕ z ) 2 + E r 2 r z 2 + E ϕ 2 ϕ z 2 ,
2 E ψ 2 = R 2 2 E z 2 .
r ρ = cos ϕ ; r z = sin ϕ ,
2 r ρ 2 = 1 r sin 2 ϕ ; 2 r z 2 = 1 r cos 2 ϕ ,
ϕ ρ = 1 r sin ϕ ; ϕ z = 1 r cos 2 ϕ ,
2 ϕ ρ 2 = 2 r 2 sin ϕ cos ϕ ; 2 ϕ z 2 = 2 r 2 sin ϕ cos ϕ .
1 ρ E ρ + 2 E r 2 + 1 r E r + 1 r 2 2 E ϕ 2 + 1 1 + 2 r R cos ϕ 2 E z 2 + n 2 k 2 E = 0
2 E r 2 + 1 r E r + 1 r 2 2 E ϕ 2 + 2 E z 2 + n 2 ( 1 + 2 r R cos ϕ ) k 2 E = 0.

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