Abstract

We investigate the influence of cross-sectional shape on propagation in optical fibers. Remarkably simple, analytical expressions are derived for modes on multimoded fibers, including the fraction of power in the cladding, the group velocity, and the modal birefringence. The fibers have arbitrary cross-sectional shape and arbitrary refractive-index difference. New analytical results are given for the rectangular, elliptical, triangular, and wedge-shaped cross sections.

© 1986 Optical Society of America

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References

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  1. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).
  2. A. W. Snyder, W. R. Young, “Modes of optical waveguides,”J. Opt. Soc. Am. 68, 297–309 (1978).
    [CrossRef]
  3. M. S. Yataki, D. N. Payne, “All-fiber polarizing beam splitter,” Electron. Lett. 2, 249–251 (1985).
    [CrossRef]
  4. T. Bricheno, V. Baker, “All-fibre polarisation splitter/combiner,” Electron. Lett. 21, 251–253 (1985).
    [CrossRef]
  5. I. Yokohama, K. Okamoto, J. Noda, “Fiber-optic polarizing beam splitter employing birefringent fiber coupler,” Electron. Lett. 21, 415–416 (1985).
    [CrossRef]
  6. A. W. Snyder, “Polarizing beamsplitters from fused-taper couplers,” Electron. Lett. 21, 623–625 (1985); A. W. Snyder, X. Zheng, “Fused couplers of arbitrary cross section,” Electron. Lett. 21, 1079–1080 (1985).
    [CrossRef]
  7. A. W. Snyder, R. Menzel, Photoreceptor Optics (Springer-Verlag, New York, 1975).
    [CrossRef]
  8. J. M. Enoch, Vertebrate Photoreceptor Optics (Springer-Verlag, New York, 1981).
    [CrossRef]
  9. K. Kirschfeld, A. W. Snyder, “Measurement of a photoreceptor’s characteristic waveguide parameter,” Vision Res. 16, 775–778 (1976).
    [CrossRef]
  10. A. W. Snyder, R. De La Rue, “Asymptotic solution of eigenvalue equations for surface waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-18, 650–651 (1970).
    [CrossRef]
  11. A. W. Snyder, “Approximate eigenvalue for a circular rod of arbitrary relative permittivity,” Electron. Lett. 7, 105–106 (1971).
    [CrossRef]
  12. E. J. Marcatelli, “Dielectric rectangular waveguide and directional coupler for integraded optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
  13. J. D. Love, R. A. Sammut, A. W. Snyder, “Birefringence in elliptically deformed fibers,” Electron. Lett. 15, 615–616 (1979).
    [CrossRef]

1985 (4)

M. S. Yataki, D. N. Payne, “All-fiber polarizing beam splitter,” Electron. Lett. 2, 249–251 (1985).
[CrossRef]

T. Bricheno, V. Baker, “All-fibre polarisation splitter/combiner,” Electron. Lett. 21, 251–253 (1985).
[CrossRef]

I. Yokohama, K. Okamoto, J. Noda, “Fiber-optic polarizing beam splitter employing birefringent fiber coupler,” Electron. Lett. 21, 415–416 (1985).
[CrossRef]

A. W. Snyder, “Polarizing beamsplitters from fused-taper couplers,” Electron. Lett. 21, 623–625 (1985); A. W. Snyder, X. Zheng, “Fused couplers of arbitrary cross section,” Electron. Lett. 21, 1079–1080 (1985).
[CrossRef]

1979 (1)

J. D. Love, R. A. Sammut, A. W. Snyder, “Birefringence in elliptically deformed fibers,” Electron. Lett. 15, 615–616 (1979).
[CrossRef]

1978 (1)

1976 (1)

K. Kirschfeld, A. W. Snyder, “Measurement of a photoreceptor’s characteristic waveguide parameter,” Vision Res. 16, 775–778 (1976).
[CrossRef]

1971 (1)

A. W. Snyder, “Approximate eigenvalue for a circular rod of arbitrary relative permittivity,” Electron. Lett. 7, 105–106 (1971).
[CrossRef]

1970 (1)

A. W. Snyder, R. De La Rue, “Asymptotic solution of eigenvalue equations for surface waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-18, 650–651 (1970).
[CrossRef]

1969 (1)

E. J. Marcatelli, “Dielectric rectangular waveguide and directional coupler for integraded optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Baker, V.

