Abstract

The loss formula for optical fibers with constant radius of curvature of their axes is derived by expressing the field outside of the fiber in terms of a superposition of cylindrical outgoing waves. The expansion coefficients are determined by matching the superposition field to the field of the fiber along a cylindrical surface that is tangential to the outer perimeter of the curved fiber. This method is a direct extension of my derivation of the curvature-loss formula for slab guides.

© 1976 Optical Society of America

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References

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  1. M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).
  2. E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
    [Crossref]
  3. L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
    [Crossref]
  4. J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
    [Crossref]
  5. A. W. Snyder (private communication).
  6. A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett. 11, 332–333 (1975).
    [Crossref]
  7. V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
    [Crossref]
  8. D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., Univ. of Colo., Boulder, Colo.; also, IEEE J. Quantum Electron. QE-11, 903–907 (1975).
  9. D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972), 398–406.
  10. D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  11. D. Marcuse, Theory of Dielectric Optical Waveguides, (Academic, New York, 1974).
  12. Reference 11, Eq. (2.2-23), p. 65, and Eq. (2.2-25), p. 66.
  13. Reference 9, Eqs. (8.2-7) through (8.2-10), p. 290.
  14. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Department of Commerce, National Bureau of Standards, Appl. Math. Ser., 55.
  15. Reference 11, Eq. (2.2-38), p. 68.
  16. Reference 11, Eq. (2.2-69), p. 73.
  17. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).
  18. Reference 17,Eq. 8.468, p. 967.
  19. Reference 14, Eq. 6.1. 18, p. 256.

1975 (1)

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

1974 (2)

L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[Crossref]

J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
[Crossref]

1973 (1)

V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[Crossref]

1971 (1)

1969 (1)

E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[Crossref]

1956 (1)

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Department of Commerce, National Bureau of Standards, Appl. Math. Ser., 55.

Arnaud, J. A.

J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
[Crossref]

Chang, D. C.

D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., Univ. of Colo., Boulder, Colo.; also, IEEE J. Quantum Electron. QE-11, 903–907 (1975).

Gloge, D.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Kuester, E. F.

D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., Univ. of Colo., Boulder, Colo.; also, IEEE J. Quantum Electron. QE-11, 903–907 (1975).

Lewin, L.

L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972), 398–406.

D. Marcuse, Theory of Dielectric Optical Waveguides, (Academic, New York, 1974).

Miller, M. A.

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Mitchell, D. J.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Shevchenko, V. V.

V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[Crossref]

Snyder, A. W.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

A. W. Snyder (private communication).

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Department of Commerce, National Bureau of Standards, Appl. Math. Ser., 55.

Talanov, V. I.

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

White, I.

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

E. A. J. Marcatili, “Bends in Optical Dielectric Guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969).
[Crossref]

J. A. Arnaud, “Transverse Coupling in Fiber Optics Part III: Bending Losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
[Crossref]

Electron. Lett. (1)

A. W. Snyder, I. White, and D. J. Mitchell, “Radiation from Bent Optical Waveguides,” Electron. Lett. 11, 332–333 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

L. Lewin, “Radiation from Curved Dielectric Slabs and Fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[Crossref]

Radiophys. Quantum Electron. (1)

V. V. Shevchenko, “Radiation Losses in Bent Waveguides for Surface Waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[Crossref]

Zh. Tekh. Fiz. (1)

M. A. Miller and V. I. Talanov, “Electromagnetic Surface Waves Guided by a Boundary with Small Curvature,” Zh. Tekh. Fiz. 26, 2755 (1956).

Other (12)

D. C. Chang and E. F. Kuester, “General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbitrary Cross Section,” , Electromagnetics Laboratory, Dept. Electr. Eng., Univ. of Colo., Boulder, Colo.; also, IEEE J. Quantum Electron. QE-11, 903–907 (1975).

D. Marcuse, Light Transmission Optics (Van Nostrand, Princeton, 1972), 398–406.

A. W. Snyder (private communication).

D. Marcuse, Theory of Dielectric Optical Waveguides, (Academic, New York, 1974).

Reference 11, Eq. (2.2-23), p. 65, and Eq. (2.2-25), p. 66.

Reference 9, Eqs. (8.2-7) through (8.2-10), p. 290.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Eq. 9.2.4, p. 364. U. S. Department of Commerce, National Bureau of Standards, Appl. Math. Ser., 55.