T. Bricheno, V. Baker, “All-fibre polarisation splitter/combiner,” Electron. Lett. 21, 251–253 (1985).
[CrossRef]

Bricheno, T.

T. Bricheno, V. Baker, “All-fibre polarisation splitter/combiner,” Electron. Lett. 21, 251–253 (1985).
[CrossRef]

De La Rue, R.

A. W. Snyder, R. De La Rue, “Asymptotic solution of eigenvalue equations for surface waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-18, 650–651 (1970).
[CrossRef]

Enoch, J. M.

J. M. Enoch, Vertebrate Photoreceptor Optics (Springer-Verlag, New York, 1981).
[CrossRef]

Kirschfeld, K.

K. Kirschfeld, A. W. Snyder, “Measurement of a photoreceptor’s characteristic waveguide parameter,” Vision Res. 16, 775–778 (1976).
[CrossRef]

Love, J. D.

J. D. Love, R. A. Sammut, A. W. Snyder, “Birefringence in elliptically deformed fibers,” Electron. Lett. 15, 615–616 (1979).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Marcatelli, E. J.

E. J. Marcatelli, “Dielectric rectangular waveguide and directional coupler for integraded optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Menzel, R.

A. W. Snyder, R. Menzel, Photoreceptor Optics (Springer-Verlag, New York, 1975).
[CrossRef]

Noda, J.

I. Yokohama, K. Okamoto, J. Noda, “Fiber-optic polarizing beam splitter employing birefringent fiber coupler,” Electron. Lett. 21, 415–416 (1985).
[CrossRef]

Okamoto, K.

I. Yokohama, K. Okamoto, J. Noda, “Fiber-optic polarizing beam splitter employing birefringent fiber coupler,” Electron. Lett. 21, 415–416 (1985).
[CrossRef]

Payne, D. N.

M. S. Yataki, D. N. Payne, “All-fiber polarizing beam splitter,” Electron. Lett. 2, 249–251 (1985).
[CrossRef]

Sammut, R. A.

J. D. Love, R. A. Sammut, A. W. Snyder, “Birefringence in elliptically deformed fibers,” Electron. Lett. 15, 615–616 (1979).
[CrossRef]

Snyder, A. W.

A. W. Snyder, “Polarizing beamsplitters from fused-taper couplers,” Electron. Lett. 21, 623–625 (1985); A. W. Snyder, X. Zheng, “Fused couplers of arbitrary cross section,” Electron. Lett. 21, 1079–1080 (1985).
[CrossRef]

J. D. Love, R. A. Sammut, A. W. Snyder, “Birefringence in elliptically deformed fibers,” Electron. Lett. 15, 615–616 (1979).
[CrossRef]

A. W. Snyder, W. R. Young, “Modes of optical waveguides,”J. Opt. Soc. Am. 68, 297–309 (1978).
[CrossRef]

K. Kirschfeld, A. W. Snyder, “Measurement of a photoreceptor’s characteristic waveguide parameter,” Vision Res. 16, 775–778 (1976).
[CrossRef]

A. W. Snyder, “Approximate eigenvalue for a circular rod of arbitrary relative permittivity,” Electron. Lett. 7, 105–106 (1971).
[CrossRef]

A. W. Snyder, R. De La Rue, “Asymptotic solution of eigenvalue equations for surface waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-18, 650–651 (1970).
[CrossRef]

A. W. Snyder, R. Menzel, Photoreceptor Optics (Springer-Verlag, New York, 1975).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Yataki, M. S.

M. S. Yataki, D. N. Payne, “All-fiber polarizing beam splitter,” Electron. Lett. 2, 249–251 (1985).
[CrossRef]

Yokohama, I.

I. Yokohama, K. Okamoto, J. Noda, “Fiber-optic polarizing beam splitter employing birefringent fiber coupler,” Electron. Lett. 21, 415–416 (1985).
[CrossRef]

Young, W. R.

Bell Syst. Tech. J. (1)

E. J. Marcatelli, “Dielectric rectangular waveguide and directional coupler for integraded optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Electron. Lett. (6)

J. D. Love, R. A. Sammut, A. W. Snyder, “Birefringence in elliptically deformed fibers,” Electron. Lett. 15, 615–616 (1979).
[CrossRef]

A. W. Snyder, “Approximate eigenvalue for a circular rod of arbitrary relative permittivity,” Electron. Lett. 7, 105–106 (1971).
[CrossRef]

M. S. Yataki, D. N. Payne, “All-fiber polarizing beam splitter,” Electron. Lett. 2, 249–251 (1985).
[CrossRef]

T. Bricheno, V. Baker, “All-fibre polarisation splitter/combiner,” Electron. Lett. 21, 251–253 (1985).
[CrossRef]

I. Yokohama, K. Okamoto, J. Noda, “Fiber-optic polarizing beam splitter employing birefringent fiber coupler,” Electron. Lett. 21, 415–416 (1985).
[CrossRef]

A. W. Snyder, “Polarizing beamsplitters from fused-taper couplers,” Electron. Lett. 21, 623–625 (1985); A. W. Snyder, X. Zheng, “Fused couplers of arbitrary cross section,” Electron. Lett. 21, 1079–1080 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, R. De La Rue, “Asymptotic solution of eigenvalue equations for surface waveguide structures,” IEEE Trans. Microwave Theory Tech. MTT-18, 650–651 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

Vision Res. (1)

K. Kirschfeld, A. W. Snyder, “Measurement of a photoreceptor’s characteristic waveguide parameter,” Vision Res. 16, 775–778 (1976).
[CrossRef]

Other (3)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

A. W. Snyder, R. Menzel, Photoreceptor Optics (Springer-Verlag, New York, 1975).
[CrossRef]

J. M. Enoch, Vertebrate Photoreceptor Optics (Springer-Verlag, New York, 1981).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of fiber cross section with core and cladding refractive indices nco and ncl, respectively. In general the modes are polarized along only two orthogonal optical axes, which are taken here to be in the x or y direction. The fiber is assumed to be uniform along its length in the z direction. Angles θx and θy are the inclination to the normal of the x and y axes, respectively, while r ^ is the unit vector along the normal to the core–cladding interface.

Fig. 2
Fig. 2

The exact modal constants of Eqs. (11)(13) derived from known analytical forms in Appendix A.

Fig. 3
Fig. 3

Approximate modal constants for the elliptical cross section. Our numerical evaluations of the Mathieu functions given in Appendix A show that the results are exact for small eccentricity and are less than 0.1% in error for an eccentricity e = (1 − b2/a2)1/2 smaller than 0.5; less than 1% in error for e < 0.85. The maximum error is e = 1 when the ellipse LP01, and LP11 odd modes must equal those modes of the planar waveguide in Fig. 2. Comparing Figs. 2 and 3, the error for the LP01 and LP11 modes is, respectively, 8.3% and 5.6% for U, 0.7% and 0.3% for A, and 0.6% and 0.1% for Sy. The LP11 even mode is in error by 7.1% for U, 1.9% for A, and 0.6% for Sy when e <0.98. When e = 1, the LP11 even mode merges into the LP01 mode, so that U. is in error by 22%.

Fig. 4
Fig. 4

The ê-field lines of a fundamental mode on a multimoded (V ≫ 1) optical fiber with a large refractive-index difference (Δ ~ 0.5).

Fig. 5
Fig. 5

Comparison of the exact eigenvalue Ui, determined numerically from the eigenvalue equation, versus the asymptotic approximation given in Eq. (5), together with results of Fig. 2, for the LP01 (HE11) mode of the fiber with a circular cross section.

Fig. 6
Fig. 6

Local Cartesian coordinates (r, l) where r is the normal to the core–cladding boundary with l parallel to the boundary.