Reference 11, Eq. (2.2-38), p. 68.

Reference 11, Eq. (2.2-69), p. 73.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Reference 17,Eq. 8.468, p. 967.

Reference 14, Eq. 6.1. 18, p. 256.

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Figures (1)

FIG. 1
FIG. 1

The fiber is bent into a torus with radius of curvature R. The dotted vertical lines indicate a cylinder that is tangential to the torus at radius r = R + a.

Equations (46)

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μ = β g R ,
2 α = Δ P / P .
E z = A [ J ν ( k a ) / H ν ( 1 ) ( i γ a ) ] H ν ( 1 ) ( i γ r ) cos ν θ e - i β g R ϕ
E ϕ = A γ 2 β g J ν ( κ a ) H ν ( 1 ) ( i γ a ) [ H ν + 1 ( 1 ) ( i γ r ) sin ( ν + 1 ) θ + H ν - 1 ( 1 ) ( i γ r ) sin ( ν - 1 ) θ ] e - i β g R ϕ ,
γ = ( β g 2 - n 2 2 k 2 ) 1 / 2 ,
κ = ( n 1 2 k 2 - β g 2 ) 1 / 2 .
A = ( 4 ( μ 0 / 0 ) 1 / 2 γ 2 P e ν π n 2 V 2 J ν - 1 ( κ a ) J ν + 1 ( κ a ) ) 1 / 2 ,
e ν = { 2 , ν = 0 1 , ν 0
V 2 = k 2 a 2 ( n 1 2 - n 2 2 ) .
E z μ j = B j H μ ( 2 ) ( ρ r ) e - i μ ϕ e - i β z ,
K z μ j = B j F j H μ ( 2 ) ( ρ r ) e - i μ ϕ e - i β z .
ρ = ( n 2 2 k 2 - β 2 ) 1 / 2 .
F 1 = n 2 ( 0 / μ 0 ) 1 / 2 { H μ ( 2 ) [ ρ ( R + a ) ] / H μ ( 2 ) [ ρ ( R + a ) ] }
F 2 = n 2 ( 0 / μ 0 ) 1 / 2 { H μ ( 2 ) [ ρ ( R + a ) ] / H μ ( 2 ) [ ρ ( R + a ) ] } .
0 2 π d ϕ - d z [ E ϕ μ j ( ρ ) K z μ j * ( ρ ) - E z μ j ( ρ ) K ϕ μ j * ( ρ ) ] r = R + a = ( 2 π ) 2 δ μ μ δ j j δ ( β - β ) N μ β j .
N μ β 1 = i n 2 2 ω 0 ρ [ ( H μ ( 2 ) * H μ ( 2 ) * ) 2 - 1 ] H μ ( 2 ) H μ ( 2 ) * B 1 2 ,
N μ β 2 = i n 2 2 ω 0 ρ [ ( H μ ( 2 ) H μ ( 2 ) ) 2 - 1 ] H μ ( 2 ) H μ ( 2 ) / * B 2 2 .
E z = - [ c 1 ( β ) E z μ 1 + c 2 ( β ) E z μ 2 ] d β ,
E ϕ = - [ c 1 ( β ) E ϕ μ 1 + c 2 ( β ) E ϕ μ 2 ] d β .
r = ( a 2 + z 2 ) 1 / 2
Θ = arctan ( z / a ) .
c j = 1 2 π N μ β j - [ E ϕ K z μ j * - E z K ϕ μ j * ] r = R + a d z = - i A B j * 2 π N μ β j J ν ( κ a ) H ν ( 1 ) ( i γ a ) ω n 2 2 0 ρ H μ ( 2 ) * [ ρ ( R + a ) ] I ν β ,
I ν β = - H ν ( 1 ) [ i γ ( a 2 + z 2 ) 1 / 2 ] cos [ ν arctan ( z / a ) ] e i β z d z .