Fig. 7
Fig. 7

Wedge-shaped cross section.

Fig. 8
Fig. 8

Slight distortion of circular cross section into an elliptical cross section required for a perturbation analysis: (a) ellipse and circle with equal areas, (b) ellipse immersed in larger circle.

Equations (75)

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Δ = ( n co 2 - n cl 2 ) / 2 n co 2 ,
V = k ρ n co ( 2 Δ ) 1 / 2
E i ~ ψ ( U i ) exp ( j β i z ) e ^ i ,
( ρ β i ) 2 = V 2 2 Δ - ( U i ) 2 ,
U i = U [ V V + A ( 1 - 2 S i Δ ) , ]
v g i = ω / β i = α / ( 1 - 2 P cl i Δ ) ,
P cl i = U i V d U i d V ~ U 2 A ( 1 - 2 S i Δ ) [ V + A ( 1 - 2 S i Δ ) ] 3 ,
P cl y P cl x ~ 1 - 2 S y Δ 1 - 2 S x Δ ,
β x - β y β x ~ 4 A U 2 ( S y - S x ) Δ 2 { V + A ( 1 - Δ ) } 3 ,
V [ 1 - ( b / a ) 2 ] 4 Δ
( ρ 2 t 2 + U 2 ) ψ = 0 ,
A = ρ 3 0 ( ψ r ) 2 d l 2 U 2 A c o ψ 2 d A ,
S i = { ψ r } 2 cos 2 θ i d l { ψ r } 2 d l ,
U ( V ) ~ U + 1 V d U d ( 1 / V ) | V = .
P cl ( V ) = A cl e × h · z ^ d A A e × h · z ^ d A ,
d U / d V = ( V / U ) P cl ( V ) ,
U ( V ) ~ U - 1 V [ V 3 U P cl ( V ) ] V = .
e cl ( x , y ) ~ e cl ( l ) exp ( - V r / ρ ) ,
h cl ( x , y ) ~ h ( l ) exp ( - V r / ρ ) .
P cl ( V ) ~ ( ρ 2 V ) e cl ( l ) × h ( l ) · z ^ d l A co e × h · z ^ d A ,
h t = γ n co z ^ × e t ,
( t 2 + k 2 n 2 co - β 2 ) e t = 0
e t = ψ ( U ) x ^ ,
e z = j ( t · e t ) / β ,
e z c l ~ j k n c o e r cl r ~ j V ρ k n c o e r c l ,
e z c o ~ j k n co ψ r cos θ i ,
e r cl ( l ) ~ - ρ V ψ r cos θ i ,
e z cl ~ j k n co ψ r cos θ i exp ( - V r / ρ ) .
h z = - j γ z ^ · t × e t / k .
h z cl ~ γ j V ρ k e l cl ,
h z c o ~ ± γ j k ψ r sin θ i ,
e l cl ( l ) ~ ± ρ V ψ r sin θ i ,
h z cl ~ ± γ j k ψ r sin θ i exp ( - V r / ρ ) .
h l c l ~ - γ j ρ 2 V 2 k n cl 2 e z r ,
h r c l ~ - j ρ 2 V 2 k n co h z r ;
h l ( l ) ~ - γ ρ V n cl 2 n co ψ r cos θ i ,
h r ( l ) ~ n co γ ρ V ψ r sin θ i .
P c l ~ ρ 3 ( 1 - 2 Δ cos 2 θ i ) ( ψ r ) 2 d l 2 V 3 A co ( ψ ) 2 d A .
P cl ~ U 2 A ( 1 - 2 S i Δ V 3 ) .
U i ~ U [ 1 - A ( 1 - 2 S i Δ V ) ] .
d U i d V ~ U 1 - A ( 1 - 2 S i Δ ) A ( 1 - 2 S i Δ ) V 2 .
U i ~ U { 1 + ln [ 1 - A ( 1 - 2 S i Δ ) V ] }
~ U { 1 - A ( 1 - 2 S i Δ ) V + [ A ( 1 - 2 S i Δ ) V ] 2 + } .
U = U ¯ + δ U .
ψ cl ~ - ( ρ / V ) ψ co / r .
δ U = ρ 2 2 U ¯ A [ ( ψ r ) cos θ i e r d ln ( n 2 ) dr ] d A A co ψ 2 d A .
- δ δ e r d ln ( n 2 ) d r d r = n cl 2 e r cl ( l ) - δ δ d ( n - 2 ) d r d r = - 2 Δ e r cl ( l ) ,
ψ = J l ( U ρ ¯ r ) g l ( ϕ ) ,
ψ = sin m π x 2 a sin n π y 2 b ,
U = ( π / 2 ) [ ( b / a ) 2 m 2 + n 2 ] 1 / 2 ,
A = 1 + ( m / n ) 2 ( b / a ) 3 1 + ( m / n ) 2 ( b / a ) 2 ,
S x = 1 1 + ( n / m ) 2 ( a / b ) 3 ,             S y = 1 1 + ( m / n ) 2 ( b / a ) 3 .
ψ = sin m π x a sin n π y a - sin n π x a sin m π y a ,
U = π ( ρ a ) ( m 2 + n 2 ) 1 / 2 ,             0 < m < n ,
S x = 1 - 2 / 2 ,             S y = 2 / 2
ψ = J v ( U r ρ ) sin v θ ,
ψ 2 d A = 1 4 α ρ 2 [ J ν ( U ) ] 2 ,
( ψ r ) 2 d l = α U 2 2 ρ [ J v ( U ) ] 2 + v 2 ρ 0 1 J v 2 ( U z ) d z z 2 .
A = 1 + 2 / α 1 - ( α / 2 m π ) 2 .
S x = 1 / 2 [ α - sin α 1 - ( α / m π ) 2 + 4 cos 2 ( α / 2 ) 1 - ( α / 2 m π ) 2 ] / × [ α + 2 1 - ( α / 2 m π ) 2 ] - 1 ,
e 2 = 1 - ( b / a ) 2 ,
A = 1 2 U 2 { 1 - e 2 2 e 2 } 3 / 2 [ F ( ξ 0 ) ] 2 0 2 π f 2 ( η ) { cosh 2 ξ 0 - cos 2 η } 1 / 2 d η Θ 0 ξ 0 F 2 ( ξ ) cosh 2 ξ d ξ + Ψ 0 ξ 0 F 2 ( ξ ) d ξ
S x = 2 ( 1 e 2 - 1 ) 0 2 π f 2 ( η ) cos 2 η { cosh 2 ξ 0 - cos 2 η } 3 / 2 d η 0 2 π f 2 ( η ) { cosh 2 ξ 0 - cos 2 η } 1 / 2 d η ,
U = 2 q e · ρ a ,
Θ = 0 2 π f 2 ( η ) d η ,             Ψ = 0 2 π f 2 ( η ) cos 2 η d η
ξ 0 = cosh - 1 ( 1 / e ) ,
f ( η ) 1 - ( q / 2 ) cos 2 η ,
S x ( 1 - e 2 ) 0 2 π ( 1 - 0.5 q cos 2 η ) 2 cos 2 η ( 1 - e 2 cos 2 η ) 3 / 2 d η 0 2 π ( 1 - 0.5 q cos 2 η ) 2 ( 1 - e 2 cos 2 η ) 1 / 2 d η .
S x = 0.5 - 0.4240 e 2 ,             S y = 0.5 + 0.4240 e 2
S x = 0.75 - 0.3231 e 2 ,             S y = 0.25 + 0.3231 e 2
S x = 0.25 - 0.3231 e 2 ,             S y = 0.75 + 0.3231 e 2
S y = 1 1 + ( 1 - e 2 ) 1.5
S y = 1 1 + ( 1 - e 2 ) 1.6958 .
ψ = ψ ˜ ( b a x , y ) ,
U = U ˜ [ 1 - e 2 ( ψ ˜ / x ) 2 d A [ ( ψ ˜ / x ) 2 + ( ψ ˜ / y ) 2 ] d A ] .

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