H μ ( 2 ) [ ρ ( R + a ) ] i ( 2 π γ R ) 1 / 2 e - γ a × exp ( 1 3 γ 3 β g 2 R ) exp ( γ R 2 n 2 2 k 2 β 2 )
H μ ( 2 ) - ( γ / n 2 k ) H μ ( 2 ) .
2 α = π P r R j = 1 2 - c j 2 ( E ϕ μ j K z μ j * - E z μ j K ϕ μ j * ) d β .
2 α = γ 5 / 2 I ν 0 2 e 2 γ a exp [ - 2 3 ( γ 3 / β g 2 ) R ] J ν 2 ( κ a ) e ν π 3 / 2 V 2 J ν - 1 ( κ a ) J ν + 1 ( κ a ) H ν ( 1 ) ( i γ a ) 2 R .
- exp [ - ( γ R / n 2 2 k 2 ) β 2 ] d β = ( π / γ R ) 1 / 2 n 2 k .
β g n 2 k
H ν ( 1 ) ( i γ a ) = ( 2 / i π ) e - i ν ( π / 2 ) K ν ( γ a ) .
J ν 2 J ν - 1 J ν + 1 H ν ( 1 ) 2 = κ 2 γ 2 H ν - 1 ( 1 ) H ν + 2 ( 1 ) = π 2 4 κ 2 γ 2 K ν - 1 ( γ a ) K ν + 1 ( γ a ) .
I ν 0 = e - i ν ( π / 2 ) ( 2 / i γ ) e - γ a .
2 α = π κ 2 exp [ - 2 3 ( γ 3 / β g 2 ) R ] e ν γ 3 / 2 V 2 R K ν - 1 ( γ a ) K ν + 1 ( γ a ) .
2 α = 2 a κ 2 e 2 γ a exp [ - 2 3 ( γ 3 / β g 2 ) R ] e ν π γ R V 2 .
J ν ( κ a ) = 0.
2 α = π κ 2 exp [ - 2 3 ( γ 3 / β g 2 ) R ] 2 γ 3 / 2 V 2 R ( ln γ a ) 2
2 α = π κ 2 ( γ a ) 2 ν - 3 / 2 a 3 / 2 exp [ - 2 3 ( γ 3 / β g 2 ) R ] 2 2 ( ν - 1 ) ( ν - 1 ) ! ( ν + 1 ) ! V 2 R .
I ν 0 = a - H ν ( 1 ) [ i γ a ( 1 + x 2 ) 1 / 2 ] cos ( ν arctan x ) d x = 2 a i π e - i ν ( π / 2 ) - K ν [ γ a ( 1 + x 2 ) 1 / 2 ] ( e i arctan x ) ν d x = 2 a i π e - i ν ( π / 2 ) - K ν [ γ a ( 1 + x 2 ) 1 / 2 ] ( 1 + i x ( 1 + x 2 ) 1 / 2 ) ν d x .
I ν 0 = 4 a i π e - i ν ( π / 2 ) k = 0 [ ν 2 ] ( - 1 ) k ( ν 2 k ) 2 k - ( 1 / 2 ) Γ ( k + 1 2 ) ( a γ ) k + ( 1 / 2 ) K ν - k - ( 1 / 2 ) ( a γ ) .
K ν - k - ( 1 / 2 ) ( a γ ) = ( π 2 a γ ) 1 / 2 e - a γ μ = 0 ν - k - 1 ( ν - k + μ - 1 ) ! μ ! ( ν - k - μ - 1 ) ! ( a γ ) μ .
Γ ( k + 1 2 ) = 2 π ( 2 k - 1 ) ! ( k - 1 ) ! 2 2 k
I ν 0 = 2 i γ e - i ν ( π / 2 ) e - γ a ( 1 + n = 1 ν - 1 ν ! ( ν - n - 1 ) ! ( 2 a γ ) n × k = 0 f ( n ) ( - 1 ) k ( ν + n - 1 - 2 k ) ! k ! ( n - k ) ! ( ν - 2 k ) ! ) ,
f ( n ) = { [ ν 2 ] if n > [ ν 2 ] n if n [ ν 2 ] .
x ν + n - 1 ( 1 - x - 2 ) n = k = 0 n ( - 1 ) k n ! k ! ( n - k ) ! x ν + n - 1 - 2 k .
d n - 1 d x n - 1 x ν + n - 1 ( 1 - x - 2 ) n = n ! k = 0 f ( n ) ( - 1 ) k ( ν + n - 1 - 2 k ) ! k ! ( n - k ) ! ( ν - 2 k ) ! x ν - 2 k .
I ν 0 = ( 2 / i γ ) e - i ν ( π / 2 ) e - γ a .